diff --git a/tasks/construct-ae-p38-factorial-factors-l2-s6/environment/Dockerfile b/tasks/construct-ae-p38-factorial-factors-l2-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l2-s6/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-ae-p38-factorial-factors-l2-s6/environment/instance.json b/tasks/construct-ae-p38-factorial-factors-l2-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..cf04d836d6fb8d76f5d936011aa76491a0a44264
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l2-s6/environment/instance.json
@@ -0,0 +1 @@
+{"N": 120, "t": 35, "family": "ae_p38_factorial_factors", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-ae-p38-factorial-factors-l2-s6/solution/solve.sh b/tasks/construct-ae-p38-factorial-factors-l2-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..2a8f0e7a40bd84566084d72fba5282f183defd09
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l2-s6/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"factors": [245, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 37, 37, 37, 38, 38, 38, 38, 38, 38, 39, 39, 39, 39, 39, 39, 39, 39, 39, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 41, 41, 43, 43, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 46, 46, 46, 46, 46, 47, 47, 49, 51, 51, 51, 51, 51, 51, 51, 53, 53, 58, 58, 58, 58, 59, 59, 61, 62, 62, 62, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113]}
+RLMATH_EOF
diff --git a/tasks/construct-ae-p38-factorial-factors-l2-s6/tests/check.py b/tasks/construct-ae-p38-factorial-factors-l2-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..93ec67a19039afa8b4ce6e39811691b802132562
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l2-s6/tests/check.py
@@ -0,0 +1,12 @@
+"""Factor N! into N factors, each >= t (AlphaEvolve problem 38 / OEIS A034258). Exact big-integer check."""
+import math
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, t = inst["N"], inst["t"]
+    f = int_list(ans.get("factors"), n, t, None, "factors")
+    if math.prod(f) != math.factorial(n):
+        raise Invalid(f"the product of the factors is not {n}!")
+    return None, {"min_factor": min(f)}
diff --git a/tasks/construct-ae-p38-factorial-factors-l2-s6/tests/instance.json b/tasks/construct-ae-p38-factorial-factors-l2-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..cf04d836d6fb8d76f5d936011aa76491a0a44264
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l2-s6/tests/instance.json
@@ -0,0 +1 @@
+{"N": 120, "t": 35, "family": "ae_p38_factorial_factors", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-ae-p38-factorial-factors-l2-s6/tests/run.py b/tasks/construct-ae-p38-factorial-factors-l2-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l2-s6/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-ae-p38-factorial-factors-l2-s6/tests/test.sh b/tasks/construct-ae-p38-factorial-factors-l2-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l2-s6/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-ae-p38-factorial-factors-l6-s8/environment/Dockerfile b/tasks/construct-ae-p38-factorial-factors-l6-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l6-s8/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-ae-p38-factorial-factors-l6-s8/environment/instance.json b/tasks/construct-ae-p38-factorial-factors-l6-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..cc4910e584ad80196338b72722ab44b7f0925bbc
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l6-s8/environment/instance.json
@@ -0,0 +1 @@
+{"N": 8823, "t": 2881, "family": "ae_p38_factorial_factors", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-ae-p38-factorial-factors-l6-s8/solution/solve.sh b/tasks/construct-ae-p38-factorial-factors-l6-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..a96c38bd43b34a25a7da2896556b07c62c16a947
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l6-s8/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"factors": [69144, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2881, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2882, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2883, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2884, 2885, 2885, 2885, 2885, 2885, 2885, 2885, 2885, 2885, 2885, 2885, 2885, 2885, 2885, 2885, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2886, 2887, 2887, 2887, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2888, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2889, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 2890, 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4649, 4651, 4657, 4663, 4666, 4666, 4666, 4673, 4678, 4678, 4678, 4679, 4682, 4682, 4682, 4691, 4694, 4694, 4694, 4702, 4702, 4702, 4703, 4714, 4714, 4714, 4721, 4723, 4729, 4733, 4742, 4742, 4742, 4751, 4754, 4754, 4754, 4759, 4762, 4762, 4762, 4766, 4766, 4766, 4778, 4778, 4778, 4783, 4786, 4786, 4786, 4787, 4789, 4793, 4798, 4798, 4798, 4799, 4801, 4813, 4817, 4822, 4822, 4822, 4831, 4834, 4834, 4834, 4846, 4846, 4846, 4861, 4871, 4874, 4874, 4874, 4877, 4882, 4882, 4882, 4889, 4894, 4894, 4894, 4903, 4909, 4918, 4918, 4918, 4919, 4931, 4933, 4934, 4934, 4934, 4937, 4943, 4946, 4946, 4946, 4951, 4954, 4954, 4954, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, 5006, 5006, 5006, 5009, 5011, 5021, 5023, 5039, 5042, 5042, 5042, 5051, 5059, 5062, 5062, 5062, 5077, 5078, 5078, 5078, 5081, 5086, 5086, 5086, 5087, 5098, 5098, 5098, 5099, 5101, 5102, 5102, 5102, 5107, 5113, 5114, 5114, 5114, 5119, 5147, 5153, 5158, 5158, 5158, 5167, 5171, 5179, 5182, 5182, 5182, 5186, 5186, 5186, 5189, 5197, 5209, 5218, 5218, 5218, 5227, 5231, 5233, 5234, 5234, 5234, 5237, 5242, 5242, 5242, 5261, 5266, 5266, 5266, 5273, 5279, 5281, 5294, 5294, 5294, 5297, 5303, 5309, 5314, 5314, 5314, 5318, 5318, 5318, 5323, 5326, 5326, 5326, 5333, 5342, 5342, 5342, 5347, 5351, 5354, 5354, 5354, 5366, 5366, 5366, 5374, 5374, 5374, 5378, 5378, 5378, 5381, 5386, 5386, 5386, 5387, 5393, 5398, 5398, 5398, 5399, 5407, 5413, 5414, 5414, 5414, 5417, 5419, 5422, 5422, 5422, 5426, 5426, 5426, 5431, 5437, 5438, 5438, 5438, 5441, 5443, 5449, 5458, 5458, 5458, 5462, 5462, 5462, 5471, 5477, 5479, 5482, 5482, 5482, 5483, 5498, 5498, 5498, 5501, 5503, 5506, 5506, 5506, 5507, 5519, 5521, 5527, 5531, 5534, 5534, 5534, 5554, 5554, 5554, 5557, 5563, 5569, 5573, 5578, 5578, 5578, 5581, 5582, 5582, 5582, 5591, 5594, 5594, 5594, 5602, 5602, 5602, 5606, 5606, 5606, 5623, 5638, 5638, 5638, 5639, 5641, 5647, 5651, 5653, 5657, 5659, 5666, 5666, 5666, 5669, 5674, 5674, 5674, 5683, 5686, 5686, 5686, 5689, 5693, 5701, 5702, 5702, 5702, 5711, 5714, 5714, 5714, 5717, 5722, 5722, 5722, 5737, 5741, 5743, 5749, 5758, 5758, 5758, 5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301, 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, 6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, 6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, 6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917, 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, 7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297, 7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411, 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499, 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723, 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, 7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919, 7927, 7933, 7937, 7949, 7951, 7963, 7993, 8009, 8011, 8017, 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, 8117, 8123, 8147, 8161, 8167, 8171, 8179, 8191, 8209, 8219, 8221, 8231, 8233, 8237, 8243, 8263, 8269, 8273, 8287, 8291, 8293, 8297, 8311, 8317, 8329, 8353, 8363, 8369, 8377, 8387, 8389, 8419, 8423, 8429, 8431, 8443, 8447, 8461, 8467, 8501, 8513, 8521, 8527, 8537, 8539, 8543, 8563, 8573, 8581, 8597, 8599, 8609, 8623, 8627, 8629, 8641, 8647, 8663, 8669, 8677, 8681, 8689, 8693, 8699, 8707, 8713, 8719, 8731, 8737, 8741, 8747, 8753, 8761, 8779, 8783, 8803, 8807, 8819, 8821]}
+RLMATH_EOF
diff --git a/tasks/construct-ae-p38-factorial-factors-l6-s8/tests/check.py b/tasks/construct-ae-p38-factorial-factors-l6-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..93ec67a19039afa8b4ce6e39811691b802132562
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l6-s8/tests/check.py
@@ -0,0 +1,12 @@
+"""Factor N! into N factors, each >= t (AlphaEvolve problem 38 / OEIS A034258). Exact big-integer check."""
+import math
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, t = inst["N"], inst["t"]
+    f = int_list(ans.get("factors"), n, t, None, "factors")
+    if math.prod(f) != math.factorial(n):
+        raise Invalid(f"the product of the factors is not {n}!")
+    return None, {"min_factor": min(f)}
diff --git a/tasks/construct-ae-p38-factorial-factors-l6-s8/tests/instance.json b/tasks/construct-ae-p38-factorial-factors-l6-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..cc4910e584ad80196338b72722ab44b7f0925bbc
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l6-s8/tests/instance.json
@@ -0,0 +1 @@
+{"N": 8823, "t": 2881, "family": "ae_p38_factorial_factors", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-ae-p38-factorial-factors-l6-s8/tests/run.py b/tasks/construct-ae-p38-factorial-factors-l6-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l6-s8/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-ae-p38-factorial-factors-l6-s8/tests/test.sh b/tasks/construct-ae-p38-factorial-factors-l6-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l6-s8/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-ae-p38-factorial-factors-l6-s9/environment/Dockerfile b/tasks/construct-ae-p38-factorial-factors-l6-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l6-s9/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-ae-p38-factorial-factors-l6-s9/environment/instance.json b/tasks/construct-ae-p38-factorial-factors-l6-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..dfa0a626f771f4ab1622987d8a2f8ddef969997c
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l6-s9/environment/instance.json
@@ -0,0 +1 @@
+{"N": 9545, "t": 3118, "family": "ae_p38_factorial_factors", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-ae-p38-factorial-factors-l6-s9/solution/solve.sh b/tasks/construct-ae-p38-factorial-factors-l6-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..f93ada1f985f59406e1869b38c1b8be77851975f
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l6-s9/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"factors": [18708, 3118, 3118, 3118, 3118, 3118, 3119, 3119, 3119, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 3120, 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+RLMATH_EOF
diff --git a/tasks/construct-ae-p38-factorial-factors-l6-s9/tests/check.py b/tasks/construct-ae-p38-factorial-factors-l6-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..93ec67a19039afa8b4ce6e39811691b802132562
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l6-s9/tests/check.py
@@ -0,0 +1,12 @@
+"""Factor N! into N factors, each >= t (AlphaEvolve problem 38 / OEIS A034258). Exact big-integer check."""
+import math
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, t = inst["N"], inst["t"]
+    f = int_list(ans.get("factors"), n, t, None, "factors")
+    if math.prod(f) != math.factorial(n):
+        raise Invalid(f"the product of the factors is not {n}!")
+    return None, {"min_factor": min(f)}
diff --git a/tasks/construct-ae-p38-factorial-factors-l6-s9/tests/instance.json b/tasks/construct-ae-p38-factorial-factors-l6-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..dfa0a626f771f4ab1622987d8a2f8ddef969997c
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l6-s9/tests/instance.json
@@ -0,0 +1 @@
+{"N": 9545, "t": 3118, "family": "ae_p38_factorial_factors", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-ae-p38-factorial-factors-l6-s9/tests/run.py b/tasks/construct-ae-p38-factorial-factors-l6-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l6-s9/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-ae-p38-factorial-factors-l6-s9/tests/test.sh b/tasks/construct-ae-p38-factorial-factors-l6-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l6-s9/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-ae-p58-happy-ending-l1-s0/environment/Dockerfile b/tasks/construct-ae-p58-happy-ending-l1-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-ae-p58-happy-ending-l1-s0/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-ae-p58-happy-ending-l1-s0/environment/instance.json b/tasks/construct-ae-p58-happy-ending-l1-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..441e1e37e4213f67c45ccf9b3a9a60401bba84a6
--- /dev/null
+++ b/tasks/construct-ae-p58-happy-ending-l1-s0/environment/instance.json
@@ -0,0 +1 @@
+{"n": 8, "k": 5, "max_abs": 1000000000000000000, "family": "ae_p58_happy_ending", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-ae-p58-happy-ending-l1-s0/solution/solve.sh b/tasks/construct-ae-p58-happy-ending-l1-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..03b207ac090f5d54ef16fb74ca9457e449b79b82
--- /dev/null
+++ b/tasks/construct-ae-p58-happy-ending-l1-s0/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"points": [[0, 0], [15000, -15000000], [15001, -14999998], [15002, -14999990], [30000, -60000000], [30001, -59999990], [30002, -59999988], [45000, -135000000]]}
+RLMATH_EOF
diff --git a/tasks/construct-ae-p58-happy-ending-l1-s0/tests/check.py b/tasks/construct-ae-p58-happy-ending-l1-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..47e4606e109bc165111a8138a5a2f60870d563c8
--- /dev/null
+++ b/tasks/construct-ae-p58-happy-ending-l1-s0/tests/check.py
@@ -0,0 +1,69 @@
+"""Erdos-Szekeres happy ending (AlphaEvolve problem 58): n = 2^(k-2) integer points in general position with no
+k points in convex position. Exact integer orientation tests; the largest convex-position subset is computed by
+the O(n^4) bottom-vertex / angular-order dynamic programme.
+
+Upstream bug: the notebook's verifier uses float orientations with a 1e-9 threshold (and numba fastmath); its
+33- and 65-point sets (claimed to have "0 convex k-gons") contain a convex 7-gon / 8-gon under exact arithmetic."""
+import functools
+
+import numpy as np
+
+from run import Invalid, int_list
+
+
+def orient_table(p):
+    n = len(p)
+    o = np.zeros((n, n, n), dtype=np.int8)
+    for i in range(n):
+        xi, yi = p[i]
+        for j in range(i + 1, n):
+            dx, dy = p[j][0] - xi, p[j][1] - yi
+            for k in range(j + 1, n):
+                c = dx * (p[k][1] - yi) - dy * (p[k][0] - xi)
+                if c == 0:
+                    raise Invalid(f"points {p[i]}, {p[j]}, {p[k]} are collinear")
+                s = 1 if c > 0 else -1
+                o[i, j, k] = o[j, k, i] = o[k, i, j] = s
+                o[i, k, j] = o[k, j, i] = o[j, i, k] = -s
+    return o
+
+
+def max_convex(p, o):
+    n = len(p)
+    best = min(n, 3)
+    order = sorted(range(n), key=lambda i: (p[i][1], p[i][0]))
+    for bi, b in enumerate(order):
+        rest = order[bi + 1:]
+        m = len(rest)
+        if m < 2:
+            continue
+        rest = sorted(rest, key=functools.cmp_to_key(lambda u, v: -int(o[b, u, v])))
+        r = np.array(rest)
+        chain = np.zeros((m, m), dtype=np.int32)  # chain[i, j]: longest convex chain b .. r_i -> r_j
+        for j in range(1, m):
+            for i in range(j):
+                v = 3
+                if i > 0:
+                    ok = (chain[:i, i] > 0) & (o[r[:i], r[i], r[j]] > 0)
+                    if ok.any():
+                        v = max(v, int(chain[:i, i][ok].max()) + 1)
+                chain[i, j] = v
+            close = o[r[:j], r[j], b] > 0
+            if close.any():
+                best = max(best, int(chain[:j, j][close].max()))
+    return best
+
+
+def check(inst, ans):
+    n, k, lim = inst["n"], inst["k"], inst["max_abs"]
+    pts = ans.get("points")
+    if not isinstance(pts, list) or len(pts) != n:
+        raise Invalid(f"points must be a list of {n} [x, y] pairs")
+    p = [tuple(int_list(q, 2, -lim, lim, "point")) for q in pts]
+    if len(set(p)) != n:
+        raise Invalid("points must be distinct")
+    o = orient_table(p)
+    mc = max_convex(p, o)
+    if mc >= k:
+        raise Invalid(f"the set contains {mc} points in convex position (>= {k})")
+    return None, {"max_convex_subset": mc}
diff --git a/tasks/construct-ae-p58-happy-ending-l1-s0/tests/instance.json b/tasks/construct-ae-p58-happy-ending-l1-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..441e1e37e4213f67c45ccf9b3a9a60401bba84a6
--- /dev/null
+++ b/tasks/construct-ae-p58-happy-ending-l1-s0/tests/instance.json
@@ -0,0 +1 @@
+{"n": 8, "k": 5, "max_abs": 1000000000000000000, "family": "ae_p58_happy_ending", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-ae-p58-happy-ending-l1-s0/tests/run.py b/tasks/construct-ae-p58-happy-ending-l1-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-ae-p58-happy-ending-l1-s0/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-ae-p58-happy-ending-l1-s0/tests/test.sh b/tasks/construct-ae-p58-happy-ending-l1-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-ae-p58-happy-ending-l1-s0/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-all-interval-l6-s0/environment/Dockerfile b/tasks/construct-mc-all-interval-l6-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-all-interval-l6-s0/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-all-interval-l6-s0/environment/instance.json b/tasks/construct-mc-all-interval-l6-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a3b9a7ddd9aac9fec27db1f35e34af7d47d3d536
--- /dev/null
+++ b/tasks/construct-mc-all-interval-l6-s0/environment/instance.json
@@ -0,0 +1 @@
+{"n": 72, "fixed": [], "family": "mc_all_interval", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-all-interval-l6-s0/solution/solve.sh b/tasks/construct-mc-all-interval-l6-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..1017432de26ee70c89d2cf4b967f574093bca8b5
--- /dev/null
+++ b/tasks/construct-mc-all-interval-l6-s0/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": [0, 71, 1, 70, 2, 69, 3, 68, 4, 67, 5, 66, 6, 65, 7, 64, 8, 63, 9, 62, 10, 61, 11, 60, 12, 59, 13, 58, 14, 57, 15, 56, 16, 55, 17, 54, 18, 53, 19, 52, 20, 51, 21, 50, 22, 49, 23, 48, 24, 47, 25, 46, 26, 45, 27, 44, 28, 43, 29, 42, 30, 41, 31, 40, 32, 39, 33, 38, 34, 37, 35, 36]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-all-interval-l6-s0/tests/check.py b/tasks/construct-mc-all-interval-l6-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2e39eec264b6422a47e35921b8cdbe1fa1748307
--- /dev/null
+++ b/tasks/construct-mc-all-interval-l6-s0/tests/check.py
@@ -0,0 +1,17 @@
+"""All-interval series of size n (MathConstraint all_interval, CC-BY-4.0). Accepts any valid series."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n = inst["n"]
+    x = int_list(ans.get("x"), n, 0, n - 1, "x")
+    if sorted(x) != list(range(n)):
+        raise Invalid("x is not a permutation of 0..n-1")
+    d = [abs(x[i + 1] - x[i]) for i in range(n - 1)]
+    if sorted(d) != list(range(1, n)):
+        raise Invalid("consecutive differences are not a permutation of 1..n-1")
+    seqs = {"x": x, "d": d}
+    for name, i, v in inst.get("fixed", []):
+        if seqs[name][i] != v:
+            raise Invalid(f"fixed value {name}[{i}] = {v} not respected")
+    return None, {}
diff --git a/tasks/construct-mc-all-interval-l6-s0/tests/instance.json b/tasks/construct-mc-all-interval-l6-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a3b9a7ddd9aac9fec27db1f35e34af7d47d3d536
--- /dev/null
+++ b/tasks/construct-mc-all-interval-l6-s0/tests/instance.json
@@ -0,0 +1 @@
+{"n": 72, "fixed": [], "family": "mc_all_interval", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-all-interval-l6-s0/tests/test.sh b/tasks/construct-mc-all-interval-l6-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-all-interval-l6-s0/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-all-interval-l6-s7/environment/Dockerfile b/tasks/construct-mc-all-interval-l6-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-all-interval-l6-s7/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-all-interval-l6-s7/environment/instance.json b/tasks/construct-mc-all-interval-l6-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8122102536afa159ddb701212c7eddb68800e431
--- /dev/null
+++ b/tasks/construct-mc-all-interval-l6-s7/environment/instance.json
@@ -0,0 +1 @@
+{"n": 120, "fixed": [], "family": "mc_all_interval", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-all-interval-l6-s7/solution/solve.sh b/tasks/construct-mc-all-interval-l6-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..bea5adf3ed7b683e5853a362c546103cb6535aab
--- /dev/null
+++ b/tasks/construct-mc-all-interval-l6-s7/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": [0, 119, 1, 118, 2, 117, 3, 116, 4, 115, 5, 114, 6, 113, 7, 112, 8, 111, 9, 110, 10, 109, 11, 108, 12, 107, 13, 106, 14, 105, 15, 104, 16, 103, 17, 102, 18, 101, 19, 100, 20, 99, 21, 98, 22, 97, 23, 96, 24, 95, 25, 94, 26, 93, 27, 92, 28, 91, 29, 90, 30, 89, 31, 88, 32, 87, 33, 86, 34, 85, 35, 84, 36, 83, 37, 82, 38, 81, 39, 80, 40, 79, 41, 78, 42, 77, 43, 76, 44, 75, 45, 74, 46, 73, 47, 72, 48, 71, 49, 70, 50, 69, 51, 68, 52, 67, 53, 66, 54, 65, 55, 64, 56, 63, 57, 62, 58, 61, 59, 60]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-all-interval-l6-s7/tests/check.py b/tasks/construct-mc-all-interval-l6-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2e39eec264b6422a47e35921b8cdbe1fa1748307
--- /dev/null
+++ b/tasks/construct-mc-all-interval-l6-s7/tests/check.py
@@ -0,0 +1,17 @@
+"""All-interval series of size n (MathConstraint all_interval, CC-BY-4.0). Accepts any valid series."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n = inst["n"]
+    x = int_list(ans.get("x"), n, 0, n - 1, "x")
+    if sorted(x) != list(range(n)):
+        raise Invalid("x is not a permutation of 0..n-1")
+    d = [abs(x[i + 1] - x[i]) for i in range(n - 1)]
+    if sorted(d) != list(range(1, n)):
+        raise Invalid("consecutive differences are not a permutation of 1..n-1")
+    seqs = {"x": x, "d": d}
+    for name, i, v in inst.get("fixed", []):
+        if seqs[name][i] != v:
+            raise Invalid(f"fixed value {name}[{i}] = {v} not respected")
+    return None, {}
diff --git a/tasks/construct-mc-all-interval-l6-s7/tests/instance.json b/tasks/construct-mc-all-interval-l6-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8122102536afa159ddb701212c7eddb68800e431
--- /dev/null
+++ b/tasks/construct-mc-all-interval-l6-s7/tests/instance.json
@@ -0,0 +1 @@
+{"n": 120, "fixed": [], "family": "mc_all_interval", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-all-interval-l6-s7/tests/run.py b/tasks/construct-mc-all-interval-l6-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-all-interval-l6-s7/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-all-interval-l6-s7/tests/test.sh b/tasks/construct-mc-all-interval-l6-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-all-interval-l6-s7/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-bibd-l5-s3/environment/Dockerfile b/tasks/construct-mc-bibd-l5-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-bibd-l5-s3/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-bibd-l5-s3/environment/instance.json b/tasks/construct-mc-bibd-l5-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b84a5d4159c814b7f1e347d36311d4b126693591
--- /dev/null
+++ b/tasks/construct-mc-bibd-l5-s3/environment/instance.json
@@ -0,0 +1 @@
+{"v": 18, "k": 3, "lambda": 2, "b": 102, "family": "mc_bibd", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-bibd-l5-s3/solution/solve.sh b/tasks/construct-mc-bibd-l5-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..1568da9d9b1de2941a6ccf1fc13315259b56623a
--- /dev/null
+++ b/tasks/construct-mc-bibd-l5-s3/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"blocks": [[4, 15, 16], [3, 13, 14], [0, 3, 7], [4, 5, 13], [5, 10, 17], [4, 7, 10], [6, 16, 17], [3, 4, 17], [0, 4, 10], [1, 10, 16], [1, 13, 17], [2, 7, 10], [10, 12, 16], [11, 13, 15], [8, 13, 17], [1, 4, 9], [0, 4, 16], [6, 7, 11], [4, 8, 12], [5, 11, 16], [4, 8, 15], [1, 2, 11], [2, 4, 13], [2, 6, 17], [1, 2, 16], [11, 14, 17], [4, 6, 11], [1, 3, 4], [3, 6, 13], [1, 14, 15], [1, 3, 12], [1, 12, 13], [0, 12, 17], [0, 3, 17], [1, 5, 7], [1, 5, 6], [5, 12, 13], [7, 13, 15], [1, 6, 8], [7, 8, 11], [1, 7, 8], [8, 9, 12], [1, 9, 17], [0, 1, 14], [6, 9, 15], [3, 6, 7], [3, 10, 15], [3, 5, 9], [3, 9, 11], [9, 12, 15], [9, 13, 16], [0, 13, 16], [0, 10, 14], [2, 7, 12], [6, 10, 13], [10, 11, 12], [8, 10, 15], [6, 12, 15], [1, 10, 11], [6, 8, 14], [3, 8, 16], [11, 12, 16], [11, 15, 17], [2, 6, 16], [2, 3, 8], [3, 15, 16], [8, 14, 16], [7, 13, 14], [0, 11, 13], [0, 6, 12], [8, 9, 11], [0, 5, 11], [6, 10, 14], [0, 1, 15], [7, 15, 17], [7, 12, 14], [5, 14, 15], [3, 5, 10], [3, 12, 14], [5, 7, 16], [4, 7, 9], [5, 9, 14], [4, 5, 6], [9, 14, 16], [0, 7, 9], [0, 2, 15], [0, 2, 8], [2, 14, 17], [5, 8, 17], [7, 16, 17], [4, 11, 14], [0, 6, 9], [2, 4, 14], [2, 5, 12], [8, 10, 13], [2, 9, 13], [9, 10, 17], [0, 5, 8], [4, 12, 17], [2, 9, 10], [2, 5, 15], [2, 3, 11]]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-bibd-l5-s3/tests/check.py b/tasks/construct-mc-bibd-l5-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..c4b5da62477026f27ff9869fe33602fcc41e2ea9
--- /dev/null
+++ b/tasks/construct-mc-bibd-l5-s3/tests/check.py
@@ -0,0 +1,25 @@
+"""Balanced incomplete block design BIBD(v, k, lambda) (MathConstraint bibd, CC-BY-4.0).
+Answer = list of b blocks; every block is a k-subset of {0..v-1}, every pair of points lies in exactly
+lambda blocks (which forces every point to lie in r blocks). Repeated blocks are allowed, as upstream
+(its 0/1 incidence-matrix model does not forbid equal columns)."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    v, k, lam, b = inst["v"], inst["k"], inst["lambda"], inst["b"]
+    blocks = ans.get("blocks")
+    if not isinstance(blocks, list) or len(blocks) != b:
+        raise Invalid(f"blocks must be a list of exactly {b} blocks")
+    pair = {}
+    for B in blocks:
+        B = int_list(B, k, 0, v - 1, "block")
+        if len(set(B)) != k:
+            raise Invalid("a block contains a repeated point")
+        for p in itertools.combinations(sorted(B), 2):
+            pair[p] = pair.get(p, 0) + 1
+    for p in itertools.combinations(range(v), 2):
+        if pair.get(p, 0) != lam:
+            raise Invalid(f"points {p} lie together in {pair.get(p, 0)} blocks, need {lam}")
+    return None, {}
diff --git a/tasks/construct-mc-bibd-l5-s3/tests/instance.json b/tasks/construct-mc-bibd-l5-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b84a5d4159c814b7f1e347d36311d4b126693591
--- /dev/null
+++ b/tasks/construct-mc-bibd-l5-s3/tests/instance.json
@@ -0,0 +1 @@
+{"v": 18, "k": 3, "lambda": 2, "b": 102, "family": "mc_bibd", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-bibd-l5-s3/tests/run.py b/tasks/construct-mc-bibd-l5-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-bibd-l5-s3/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-bibd-l5-s3/tests/test.sh b/tasks/construct-mc-bibd-l5-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-bibd-l5-s3/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-bibd-l5-s4/environment/Dockerfile b/tasks/construct-mc-bibd-l5-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-bibd-l5-s4/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-bibd-l5-s4/environment/instance.json b/tasks/construct-mc-bibd-l5-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c054da1cb0164f024c5808cb0797b6440ee4323a
--- /dev/null
+++ b/tasks/construct-mc-bibd-l5-s4/environment/instance.json
@@ -0,0 +1 @@
+{"v": 23, "k": 11, "lambda": 5, "b": 23, "family": "mc_bibd", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-bibd-l5-s4/solution/solve.sh b/tasks/construct-mc-bibd-l5-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..bf6d9f86950dc3d10a7cf30553f27768578ec541
--- /dev/null
+++ b/tasks/construct-mc-bibd-l5-s4/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"blocks": [[1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18], [2, 3, 4, 5, 7, 9, 10, 13, 14, 17, 19], [3, 4, 5, 6, 8, 10, 11, 14, 15, 18, 20], [4, 5, 6, 7, 9, 11, 12, 15, 16, 19, 21], [5, 6, 7, 8, 10, 12, 13, 16, 17, 20, 22], [0, 6, 7, 8, 9, 11, 13, 14, 17, 18, 21], [1, 7, 8, 9, 10, 12, 14, 15, 18, 19, 22], [0, 2, 8, 9, 10, 11, 13, 15, 16, 19, 20], [1, 3, 9, 10, 11, 12, 14, 16, 17, 20, 21], [2, 4, 10, 11, 12, 13, 15, 17, 18, 21, 22], [0, 3, 5, 11, 12, 13, 14, 16, 18, 19, 22], [0, 1, 4, 6, 12, 13, 14, 15, 17, 19, 20], [1, 2, 5, 7, 13, 14, 15, 16, 18, 20, 21], [2, 3, 6, 8, 14, 15, 16, 17, 19, 21, 22], [0, 3, 4, 7, 9, 15, 16, 17, 18, 20, 22], [0, 1, 4, 5, 8, 10, 16, 17, 18, 19, 21], [1, 2, 5, 6, 9, 11, 17, 18, 19, 20, 22], [0, 2, 3, 6, 7, 10, 12, 18, 19, 20, 21], [1, 3, 4, 7, 8, 11, 13, 19, 20, 21, 22], [0, 2, 4, 5, 8, 9, 12, 14, 20, 21, 22], [0, 1, 3, 5, 6, 9, 10, 13, 15, 21, 22], [0, 1, 2, 4, 6, 7, 10, 11, 14, 16, 22], [0, 1, 2, 3, 5, 7, 8, 11, 12, 15, 17]]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-bibd-l5-s4/tests/check.py b/tasks/construct-mc-bibd-l5-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..c4b5da62477026f27ff9869fe33602fcc41e2ea9
--- /dev/null
+++ b/tasks/construct-mc-bibd-l5-s4/tests/check.py
@@ -0,0 +1,25 @@
+"""Balanced incomplete block design BIBD(v, k, lambda) (MathConstraint bibd, CC-BY-4.0).
+Answer = list of b blocks; every block is a k-subset of {0..v-1}, every pair of points lies in exactly
+lambda blocks (which forces every point to lie in r blocks). Repeated blocks are allowed, as upstream
+(its 0/1 incidence-matrix model does not forbid equal columns)."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    v, k, lam, b = inst["v"], inst["k"], inst["lambda"], inst["b"]
+    blocks = ans.get("blocks")
+    if not isinstance(blocks, list) or len(blocks) != b:
+        raise Invalid(f"blocks must be a list of exactly {b} blocks")
+    pair = {}
+    for B in blocks:
+        B = int_list(B, k, 0, v - 1, "block")
+        if len(set(B)) != k:
+            raise Invalid("a block contains a repeated point")
+        for p in itertools.combinations(sorted(B), 2):
+            pair[p] = pair.get(p, 0) + 1
+    for p in itertools.combinations(range(v), 2):
+        if pair.get(p, 0) != lam:
+            raise Invalid(f"points {p} lie together in {pair.get(p, 0)} blocks, need {lam}")
+    return None, {}
diff --git a/tasks/construct-mc-bibd-l5-s4/tests/instance.json b/tasks/construct-mc-bibd-l5-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c054da1cb0164f024c5808cb0797b6440ee4323a
--- /dev/null
+++ b/tasks/construct-mc-bibd-l5-s4/tests/instance.json
@@ -0,0 +1 @@
+{"v": 23, "k": 11, "lambda": 5, "b": 23, "family": "mc_bibd", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-bibd-l5-s4/tests/run.py b/tasks/construct-mc-bibd-l5-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-bibd-l5-s4/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-bibd-l5-s4/tests/test.sh b/tasks/construct-mc-bibd-l5-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-bibd-l5-s4/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-graceful-graph-l5-s1/environment/Dockerfile b/tasks/construct-mc-graceful-graph-l5-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-graceful-graph-l5-s1/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-graceful-graph-l5-s1/environment/instance.json b/tasks/construct-mc-graceful-graph-l5-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..96a594eca3a3488a5c7cd66192f61a832d85597e
--- /dev/null
+++ b/tasks/construct-mc-graceful-graph-l5-s1/environment/instance.json
@@ -0,0 +1 @@
+{"k": 4, "p": 5, "num_vertices": 20, "edges": [[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3], [4, 5], [4, 6], [4, 7], [5, 6], [5, 7], [6, 7], [8, 9], [8, 10], [8, 11], [9, 10], [9, 11], [10, 11], [12, 13], [12, 14], [12, 15], [13, 14], [13, 15], [14, 15], [16, 17], [16, 18], [16, 19], [17, 18], [17, 19], [18, 19], [0, 4], [1, 5], [2, 6], [3, 7], [4, 8], [5, 9], [6, 10], [7, 11], [8, 12], [9, 13], [10, 14], [11, 15], [12, 16], [13, 17], [14, 18], [15, 19]], "fixed": [], "family": "mc_graceful_graph", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-graceful-graph-l5-s1/solution/solve.sh b/tasks/construct-mc-graceful-graph-l5-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..2f0a975a227978fe405fa2d1e2b8d7db10b93994
--- /dev/null
+++ b/tasks/construct-mc-graceful-graph-l5-s1/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"labels": [8, 1, 26, 45, 40, 6, 2, 0, 4, 3, 31, 46, 27, 44, 5, 13, 39, 9, 18, 29]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-graceful-graph-l5-s1/tests/check.py b/tasks/construct-mc-graceful-graph-l5-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..f4d33af1d63c0495c3b32d34db6ae10f0fb45923
--- /dev/null
+++ b/tasks/construct-mc-graceful-graph-l5-s1/tests/check.py
@@ -0,0 +1,19 @@
+"""Graceful labelling of K_k x P_p (MathConstraint graceful_graph, CC-BY-4.0): distinct vertex labels in
+0..m (m = number of edges) whose edge differences |l(u) - l(v)| are all distinct."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    nv, edges = inst["num_vertices"], inst["edges"]
+    m = len(edges)
+    x = int_list(ans.get("labels"), nv, 0, m, "labels")
+    if len(set(x)) != nv:
+        raise Invalid("vertex labels must be distinct")
+    d = [abs(x[u] - x[v]) for u, v in edges]
+    if len(set(d)) != m:
+        raise Invalid("edge labels are not distinct")
+    seqs = {"labels": x, "d": d}
+    for name, i, v in inst.get("fixed", []):
+        if seqs[name][i] != v:
+            raise Invalid(f"fixed value {name}[{i}] = {v} not respected")
+    return None, {}
diff --git a/tasks/construct-mc-graceful-graph-l5-s1/tests/instance.json b/tasks/construct-mc-graceful-graph-l5-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..96a594eca3a3488a5c7cd66192f61a832d85597e
--- /dev/null
+++ b/tasks/construct-mc-graceful-graph-l5-s1/tests/instance.json
@@ -0,0 +1 @@
+{"k": 4, "p": 5, "num_vertices": 20, "edges": [[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3], [4, 5], [4, 6], [4, 7], [5, 6], [5, 7], [6, 7], [8, 9], [8, 10], [8, 11], [9, 10], [9, 11], [10, 11], [12, 13], [12, 14], [12, 15], [13, 14], [13, 15], [14, 15], [16, 17], [16, 18], [16, 19], [17, 18], [17, 19], [18, 19], [0, 4], [1, 5], [2, 6], [3, 7], [4, 8], [5, 9], [6, 10], [7, 11], [8, 12], [9, 13], [10, 14], [11, 15], [12, 16], [13, 17], [14, 18], [15, 19]], "fixed": [], "family": "mc_graceful_graph", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-graceful-graph-l5-s1/tests/run.py b/tasks/construct-mc-graceful-graph-l5-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-graceful-graph-l5-s1/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-graceful-graph-l5-s1/tests/test.sh b/tasks/construct-mc-graceful-graph-l5-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-graceful-graph-l5-s1/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-graph-k-coloring-l1-s1/environment/Dockerfile b/tasks/construct-mc-graph-k-coloring-l1-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l1-s1/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-graph-k-coloring-l1-s1/environment/instance.json b/tasks/construct-mc-graph-k-coloring-l1-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..978e9d809e4524893bce20c656e9b9c49414ad1d
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l1-s1/environment/instance.json
@@ -0,0 +1 @@
+{"n": 20, "k": 3, "edges": [[0, 9], [1, 10], [1, 11], [1, 19], [2, 3], [2, 6], [3, 4], [3, 9], [3, 19], [4, 10], [4, 12], [4, 13], [4, 14], [4, 15], [5, 10], [5, 13], [6, 8], [7, 8], [8, 12], [8, 19], [9, 11], [10, 13], [10, 14], [10, 19], [11, 14], [12, 13], [12, 18], [12, 19], [13, 18], [15, 17]], "family": "mc_graph_k_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-graph-k-coloring-l1-s1/solution/solve.sh b/tasks/construct-mc-graph-k-coloring-l1-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..fee6516f6cf7a2f1f66f734018843be68adf07e5
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l1-s1/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"coloring": [2, 1, 0, 2, 1, 1, 2, 0, 1, 0, 2, 2, 2, 0, 0, 0, 1, 1, 1, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-graph-k-coloring-l1-s1/tests/check.py b/tasks/construct-mc-graph-k-coloring-l1-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2b5f38cd0c22ac1b4ba37c0d3ff0a34635f08854
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l1-s1/tests/check.py
@@ -0,0 +1,11 @@
+"""Proper k-colouring of a given graph (MathConstraint graph_k_coloring, CC-BY-4.0). Any valid colouring."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    c = int_list(ans.get("coloring"), n, 0, k - 1, "coloring")
+    for u, v in inst["edges"]:
+        if c[u] == c[v]:
+            raise Invalid(f"edge ({u},{v}) is monochromatic")
+    return None, {}
diff --git a/tasks/construct-mc-graph-k-coloring-l1-s1/tests/instance.json b/tasks/construct-mc-graph-k-coloring-l1-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..978e9d809e4524893bce20c656e9b9c49414ad1d
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l1-s1/tests/instance.json
@@ -0,0 +1 @@
+{"n": 20, "k": 3, "edges": [[0, 9], [1, 10], [1, 11], [1, 19], [2, 3], [2, 6], [3, 4], [3, 9], [3, 19], [4, 10], [4, 12], [4, 13], [4, 14], [4, 15], [5, 10], [5, 13], [6, 8], [7, 8], [8, 12], [8, 19], [9, 11], [10, 13], [10, 14], [10, 19], [11, 14], [12, 13], [12, 18], [12, 19], [13, 18], [15, 17]], "family": "mc_graph_k_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-graph-k-coloring-l1-s1/tests/run.py b/tasks/construct-mc-graph-k-coloring-l1-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l1-s1/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-graph-k-coloring-l1-s1/tests/test.sh b/tasks/construct-mc-graph-k-coloring-l1-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l1-s1/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-graph-k-coloring-l1-s6/environment/Dockerfile b/tasks/construct-mc-graph-k-coloring-l1-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l1-s6/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-graph-k-coloring-l1-s6/environment/instance.json b/tasks/construct-mc-graph-k-coloring-l1-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..83cd83ffce53baf06160377ff9b4f96c8a3dd8f0
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l1-s6/environment/instance.json
@@ -0,0 +1 @@
+{"n": 20, "k": 3, "edges": [[0, 2], [0, 4], [0, 9], [0, 15], [0, 19], [1, 17], [1, 19], [2, 16], [3, 9], [3, 11], [3, 19], [4, 8], [4, 14], [4, 17], [5, 7], [5, 15], [6, 11], [6, 16], [7, 10], [7, 12], [7, 15], [7, 16], [7, 19], [10, 14], [10, 16], [10, 18], [11, 19], [12, 16], [13, 14], [15, 18]], "family": "mc_graph_k_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-graph-k-coloring-l1-s6/solution/solve.sh b/tasks/construct-mc-graph-k-coloring-l1-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..fbd14998271dd5ba1726ab200f11e2e7e3c6b9c8
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l1-s6/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"coloring": [1, 0, 2, 1, 0, 0, 2, 1, 1, 0, 2, 0, 2, 0, 1, 2, 0, 1, 1, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-graph-k-coloring-l1-s6/tests/check.py b/tasks/construct-mc-graph-k-coloring-l1-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2b5f38cd0c22ac1b4ba37c0d3ff0a34635f08854
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l1-s6/tests/check.py
@@ -0,0 +1,11 @@
+"""Proper k-colouring of a given graph (MathConstraint graph_k_coloring, CC-BY-4.0). Any valid colouring."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    c = int_list(ans.get("coloring"), n, 0, k - 1, "coloring")
+    for u, v in inst["edges"]:
+        if c[u] == c[v]:
+            raise Invalid(f"edge ({u},{v}) is monochromatic")
+    return None, {}
diff --git a/tasks/construct-mc-graph-k-coloring-l1-s6/tests/instance.json b/tasks/construct-mc-graph-k-coloring-l1-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..83cd83ffce53baf06160377ff9b4f96c8a3dd8f0
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l1-s6/tests/instance.json
@@ -0,0 +1 @@
+{"n": 20, "k": 3, "edges": [[0, 2], [0, 4], [0, 9], [0, 15], [0, 19], [1, 17], [1, 19], [2, 16], [3, 9], [3, 11], [3, 19], [4, 8], [4, 14], [4, 17], [5, 7], [5, 15], [6, 11], [6, 16], [7, 10], [7, 12], [7, 15], [7, 16], [7, 19], [10, 14], [10, 16], [10, 18], [11, 19], [12, 16], [13, 14], [15, 18]], "family": "mc_graph_k_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-graph-k-coloring-l1-s6/tests/run.py b/tasks/construct-mc-graph-k-coloring-l1-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l1-s6/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-graph-k-coloring-l1-s6/tests/test.sh b/tasks/construct-mc-graph-k-coloring-l1-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l1-s6/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-graph-k-coloring-l5-s0/environment/Dockerfile b/tasks/construct-mc-graph-k-coloring-l5-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l5-s0/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-graph-k-coloring-l5-s0/environment/instance.json b/tasks/construct-mc-graph-k-coloring-l5-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c8613a9fcf3ef27219325b59a4dcc69fa52ec85c
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l5-s0/environment/instance.json
@@ -0,0 +1 @@
+{"n": 100, "k": 4, "edges": [[0, 5], [0, 11], [0, 16], [0, 35], [0, 45], [0, 64], [0, 89], [0, 90], [0, 91], [1, 30], [1, 38], [1, 40], [1, 44], [1, 93], [2, 7], [2, 9], [2, 10], [2, 12], [2, 26], [2, 37], [2, 42], [2, 83], [3, 24], [3, 30], [3, 47], [3, 55], [3, 58], [3, 78], [3, 87], [4, 16], [4, 20], [4, 36], [4, 51], [4, 52], [4, 64], [4, 75], [4, 80], [4, 93], [4, 94], [5, 8], [5, 47], [5, 52], [5, 58], [5, 84], [5, 92], [5, 99], [6, 11], [6, 16], [6, 22], [6, 28], [6, 37], [6, 70], [6, 82], [6, 83], [6, 88], [6, 97], [7, 8], [7, 12], [7, 15], [7, 23], [7, 86], [7, 87], [7, 95], [8, 12], [8, 14], [8, 31], [8, 32], [8, 71], [8, 89], [8, 93], [9, 12], [9, 15], [9, 18], [9, 39], [9, 42], [9, 55], [9, 57], [9, 69], [9, 96], [10, 23], [10, 49], [10, 57], [11, 20], [11, 22], [11, 26], [11, 32], [11, 43], [11, 46], [11, 55], [11, 66], [11, 79], [11, 84], [12, 16], [12, 22], [12, 30], [12, 44], [12, 58], [12, 73], [12, 90], [12, 98], [13, 40], [13, 53], [13, 63], [13, 69], [13, 77], [14, 24], [14, 48], [14, 54], [14, 57], [14, 71], [14, 75], [14, 83], [15, 87], [16, 26], [16, 33], [16, 41], [16, 56], [16, 58], [16, 61], [16, 91], [16, 92], [17, 45], [17, 69], [18, 23], [18, 26], [18, 27], [18, 33], [18, 36], [18, 40], [18, 64], [18, 66], [18, 67], [18, 73], [18, 75], [18, 82], [18, 94], [19, 22], [19, 38], [19, 48], [19, 58], [19, 71], [19, 73], [19, 87], [20, 35], [20, 48], [20, 60], [20, 66], [20, 76], [21, 24], [21, 34], [21, 39], [21, 63], [22, 35], [22, 36], [22, 38], [22, 56], [22, 59], [22, 61], [22, 62], [22, 63], [22, 71], [22, 72], [22, 81], [23, 26], [23, 65], [23, 71], [23, 74], [23, 76], [23, 78], [23, 87], [24, 32], [24, 38], [24, 52], [25, 37], [25, 45], [25, 46], [25, 53], [25, 60], [25, 82], [26, 44], [26, 51], [26, 53], [26, 64], [26, 80], [26, 86], [26, 95], [27, 43], [27, 59], [27, 76], [28, 47], [28, 51], [28, 76], [28, 85], [28, 90], [28, 93], [29, 42], [29, 47], [29, 55], [29, 57], [29, 77], [29, 88], [29, 89], [29, 99], [30, 36], [30, 59], [30, 60], [30, 61], [30, 66], [30, 67], [30, 86], [30, 87], [30, 89], [30, 90], [30, 93], [30, 97], [30, 99], [31, 33], [31, 42], [31, 58], [31, 59], [31, 62], [31, 89], [31, 90], [32, 40], [32, 89], [32, 90], [33, 37], [33, 79], [34, 48], [34, 91], [35, 52], [35, 58], [35, 66], [35, 73], [35, 78], [36, 44], [36, 45], [36, 70], [36, 81], [36, 96], [36, 98], [37, 38], [37, 57], [37, 59], [37, 68], [37, 83], [37, 94], [37, 99], [38, 62], [38, 68], [38, 78], [38, 79], [39, 45], [39, 50], [39, 55], [39, 72], [39, 77], [41, 49], [41, 55], [41, 60], [41, 71], [41, 96], [42, 53], [42, 57], [42, 59], [42, 75], [42, 77], [43, 57], [43, 60], [43, 62], [43, 66], [43, 72], [43, 77], [43, 89], [43, 94], [43, 99], [44, 45], [44, 46], [44, 65], [44, 78], [44, 93], [44, 94], [45, 52], [45, 60], [45, 74], [45, 79], [45, 86], [45, 92], [45, 95], [45, 96], [46, 56], [47, 66], [47, 72], [47, 73], [47, 77], [47, 82], [48, 58], [48, 62], [48, 78], [48, 85], [48, 99], [49, 56], [49, 69], [49, 81], [49, 89], [50, 57], [50, 60], [50, 63], [50, 65], [50, 66], [50, 96], [50, 98], [51, 53], [51, 89], [51, 96], [52, 54], [52, 80], [53, 67], [53, 96], [53, 97], [54, 88], [54, 96], [55, 91], [55, 96], [55, 97], [56, 57], [56, 71], [56, 77], [57, 74], [57, 78], [57, 87], [58, 67], [58, 88], [58, 96], [58, 99], [59, 86], [59, 99], [60, 67], [60, 68], [60, 72], [60, 91], [60, 92], [61, 67], [62, 63], [62, 68], [62, 78], [62, 95], [63, 79], [63, 90], [63, 92], [64, 79], [65, 80], [65, 81], [65, 86], [65, 88], [65, 99], [66, 80], [67, 72], [67, 83], [67, 94], [67, 96], [68, 71], [68, 75], [68, 80], [68, 98], [69, 72], [70, 85], [70, 93], [70, 96], [71, 86], [72, 86], [72, 87], [72, 90], [73, 82], [73, 93], [73, 95], [73, 96], [73, 99], [74, 77], [75, 83], [76, 78], [78, 80], [78, 81], [78, 96], [79, 81], [80, 86], [82, 86], [84, 86], [85, 93], [86, 89], [86, 96], [86, 99], [87, 89], [88, 96], [90, 98], [92, 97], [93, 98], [94, 95], [95, 98], [97, 99]], "family": "mc_graph_k_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-graph-k-coloring-l5-s0/solution/solve.sh b/tasks/construct-mc-graph-k-coloring-l5-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..1772ee6196e20ca7d0822de9e805e56d72bedaef
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l5-s0/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"coloring": [3, 2, 2, 3, 0, 2, 0, 0, 1, 1, 0, 1, 3, 0, 2, 3, 2, 2, 0, 0, 2, 3, 2, 1, 1, 0, 3, 1, 1, 1, 1, 2, 2, 3, 1, 0, 3, 1, 3, 0, 3, 1, 0, 2, 0, 1, 2, 0, 3, 2, 2, 2, 3, 1, 1, 2, 1, 3, 1, 3, 3, 0, 0, 1, 1, 3, 3, 2, 2, 3, 2, 3, 1, 1, 0, 1, 0, 3, 2, 2, 1, 1, 3, 3, 0, 0, 2, 2, 2, 0, 0, 0, 0, 3, 3, 2, 0, 3, 1, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-graph-k-coloring-l5-s0/tests/check.py b/tasks/construct-mc-graph-k-coloring-l5-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2b5f38cd0c22ac1b4ba37c0d3ff0a34635f08854
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l5-s0/tests/check.py
@@ -0,0 +1,11 @@
+"""Proper k-colouring of a given graph (MathConstraint graph_k_coloring, CC-BY-4.0). Any valid colouring."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    c = int_list(ans.get("coloring"), n, 0, k - 1, "coloring")
+    for u, v in inst["edges"]:
+        if c[u] == c[v]:
+            raise Invalid(f"edge ({u},{v}) is monochromatic")
+    return None, {}
diff --git a/tasks/construct-mc-graph-k-coloring-l5-s0/tests/instance.json b/tasks/construct-mc-graph-k-coloring-l5-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c8613a9fcf3ef27219325b59a4dcc69fa52ec85c
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l5-s0/tests/instance.json
@@ -0,0 +1 @@
+{"n": 100, "k": 4, "edges": [[0, 5], [0, 11], [0, 16], [0, 35], [0, 45], [0, 64], [0, 89], [0, 90], [0, 91], [1, 30], [1, 38], [1, 40], [1, 44], [1, 93], [2, 7], [2, 9], [2, 10], [2, 12], [2, 26], [2, 37], [2, 42], [2, 83], [3, 24], [3, 30], [3, 47], [3, 55], [3, 58], [3, 78], [3, 87], [4, 16], [4, 20], [4, 36], [4, 51], [4, 52], [4, 64], [4, 75], [4, 80], [4, 93], [4, 94], [5, 8], [5, 47], [5, 52], [5, 58], [5, 84], [5, 92], [5, 99], [6, 11], [6, 16], [6, 22], [6, 28], [6, 37], [6, 70], [6, 82], [6, 83], [6, 88], [6, 97], [7, 8], [7, 12], [7, 15], [7, 23], [7, 86], [7, 87], [7, 95], [8, 12], [8, 14], [8, 31], [8, 32], [8, 71], [8, 89], [8, 93], [9, 12], [9, 15], [9, 18], [9, 39], [9, 42], [9, 55], [9, 57], [9, 69], [9, 96], [10, 23], [10, 49], [10, 57], [11, 20], [11, 22], [11, 26], [11, 32], [11, 43], [11, 46], [11, 55], [11, 66], [11, 79], [11, 84], [12, 16], [12, 22], [12, 30], [12, 44], [12, 58], [12, 73], [12, 90], [12, 98], [13, 40], [13, 53], [13, 63], [13, 69], [13, 77], [14, 24], [14, 48], [14, 54], [14, 57], [14, 71], [14, 75], [14, 83], [15, 87], [16, 26], [16, 33], [16, 41], [16, 56], [16, 58], [16, 61], [16, 91], [16, 92], [17, 45], [17, 69], [18, 23], [18, 26], [18, 27], [18, 33], [18, 36], [18, 40], [18, 64], [18, 66], [18, 67], [18, 73], [18, 75], [18, 82], [18, 94], [19, 22], [19, 38], [19, 48], [19, 58], [19, 71], [19, 73], [19, 87], [20, 35], [20, 48], [20, 60], [20, 66], [20, 76], [21, 24], [21, 34], [21, 39], [21, 63], [22, 35], [22, 36], [22, 38], [22, 56], [22, 59], [22, 61], [22, 62], [22, 63], [22, 71], [22, 72], [22, 81], [23, 26], [23, 65], [23, 71], [23, 74], [23, 76], [23, 78], [23, 87], [24, 32], [24, 38], [24, 52], [25, 37], [25, 45], [25, 46], [25, 53], [25, 60], [25, 82], [26, 44], [26, 51], [26, 53], [26, 64], [26, 80], [26, 86], [26, 95], [27, 43], [27, 59], [27, 76], [28, 47], [28, 51], [28, 76], [28, 85], [28, 90], [28, 93], [29, 42], [29, 47], [29, 55], [29, 57], [29, 77], [29, 88], [29, 89], [29, 99], [30, 36], [30, 59], [30, 60], [30, 61], [30, 66], [30, 67], [30, 86], [30, 87], [30, 89], [30, 90], [30, 93], [30, 97], [30, 99], [31, 33], [31, 42], [31, 58], [31, 59], [31, 62], [31, 89], [31, 90], [32, 40], [32, 89], [32, 90], [33, 37], [33, 79], [34, 48], [34, 91], [35, 52], [35, 58], [35, 66], [35, 73], [35, 78], [36, 44], [36, 45], [36, 70], [36, 81], [36, 96], [36, 98], [37, 38], [37, 57], [37, 59], [37, 68], [37, 83], [37, 94], [37, 99], [38, 62], [38, 68], [38, 78], [38, 79], [39, 45], [39, 50], [39, 55], [39, 72], [39, 77], [41, 49], [41, 55], [41, 60], [41, 71], [41, 96], [42, 53], [42, 57], [42, 59], [42, 75], [42, 77], [43, 57], [43, 60], [43, 62], [43, 66], [43, 72], [43, 77], [43, 89], [43, 94], [43, 99], [44, 45], [44, 46], [44, 65], [44, 78], [44, 93], [44, 94], [45, 52], [45, 60], [45, 74], [45, 79], [45, 86], [45, 92], [45, 95], [45, 96], [46, 56], [47, 66], [47, 72], [47, 73], [47, 77], [47, 82], [48, 58], [48, 62], [48, 78], [48, 85], [48, 99], [49, 56], [49, 69], [49, 81], [49, 89], [50, 57], [50, 60], [50, 63], [50, 65], [50, 66], [50, 96], [50, 98], [51, 53], [51, 89], [51, 96], [52, 54], [52, 80], [53, 67], [53, 96], [53, 97], [54, 88], [54, 96], [55, 91], [55, 96], [55, 97], [56, 57], [56, 71], [56, 77], [57, 74], [57, 78], [57, 87], [58, 67], [58, 88], [58, 96], [58, 99], [59, 86], [59, 99], [60, 67], [60, 68], [60, 72], [60, 91], [60, 92], [61, 67], [62, 63], [62, 68], [62, 78], [62, 95], [63, 79], [63, 90], [63, 92], [64, 79], [65, 80], [65, 81], [65, 86], [65, 88], [65, 99], [66, 80], [67, 72], [67, 83], [67, 94], [67, 96], [68, 71], [68, 75], [68, 80], [68, 98], [69, 72], [70, 85], [70, 93], [70, 96], [71, 86], [72, 86], [72, 87], [72, 90], [73, 82], [73, 93], [73, 95], [73, 96], [73, 99], [74, 77], [75, 83], [76, 78], [78, 80], [78, 81], [78, 96], [79, 81], [80, 86], [82, 86], [84, 86], [85, 93], [86, 89], [86, 96], [86, 99], [87, 89], [88, 96], [90, 98], [92, 97], [93, 98], [94, 95], [95, 98], [97, 99]], "family": "mc_graph_k_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-graph-k-coloring-l5-s0/tests/run.py b/tasks/construct-mc-graph-k-coloring-l5-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l5-s0/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-graph-k-coloring-l5-s0/tests/test.sh b/tasks/construct-mc-graph-k-coloring-l5-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l5-s0/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-graph-k-coloring-l5-s9/environment/Dockerfile b/tasks/construct-mc-graph-k-coloring-l5-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l5-s9/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-graph-k-coloring-l5-s9/environment/instance.json b/tasks/construct-mc-graph-k-coloring-l5-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..9a87a27f08e8090645a07921f7595b90e87353e8
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l5-s9/environment/instance.json
@@ -0,0 +1 @@
+{"n": 100, "k": 4, "edges": [[0, 11], [0, 24], [0, 42], [0, 53], [0, 59], [0, 71], [0, 76], [0, 84], [0, 85], [0, 96], [1, 2], [1, 18], [1, 24], [1, 39], [1, 42], [1, 53], [1, 58], [1, 71], [1, 90], [1, 94], [2, 5], [2, 32], [2, 38], [2, 41], [2, 73], [2, 96], [3, 8], [3, 9], [3, 12], [3, 26], [3, 31], [3, 50], [3, 61], [3, 63], [3, 64], [3, 69], [4, 6], [4, 29], [4, 49], [4, 64], [4, 80], [5, 18], [5, 19], [5, 41], [5, 67], [5, 74], [5, 93], [5, 94], [6, 16], [6, 18], [6, 34], [6, 41], [6, 68], [6, 83], [6, 87], [7, 34], [7, 43], [7, 45], [7, 50], [7, 54], [7, 56], [7, 58], [7, 69], [7, 84], [7, 90], [8, 21], [8, 25], [8, 30], [8, 41], [8, 42], [8, 65], [8, 69], [8, 75], [8, 97], [8, 98], [9, 20], [9, 33], [9, 77], [9, 80], [10, 22], [10, 54], [10, 56], [10, 66], [10, 81], [10, 86], [10, 96], [11, 15], [11, 25], [11, 33], [11, 37], [11, 39], [11, 68], [11, 84], [12, 25], [12, 30], [12, 65], [12, 67], [12, 77], [12, 93], [13, 16], [13, 20], [13, 21], [13, 23], [13, 72], [13, 76], [13, 83], [14, 22], [14, 44], [14, 89], [14, 91], [15, 21], [15, 24], [15, 28], [15, 32], [15, 59], [15, 63], [15, 64], [15, 85], [15, 86], [15, 97], [16, 29], [16, 59], [16, 67], [16, 76], [16, 90], [16, 91], [16, 92], [17, 22], [17, 37], [17, 40], [17, 57], [17, 73], [17, 77], [18, 48], [18, 58], [18, 79], [18, 99], [19, 22], [19, 35], [19, 47], [19, 55], [19, 57], [19, 67], [19, 88], [20, 30], [20, 33], [20, 37], [20, 62], [20, 83], [21, 22], [21, 42], [21, 56], [21, 61], [21, 62], [21, 84], [21, 87], [22, 27], [22, 33], [22, 44], [22, 56], [22, 66], [22, 76], [22, 82], [22, 88], [22, 89], [22, 95], [23, 30], [23, 32], [23, 40], [23, 47], [23, 84], [23, 91], [23, 94], [24, 37], [24, 57], [24, 64], [24, 88], [25, 29], [25, 54], [25, 59], [25, 70], [25, 94], [26, 32], [26, 36], [26, 74], [26, 90], [26, 92], [27, 45], [27, 58], [28, 30], [28, 34], [28, 35], [28, 42], [28, 51], [28, 74], [28, 96], [29, 37], [29, 46], [29, 51], [29, 55], [29, 68], [29, 74], [29, 86], [29, 88], [30, 37], [30, 69], [30, 95], [31, 38], [31, 40], [31, 53], [31, 56], [31, 78], [31, 82], [31, 85], [32, 36], [32, 47], [32, 51], [32, 53], [32, 59], [32, 66], [32, 71], [32, 84], [32, 85], [33, 58], [33, 75], [33, 77], [33, 78], [33, 97], [34, 37], [34, 50], [34, 61], [35, 41], [35, 43], [35, 47], [35, 77], [35, 86], [35, 93], [36, 52], [36, 65], [36, 66], [36, 95], [36, 98], [37, 41], [37, 46], [37, 50], [37, 58], [37, 68], [37, 71], [37, 89], [38, 80], [39, 51], [39, 66], [40, 45], [40, 90], [40, 95], [40, 98], [41, 46], [41, 48], [41, 50], [41, 56], [41, 61], [41, 63], [41, 68], [41, 69], [41, 72], [41, 78], [41, 98], [42, 44], [42, 45], [42, 58], [42, 59], [42, 62], [43, 76], [43, 92], [43, 93], [43, 99], [44, 68], [44, 83], [45, 47], [45, 70], [45, 73], [45, 98], [45, 99], [46, 55], [46, 57], [46, 65], [46, 77], [47, 59], [47, 62], [48, 50], [48, 59], [48, 95], [49, 53], [49, 67], [49, 90], [49, 92], [50, 68], [50, 74], [50, 94], [51, 74], [51, 81], [51, 89], [52, 54], [52, 73], [52, 85], [52, 96], [52, 97], [53, 97], [54, 55], [54, 77], [54, 94], [55, 58], [55, 65], [55, 72], [55, 83], [55, 94], [55, 97], [56, 57], [56, 78], [56, 87], [57, 68], [57, 90], [58, 62], [58, 73], [58, 96], [59, 75], [59, 87], [60, 68], [60, 73], [60, 78], [60, 82], [60, 89], [60, 96], [61, 64], [61, 68], [61, 72], [61, 87], [62, 63], [62, 79], [63, 68], [63, 70], [63, 76], [63, 80], [63, 87], [64, 68], [64, 83], [64, 91], [64, 96], [65, 67], [65, 88], [66, 73], [66, 74], [66, 77], [66, 79], [67, 69], [67, 73], [67, 80], [67, 87], [68, 93], [68, 94], [68, 98], [69, 77], [69, 94], [70, 71], [70, 83], [70, 90], [71, 86], [72, 78], [72, 86], [73, 86], [74, 94], [75, 88], [75, 94], [76, 83], [77, 87], [77, 89], [77, 92], [78, 80], [78, 88], [79, 82], [79, 91], [80, 89], [80, 91], [80, 98], [82, 86], [83, 99], [84, 93], [84, 94], [85, 99], [87, 88], [87, 95], [87, 98], [87, 99], [88, 89], [88, 92], [88, 97], [89, 98], [90, 97], [93, 96], [94, 98]], "family": "mc_graph_k_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-graph-k-coloring-l5-s9/solution/solve.sh b/tasks/construct-mc-graph-k-coloring-l5-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..a73079e0df503adf5e484f15ff96eb93aeace0b4
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l5-s9/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"coloring": [0, 0, 3, 1, 2, 0, 3, 1, 3, 2, 1, 3, 0, 0, 1, 0, 2, 3, 2, 1, 1, 1, 2, 3, 1, 2, 0, 0, 1, 1, 2, 0, 1, 0, 2, 3, 2, 0, 2, 2, 1, 1, 2, 0, 3, 3, 3, 2, 0, 1, 3, 0, 0, 2, 3, 0, 3, 0, 3, 3, 0, 0, 0, 3, 3, 1, 0, 3, 2, 0, 1, 3, 3, 1, 2, 2, 1, 1, 2, 1, 1, 2, 3, 2, 2, 2, 2, 2, 0, 3, 3, 0, 3, 1, 1, 3, 2, 1, 0, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-graph-k-coloring-l5-s9/tests/check.py b/tasks/construct-mc-graph-k-coloring-l5-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2b5f38cd0c22ac1b4ba37c0d3ff0a34635f08854
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l5-s9/tests/check.py
@@ -0,0 +1,11 @@
+"""Proper k-colouring of a given graph (MathConstraint graph_k_coloring, CC-BY-4.0). Any valid colouring."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    c = int_list(ans.get("coloring"), n, 0, k - 1, "coloring")
+    for u, v in inst["edges"]:
+        if c[u] == c[v]:
+            raise Invalid(f"edge ({u},{v}) is monochromatic")
+    return None, {}
diff --git a/tasks/construct-mc-graph-k-coloring-l5-s9/tests/instance.json b/tasks/construct-mc-graph-k-coloring-l5-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..9a87a27f08e8090645a07921f7595b90e87353e8
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l5-s9/tests/instance.json
@@ -0,0 +1 @@
+{"n": 100, "k": 4, "edges": [[0, 11], [0, 24], [0, 42], [0, 53], [0, 59], [0, 71], [0, 76], [0, 84], [0, 85], [0, 96], [1, 2], [1, 18], [1, 24], [1, 39], [1, 42], [1, 53], [1, 58], [1, 71], [1, 90], [1, 94], [2, 5], [2, 32], [2, 38], [2, 41], [2, 73], [2, 96], [3, 8], [3, 9], [3, 12], [3, 26], [3, 31], [3, 50], [3, 61], [3, 63], [3, 64], [3, 69], [4, 6], [4, 29], [4, 49], [4, 64], [4, 80], [5, 18], [5, 19], [5, 41], [5, 67], [5, 74], [5, 93], [5, 94], [6, 16], [6, 18], [6, 34], [6, 41], [6, 68], [6, 83], [6, 87], [7, 34], [7, 43], [7, 45], [7, 50], [7, 54], [7, 56], [7, 58], [7, 69], [7, 84], [7, 90], [8, 21], [8, 25], [8, 30], [8, 41], [8, 42], [8, 65], [8, 69], [8, 75], [8, 97], [8, 98], [9, 20], [9, 33], [9, 77], [9, 80], [10, 22], [10, 54], [10, 56], [10, 66], [10, 81], [10, 86], [10, 96], [11, 15], [11, 25], [11, 33], [11, 37], [11, 39], [11, 68], [11, 84], [12, 25], [12, 30], [12, 65], [12, 67], [12, 77], [12, 93], [13, 16], [13, 20], [13, 21], [13, 23], [13, 72], [13, 76], [13, 83], [14, 22], [14, 44], [14, 89], [14, 91], [15, 21], [15, 24], [15, 28], [15, 32], [15, 59], [15, 63], [15, 64], [15, 85], [15, 86], [15, 97], [16, 29], [16, 59], [16, 67], [16, 76], [16, 90], [16, 91], [16, 92], [17, 22], [17, 37], [17, 40], [17, 57], [17, 73], [17, 77], [18, 48], [18, 58], [18, 79], [18, 99], [19, 22], [19, 35], [19, 47], [19, 55], [19, 57], [19, 67], [19, 88], [20, 30], [20, 33], [20, 37], [20, 62], [20, 83], [21, 22], [21, 42], [21, 56], [21, 61], [21, 62], [21, 84], [21, 87], [22, 27], [22, 33], [22, 44], [22, 56], [22, 66], [22, 76], [22, 82], [22, 88], [22, 89], [22, 95], [23, 30], [23, 32], [23, 40], [23, 47], [23, 84], [23, 91], [23, 94], [24, 37], [24, 57], [24, 64], [24, 88], [25, 29], [25, 54], [25, 59], [25, 70], [25, 94], [26, 32], [26, 36], [26, 74], [26, 90], [26, 92], [27, 45], [27, 58], [28, 30], [28, 34], [28, 35], [28, 42], [28, 51], [28, 74], [28, 96], [29, 37], [29, 46], [29, 51], [29, 55], [29, 68], [29, 74], [29, 86], [29, 88], [30, 37], [30, 69], [30, 95], [31, 38], [31, 40], [31, 53], [31, 56], [31, 78], [31, 82], [31, 85], [32, 36], [32, 47], [32, 51], [32, 53], [32, 59], [32, 66], [32, 71], [32, 84], [32, 85], [33, 58], [33, 75], [33, 77], [33, 78], [33, 97], [34, 37], [34, 50], [34, 61], [35, 41], [35, 43], [35, 47], [35, 77], [35, 86], [35, 93], [36, 52], [36, 65], [36, 66], [36, 95], [36, 98], [37, 41], [37, 46], [37, 50], [37, 58], [37, 68], [37, 71], [37, 89], [38, 80], [39, 51], [39, 66], [40, 45], [40, 90], [40, 95], [40, 98], [41, 46], [41, 48], [41, 50], [41, 56], [41, 61], [41, 63], [41, 68], [41, 69], [41, 72], [41, 78], [41, 98], [42, 44], [42, 45], [42, 58], [42, 59], [42, 62], [43, 76], [43, 92], [43, 93], [43, 99], [44, 68], [44, 83], [45, 47], [45, 70], [45, 73], [45, 98], [45, 99], [46, 55], [46, 57], [46, 65], [46, 77], [47, 59], [47, 62], [48, 50], [48, 59], [48, 95], [49, 53], [49, 67], [49, 90], [49, 92], [50, 68], [50, 74], [50, 94], [51, 74], [51, 81], [51, 89], [52, 54], [52, 73], [52, 85], [52, 96], [52, 97], [53, 97], [54, 55], [54, 77], [54, 94], [55, 58], [55, 65], [55, 72], [55, 83], [55, 94], [55, 97], [56, 57], [56, 78], [56, 87], [57, 68], [57, 90], [58, 62], [58, 73], [58, 96], [59, 75], [59, 87], [60, 68], [60, 73], [60, 78], [60, 82], [60, 89], [60, 96], [61, 64], [61, 68], [61, 72], [61, 87], [62, 63], [62, 79], [63, 68], [63, 70], [63, 76], [63, 80], [63, 87], [64, 68], [64, 83], [64, 91], [64, 96], [65, 67], [65, 88], [66, 73], [66, 74], [66, 77], [66, 79], [67, 69], [67, 73], [67, 80], [67, 87], [68, 93], [68, 94], [68, 98], [69, 77], [69, 94], [70, 71], [70, 83], [70, 90], [71, 86], [72, 78], [72, 86], [73, 86], [74, 94], [75, 88], [75, 94], [76, 83], [77, 87], [77, 89], [77, 92], [78, 80], [78, 88], [79, 82], [79, 91], [80, 89], [80, 91], [80, 98], [82, 86], [83, 99], [84, 93], [84, 94], [85, 99], [87, 88], [87, 95], [87, 98], [87, 99], [88, 89], [88, 92], [88, 97], [89, 98], [90, 97], [93, 96], [94, 98]], "family": "mc_graph_k_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-graph-k-coloring-l5-s9/tests/run.py b/tasks/construct-mc-graph-k-coloring-l5-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l5-s9/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-graph-k-coloring-l5-s9/tests/test.sh b/tasks/construct-mc-graph-k-coloring-l5-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l5-s9/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-hadamard-l5-s1/environment/Dockerfile b/tasks/construct-mc-hadamard-l5-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-hadamard-l5-s1/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-hadamard-l5-s1/environment/instance.json b/tasks/construct-mc-hadamard-l5-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b3bcdacf87e49731f33d502108ea56a6a05abdb2
--- /dev/null
+++ b/tasks/construct-mc-hadamard-l5-s1/environment/instance.json
@@ -0,0 +1 @@
+{"n": 53, "family": "mc_hadamard", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-hadamard-l5-s1/solution/solve.sh b/tasks/construct-mc-hadamard-l5-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..4d20139aaf6bb5d8997b043c499471b47414e31d
--- /dev/null
+++ b/tasks/construct-mc-hadamard-l5-s1/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": [1, 1, -1, -1, 1, -1, 1, 1, -1, 1, 1, 1, -1, 1, -1, 1, 1, 1, -1, -1, -1, -1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, -1, -1, -1, -1, 1, 1, 1, -1, 1, -1, 1, 1, 1, -1, 1, 1, -1, 1, -1, -1, 1], "y": [1, -1, 1, 1, -1, 1, -1, -1, 1, -1, -1, -1, 1, -1, 1, -1, -1, -1, 1, 1, 1, 1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, 1, 1, 1, 1, -1, -1, -1, 1, -1, 1, -1, -1, -1, 1, -1, -1, 1, -1, 1, 1, -1]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-hadamard-l5-s1/tests/check.py b/tasks/construct-mc-hadamard-l5-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..fc4b93160fd7473dd14935db9d712e52abdd00c2
--- /dev/null
+++ b/tasks/construct-mc-hadamard-l5-s1/tests/check.py
@@ -0,0 +1,18 @@
+"""Legendre pair of odd length n (MathConstraint hadamard, CC-BY-4.0): x, y in {-1,+1}^n with
+sum(x) = sum(y) = 1 and PAF_x(k) + PAF_y(k) = -2 for k = 1..(n-1)/2 (periodic autocorrelation)."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n = inst["n"]
+    x = int_list(ans.get("x"), n, -1, 1, "x")
+    y = int_list(ans.get("y"), n, -1, 1, "y")
+    if 0 in x or 0 in y:
+        raise Invalid("entries must be -1 or +1")
+    if sum(x) != 1 or sum(y) != 1:
+        raise Invalid("sum(x) and sum(y) must both equal 1")
+    for k in range(1, (n - 1) // 2 + 1):
+        p = sum(x[i] * x[(i + k) % n] for i in range(n)) + sum(y[i] * y[(i + k) % n] for i in range(n))
+        if p != -2:
+            raise Invalid(f"PAF_x({k}) + PAF_y({k}) = {p}, need -2")
+    return None, {}
diff --git a/tasks/construct-mc-hadamard-l5-s1/tests/instance.json b/tasks/construct-mc-hadamard-l5-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b3bcdacf87e49731f33d502108ea56a6a05abdb2
--- /dev/null
+++ b/tasks/construct-mc-hadamard-l5-s1/tests/instance.json
@@ -0,0 +1 @@
+{"n": 53, "family": "mc_hadamard", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-hadamard-l5-s1/tests/run.py b/tasks/construct-mc-hadamard-l5-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-hadamard-l5-s1/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-hadamard-l5-s1/tests/test.sh b/tasks/construct-mc-hadamard-l5-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-hadamard-l5-s1/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-low-autocorrelation-l3-s3/environment/Dockerfile b/tasks/construct-mc-low-autocorrelation-l3-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-l3-s3/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-low-autocorrelation-l3-s3/environment/instance.json b/tasks/construct-mc-low-autocorrelation-l3-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..00228985f8dbbece2b12a66a0844017459eff8fe
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-l3-s3/environment/instance.json
@@ -0,0 +1 @@
+{"n": 21, "bound": 52, "fixed": [["seq", 5, -1], ["seq", 12, -1], ["seq", 17, 1], ["seq", 20, 1], ["c", 0, 2], ["c", 1, -1], ["c", 8, 2], ["c", 19, -1]], "family": "mc_low_autocorrelation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-low-autocorrelation-l3-s3/solution/solve.sh b/tasks/construct-mc-low-autocorrelation-l3-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..86b8a40ede7e1971728cebdc536634f6aa4889f5
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-l3-s3/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"seq": [-1, 1, 1, -1, -1, -1, 1, 1, 1, -1, 1, -1, -1, -1, -1, -1, -1, 1, -1, -1, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-low-autocorrelation-l3-s3/tests/check.py b/tasks/construct-mc-low-autocorrelation-l3-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..8456f7b1c0ff0c9fb4bc9ae5a49074c8afa7d1aa
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-l3-s3/tests/check.py
@@ -0,0 +1,20 @@
+"""+-1 sequence of length n with aperiodic autocorrelation energy sum_k C_k^2 <= bound
+(MathConstraint low_autocorrelation, CC-BY-4.0). Exact integer arithmetic."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, bound = inst["n"], inst["bound"]
+    s = int_list(ans.get("seq"), n, -1, 1, "seq")
+    if any(v == 0 for v in s):
+        raise Invalid("entries must be -1 or +1")
+    c = [sum(s[i] * s[i + k] for i in range(n - k)) for k in range(1, n)]
+    e = sum(v * v for v in c)
+    # fixed values: seq[i] or c[k] (c[k] is the autocorrelation at shift k+1, as upstream)
+    seqs = {"seq": s, "c": c}
+    for name, i, v in inst.get("fixed", []):
+        if seqs[name][i] != v:
+            raise Invalid(f"fixed value {name}[{i}] = {v} not respected")
+    if e > bound:
+        raise Invalid(f"energy {e} exceeds the bound {bound}")
+    return None, {"energy": e}
diff --git a/tasks/construct-mc-low-autocorrelation-l3-s3/tests/instance.json b/tasks/construct-mc-low-autocorrelation-l3-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..00228985f8dbbece2b12a66a0844017459eff8fe
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-l3-s3/tests/instance.json
@@ -0,0 +1 @@
+{"n": 21, "bound": 52, "fixed": [["seq", 5, -1], ["seq", 12, -1], ["seq", 17, 1], ["seq", 20, 1], ["c", 0, 2], ["c", 1, -1], ["c", 8, 2], ["c", 19, -1]], "family": "mc_low_autocorrelation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-low-autocorrelation-l3-s3/tests/run.py b/tasks/construct-mc-low-autocorrelation-l3-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-l3-s3/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-low-autocorrelation-l3-s3/tests/test.sh b/tasks/construct-mc-low-autocorrelation-l3-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-l3-s3/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-low-autocorrelation-l3-s4/environment/Dockerfile b/tasks/construct-mc-low-autocorrelation-l3-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-l3-s4/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-low-autocorrelation-l3-s4/environment/instance.json b/tasks/construct-mc-low-autocorrelation-l3-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1060df4925dcc01dd59001d1a733d674d1cb0e38
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-l3-s4/environment/instance.json
@@ -0,0 +1 @@
+{"n": 22, "bound": 56, "fixed": [], "family": "mc_low_autocorrelation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-low-autocorrelation-l3-s4/solution/solve.sh b/tasks/construct-mc-low-autocorrelation-l3-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..b636d86174f7f16caff73e8b43c6d55d9db2b133
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-l3-s4/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"seq": [1, -1, 1, -1, 1, 1, -1, -1, 1, -1, -1, 1, 1, 1, -1, -1, -1, -1, 1, -1, -1, -1]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-low-autocorrelation-l3-s4/tests/check.py b/tasks/construct-mc-low-autocorrelation-l3-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..8456f7b1c0ff0c9fb4bc9ae5a49074c8afa7d1aa
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-l3-s4/tests/check.py
@@ -0,0 +1,20 @@
+"""+-1 sequence of length n with aperiodic autocorrelation energy sum_k C_k^2 <= bound
+(MathConstraint low_autocorrelation, CC-BY-4.0). Exact integer arithmetic."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, bound = inst["n"], inst["bound"]
+    s = int_list(ans.get("seq"), n, -1, 1, "seq")
+    if any(v == 0 for v in s):
+        raise Invalid("entries must be -1 or +1")
+    c = [sum(s[i] * s[i + k] for i in range(n - k)) for k in range(1, n)]
+    e = sum(v * v for v in c)
+    # fixed values: seq[i] or c[k] (c[k] is the autocorrelation at shift k+1, as upstream)
+    seqs = {"seq": s, "c": c}
+    for name, i, v in inst.get("fixed", []):
+        if seqs[name][i] != v:
+            raise Invalid(f"fixed value {name}[{i}] = {v} not respected")
+    if e > bound:
+        raise Invalid(f"energy {e} exceeds the bound {bound}")
+    return None, {"energy": e}
diff --git a/tasks/construct-mc-low-autocorrelation-l3-s4/tests/instance.json b/tasks/construct-mc-low-autocorrelation-l3-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1060df4925dcc01dd59001d1a733d674d1cb0e38
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-l3-s4/tests/instance.json
@@ -0,0 +1 @@
+{"n": 22, "bound": 56, "fixed": [], "family": "mc_low_autocorrelation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-low-autocorrelation-l3-s4/tests/run.py b/tasks/construct-mc-low-autocorrelation-l3-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-l3-s4/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-low-autocorrelation-l3-s4/tests/test.sh b/tasks/construct-mc-low-autocorrelation-l3-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-l3-s4/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-magic-sequence-l4-s2/environment/Dockerfile b/tasks/construct-mc-magic-sequence-l4-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-l4-s2/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-magic-sequence-l4-s2/environment/instance.json b/tasks/construct-mc-magic-sequence-l4-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..66ea05d9fc3a624caea719c60c8c179f1697f793
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-l4-s2/environment/instance.json
@@ -0,0 +1 @@
+{"n": 28, "fixed": [], "family": "mc_magic_sequence", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-magic-sequence-l4-s2/solution/solve.sh b/tasks/construct-mc-magic-sequence-l4-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..858624f34f0251e71566a274dac717eae5122cda
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-l4-s2/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": [24, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-magic-sequence-l4-s2/tests/check.py b/tasks/construct-mc-magic-sequence-l4-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..27f636ddd6cfe523924ae01c6b9f37ee392f0332
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-l4-s2/tests/check.py
@@ -0,0 +1,18 @@
+"""Magic sequence of length n: x[i] = number of occurrences of i in x (MathConstraint magic_sequence,
+CC-BY-4.0). The property itself is checked (no stored answer)."""
+from collections import Counter
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n = inst["n"]
+    x = int_list(ans.get("x"), n, 0, n, "x")
+    cnt = Counter(x)
+    for i in range(n):
+        if x[i] != cnt.get(i, 0):
+            raise Invalid(f"x[{i}] = {x[i]} but {i} occurs {cnt.get(i, 0)} times")
+    for name, i, v in inst.get("fixed", []):
+        if x[i] != v:
+            raise Invalid(f"fixed value {name}[{i}] = {v} not respected")
+    return None, {}
diff --git a/tasks/construct-mc-magic-sequence-l4-s2/tests/instance.json b/tasks/construct-mc-magic-sequence-l4-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..66ea05d9fc3a624caea719c60c8c179f1697f793
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-l4-s2/tests/instance.json
@@ -0,0 +1 @@
+{"n": 28, "fixed": [], "family": "mc_magic_sequence", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-magic-sequence-l4-s2/tests/run.py b/tasks/construct-mc-magic-sequence-l4-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-l4-s2/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-magic-sequence-l4-s2/tests/test.sh b/tasks/construct-mc-magic-sequence-l4-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-l4-s2/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-magic-sequence-l4-s5/environment/Dockerfile b/tasks/construct-mc-magic-sequence-l4-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-l4-s5/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-magic-sequence-l4-s5/environment/instance.json b/tasks/construct-mc-magic-sequence-l4-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4289e70f60916781f6c4797c0e64214019067860
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-l4-s5/environment/instance.json
@@ -0,0 +1 @@
+{"n": 32, "fixed": [], "family": "mc_magic_sequence", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-magic-sequence-l4-s5/solution/solve.sh b/tasks/construct-mc-magic-sequence-l4-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..fe970c54d32b5d6758d42decf7c466a90686d6a5
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-l4-s5/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": [28, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-magic-sequence-l4-s5/tests/check.py b/tasks/construct-mc-magic-sequence-l4-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..27f636ddd6cfe523924ae01c6b9f37ee392f0332
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-l4-s5/tests/check.py
@@ -0,0 +1,18 @@
+"""Magic sequence of length n: x[i] = number of occurrences of i in x (MathConstraint magic_sequence,
+CC-BY-4.0). The property itself is checked (no stored answer)."""
+from collections import Counter
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n = inst["n"]
+    x = int_list(ans.get("x"), n, 0, n, "x")
+    cnt = Counter(x)
+    for i in range(n):
+        if x[i] != cnt.get(i, 0):
+            raise Invalid(f"x[{i}] = {x[i]} but {i} occurs {cnt.get(i, 0)} times")
+    for name, i, v in inst.get("fixed", []):
+        if x[i] != v:
+            raise Invalid(f"fixed value {name}[{i}] = {v} not respected")
+    return None, {}
diff --git a/tasks/construct-mc-magic-sequence-l4-s5/tests/instance.json b/tasks/construct-mc-magic-sequence-l4-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4289e70f60916781f6c4797c0e64214019067860
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-l4-s5/tests/instance.json
@@ -0,0 +1 @@
+{"n": 32, "fixed": [], "family": "mc_magic_sequence", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-magic-sequence-l4-s5/tests/run.py b/tasks/construct-mc-magic-sequence-l4-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-l4-s5/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-magic-sequence-l4-s5/tests/test.sh b/tasks/construct-mc-magic-sequence-l4-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-l4-s5/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-max-clique-l3-s3/environment/Dockerfile b/tasks/construct-mc-max-clique-l3-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-max-clique-l3-s3/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-max-clique-l3-s3/environment/instance.json b/tasks/construct-mc-max-clique-l3-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..649d89184b4fed9c92964e08f7a75a63c813aa8c
--- /dev/null
+++ b/tasks/construct-mc-max-clique-l3-s3/environment/instance.json
@@ -0,0 +1 @@
+{"n": 50, "k": 5, "edges": [[0, 2], [0, 4], [0, 5], [0, 7], [0, 8], [0, 9], [0, 10], [0, 12], [0, 13], [0, 14], [0, 21], [0, 22], [0, 23], [0, 24], [0, 27], [0, 30], [0, 33], [0, 35], [0, 36], [0, 40], [0, 41], [0, 42], [0, 43], [0, 45], [0, 46], [0, 47], [1, 2], [1, 3], [1, 4], [1, 5], [1, 7], [1, 9], [1, 10], [1, 12], [1, 15], [1, 16], [1, 19], [1, 20], [1, 22], [1, 23], [1, 24], [1, 25], [1, 26], [1, 28], [1, 29], [1, 30], [1, 31], [1, 33], [1, 34], [1, 35], [1, 38], [1, 39], [1, 40], [1, 45], [1, 46], [1, 48], [1, 49], [2, 5], [2, 6], [2, 7], [2, 9], [2, 11], [2, 12], [2, 15], [2, 17], [2, 19], [2, 20], [2, 21], [2, 22], [2, 24], [2, 27], [2, 30], [2, 32], [2, 33], [2, 35], [2, 39], [2, 40], [2, 42], [2, 43], [2, 47], [2, 48], [3, 4], [3, 5], [3, 8], [3, 9], [3, 12], [3, 13], [3, 16], [3, 17], [3, 18], [3, 20], [3, 21], [3, 22], [3, 23], [3, 24], [3, 25], [3, 26], [3, 29], [3, 30], [3, 34], [3, 37], [3, 42], [3, 43], [3, 45], [3, 49], [4, 6], [4, 7], [4, 9], [4, 10], [4, 11], [4, 12], [4, 14], [4, 17], [4, 18], [4, 19], [4, 22], [4, 23], [4, 24], [4, 25], [4, 26], [4, 27], [4, 28], [4, 32], [4, 35], [4, 36], [4, 38], [4, 41], [4, 42], [4, 43], [4, 46], [4, 49], [5, 7], [5, 9], [5, 10], [5, 11], [5, 12], [5, 14], [5, 15], [5, 16], [5, 17], [5, 18], [5, 19], [5, 20], [5, 21], [5, 22], [5, 23], [5, 24], [5, 26], [5, 27], [5, 28], [5, 29], [5, 31], [5, 32], [5, 33], [5, 34], [5, 38], [5, 41], [5, 42], [5, 43], [5, 44], [5, 45], [5, 49], [6, 8], [6, 9], [6, 10], [6, 11], [6, 13], [6, 15], [6, 16], [6, 17], [6, 20], [6, 21], [6, 22], [6, 24], [6, 26], [6, 28], [6, 29], [6, 31], [6, 34], [6, 35], [6, 36], [6, 38], [6, 41], [6, 43], [6, 44], [6, 47], [6, 48], [7, 11], [7, 12], [7, 14], [7, 16], [7, 17], [7, 21], [7, 27], [7, 31], [7, 34], [7, 35], [7, 36], [7, 37], [7, 38], [7, 41], [7, 43], [7, 44], [7, 45], [7, 46], [8, 10], [8, 11], [8, 12], [8, 13], [8, 14], [8, 17], [8, 18], [8, 20], [8, 22], [8, 23], [8, 24], [8, 25], [8, 28], [8, 29], [8, 30], [8, 31], [8, 32], [8, 33], [8, 35], [8, 36], [8, 38], [8, 39], [8, 41], [8, 43], [8, 46], [8, 47], [8, 49], [9, 10], [9, 11], [9, 12], [9, 13], [9, 15], [9, 17], [9, 18], [9, 19], [9, 23], [9, 26], [9, 27], [9, 28], [9, 29], [9, 30], [9, 32], [9, 34], [9, 36], [9, 39], [9, 40], [9, 43], [9, 45], [9, 48], [9, 49], [10, 12], [10, 14], [10, 16], [10, 17], [10, 18], [10, 23], [10, 25], [10, 27], [10, 28], [10, 32], [10, 33], [10, 35], [10, 37], [10, 39], [10, 41], [10, 43], [10, 45], [10, 47], [10, 48], [10, 49], [11, 12], [11, 14], [11, 16], [11, 17], [11, 19], [11, 21], [11, 22], [11, 23], [11, 24], [11, 25], [11, 26], [11, 27], [11, 28], [11, 29], [11, 30], [11, 32], [11, 33], [11, 34], [11, 37], [11, 38], [11, 42], [11, 43], [11, 44], [11, 45], [11, 46], [11, 49], [12, 14], [12, 15], [12, 17], [12, 19], [12, 22], [12, 23], [12, 24], [12, 25], [12, 30], [12, 31], [12, 32], [12, 33], [12, 34], [12, 37], [12, 38], [12, 40], [12, 41], [12, 42], [12, 43], [12, 45], [12, 46], [12, 47], [12, 48], [13, 17], [13, 18], [13, 19], [13, 20], [13, 22], [13, 25], [13, 26], [13, 29], [13, 34], [13, 35], [13, 36], [13, 37], [13, 38], [13, 40], [13, 43], [13, 44], [13, 45], [13, 46], [14, 18], [14, 19], [14, 20], [14, 24], [14, 26], [14, 28], [14, 30], [14, 31], [14, 33], [14, 34], [14, 35], [14, 36], [14, 38], [14, 43], [14, 45], [14, 46], [14, 47], [14, 48], [14, 49], [15, 17], [15, 18], [15, 19], [15, 20], [15, 22], [15, 25], [15, 28], [15, 29], [15, 31], [15, 32], [15, 33], [15, 35], [15, 37], [15, 38], [15, 39], [15, 40], [15, 41], [15, 45], [15, 49], [16, 20], [16, 21], [16, 23], [16, 24], [16, 26], [16, 27], [16, 29], [16, 32], [16, 33], [16, 36], [16, 37], [16, 38], [16, 40], [16, 45], [16, 48], [16, 49], [17, 20], [17, 21], [17, 22], [17, 24], [17, 27], [17, 29], [17, 31], [17, 32], [17, 34], [17, 35], [17, 36], [17, 39], [17, 40], [17, 43], [17, 45], [17, 46], [17, 47], [17, 48], [17, 49], [18, 20], [18, 21], [18, 23], [18, 28], [18, 29], [18, 31], [18, 33], [18, 34], [18, 36], [18, 37], [18, 38], [18, 40], [18, 42], [18, 43], [18, 45], [18, 46], [18, 47], [18, 48], [19, 23], [19, 25], [19, 26], [19, 27], [19, 28], [19, 30], [19, 31], [19, 33], [19, 34], [19, 36], [19, 37], [19, 38], [19, 39], [19, 41], [19, 42], [19, 43], [19, 44], [19, 45], [19, 46], [19, 47], [19, 48], [20, 21], [20, 22], [20, 23], [20, 24], [20, 25], [20, 26], [20, 28], [20, 29], [20, 30], [20, 33], [20, 35], [20, 36], [20, 37], [20, 38], [20, 39], [20, 40], [20, 41], [20, 43], [20, 45], [20, 47], [20, 48], [20, 49], [21, 22], [21, 26], [21, 29], [21, 32], [21, 33], [21, 34], [21, 36], [21, 37], [21, 38], [21, 40], [21, 41], [21, 42], [21, 43], [21, 44], [21, 45], [21, 49], [22, 25], [22, 26], [22, 28], [22, 30], [22, 31], [22, 32], [22, 34], [22, 37], [22, 38], [22, 39], [22, 40], [22, 41], [22, 42], [22, 43], [22, 45], [22, 47], [22, 48], [23, 26], [23, 27], [23, 28], [23, 29], [23, 30], [23, 34], [23, 37], [23, 39], [23, 40], [23, 42], [23, 43], [23, 45], [23, 47], [23, 48], [23, 49], [24, 26], [24, 29], [24, 30], [24, 31], [24, 32], [24, 33], [24, 34], [24, 36], [24, 37], [24, 38], [24, 39], [24, 40], [24, 43], [24, 45], [24, 46], [24, 48], [24, 49], [25, 26], [25, 27], [25, 31], [25, 32], [25, 33], [25, 34], [25, 35], [25, 36], [25, 37], [25, 39], [25, 40], [25, 41], [25, 45], [25, 46], [25, 47], [25, 48], [25, 49], [26, 30], [26, 31], [26, 32], [26, 33], [26, 35], [26, 38], [26, 39], [26, 40], [26, 44], [26, 46], [26, 48], [26, 49], [27, 29], [27, 30], [27, 33], [27, 38], [27, 39], [27, 42], [27, 45], [27, 46], [27, 47], [27, 48], [27, 49], [28, 32], [28, 33], [28, 34], [28, 35], [28, 36], [28, 39], [28, 40], [28, 41], [28, 42], [28, 44], [28, 47], [28, 49], [29, 33], [29, 34], [29, 35], [29, 36], [29, 38], [29, 39], [29, 40], [29, 42], [29, 43], [29, 45], [29, 46], [29, 47], [30, 31], [30, 32], [30, 33], [30, 34], [30, 35], [30, 36], [30, 38], [30, 39], [30, 40], [30, 42], [30, 44], [30, 45], [30, 46], [30, 49], [31, 32], [31, 34], [31, 35], [31, 37], [31, 38], [31, 39], [31, 41], [31, 42], [31, 43], [31, 44], [31, 48], [31, 49], [32, 33], [32, 34], [32, 35], [32, 36], [32, 37], [32, 42], [32, 45], [32, 46], [32, 48], [33, 36], [33, 37], [33, 39], [33, 40], [33, 41], [33, 43], [33, 44], [33, 45], [33, 49], [34, 35], [34, 36], [34, 41], [34, 44], [34, 47], [34, 48], [34, 49], [35, 39], [35, 40], [35, 42], [35, 43], [35, 46], [35, 47], [35, 48], [36, 37], [36, 38], [36, 39], [36, 40], [36, 41], [36, 43], [36, 44], [36, 45], [36, 48], [37, 39], [37, 40], [37, 41], [37, 43], [37, 44], [37, 45], [37, 48], [38, 41], [38, 42], [38, 43], [38, 44], [38, 45], [38, 48], [39, 40], [39, 41], [39, 42], [39, 45], [39, 47], [39, 48], [40, 41], [40, 43], [40, 45], [40, 46], [40, 47], [40, 49], [41, 42], [41, 45], [41, 46], [41, 47], [41, 49], [42, 44], [42, 45], [42, 47], [42, 48], [42, 49], [43, 44], [43, 47], [43, 48], [43, 49], [44, 45], [44, 47], [44, 48], [44, 49], [45, 47], [45, 48], [46, 47], [46, 48], [47, 48]], "family": "mc_max_clique", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-max-clique-l3-s3/solution/solve.sh b/tasks/construct-mc-max-clique-l3-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..a77a2b2ff0430db6ab59602b6f27272c964f9470
--- /dev/null
+++ b/tasks/construct-mc-max-clique-l3-s3/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"clique": [42, 44, 45, 47, 48]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-max-clique-l3-s3/tests/check.py b/tasks/construct-mc-max-clique-l3-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..e3f70dbd5b155a3514c2102aa4ad460201495210
--- /dev/null
+++ b/tasks/construct-mc-max-clique-l3-s3/tests/check.py
@@ -0,0 +1,18 @@
+"""Clique of size >= k in a given graph (MathConstraint max_clique decision version, CC-BY-4.0)."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    S = int_list(ans.get("clique"), None, 0, n - 1, "clique")
+    if len(set(S)) != len(S):
+        raise Invalid("repeated vertex")
+    if len(S) < k:
+        raise Invalid(f"clique has {len(S)} vertices, need at least {k}")
+    E = {(u, v) for u, v in inst["edges"]}
+    for u, v in itertools.combinations(sorted(S), 2):
+        if (u, v) not in E:
+            raise Invalid(f"vertices {u} and {v} are not adjacent")
+    return None, {"size": len(S)}
diff --git a/tasks/construct-mc-max-clique-l3-s3/tests/instance.json b/tasks/construct-mc-max-clique-l3-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..649d89184b4fed9c92964e08f7a75a63c813aa8c
--- /dev/null
+++ b/tasks/construct-mc-max-clique-l3-s3/tests/instance.json
@@ -0,0 +1 @@
+{"n": 50, "k": 5, "edges": [[0, 2], [0, 4], [0, 5], [0, 7], [0, 8], [0, 9], [0, 10], [0, 12], [0, 13], [0, 14], [0, 21], [0, 22], [0, 23], [0, 24], [0, 27], [0, 30], [0, 33], [0, 35], [0, 36], [0, 40], [0, 41], [0, 42], [0, 43], [0, 45], [0, 46], [0, 47], [1, 2], [1, 3], [1, 4], [1, 5], [1, 7], [1, 9], [1, 10], [1, 12], [1, 15], [1, 16], [1, 19], [1, 20], [1, 22], [1, 23], [1, 24], [1, 25], [1, 26], [1, 28], [1, 29], [1, 30], [1, 31], [1, 33], [1, 34], [1, 35], [1, 38], [1, 39], [1, 40], [1, 45], [1, 46], [1, 48], [1, 49], [2, 5], [2, 6], [2, 7], [2, 9], [2, 11], [2, 12], [2, 15], [2, 17], [2, 19], [2, 20], [2, 21], [2, 22], [2, 24], [2, 27], [2, 30], [2, 32], [2, 33], [2, 35], [2, 39], [2, 40], [2, 42], [2, 43], [2, 47], [2, 48], [3, 4], [3, 5], [3, 8], [3, 9], [3, 12], [3, 13], [3, 16], [3, 17], [3, 18], [3, 20], [3, 21], [3, 22], [3, 23], [3, 24], [3, 25], [3, 26], [3, 29], [3, 30], [3, 34], [3, 37], [3, 42], [3, 43], [3, 45], [3, 49], [4, 6], [4, 7], [4, 9], [4, 10], [4, 11], [4, 12], [4, 14], [4, 17], [4, 18], [4, 19], [4, 22], [4, 23], [4, 24], [4, 25], [4, 26], [4, 27], [4, 28], [4, 32], [4, 35], [4, 36], [4, 38], [4, 41], [4, 42], [4, 43], [4, 46], [4, 49], [5, 7], [5, 9], [5, 10], [5, 11], [5, 12], [5, 14], [5, 15], [5, 16], [5, 17], [5, 18], [5, 19], [5, 20], [5, 21], [5, 22], [5, 23], [5, 24], [5, 26], [5, 27], [5, 28], [5, 29], [5, 31], [5, 32], [5, 33], [5, 34], [5, 38], [5, 41], [5, 42], [5, 43], [5, 44], [5, 45], [5, 49], [6, 8], [6, 9], [6, 10], [6, 11], [6, 13], [6, 15], [6, 16], [6, 17], [6, 20], [6, 21], [6, 22], [6, 24], [6, 26], [6, 28], [6, 29], [6, 31], [6, 34], [6, 35], [6, 36], [6, 38], [6, 41], [6, 43], [6, 44], [6, 47], [6, 48], [7, 11], [7, 12], [7, 14], [7, 16], [7, 17], [7, 21], [7, 27], [7, 31], [7, 34], [7, 35], [7, 36], [7, 37], [7, 38], [7, 41], [7, 43], [7, 44], [7, 45], [7, 46], [8, 10], [8, 11], [8, 12], [8, 13], [8, 14], [8, 17], [8, 18], [8, 20], [8, 22], [8, 23], [8, 24], [8, 25], [8, 28], [8, 29], [8, 30], [8, 31], [8, 32], [8, 33], [8, 35], [8, 36], [8, 38], [8, 39], [8, 41], [8, 43], [8, 46], [8, 47], [8, 49], [9, 10], [9, 11], [9, 12], [9, 13], [9, 15], [9, 17], [9, 18], [9, 19], [9, 23], [9, 26], [9, 27], [9, 28], [9, 29], [9, 30], [9, 32], [9, 34], [9, 36], [9, 39], [9, 40], [9, 43], [9, 45], [9, 48], [9, 49], [10, 12], [10, 14], [10, 16], [10, 17], [10, 18], [10, 23], [10, 25], [10, 27], [10, 28], [10, 32], [10, 33], [10, 35], [10, 37], [10, 39], [10, 41], [10, 43], [10, 45], [10, 47], [10, 48], [10, 49], [11, 12], [11, 14], [11, 16], [11, 17], [11, 19], [11, 21], [11, 22], [11, 23], [11, 24], [11, 25], [11, 26], [11, 27], [11, 28], [11, 29], [11, 30], [11, 32], [11, 33], [11, 34], [11, 37], [11, 38], [11, 42], [11, 43], [11, 44], [11, 45], [11, 46], [11, 49], [12, 14], [12, 15], [12, 17], [12, 19], [12, 22], [12, 23], [12, 24], [12, 25], [12, 30], [12, 31], [12, 32], [12, 33], [12, 34], [12, 37], [12, 38], [12, 40], [12, 41], [12, 42], [12, 43], [12, 45], [12, 46], [12, 47], [12, 48], [13, 17], 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"construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-max-clique-l3-s3/tests/run.py b/tasks/construct-mc-max-clique-l3-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-max-clique-l3-s3/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-max-clique-l3-s3/tests/test.sh b/tasks/construct-mc-max-clique-l3-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-max-clique-l3-s3/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-max-clique-l3-s4/environment/Dockerfile b/tasks/construct-mc-max-clique-l3-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-max-clique-l3-s4/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-max-clique-l3-s4/environment/instance.json b/tasks/construct-mc-max-clique-l3-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..144d0be5c3966670f2c406a1e581328342e44c65
--- /dev/null
+++ b/tasks/construct-mc-max-clique-l3-s4/environment/instance.json
@@ -0,0 +1 @@
+{"n": 56, "k": 5, "edges": [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [0, 8], [0, 11], [0, 12], [0, 15], [0, 17], [0, 18], [0, 19], [0, 20], [0, 22], [0, 23], [0, 24], [0, 25], [0, 26], [0, 30], [0, 32], [0, 34], [0, 36], [0, 38], [0, 39], [0, 40], [0, 41], [0, 43], [0, 44], [0, 45], [0, 46], [0, 47], [0, 49], [0, 52], [0, 54], [0, 55], [1, 2], [1, 7], [1, 8], [1, 9], [1, 10], [1, 12], [1, 13], [1, 14], [1, 15], [1, 17], [1, 20], [1, 22], [1, 23], [1, 26], [1, 27], [1, 28], [1, 29], [1, 30], [1, 33], [1, 36], [1, 37], [1, 38], [1, 39], [1, 41], [1, 44], [1, 46], [1, 47], [1, 48], [1, 49], [1, 51], [1, 52], [1, 53], [1, 54], [2, 5], [2, 7], [2, 11], [2, 13], [2, 14], [2, 16], [2, 19], [2, 21], [2, 23], [2, 25], [2, 26], [2, 28], [2, 29], [2, 30], [2, 31], [2, 32], [2, 33], [2, 34], [2, 36], [2, 40], [2, 42], [2, 43], [2, 44], [2, 45], [2, 46], [2, 48], [2, 49], [2, 50], [2, 53], [2, 54], [2, 55], [3, 4], [3, 5], [3, 7], [3, 8], [3, 9], [3, 10], [3, 11], [3, 14], [3, 15], [3, 20], [3, 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[51, 53], [51, 54], [51, 55], [52, 53], [53, 54]], "family": "mc_max_clique", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-max-clique-l3-s4/solution/solve.sh b/tasks/construct-mc-max-clique-l3-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..ecff296790066e735a17c20fa68616ac2ee86f46
--- /dev/null
+++ b/tasks/construct-mc-max-clique-l3-s4/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"clique": [43, 44, 46, 48, 51]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-max-clique-l3-s4/tests/check.py b/tasks/construct-mc-max-clique-l3-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..e3f70dbd5b155a3514c2102aa4ad460201495210
--- /dev/null
+++ b/tasks/construct-mc-max-clique-l3-s4/tests/check.py
@@ -0,0 +1,18 @@
+"""Clique of size >= k in a given graph (MathConstraint max_clique decision version, CC-BY-4.0)."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    S = int_list(ans.get("clique"), None, 0, n - 1, "clique")
+    if len(set(S)) != len(S):
+        raise Invalid("repeated vertex")
+    if len(S) < k:
+        raise Invalid(f"clique has {len(S)} vertices, need at least {k}")
+    E = {(u, v) for u, v in inst["edges"]}
+    for u, v in itertools.combinations(sorted(S), 2):
+        if (u, v) not in E:
+            raise Invalid(f"vertices {u} and {v} are not adjacent")
+    return None, {"size": len(S)}
diff --git a/tasks/construct-mc-max-clique-l3-s4/tests/instance.json b/tasks/construct-mc-max-clique-l3-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..144d0be5c3966670f2c406a1e581328342e44c65
--- /dev/null
+++ b/tasks/construct-mc-max-clique-l3-s4/tests/instance.json
@@ -0,0 +1 @@
+{"n": 56, "k": 5, "edges": [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [0, 8], [0, 11], [0, 12], [0, 15], [0, 17], [0, 18], [0, 19], [0, 20], [0, 22], [0, 23], [0, 24], [0, 25], [0, 26], [0, 30], [0, 32], [0, 34], [0, 36], [0, 38], [0, 39], [0, 40], [0, 41], [0, 43], [0, 44], [0, 45], [0, 46], [0, 47], [0, 49], [0, 52], [0, 54], [0, 55], [1, 2], [1, 7], [1, 8], [1, 9], [1, 10], [1, 12], [1, 13], [1, 14], [1, 15], [1, 17], [1, 20], [1, 22], [1, 23], [1, 26], [1, 27], [1, 28], [1, 29], [1, 30], [1, 33], [1, 36], [1, 37], [1, 38], [1, 39], [1, 41], [1, 44], [1, 46], [1, 47], [1, 48], [1, 49], [1, 51], [1, 52], [1, 53], [1, 54], [2, 5], [2, 7], [2, 11], [2, 13], [2, 14], [2, 16], [2, 19], [2, 21], [2, 23], [2, 25], [2, 26], [2, 28], [2, 29], [2, 30], [2, 31], [2, 32], [2, 33], [2, 34], [2, 36], [2, 40], [2, 42], [2, 43], [2, 44], [2, 45], [2, 46], [2, 48], [2, 49], [2, 50], [2, 53], [2, 54], [2, 55], [3, 4], [3, 5], [3, 7], [3, 8], [3, 9], [3, 10], [3, 11], [3, 14], [3, 15], [3, 20], [3, 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[29, 31], [29, 35], [29, 36], [29, 37], [29, 39], [29, 41], [29, 42], [29, 49], [29, 51], [29, 53], [30, 31], [30, 32], [30, 33], [30, 34], [30, 36], [30, 37], [30, 38], [30, 39], [30, 42], [30, 45], [30, 46], [30, 49], [30, 50], [30, 53], [31, 33], [31, 34], [31, 35], [31, 37], [31, 40], [31, 41], [31, 43], [31, 44], [31, 48], [31, 50], [31, 51], [31, 52], [31, 53], [31, 54], [31, 55], [32, 37], [32, 38], [32, 40], [32, 41], [32, 44], [32, 47], [32, 49], [32, 50], [32, 51], [32, 52], [32, 53], [32, 55], [33, 34], [33, 35], [33, 36], [33, 37], [33, 41], [33, 42], [33, 43], [33, 44], [33, 46], [33, 48], [33, 50], [33, 51], [33, 52], [33, 53], [34, 35], [34, 36], [34, 37], [34, 38], [34, 40], [34, 42], [34, 44], [34, 45], [34, 48], [34, 49], [34, 50], [34, 53], [34, 54], [35, 36], [35, 37], [35, 38], [35, 39], [35, 41], [35, 42], [35, 46], [35, 47], [35, 49], [35, 52], [35, 54], [35, 55], [36, 37], [36, 42], [36, 43], [36, 46], [36, 48], [36, 49], [36, 50], [36, 51], [36, 52], [36, 53], [36, 54], [36, 55], [37, 41], [37, 42], [37, 43], [37, 44], [37, 45], [37, 46], [37, 47], [37, 49], [37, 50], [37, 51], [37, 52], [37, 55], [38, 40], [38, 42], [38, 43], [38, 44], [38, 45], [38, 46], [38, 48], [38, 49], [38, 50], [38, 54], [39, 40], [39, 41], [39, 43], [39, 46], [39, 47], [39, 48], [39, 49], [39, 50], [39, 52], [39, 54], [39, 55], [40, 42], [40, 43], [40, 44], [40, 45], [40, 47], [40, 48], [40, 50], [40, 52], [40, 54], [41, 44], [41, 45], [41, 49], [41, 50], [41, 52], [41, 53], [41, 54], [42, 44], [42, 45], [42, 47], [42, 49], [42, 55], [43, 44], [43, 45], [43, 46], [43, 48], [43, 49], [43, 50], [43, 51], [43, 53], [43, 54], [43, 55], [44, 46], [44, 47], [44, 48], [44, 50], [44, 51], [44, 54], [45, 46], [45, 47], [45, 48], [45, 50], [45, 53], [45, 54], [46, 47], [46, 48], [46, 49], [46, 51], [46, 53], [46, 54], [47, 48], [47, 49], [47, 50], [47, 52], [48, 50], [48, 51], [48, 52], [49, 50], [49, 53], [49, 54], [49, 55], [50, 51], [50, 52], [50, 53], [50, 55], [51, 52], [51, 53], [51, 54], [51, 55], [52, 53], [53, 54]], "family": "mc_max_clique", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-max-clique-l3-s4/tests/run.py b/tasks/construct-mc-max-clique-l3-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-max-clique-l3-s4/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-max-clique-l3-s4/tests/test.sh b/tasks/construct-mc-max-clique-l3-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-max-clique-l3-s4/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-max-independent-set-l4-s0/environment/Dockerfile b/tasks/construct-mc-max-independent-set-l4-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-l4-s0/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-max-independent-set-l4-s0/environment/instance.json b/tasks/construct-mc-max-independent-set-l4-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..33433c0aa56e9e07e42d2bef1dbed5bd79ce54fc
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-l4-s0/environment/instance.json
@@ -0,0 +1 @@
+{"n": 60, "k": 14, "edges": [[0, 3], [0, 4], [0, 6], [0, 8], [0, 10], [0, 18], [0, 19], [0, 20], [0, 21], [0, 26], [0, 29], [0, 34], [0, 36], [0, 41], [0, 45], [0, 49], [0, 53], [0, 56], [0, 57], [0, 58], [0, 59], [1, 2], [1, 5], [1, 7], [1, 9], [1, 15], [1, 16], [1, 17], [1, 21], [1, 28], [1, 31], [1, 35], [1, 39], [1, 48], [1, 51], [1, 59], [2, 6], [2, 8], [2, 9], [2, 10], [2, 13], [2, 19], [2, 24], [2, 29], [2, 30], [2, 31], [2, 32], [2, 33], [2, 36], [2, 41], [2, 43], [2, 46], [2, 48], [2, 59], [3, 6], [3, 9], [3, 10], [3, 20], [3, 27], [3, 31], [3, 34], [3, 40], [3, 42], [3, 45], [3, 46], [3, 47], [3, 50], [3, 53], [4, 8], [4, 10], [4, 13], [4, 16], [4, 26], [4, 31], [4, 32], [4, 35], [4, 40], [4, 41], [4, 42], [4, 45], [4, 48], [4, 51], [4, 54], [4, 57], [4, 59], [5, 7], [5, 9], [5, 10], [5, 13], [5, 14], [5, 22], [5, 23], [5, 33], [5, 34], [5, 40], [5, 41], [5, 43], [5, 47], [5, 48], [5, 52], [6, 11], [6, 15], [6, 16], [6, 24], [6, 32], [6, 37], [6, 41], [6, 42], [6, 43], [6, 46], [6, 48], [6, 52], [6, 53], [6, 59], [7, 8], [7, 9], [7, 17], [7, 19], [7, 20], [7, 23], [7, 26], [7, 29], [7, 30], [7, 37], [7, 43], [7, 47], [7, 50], [7, 53], [7, 55], [7, 56], [7, 58], [8, 17], [8, 24], [8, 27], [8, 28], [8, 31], [8, 35], [8, 39], [8, 49], [8, 52], [8, 53], [9, 10], [9, 11], [9, 21], [9, 24], [9, 25], [9, 28], [9, 30], [9, 31], [9, 32], [9, 37], [9, 38], [9, 42], [9, 43], [9, 44], [9, 46], [9, 47], [9, 50], [9, 52], [9, 56], [10, 15], [10, 19], [10, 29], [10, 31], [10, 32], [10, 34], [10, 42], [10, 44], [10, 47], [10, 55], [10, 56], [11, 12], [11, 18], [11, 22], [11, 24], [11, 31], [11, 32], [11, 37], [11, 38], [11, 39], [11, 42], [11, 47], [11, 53], [11, 54], [12, 13], [12, 15], [12, 17], [12, 20], [12, 22], [12, 23], [12, 25], [12, 30], [12, 33], [12, 35], [12, 36], [12, 45], [12, 47], [12, 48], [12, 50], [12, 52], [12, 56], [12, 57], [12, 59], [13, 17], [13, 18], [13, 20], [13, 40], [13, 41], [13, 48], [14, 22], [14, 23], [14, 25], [14, 40], [14, 41], [14, 44], [14, 46], [14, 48], [14, 49], [14, 53], [14, 54], [14, 55], [14, 57], [15, 18], [15, 21], [15, 22], [15, 27], [15, 30], [15, 34], [15, 38], [15, 44], [15, 45], [15, 48], [15, 54], [15, 55], [15, 57], [15, 59], [16, 17], [16, 19], [16, 25], [16, 26], [16, 27], [16, 29], [16, 31], [16, 34], [16, 39], [16, 43], [16, 48], [16, 50], [16, 51], [16, 52], [16, 59], [17, 19], [17, 24], [17, 31], [17, 37], [17, 43], [17, 45], [17, 47], [17, 55], [17, 56], [17, 59], [18, 23], [18, 26], [18, 29], [18, 30], [18, 35], [18, 43], [18, 44], [18, 48], [18, 58], [19, 21], [19, 31], [19, 34], [19, 35], [19, 36], [19, 40], [19, 43], [19, 46], [19, 47], [19, 51], [19, 53], [19, 55], [19, 58], [20, 22], [20, 23], [20, 26], [20, 27], [20, 29], [20, 31], [20, 32], [20, 33], [20, 35], [20, 37], [20, 40], [20, 41], [20, 44], [20, 46], [20, 51], [20, 53], [20, 54], [20, 57], [21, 23], [21, 24], [21, 30], [21, 32], [21, 34], [21, 35], [21, 41], [21, 45], [21, 47], [21, 48], [21, 52], [21, 59], [22, 31], [22, 41], [23, 29], [23, 30], [23, 31], [23, 39], [23, 43], [23, 47], [23, 48], [23, 49], [23, 51], [23, 58], [24, 27], [24, 29], [24, 31], [24, 34], [24, 35], [24, 36], [24, 44], [24, 49], [25, 30], [25, 35], [25, 36], [25, 39], [25, 45], [25, 56], [25, 59], [26, 27], [26, 31], [26, 32], [26, 34], [26, 35], [26, 36], [26, 39], [26, 45], [26, 49], [26, 50], [26, 51], [26, 52], [26, 59], [27, 29], [27, 31], [27, 35], [27, 37], [27, 49], [27, 51], [27, 57], [28, 33], [28, 52], [28, 53], [29, 32], [29, 38], [29, 39], [29, 44], [29, 46], [29, 51], [29, 52], [29, 53], [29, 56], [29, 59], [30, 32], [30, 35], [30, 37], [30, 42], [30, 48], [30, 49], [30, 51], [30, 57], [31, 34], [31, 35], [31, 40], [31, 47], [31, 51], [31, 52], [31, 53], [31, 56], [31, 59], [32, 36], [32, 40], [32, 46], [32, 56], [32, 58], [33, 35], [33, 39], [33, 46], [33, 48], [33, 50], [33, 53], [33, 59], [34, 38], [34, 45], [34, 49], [34, 52], [34, 53], [34, 56], [34, 57], [34, 58], [34, 59], [35, 38], [35, 41], [35, 43], [35, 46], [35, 53], [35, 55], [35, 56], [36, 37], [36, 38], [36, 40], [36, 46], [36, 49], [36, 53], [36, 58], [37, 48], [37, 51], [37, 53], [38, 40], [38, 43], [38, 49], [38, 52], [38, 56], [38, 58], [39, 40], [39, 44], [39, 47], [39, 49], [39, 50], [39, 51], [39, 52], [39, 59], [40, 43], [40, 45], [40, 47], [40, 48], [40, 49], [40, 54], [41, 45], [41, 49], [41, 50], [41, 53], [41, 56], [41, 57], [41, 58], [42, 44], [42, 51], [42, 54], [42, 56], [42, 57], [42, 58], [42, 59], [43, 50], [43, 54], [43, 55], [43, 57], [44, 49], [44, 52], [44, 53], [44, 55], [45, 48], [45, 52], [45, 53], [45, 55], [45, 59], [46, 47], [46, 54], [46, 58], [47, 53], [47, 59], [48, 51], [48, 54], [48, 59], [49, 52], [49, 57], [50, 51], [50, 52], [50, 57], [50, 58], [51, 57], [52, 53], [52, 54], [52, 59], [53, 54], [54, 56], [55, 56], [55, 57], [56, 58], [56, 59]], "family": "mc_max_independent_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-max-independent-set-l4-s0/solution/solve.sh b/tasks/construct-mc-max-independent-set-l4-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..b07ccae79bc75d7055a3906bda9ecd5378b495ed
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-l4-s0/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"independent_set": [1, 3, 8, 13, 19, 22, 23, 25, 33, 37, 38, 44, 54, 57]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-max-independent-set-l4-s0/tests/check.py b/tasks/construct-mc-max-independent-set-l4-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..adce413e8578d996549583b990e4ff559f38ed41
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-l4-s0/tests/check.py
@@ -0,0 +1,18 @@
+"""Independent set of size >= k in a given graph (MathConstraint max_independent_set, CC-BY-4.0)."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    S = int_list(ans.get("independent_set"), None, 0, n - 1, "independent_set")
+    if len(set(S)) != len(S):
+        raise Invalid("repeated vertex")
+    if len(S) < k:
+        raise Invalid(f"set has {len(S)} vertices, need at least {k}")
+    E = {(u, v) for u, v in inst["edges"]}
+    for u, v in itertools.combinations(sorted(S), 2):
+        if (u, v) in E:
+            raise Invalid(f"vertices {u} and {v} are adjacent")
+    return None, {"size": len(S)}
diff --git a/tasks/construct-mc-max-independent-set-l4-s0/tests/instance.json b/tasks/construct-mc-max-independent-set-l4-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..33433c0aa56e9e07e42d2bef1dbed5bd79ce54fc
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-l4-s0/tests/instance.json
@@ -0,0 +1 @@
+{"n": 60, "k": 14, "edges": [[0, 3], [0, 4], [0, 6], [0, 8], [0, 10], [0, 18], [0, 19], [0, 20], [0, 21], [0, 26], [0, 29], [0, 34], [0, 36], [0, 41], [0, 45], [0, 49], [0, 53], [0, 56], [0, 57], [0, 58], [0, 59], [1, 2], [1, 5], [1, 7], [1, 9], [1, 15], [1, 16], [1, 17], [1, 21], [1, 28], [1, 31], [1, 35], [1, 39], [1, 48], [1, 51], [1, 59], [2, 6], [2, 8], [2, 9], [2, 10], [2, 13], [2, 19], [2, 24], [2, 29], [2, 30], [2, 31], [2, 32], [2, 33], [2, 36], [2, 41], [2, 43], [2, 46], [2, 48], [2, 59], [3, 6], [3, 9], [3, 10], [3, 20], [3, 27], [3, 31], [3, 34], [3, 40], [3, 42], [3, 45], [3, 46], [3, 47], [3, 50], [3, 53], [4, 8], [4, 10], [4, 13], [4, 16], [4, 26], [4, 31], [4, 32], [4, 35], [4, 40], [4, 41], [4, 42], [4, 45], [4, 48], [4, 51], [4, 54], [4, 57], [4, 59], [5, 7], [5, 9], [5, 10], [5, 13], [5, 14], [5, 22], [5, 23], [5, 33], [5, 34], [5, 40], [5, 41], [5, 43], [5, 47], [5, 48], [5, 52], [6, 11], [6, 15], [6, 16], [6, 24], [6, 32], [6, 37], [6, 41], [6, 42], [6, 43], [6, 46], [6, 48], [6, 52], [6, 53], [6, 59], [7, 8], [7, 9], [7, 17], [7, 19], [7, 20], [7, 23], [7, 26], [7, 29], [7, 30], [7, 37], [7, 43], [7, 47], [7, 50], [7, 53], [7, 55], [7, 56], [7, 58], [8, 17], [8, 24], [8, 27], [8, 28], [8, 31], [8, 35], [8, 39], [8, 49], [8, 52], [8, 53], [9, 10], [9, 11], [9, 21], [9, 24], [9, 25], [9, 28], [9, 30], [9, 31], [9, 32], [9, 37], [9, 38], [9, 42], [9, 43], [9, 44], [9, 46], [9, 47], [9, 50], [9, 52], [9, 56], [10, 15], [10, 19], [10, 29], [10, 31], [10, 32], [10, 34], [10, 42], [10, 44], [10, 47], [10, 55], [10, 56], [11, 12], [11, 18], [11, 22], [11, 24], [11, 31], [11, 32], [11, 37], [11, 38], [11, 39], [11, 42], [11, 47], [11, 53], [11, 54], [12, 13], [12, 15], [12, 17], [12, 20], [12, 22], [12, 23], [12, 25], [12, 30], [12, 33], [12, 35], [12, 36], [12, 45], [12, 47], [12, 48], [12, 50], [12, 52], [12, 56], [12, 57], [12, 59], [13, 17], [13, 18], [13, 20], [13, 40], [13, 41], [13, 48], [14, 22], [14, 23], [14, 25], [14, 40], [14, 41], [14, 44], [14, 46], [14, 48], [14, 49], [14, 53], [14, 54], [14, 55], [14, 57], [15, 18], [15, 21], [15, 22], [15, 27], [15, 30], [15, 34], [15, 38], [15, 44], [15, 45], [15, 48], [15, 54], [15, 55], [15, 57], [15, 59], [16, 17], [16, 19], [16, 25], [16, 26], [16, 27], [16, 29], [16, 31], [16, 34], [16, 39], [16, 43], [16, 48], [16, 50], [16, 51], [16, 52], [16, 59], [17, 19], [17, 24], [17, 31], [17, 37], [17, 43], [17, 45], [17, 47], [17, 55], [17, 56], [17, 59], [18, 23], [18, 26], [18, 29], [18, 30], [18, 35], [18, 43], [18, 44], [18, 48], [18, 58], [19, 21], [19, 31], [19, 34], [19, 35], [19, 36], [19, 40], [19, 43], [19, 46], [19, 47], [19, 51], [19, 53], [19, 55], [19, 58], [20, 22], [20, 23], [20, 26], [20, 27], [20, 29], [20, 31], [20, 32], [20, 33], [20, 35], [20, 37], [20, 40], [20, 41], [20, 44], [20, 46], [20, 51], [20, 53], [20, 54], [20, 57], [21, 23], [21, 24], [21, 30], [21, 32], [21, 34], [21, 35], [21, 41], [21, 45], [21, 47], [21, 48], [21, 52], [21, 59], [22, 31], [22, 41], [23, 29], [23, 30], [23, 31], [23, 39], [23, 43], [23, 47], [23, 48], [23, 49], [23, 51], [23, 58], [24, 27], [24, 29], [24, 31], [24, 34], [24, 35], [24, 36], [24, 44], [24, 49], [25, 30], [25, 35], [25, 36], [25, 39], [25, 45], [25, 56], [25, 59], [26, 27], [26, 31], [26, 32], [26, 34], [26, 35], [26, 36], [26, 39], [26, 45], [26, 49], [26, 50], [26, 51], [26, 52], [26, 59], [27, 29], [27, 31], [27, 35], [27, 37], [27, 49], [27, 51], [27, 57], [28, 33], [28, 52], [28, 53], [29, 32], [29, 38], [29, 39], [29, 44], [29, 46], [29, 51], [29, 52], [29, 53], [29, 56], [29, 59], [30, 32], [30, 35], [30, 37], [30, 42], [30, 48], [30, 49], [30, 51], [30, 57], [31, 34], [31, 35], [31, 40], [31, 47], [31, 51], [31, 52], [31, 53], [31, 56], [31, 59], [32, 36], [32, 40], [32, 46], [32, 56], [32, 58], [33, 35], [33, 39], [33, 46], [33, 48], [33, 50], [33, 53], [33, 59], [34, 38], [34, 45], [34, 49], [34, 52], [34, 53], [34, 56], [34, 57], [34, 58], [34, 59], [35, 38], [35, 41], [35, 43], [35, 46], [35, 53], [35, 55], [35, 56], [36, 37], [36, 38], [36, 40], [36, 46], [36, 49], [36, 53], [36, 58], [37, 48], [37, 51], [37, 53], [38, 40], [38, 43], [38, 49], [38, 52], [38, 56], [38, 58], [39, 40], [39, 44], [39, 47], [39, 49], [39, 50], [39, 51], [39, 52], [39, 59], [40, 43], [40, 45], [40, 47], [40, 48], [40, 49], [40, 54], [41, 45], [41, 49], [41, 50], [41, 53], [41, 56], [41, 57], [41, 58], [42, 44], [42, 51], [42, 54], [42, 56], [42, 57], [42, 58], [42, 59], [43, 50], [43, 54], [43, 55], [43, 57], [44, 49], [44, 52], [44, 53], [44, 55], [45, 48], [45, 52], [45, 53], [45, 55], [45, 59], [46, 47], [46, 54], [46, 58], [47, 53], [47, 59], [48, 51], [48, 54], [48, 59], [49, 52], [49, 57], [50, 51], [50, 52], [50, 57], [50, 58], [51, 57], [52, 53], [52, 54], [52, 59], [53, 54], [54, 56], [55, 56], [55, 57], [56, 58], [56, 59]], "family": "mc_max_independent_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-max-independent-set-l4-s0/tests/run.py b/tasks/construct-mc-max-independent-set-l4-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-l4-s0/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-max-independent-set-l4-s0/tests/test.sh b/tasks/construct-mc-max-independent-set-l4-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-l4-s0/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-max-independent-set-l4-s7/tests/check.py b/tasks/construct-mc-max-independent-set-l4-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..adce413e8578d996549583b990e4ff559f38ed41
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-l4-s7/tests/check.py
@@ -0,0 +1,18 @@
+"""Independent set of size >= k in a given graph (MathConstraint max_independent_set, CC-BY-4.0)."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    S = int_list(ans.get("independent_set"), None, 0, n - 1, "independent_set")
+    if len(set(S)) != len(S):
+        raise Invalid("repeated vertex")
+    if len(S) < k:
+        raise Invalid(f"set has {len(S)} vertices, need at least {k}")
+    E = {(u, v) for u, v in inst["edges"]}
+    for u, v in itertools.combinations(sorted(S), 2):
+        if (u, v) in E:
+            raise Invalid(f"vertices {u} and {v} are adjacent")
+    return None, {"size": len(S)}
diff --git a/tasks/construct-mc-max-independent-set-l4-s7/tests/instance.json b/tasks/construct-mc-max-independent-set-l4-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..114404eb4999b21ea80d0010efc4b152027897fd
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-l4-s7/tests/instance.json
@@ -0,0 +1 @@
+{"n": 60, "k": 14, "edges": [[0, 1], [0, 4], [0, 6], [0, 8], [0, 17], [0, 20], [0, 27], [0, 35], [0, 40], [0, 43], [0, 45], [0, 46], [0, 47], [0, 48], [0, 50], [0, 52], [0, 57], [1, 3], [1, 4], [1, 10], [1, 11], [1, 13], [1, 14], [1, 15], [1, 16], [1, 21], [1, 23], [1, 27], [1, 28], [1, 29], [1, 38], [1, 47], [1, 48], [1, 53], [1, 54], [1, 55], [1, 56], [1, 59], [2, 4], [2, 6], [2, 7], [2, 9], [2, 13], [2, 16], [2, 25], [2, 29], [2, 31], [2, 32], [2, 33], [2, 37], [2, 44], [2, 46], [2, 48], [2, 49], [3, 6], [3, 10], [3, 11], [3, 12], [3, 17], [3, 18], [3, 19], [3, 20], [3, 26], [3, 27], [3, 34], [3, 36], [3, 41], [3, 51], [3, 53], [4, 8], [4, 11], [4, 13], [4, 14], [4, 16], [4, 21], [4, 22], [4, 23], [4, 30], [4, 35], [4, 37], [4, 42], [4, 45], [4, 47], [4, 48], [5, 7], [5, 9], [5, 15], [5, 23], [5, 31], [5, 35], [5, 39], [5, 40], [5, 45], [5, 49], [5, 50], [5, 52], [5, 53], [5, 55], [5, 56], [5, 57], [6, 7], [6, 12], [6, 14], [6, 23], [6, 24], [6, 25], [6, 29], [6, 43], [6, 45], [6, 46], [6, 49], [6, 52], [6, 54], [7, 8], [7, 9], [7, 10], [7, 13], [7, 18], [7, 21], [7, 24], [7, 25], [7, 26], [7, 32], [7, 34], [7, 35], [7, 41], [7, 45], [7, 47], [8, 11], [8, 12], [8, 13], [8, 14], [8, 20], [8, 26], [8, 28], [8, 34], [8, 37], [8, 39], [8, 40], [8, 50], [8, 51], [8, 56], [9, 11], [9, 15], [9, 17], [9, 21], [9, 22], [9, 24], [9, 31], [9, 33], [9, 35], [9, 38], [9, 40], [9, 44], [9, 47], [9, 52], [9, 56], [9, 57], [10, 11], [10, 15], [10, 24], [10, 25], [10, 27], [10, 28], [10, 37], [10, 42], [10, 51], [10, 54], [10, 56], [11, 12], [11, 13], [11, 14], [11, 15], [11, 21], [11, 28], [11, 29], [11, 36], [11, 39], [11, 44], [11, 49], [11, 50], [11, 55], [12, 14], [12, 17], [12, 19], [12, 22], [12, 26], [12, 27], [12, 33], [12, 34], [12, 35], [12, 40], [12, 42], [12, 43], [12, 46], [12, 49], [13, 14], [13, 16], [13, 17], [13, 20], [13, 23], [13, 27], [13, 29], [13, 30], [13, 33], [13, 41], [13, 43], [13, 44], [13, 45], [13, 51], [13, 53], [13, 55], [13, 56], [13, 57], [13, 58], [14, 15], [14, 16], [14, 17], [14, 25], [14, 26], [14, 27], [14, 31], [14, 32], [14, 33], [14, 36], [14, 42], [14, 46], [14, 52], [14, 53], [14, 55], [14, 56], [14, 57], [14, 58], [14, 59], [15, 16], [15, 20], [15, 25], [15, 26], [15, 27], [15, 30], [15, 35], [15, 38], [15, 39], [15, 44], [15, 47], [15, 51], [15, 52], [15, 59], [16, 18], [16, 20], [16, 34], [16, 36], [16, 42], [16, 43], [16, 54], [16, 56], [17, 20], [17, 23], [17, 28], [17, 32], [17, 38], [17, 46], [17, 48], [17, 50], [17, 51], [17, 52], [17, 55], [18, 19], [18, 23], [18, 24], [18, 25], [18, 31], [18, 32], [18, 33], [18, 34], [18, 36], [18, 37], [18, 43], [18, 53], [19, 21], [19, 23], [19, 24], [19, 25], [19, 28], [19, 30], [19, 36], [19, 40], [19, 43], [19, 45], [19, 46], [19, 53], [19, 57], [20, 22], [20, 25], [20, 29], [20, 31], [20, 32], [20, 44], [20, 45], [20, 49], [20, 57], [20, 59], [21, 22], [21, 26], [21, 30], [21, 32], [21, 33], [21, 34], [21, 37], [21, 42], [21, 43], [21, 51], [21, 53], [21, 57], [21, 58], [21, 59], [22, 25], [22, 30], [22, 32], [22, 35], [22, 42], [23, 26], [23, 28], [23, 30], [23, 31], [23, 37], [23, 43], [23, 45], [23, 46], [23, 48], [23, 54], [23, 55], [23, 56], [23, 57], [23, 59], [24, 26], [24, 27], [24, 28], [24, 30], [24, 31], [24, 40], [24, 54], [24, 57], [24, 58], [25, 26], [25, 27], [25, 28], [25, 31], [25, 33], [25, 34], [25, 39], [25, 40], [25, 50], [25, 53], [26, 27], [26, 29], [26, 30], [26, 32], [26, 34], [26, 37], [26, 38], [26, 42], [26, 43], [26, 48], [26, 53], [26, 55], [27, 34], [27, 36], [27, 39], [27, 47], [27, 48], [27, 52], [27, 55], [27, 57], [27, 59], [28, 31], [28, 33], [28, 35], [28, 37], [28, 39], [28, 51], [28, 57], [29, 31], [29, 33], [29, 34], [29, 42], [29, 52], [30, 33], [30, 39], [30, 44], [30, 54], [31, 43], [32, 34], [32, 39], [32, 42], [32, 46], [32, 53], [32, 55], [32, 58], [33, 35], [33, 45], [33, 49], [33, 50], [33, 56], [33, 59], [34, 48], [34, 50], [34, 54], [34, 55], [34, 59], [35, 37], [35, 38], [35, 42], [35, 43], [35, 44], [35, 46], [35, 49], [35, 52], [35, 54], [35, 56], [35, 59], [36, 46], [36, 49], [36, 51], [36, 52], [36, 56], [36, 57], [37, 39], [37, 40], [37, 44], [37, 53], [37, 58], [37, 59], [38, 40], [38, 41], [38, 49], [38, 52], [39, 42], [39, 44], [39, 45], [39, 46], [39, 47], [39, 49], [39, 50], [39, 57], [39, 59], [40, 45], [40, 46], [40, 48], [40, 49], [40, 54], [40, 55], [40, 59], [41, 42], [41, 48], [41, 49], [41, 57], [42, 43], [42, 44], [42, 45], [42, 47], [42, 48], [42, 52], [43, 54], [43, 55], [43, 57], [44, 45], [44, 46], [44, 51], [44, 52], [44, 56], [44, 59], [45, 47], [45, 49], [46, 47], [46, 52], [46, 59], [47, 54], [47, 58], [48, 49], [48, 52], [48, 56], [49, 53], [49, 54], [49, 57], [50, 51], [50, 54], [50, 58], [50, 59], [51, 53], [51, 54], [51, 56], [51, 59], [52, 54], [52, 57], [53, 54], [53, 59], [54, 56], [54, 57], [55, 56], [56, 59], [58, 59]], "family": "mc_max_independent_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-max-independent-set-l4-s7/tests/run.py b/tasks/construct-mc-max-independent-set-l4-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-l4-s7/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-max-independent-set-l4-s7/tests/test.sh b/tasks/construct-mc-max-independent-set-l4-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-l4-s7/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-non-transitive-dice-l4-s0/environment/Dockerfile b/tasks/construct-mc-non-transitive-dice-l4-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-non-transitive-dice-l4-s0/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-non-transitive-dice-l4-s0/environment/instance.json b/tasks/construct-mc-non-transitive-dice-l4-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..5ff8e0054afb6d69b52aced3e93bcba00cf843be
--- /dev/null
+++ b/tasks/construct-mc-non-transitive-dice-l4-s0/environment/instance.json
@@ -0,0 +1 @@
+{"n": 6, "m": 6, "family": "mc_non_transitive_dice", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-non-transitive-dice-l4-s0/solution/solve.sh b/tasks/construct-mc-non-transitive-dice-l4-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..2593e796ec360991150099f11b7674a417e4655c
--- /dev/null
+++ b/tasks/construct-mc-non-transitive-dice-l4-s0/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"dice": [[2, 5, 5, 5, 5, 11], [0, 0, 4, 9, 10, 11], [0, 4, 6, 8, 8, 8], [3, 3, 3, 6, 6, 11], [1, 1, 2, 10, 10, 11], [0, 6, 8, 8, 9, 9]]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-non-transitive-dice-l4-s0/tests/check.py b/tasks/construct-mc-non-transitive-dice-l4-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..e3d5c04fa6e2819bf32f34fafe13a383437dc62a
--- /dev/null
+++ b/tasks/construct-mc-non-transitive-dice-l4-s0/tests/check.py
@@ -0,0 +1,18 @@
+"""Cyclically intransitive dice (MathConstraint non_transitive_dice, CC-BY-4.0): n dice with m faces in
+0..2m-1; die i beats die (i+1) mod n, i.e. #{(a, b): face_i(a) > face_{i+1}(b)} > floor(m^2 / 2).
+Face order is irrelevant (upstream additionally demanded sorted faces as a symmetry break; not required)."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, m = inst["n"], inst["m"]
+    D = ans.get("dice")
+    if not isinstance(D, list) or len(D) != n:
+        raise Invalid(f"dice must be a list of {n} dice")
+    D = [int_list(d, m, 0, 2 * m - 1, "die") for d in D]
+    for i in range(n):
+        A, B = D[i], D[(i + 1) % n]
+        w = sum(1 for a in A for b in B if a > b)
+        if w <= m * m // 2:
+            raise Invalid(f"die {i} wins only {w} of {m * m} face pairs against die {(i + 1) % n}")
+    return None, {}
diff --git a/tasks/construct-mc-non-transitive-dice-l4-s0/tests/instance.json b/tasks/construct-mc-non-transitive-dice-l4-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..5ff8e0054afb6d69b52aced3e93bcba00cf843be
--- /dev/null
+++ b/tasks/construct-mc-non-transitive-dice-l4-s0/tests/instance.json
@@ -0,0 +1 @@
+{"n": 6, "m": 6, "family": "mc_non_transitive_dice", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-non-transitive-dice-l4-s0/tests/run.py b/tasks/construct-mc-non-transitive-dice-l4-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-non-transitive-dice-l4-s0/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-non-transitive-dice-l4-s0/tests/test.sh b/tasks/construct-mc-non-transitive-dice-l4-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-non-transitive-dice-l4-s0/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-number-partitioning-l4-s3/environment/Dockerfile b/tasks/construct-mc-number-partitioning-l4-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l4-s3/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-number-partitioning-l4-s3/environment/instance.json b/tasks/construct-mc-number-partitioning-l4-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0e143455222d9472645ed07bbfd7a2baf493bf55
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l4-s3/environment/instance.json
@@ -0,0 +1 @@
+{"n": 39, "k": 5, "family": "mc_number_partitioning", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-number-partitioning-l4-s3/solution/solve.sh b/tasks/construct-mc-number-partitioning-l4-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..bb10d0bbdbfcf9dfb35d87b16227b882f1e5e7d5
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l4-s3/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": [1, 2, 3, 4, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 4, 3, 2, 1, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-number-partitioning-l4-s3/tests/check.py b/tasks/construct-mc-number-partitioning-l4-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..67793721db8ba7d3c3979ea0e1e07091491ffa7e
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l4-s3/tests/check.py
@@ -0,0 +1,14 @@
+"""Partition {1..n} into k subsets of equal sum (MathConstraint number_partitioning, CC-BY-4.0)."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    x = int_list(ans.get("x"), n, 0, k - 1, "x")
+    target = n * (n + 1) // 2 // k
+    sums = [0] * k
+    for i, s in enumerate(x):
+        sums[s] += i + 1
+    if any(t != target for t in sums):
+        raise Invalid(f"subset sums {sums} are not all {target}")
+    return None, {}
diff --git a/tasks/construct-mc-number-partitioning-l4-s3/tests/instance.json b/tasks/construct-mc-number-partitioning-l4-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0e143455222d9472645ed07bbfd7a2baf493bf55
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l4-s3/tests/instance.json
@@ -0,0 +1 @@
+{"n": 39, "k": 5, "family": "mc_number_partitioning", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-number-partitioning-l4-s3/tests/run.py b/tasks/construct-mc-number-partitioning-l4-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l4-s3/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-number-partitioning-l4-s3/tests/test.sh b/tasks/construct-mc-number-partitioning-l4-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l4-s3/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-number-partitioning-l4-s4/environment/Dockerfile b/tasks/construct-mc-number-partitioning-l4-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l4-s4/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-number-partitioning-l4-s4/environment/instance.json b/tasks/construct-mc-number-partitioning-l4-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..18142b9a2dcd5019ee4a72119cd8f924f4583638
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l4-s4/environment/instance.json
@@ -0,0 +1 @@
+{"n": 35, "k": 5, "family": "mc_number_partitioning", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-number-partitioning-l4-s4/solution/solve.sh b/tasks/construct-mc-number-partitioning-l4-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..53fbc5a1ebb046229acf970a4e4cf27fb0c8921f
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l4-s4/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": [0, 2, 4, 1, 3, 1, 3, 0, 2, 4, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 4, 3, 2, 1, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-number-partitioning-l4-s4/tests/check.py b/tasks/construct-mc-number-partitioning-l4-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..67793721db8ba7d3c3979ea0e1e07091491ffa7e
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l4-s4/tests/check.py
@@ -0,0 +1,14 @@
+"""Partition {1..n} into k subsets of equal sum (MathConstraint number_partitioning, CC-BY-4.0)."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    x = int_list(ans.get("x"), n, 0, k - 1, "x")
+    target = n * (n + 1) // 2 // k
+    sums = [0] * k
+    for i, s in enumerate(x):
+        sums[s] += i + 1
+    if any(t != target for t in sums):
+        raise Invalid(f"subset sums {sums} are not all {target}")
+    return None, {}
diff --git a/tasks/construct-mc-number-partitioning-l4-s4/tests/instance.json b/tasks/construct-mc-number-partitioning-l4-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..18142b9a2dcd5019ee4a72119cd8f924f4583638
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l4-s4/tests/instance.json
@@ -0,0 +1 @@
+{"n": 35, "k": 5, "family": "mc_number_partitioning", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-number-partitioning-l4-s4/tests/run.py b/tasks/construct-mc-number-partitioning-l4-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l4-s4/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-number-partitioning-l4-s4/tests/test.sh b/tasks/construct-mc-number-partitioning-l4-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l4-s4/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-ortholatin-l6-s3/environment/Dockerfile b/tasks/construct-mc-ortholatin-l6-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-ortholatin-l6-s3/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-ortholatin-l6-s3/environment/instance.json b/tasks/construct-mc-ortholatin-l6-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0096ed58887ceac78440170b4f235db544d42b0e
--- /dev/null
+++ b/tasks/construct-mc-ortholatin-l6-s3/environment/instance.json
@@ -0,0 +1 @@
+{"n": 27, "family": "mc_ortholatin", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-ortholatin-l6-s3/solution/solve.sh b/tasks/construct-mc-ortholatin-l6-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..ff8a6717452e108750f215b5db67b7f947d7374f
--- /dev/null
+++ b/tasks/construct-mc-ortholatin-l6-s3/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"X": [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]], "Y": [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], [2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 0], [3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 1], [4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 0, 2], [5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 1, 3], [6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 0, 2, 4], [7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 1, 3, 5], [8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 0, 2, 4, 6], [9, 11, 13, 15, 17, 19, 21, 23, 25, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 1, 3, 5, 7], [10, 12, 14, 16, 18, 20, 22, 24, 26, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 0, 2, 4, 6, 8], [11, 13, 15, 17, 19, 21, 23, 25, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 1, 3, 5, 7, 9], [12, 14, 16, 18, 20, 22, 24, 26, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 0, 2, 4, 6, 8, 10], [13, 15, 17, 19, 21, 23, 25, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 1, 3, 5, 7, 9, 11], [14, 16, 18, 20, 22, 24, 26, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 0, 2, 4, 6, 8, 10, 12], [15, 17, 19, 21, 23, 25, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 1, 3, 5, 7, 9, 11, 13], [16, 18, 20, 22, 24, 26, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 0, 2, 4, 6, 8, 10, 12, 14], [17, 19, 21, 23, 25, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 1, 3, 5, 7, 9, 11, 13, 15], [18, 20, 22, 24, 26, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 0, 2, 4, 6, 8, 10, 12, 14, 16], [19, 21, 23, 25, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 1, 3, 5, 7, 9, 11, 13, 15, 17], [20, 22, 24, 26, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18], [21, 23, 25, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19], [22, 24, 26, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], [23, 25, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [24, 26, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], [25, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], [26, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24]]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-ortholatin-l6-s3/tests/check.py b/tasks/construct-mc-ortholatin-l6-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..a5418a59fae1c86318eee903e90d75b36d8a3306
--- /dev/null
+++ b/tasks/construct-mc-ortholatin-l6-s3/tests/check.py
@@ -0,0 +1,25 @@
+"""Pair of orthogonal Latin squares of order n (MathConstraint ortholatin, CC-BY-4.0)."""
+from run import Invalid, int_list
+
+
+def _square(v, n, name):
+    if not isinstance(v, list) or len(v) != n:
+        raise Invalid(f"{name} must be a list of {n} rows")
+    S = [int_list(r, n, 0, n - 1, f"{name} row") for r in v]
+    full = list(range(n))
+    for i in range(n):
+        if sorted(S[i]) != full:
+            raise Invalid(f"{name}: row {i} is not a permutation of 0..{n - 1}")
+        if sorted(S[r][i] for r in range(n)) != full:
+            raise Invalid(f"{name}: column {i} is not a permutation of 0..{n - 1}")
+    return S
+
+
+def check(inst, ans):
+    n = inst["n"]
+    X = _square(ans.get("X"), n, "X")
+    Y = _square(ans.get("Y"), n, "Y")
+    pairs = {(X[i][j], Y[i][j]) for i in range(n) for j in range(n)}
+    if len(pairs) != n * n:
+        raise Invalid("X and Y are not orthogonal (some ordered pair repeats)")
+    return None, {}
diff --git a/tasks/construct-mc-ortholatin-l6-s3/tests/instance.json b/tasks/construct-mc-ortholatin-l6-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0096ed58887ceac78440170b4f235db544d42b0e
--- /dev/null
+++ b/tasks/construct-mc-ortholatin-l6-s3/tests/instance.json
@@ -0,0 +1 @@
+{"n": 27, "family": "mc_ortholatin", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-ortholatin-l6-s3/tests/run.py b/tasks/construct-mc-ortholatin-l6-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-ortholatin-l6-s3/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-ortholatin-l6-s3/tests/test.sh b/tasks/construct-mc-ortholatin-l6-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-ortholatin-l6-s3/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-pysms-chromatic-girth-l2-s2/environment/Dockerfile b/tasks/construct-mc-pysms-chromatic-girth-l2-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-pysms-chromatic-girth-l2-s2/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-pysms-chromatic-girth-l2-s2/environment/instance.json b/tasks/construct-mc-pysms-chromatic-girth-l2-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8d5a5630a359f6162a29cb3a1f5c65f1031c46b3
--- /dev/null
+++ b/tasks/construct-mc-pysms-chromatic-girth-l2-s2/environment/instance.json
@@ -0,0 +1 @@
+{"vertices": 15, "required_edges": [], "max_chromatic_number": 2, "min_edges": 16, "min_girth": 6, "family": "mc_pysms_chromatic_girth", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-chromatic-girth-l2-s2/solution/solve.sh b/tasks/construct-mc-pysms-chromatic-girth-l2-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..29b31295aabc76eea29190ef259ee5c499d715a7
--- /dev/null
+++ b/tasks/construct-mc-pysms-chromatic-girth-l2-s2/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"edges": [[0, 13], [0, 14], [1, 11], [1, 12], [2, 10], [2, 12], [2, 14], [3, 9], [3, 12], [3, 13], [4, 9], [4, 10], [4, 11], [5, 8], [5, 11], [5, 14], [6, 8], [6, 10], [6, 13], [7, 8], [7, 9]], "coloring": [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-pysms-chromatic-girth-l2-s2/tests/check.py b/tasks/construct-mc-pysms-chromatic-girth-l2-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..4ab0b12508cf59b733f007a9d1f50831aec9efdf
--- /dev/null
+++ b/tasks/construct-mc-pysms-chromatic-girth-l2-s2/tests/check.py
@@ -0,0 +1,151 @@
+"""Graph-existence constraints (MathConstraint pysms_* problem types, CC-BY-4.0). The answer is a graph on
+the stated vertex set, plus a proper colouring when a chromatic-number upper bound is asked for. Every
+constraint present in the instance is checked exactly (no stored answer is compared).
+
+Upstream note: upstream verifies by fixing the edge variables in the pysms CNF and calling CaDiCaL; the
+min-chromatic-number bound of pysms_graph_builder is only a raw smsg solver parameter, not a clause, so the
+upstream verifier ignored it. It is enforced here. Edge endpoints are also type-checked (upstream accepted
+floats/bools that compare in range)."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def _has_clique(adj, n, k):
+    def rec(cand, size):
+        if size == k:
+            return True
+        while cand:
+            if size + bin(cand).count("1") < k:
+                return False
+            v = cand.bit_length() - 1
+            cand &= ~(1 << v)
+            if rec(cand & adj[v], size + 1):
+                return True
+        return False
+    return rec((1 << n) - 1, 0)
+
+
+def _connected(adj, n, removed):
+    alive = [v for v in range(n) if not removed >> v & 1]
+    if len(alive) <= 1:
+        return True
+    mask = sum(1 << v for v in alive)
+    seen = 1 << alive[0]
+    stack = [alive[0]]
+    while stack:
+        v = stack.pop()
+        new = adj[v] & mask & ~seen
+        seen |= new
+        while new:
+            u = new.bit_length() - 1
+            new &= ~(1 << u)
+            stack.append(u)
+    return seen == mask
+
+
+def _girth(adj, n):
+    best = None
+    for r in range(n):
+        dist, par, q = {r: 0}, {r: -1}, [r]
+        for v in q:
+            nb = adj[v]
+            while nb:
+                u = nb.bit_length() - 1
+                nb &= ~(1 << u)
+                if u not in dist:
+                    dist[u], par[u] = dist[v] + 1, v
+                    q.append(u)
+                elif par[v] != u:
+                    c = dist[u] + dist[v] + 1
+                    best = c if best is None else min(best, c)
+    return best  # None = acyclic
+
+
+def _colourable(adj, n, k):
+    col = [-1] * n
+    order = sorted(range(n), key=lambda v: -bin(adj[v]).count("1"))
+
+    def rec(i):
+        if i == n:
+            return True
+        v = order[i]
+        used = {col[u] for u in range(n) if adj[v] >> u & 1 and col[u] >= 0}
+        for c in range(k):
+            if c not in used:
+                col[v] = c
+                if rec(i + 1):
+                    return True
+        col[v] = -1
+        return False
+    return rec(0)
+
+
+def check(inst, ans):
+    n = inst["vertices"]
+    E = ans.get("edges")
+    if not isinstance(E, list):
+        raise Invalid("edges must be a list of [u, v] pairs")
+    edges = set()
+    for e in E:
+        u, v = int_list(e, 2, 0, n - 1, "edge")
+        if u == v:
+            raise Invalid(f"self-loop at {u}")
+        e2 = (min(u, v), max(u, v))
+        if e2 in edges:
+            raise Invalid(f"duplicate edge {e2}")
+        edges.add(e2)
+    adj = [0] * n
+    for u, v in edges:
+        adj[u] |= 1 << v
+        adj[v] |= 1 << u
+    deg = [bin(a).count("1") for a in adj]
+    for u, v in inst.get("required_edges", []):
+        if (min(u, v), max(u, v)) not in edges:
+            raise Invalid(f"required edge ({u},{v}) is missing")
+    c = inst.get
+    if c("min_degree") is not None and min(deg) < c("min_degree"):
+        raise Invalid(f"minimum degree {min(deg)} < {c('min_degree')}")
+    if c("max_degree") is not None and max(deg) > c("max_degree"):
+        raise Invalid(f"maximum degree {max(deg)} > {c('max_degree')}")
+    if c("min_edges") is not None and len(edges) < c("min_edges"):
+        raise Invalid(f"{len(edges)} edges < {c('min_edges')}")
+    if c("max_edges") is not None and len(edges) > c("max_edges"):
+        raise Invalid(f"{len(edges)} edges > {c('max_edges')}")
+    if c("max_clique") is not None and _has_clique(adj, n, c("max_clique") + 1):
+        raise Invalid(f"graph contains a clique of size {c('max_clique') + 1}")
+    if c("max_independent_set") is not None:
+        full = (1 << n) - 1
+        comp = [full & ~adj[v] & ~(1 << v) for v in range(n)]
+        if _has_clique(comp, n, c("max_independent_set") + 1):
+            raise Invalid(f"graph contains an independent set of size {c('max_independent_set') + 1}")
+    if c("min_girth") is not None:
+        g = _girth(adj, n)
+        if g is not None and g < c("min_girth"):
+            raise Invalid(f"graph has a cycle of length {g} < {c('min_girth')}")
+    if c("min_connectivity") is not None:
+        k = c("min_connectivity")
+        if n <= k:
+            raise Invalid("too few vertices for this connectivity")
+        for s in range(k):
+            for I in itertools.combinations(range(n), s):
+                if not _connected(adj, n, sum(1 << v for v in I)):
+                    raise Invalid(f"deleting vertices {list(I)} disconnects the graph")
+    if c("mtf"):
+        for u, v in itertools.combinations(range(n), 2):
+            if adj[u] >> v & 1:
+                if adj[u] & adj[v]:
+                    raise Invalid(f"triangle on edge ({u},{v})")
+            elif not adj[u] & adj[v]:
+                raise Invalid(f"non-adjacent {u}, {v} have no common neighbour (edge could be added)")
+    if c("max_chromatic_number") is not None:
+        k = c("max_chromatic_number")
+        col = int_list(ans.get("coloring"), n, 0, k - 1, "coloring")
+        for u, v in edges:
+            if col[u] == col[v]:
+                raise Invalid(f"coloring: edge ({u},{v}) is monochromatic")
+    if c("min_chromatic_number") is not None and c("min_chromatic_number") >= 2:
+        m = c("min_chromatic_number")
+        if not edges or _colourable(adj, n, m - 1):
+            raise Invalid(f"graph is {m - 1}-colourable, chromatic number must be at least {m}")
+    return None, {"num_edges": len(edges)}
diff --git a/tasks/construct-mc-pysms-chromatic-girth-l2-s2/tests/instance.json b/tasks/construct-mc-pysms-chromatic-girth-l2-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8d5a5630a359f6162a29cb3a1f5c65f1031c46b3
--- /dev/null
+++ b/tasks/construct-mc-pysms-chromatic-girth-l2-s2/tests/instance.json
@@ -0,0 +1 @@
+{"vertices": 15, "required_edges": [], "max_chromatic_number": 2, "min_edges": 16, "min_girth": 6, "family": "mc_pysms_chromatic_girth", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-chromatic-girth-l2-s2/tests/run.py b/tasks/construct-mc-pysms-chromatic-girth-l2-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-pysms-chromatic-girth-l2-s2/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-pysms-chromatic-girth-l2-s2/tests/test.sh b/tasks/construct-mc-pysms-chromatic-girth-l2-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-pysms-chromatic-girth-l2-s2/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-pysms-clique-coloring-l1-s0/environment/Dockerfile b/tasks/construct-mc-pysms-clique-coloring-l1-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-pysms-clique-coloring-l1-s0/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-pysms-clique-coloring-l1-s0/environment/instance.json b/tasks/construct-mc-pysms-clique-coloring-l1-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8298a412db9ad82d0ec7809a81043c431331b4d4
--- /dev/null
+++ b/tasks/construct-mc-pysms-clique-coloring-l1-s0/environment/instance.json
@@ -0,0 +1 @@
+{"vertices": 8, "required_edges": [[2, 7], [0, 5]], "max_chromatic_number": 2, "max_clique": 4, "min_degree": 1, "family": "mc_pysms_clique_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-clique-coloring-l1-s0/solution/solve.sh b/tasks/construct-mc-pysms-clique-coloring-l1-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..08bf1a50f3f92d78f6044651a9d06e35672a9282
--- /dev/null
+++ b/tasks/construct-mc-pysms-clique-coloring-l1-s0/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"edges": [[0, 5], [0, 6], [0, 7], [1, 4], [1, 6], [1, 7], [2, 4], [2, 5], [2, 6], [2, 7], [3, 4], [3, 5], [3, 6], [3, 7]], "coloring": [0, 0, 0, 0, 1, 1, 1, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-pysms-clique-coloring-l1-s0/tests/check.py b/tasks/construct-mc-pysms-clique-coloring-l1-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..4ab0b12508cf59b733f007a9d1f50831aec9efdf
--- /dev/null
+++ b/tasks/construct-mc-pysms-clique-coloring-l1-s0/tests/check.py
@@ -0,0 +1,151 @@
+"""Graph-existence constraints (MathConstraint pysms_* problem types, CC-BY-4.0). The answer is a graph on
+the stated vertex set, plus a proper colouring when a chromatic-number upper bound is asked for. Every
+constraint present in the instance is checked exactly (no stored answer is compared).
+
+Upstream note: upstream verifies by fixing the edge variables in the pysms CNF and calling CaDiCaL; the
+min-chromatic-number bound of pysms_graph_builder is only a raw smsg solver parameter, not a clause, so the
+upstream verifier ignored it. It is enforced here. Edge endpoints are also type-checked (upstream accepted
+floats/bools that compare in range)."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def _has_clique(adj, n, k):
+    def rec(cand, size):
+        if size == k:
+            return True
+        while cand:
+            if size + bin(cand).count("1") < k:
+                return False
+            v = cand.bit_length() - 1
+            cand &= ~(1 << v)
+            if rec(cand & adj[v], size + 1):
+                return True
+        return False
+    return rec((1 << n) - 1, 0)
+
+
+def _connected(adj, n, removed):
+    alive = [v for v in range(n) if not removed >> v & 1]
+    if len(alive) <= 1:
+        return True
+    mask = sum(1 << v for v in alive)
+    seen = 1 << alive[0]
+    stack = [alive[0]]
+    while stack:
+        v = stack.pop()
+        new = adj[v] & mask & ~seen
+        seen |= new
+        while new:
+            u = new.bit_length() - 1
+            new &= ~(1 << u)
+            stack.append(u)
+    return seen == mask
+
+
+def _girth(adj, n):
+    best = None
+    for r in range(n):
+        dist, par, q = {r: 0}, {r: -1}, [r]
+        for v in q:
+            nb = adj[v]
+            while nb:
+                u = nb.bit_length() - 1
+                nb &= ~(1 << u)
+                if u not in dist:
+                    dist[u], par[u] = dist[v] + 1, v
+                    q.append(u)
+                elif par[v] != u:
+                    c = dist[u] + dist[v] + 1
+                    best = c if best is None else min(best, c)
+    return best  # None = acyclic
+
+
+def _colourable(adj, n, k):
+    col = [-1] * n
+    order = sorted(range(n), key=lambda v: -bin(adj[v]).count("1"))
+
+    def rec(i):
+        if i == n:
+            return True
+        v = order[i]
+        used = {col[u] for u in range(n) if adj[v] >> u & 1 and col[u] >= 0}
+        for c in range(k):
+            if c not in used:
+                col[v] = c
+                if rec(i + 1):
+                    return True
+        col[v] = -1
+        return False
+    return rec(0)
+
+
+def check(inst, ans):
+    n = inst["vertices"]
+    E = ans.get("edges")
+    if not isinstance(E, list):
+        raise Invalid("edges must be a list of [u, v] pairs")
+    edges = set()
+    for e in E:
+        u, v = int_list(e, 2, 0, n - 1, "edge")
+        if u == v:
+            raise Invalid(f"self-loop at {u}")
+        e2 = (min(u, v), max(u, v))
+        if e2 in edges:
+            raise Invalid(f"duplicate edge {e2}")
+        edges.add(e2)
+    adj = [0] * n
+    for u, v in edges:
+        adj[u] |= 1 << v
+        adj[v] |= 1 << u
+    deg = [bin(a).count("1") for a in adj]
+    for u, v in inst.get("required_edges", []):
+        if (min(u, v), max(u, v)) not in edges:
+            raise Invalid(f"required edge ({u},{v}) is missing")
+    c = inst.get
+    if c("min_degree") is not None and min(deg) < c("min_degree"):
+        raise Invalid(f"minimum degree {min(deg)} < {c('min_degree')}")
+    if c("max_degree") is not None and max(deg) > c("max_degree"):
+        raise Invalid(f"maximum degree {max(deg)} > {c('max_degree')}")
+    if c("min_edges") is not None and len(edges) < c("min_edges"):
+        raise Invalid(f"{len(edges)} edges < {c('min_edges')}")
+    if c("max_edges") is not None and len(edges) > c("max_edges"):
+        raise Invalid(f"{len(edges)} edges > {c('max_edges')}")
+    if c("max_clique") is not None and _has_clique(adj, n, c("max_clique") + 1):
+        raise Invalid(f"graph contains a clique of size {c('max_clique') + 1}")
+    if c("max_independent_set") is not None:
+        full = (1 << n) - 1
+        comp = [full & ~adj[v] & ~(1 << v) for v in range(n)]
+        if _has_clique(comp, n, c("max_independent_set") + 1):
+            raise Invalid(f"graph contains an independent set of size {c('max_independent_set') + 1}")
+    if c("min_girth") is not None:
+        g = _girth(adj, n)
+        if g is not None and g < c("min_girth"):
+            raise Invalid(f"graph has a cycle of length {g} < {c('min_girth')}")
+    if c("min_connectivity") is not None:
+        k = c("min_connectivity")
+        if n <= k:
+            raise Invalid("too few vertices for this connectivity")
+        for s in range(k):
+            for I in itertools.combinations(range(n), s):
+                if not _connected(adj, n, sum(1 << v for v in I)):
+                    raise Invalid(f"deleting vertices {list(I)} disconnects the graph")
+    if c("mtf"):
+        for u, v in itertools.combinations(range(n), 2):
+            if adj[u] >> v & 1:
+                if adj[u] & adj[v]:
+                    raise Invalid(f"triangle on edge ({u},{v})")
+            elif not adj[u] & adj[v]:
+                raise Invalid(f"non-adjacent {u}, {v} have no common neighbour (edge could be added)")
+    if c("max_chromatic_number") is not None:
+        k = c("max_chromatic_number")
+        col = int_list(ans.get("coloring"), n, 0, k - 1, "coloring")
+        for u, v in edges:
+            if col[u] == col[v]:
+                raise Invalid(f"coloring: edge ({u},{v}) is monochromatic")
+    if c("min_chromatic_number") is not None and c("min_chromatic_number") >= 2:
+        m = c("min_chromatic_number")
+        if not edges or _colourable(adj, n, m - 1):
+            raise Invalid(f"graph is {m - 1}-colourable, chromatic number must be at least {m}")
+    return None, {"num_edges": len(edges)}
diff --git a/tasks/construct-mc-pysms-clique-coloring-l1-s0/tests/instance.json b/tasks/construct-mc-pysms-clique-coloring-l1-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8298a412db9ad82d0ec7809a81043c431331b4d4
--- /dev/null
+++ b/tasks/construct-mc-pysms-clique-coloring-l1-s0/tests/instance.json
@@ -0,0 +1 @@
+{"vertices": 8, "required_edges": [[2, 7], [0, 5]], "max_chromatic_number": 2, "max_clique": 4, "min_degree": 1, "family": "mc_pysms_clique_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-clique-coloring-l1-s0/tests/run.py b/tasks/construct-mc-pysms-clique-coloring-l1-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-pysms-clique-coloring-l1-s0/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-pysms-clique-coloring-l1-s0/tests/test.sh b/tasks/construct-mc-pysms-clique-coloring-l1-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-pysms-clique-coloring-l1-s0/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-pysms-clique-coloring-l5-s1/environment/Dockerfile b/tasks/construct-mc-pysms-clique-coloring-l5-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-pysms-clique-coloring-l5-s1/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-pysms-clique-coloring-l5-s1/environment/instance.json b/tasks/construct-mc-pysms-clique-coloring-l5-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d993d87c0ef1beccc527208fa76ae696e7add9a7
--- /dev/null
+++ b/tasks/construct-mc-pysms-clique-coloring-l5-s1/environment/instance.json
@@ -0,0 +1 @@
+{"vertices": 19, "required_edges": [], "max_chromatic_number": 5, "max_clique": 5, "min_degree": 5, "family": "mc_pysms_clique_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-clique-coloring-l5-s1/solution/solve.sh b/tasks/construct-mc-pysms-clique-coloring-l5-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..8f7408debf29c73eac398520cf97c5d3e2978829
--- /dev/null
+++ b/tasks/construct-mc-pysms-clique-coloring-l5-s1/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"edges": [[0, 2], [0, 3], [0, 4], [0, 5], [0, 6], [0, 7], [0, 8], [0, 9], [0, 10], [0, 11], [0, 12], [0, 13], [0, 14], [0, 15], [0, 16], [0, 17], [0, 18], [1, 2], [1, 3], [1, 4], [1, 5], [1, 6], [1, 7], [1, 8], [1, 9], [1, 10], [1, 11], [1, 12], [1, 13], [1, 14], [1, 15], [1, 16], [1, 17], [1, 18], [2, 3], [2, 4], [2, 5], [2, 6], [2, 7], [2, 8], [2, 9], [2, 10], [2, 11], [2, 12], [2, 13], [2, 14], [2, 15], [2, 16], [2, 17], [2, 18], [3, 4], [3, 5], [3, 6], [3, 7], [3, 8], [3, 9], [3, 10], [3, 11], [3, 12], [3, 13], [3, 14], [3, 15], [3, 16], [3, 17], [3, 18], [4, 5], [4, 6], [4, 7], [4, 8], [4, 9], [4, 10], [4, 11], [4, 12], [4, 13], [4, 14], [4, 15], [4, 16], [4, 17], [4, 18]], "coloring": [3, 3, 0, 1, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-pysms-clique-coloring-l5-s1/tests/check.py b/tasks/construct-mc-pysms-clique-coloring-l5-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..4ab0b12508cf59b733f007a9d1f50831aec9efdf
--- /dev/null
+++ b/tasks/construct-mc-pysms-clique-coloring-l5-s1/tests/check.py
@@ -0,0 +1,151 @@
+"""Graph-existence constraints (MathConstraint pysms_* problem types, CC-BY-4.0). The answer is a graph on
+the stated vertex set, plus a proper colouring when a chromatic-number upper bound is asked for. Every
+constraint present in the instance is checked exactly (no stored answer is compared).
+
+Upstream note: upstream verifies by fixing the edge variables in the pysms CNF and calling CaDiCaL; the
+min-chromatic-number bound of pysms_graph_builder is only a raw smsg solver parameter, not a clause, so the
+upstream verifier ignored it. It is enforced here. Edge endpoints are also type-checked (upstream accepted
+floats/bools that compare in range)."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def _has_clique(adj, n, k):
+    def rec(cand, size):
+        if size == k:
+            return True
+        while cand:
+            if size + bin(cand).count("1") < k:
+                return False
+            v = cand.bit_length() - 1
+            cand &= ~(1 << v)
+            if rec(cand & adj[v], size + 1):
+                return True
+        return False
+    return rec((1 << n) - 1, 0)
+
+
+def _connected(adj, n, removed):
+    alive = [v for v in range(n) if not removed >> v & 1]
+    if len(alive) <= 1:
+        return True
+    mask = sum(1 << v for v in alive)
+    seen = 1 << alive[0]
+    stack = [alive[0]]
+    while stack:
+        v = stack.pop()
+        new = adj[v] & mask & ~seen
+        seen |= new
+        while new:
+            u = new.bit_length() - 1
+            new &= ~(1 << u)
+            stack.append(u)
+    return seen == mask
+
+
+def _girth(adj, n):
+    best = None
+    for r in range(n):
+        dist, par, q = {r: 0}, {r: -1}, [r]
+        for v in q:
+            nb = adj[v]
+            while nb:
+                u = nb.bit_length() - 1
+                nb &= ~(1 << u)
+                if u not in dist:
+                    dist[u], par[u] = dist[v] + 1, v
+                    q.append(u)
+                elif par[v] != u:
+                    c = dist[u] + dist[v] + 1
+                    best = c if best is None else min(best, c)
+    return best  # None = acyclic
+
+
+def _colourable(adj, n, k):
+    col = [-1] * n
+    order = sorted(range(n), key=lambda v: -bin(adj[v]).count("1"))
+
+    def rec(i):
+        if i == n:
+            return True
+        v = order[i]
+        used = {col[u] for u in range(n) if adj[v] >> u & 1 and col[u] >= 0}
+        for c in range(k):
+            if c not in used:
+                col[v] = c
+                if rec(i + 1):
+                    return True
+        col[v] = -1
+        return False
+    return rec(0)
+
+
+def check(inst, ans):
+    n = inst["vertices"]
+    E = ans.get("edges")
+    if not isinstance(E, list):
+        raise Invalid("edges must be a list of [u, v] pairs")
+    edges = set()
+    for e in E:
+        u, v = int_list(e, 2, 0, n - 1, "edge")
+        if u == v:
+            raise Invalid(f"self-loop at {u}")
+        e2 = (min(u, v), max(u, v))
+        if e2 in edges:
+            raise Invalid(f"duplicate edge {e2}")
+        edges.add(e2)
+    adj = [0] * n
+    for u, v in edges:
+        adj[u] |= 1 << v
+        adj[v] |= 1 << u
+    deg = [bin(a).count("1") for a in adj]
+    for u, v in inst.get("required_edges", []):
+        if (min(u, v), max(u, v)) not in edges:
+            raise Invalid(f"required edge ({u},{v}) is missing")
+    c = inst.get
+    if c("min_degree") is not None and min(deg) < c("min_degree"):
+        raise Invalid(f"minimum degree {min(deg)} < {c('min_degree')}")
+    if c("max_degree") is not None and max(deg) > c("max_degree"):
+        raise Invalid(f"maximum degree {max(deg)} > {c('max_degree')}")
+    if c("min_edges") is not None and len(edges) < c("min_edges"):
+        raise Invalid(f"{len(edges)} edges < {c('min_edges')}")
+    if c("max_edges") is not None and len(edges) > c("max_edges"):
+        raise Invalid(f"{len(edges)} edges > {c('max_edges')}")
+    if c("max_clique") is not None and _has_clique(adj, n, c("max_clique") + 1):
+        raise Invalid(f"graph contains a clique of size {c('max_clique') + 1}")
+    if c("max_independent_set") is not None:
+        full = (1 << n) - 1
+        comp = [full & ~adj[v] & ~(1 << v) for v in range(n)]
+        if _has_clique(comp, n, c("max_independent_set") + 1):
+            raise Invalid(f"graph contains an independent set of size {c('max_independent_set') + 1}")
+    if c("min_girth") is not None:
+        g = _girth(adj, n)
+        if g is not None and g < c("min_girth"):
+            raise Invalid(f"graph has a cycle of length {g} < {c('min_girth')}")
+    if c("min_connectivity") is not None:
+        k = c("min_connectivity")
+        if n <= k:
+            raise Invalid("too few vertices for this connectivity")
+        for s in range(k):
+            for I in itertools.combinations(range(n), s):
+                if not _connected(adj, n, sum(1 << v for v in I)):
+                    raise Invalid(f"deleting vertices {list(I)} disconnects the graph")
+    if c("mtf"):
+        for u, v in itertools.combinations(range(n), 2):
+            if adj[u] >> v & 1:
+                if adj[u] & adj[v]:
+                    raise Invalid(f"triangle on edge ({u},{v})")
+            elif not adj[u] & adj[v]:
+                raise Invalid(f"non-adjacent {u}, {v} have no common neighbour (edge could be added)")
+    if c("max_chromatic_number") is not None:
+        k = c("max_chromatic_number")
+        col = int_list(ans.get("coloring"), n, 0, k - 1, "coloring")
+        for u, v in edges:
+            if col[u] == col[v]:
+                raise Invalid(f"coloring: edge ({u},{v}) is monochromatic")
+    if c("min_chromatic_number") is not None and c("min_chromatic_number") >= 2:
+        m = c("min_chromatic_number")
+        if not edges or _colourable(adj, n, m - 1):
+            raise Invalid(f"graph is {m - 1}-colourable, chromatic number must be at least {m}")
+    return None, {"num_edges": len(edges)}
diff --git a/tasks/construct-mc-pysms-clique-coloring-l5-s1/tests/instance.json b/tasks/construct-mc-pysms-clique-coloring-l5-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d993d87c0ef1beccc527208fa76ae696e7add9a7
--- /dev/null
+++ b/tasks/construct-mc-pysms-clique-coloring-l5-s1/tests/instance.json
@@ -0,0 +1 @@
+{"vertices": 19, "required_edges": [], "max_chromatic_number": 5, "max_clique": 5, "min_degree": 5, "family": "mc_pysms_clique_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-clique-coloring-l5-s1/tests/run.py b/tasks/construct-mc-pysms-clique-coloring-l5-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-pysms-clique-coloring-l5-s1/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-pysms-clique-coloring-l5-s1/tests/test.sh b/tasks/construct-mc-pysms-clique-coloring-l5-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-pysms-clique-coloring-l5-s1/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-pysms-girth-degree-l1-s0/environment/Dockerfile b/tasks/construct-mc-pysms-girth-degree-l1-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-pysms-girth-degree-l1-s0/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-pysms-girth-degree-l1-s0/environment/instance.json b/tasks/construct-mc-pysms-girth-degree-l1-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..9ea9f7bd58978c6bed7d38a1fd2a8f4dd21b6519
--- /dev/null
+++ b/tasks/construct-mc-pysms-girth-degree-l1-s0/environment/instance.json
@@ -0,0 +1 @@
+{"vertices": 11, "required_edges": [], "max_degree": 5, "min_degree": 2, "min_girth": 6, "family": "mc_pysms_girth_degree", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-girth-degree-l1-s0/solution/solve.sh b/tasks/construct-mc-pysms-girth-degree-l1-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..4874c4e575317420207cb776e6b4d6e3e99d0ff7
--- /dev/null
+++ b/tasks/construct-mc-pysms-girth-degree-l1-s0/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"edges": [[0, 9], [0, 10], [1, 8], [1, 10], [2, 7], [2, 10], [3, 6], [3, 10], [4, 5], [4, 8], [4, 9], [5, 6], [5, 7]]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-pysms-girth-degree-l1-s0/tests/check.py b/tasks/construct-mc-pysms-girth-degree-l1-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..4ab0b12508cf59b733f007a9d1f50831aec9efdf
--- /dev/null
+++ b/tasks/construct-mc-pysms-girth-degree-l1-s0/tests/check.py
@@ -0,0 +1,151 @@
+"""Graph-existence constraints (MathConstraint pysms_* problem types, CC-BY-4.0). The answer is a graph on
+the stated vertex set, plus a proper colouring when a chromatic-number upper bound is asked for. Every
+constraint present in the instance is checked exactly (no stored answer is compared).
+
+Upstream note: upstream verifies by fixing the edge variables in the pysms CNF and calling CaDiCaL; the
+min-chromatic-number bound of pysms_graph_builder is only a raw smsg solver parameter, not a clause, so the
+upstream verifier ignored it. It is enforced here. Edge endpoints are also type-checked (upstream accepted
+floats/bools that compare in range)."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def _has_clique(adj, n, k):
+    def rec(cand, size):
+        if size == k:
+            return True
+        while cand:
+            if size + bin(cand).count("1") < k:
+                return False
+            v = cand.bit_length() - 1
+            cand &= ~(1 << v)
+            if rec(cand & adj[v], size + 1):
+                return True
+        return False
+    return rec((1 << n) - 1, 0)
+
+
+def _connected(adj, n, removed):
+    alive = [v for v in range(n) if not removed >> v & 1]
+    if len(alive) <= 1:
+        return True
+    mask = sum(1 << v for v in alive)
+    seen = 1 << alive[0]
+    stack = [alive[0]]
+    while stack:
+        v = stack.pop()
+        new = adj[v] & mask & ~seen
+        seen |= new
+        while new:
+            u = new.bit_length() - 1
+            new &= ~(1 << u)
+            stack.append(u)
+    return seen == mask
+
+
+def _girth(adj, n):
+    best = None
+    for r in range(n):
+        dist, par, q = {r: 0}, {r: -1}, [r]
+        for v in q:
+            nb = adj[v]
+            while nb:
+                u = nb.bit_length() - 1
+                nb &= ~(1 << u)
+                if u not in dist:
+                    dist[u], par[u] = dist[v] + 1, v
+                    q.append(u)
+                elif par[v] != u:
+                    c = dist[u] + dist[v] + 1
+                    best = c if best is None else min(best, c)
+    return best  # None = acyclic
+
+
+def _colourable(adj, n, k):
+    col = [-1] * n
+    order = sorted(range(n), key=lambda v: -bin(adj[v]).count("1"))
+
+    def rec(i):
+        if i == n:
+            return True
+        v = order[i]
+        used = {col[u] for u in range(n) if adj[v] >> u & 1 and col[u] >= 0}
+        for c in range(k):
+            if c not in used:
+                col[v] = c
+                if rec(i + 1):
+                    return True
+        col[v] = -1
+        return False
+    return rec(0)
+
+
+def check(inst, ans):
+    n = inst["vertices"]
+    E = ans.get("edges")
+    if not isinstance(E, list):
+        raise Invalid("edges must be a list of [u, v] pairs")
+    edges = set()
+    for e in E:
+        u, v = int_list(e, 2, 0, n - 1, "edge")
+        if u == v:
+            raise Invalid(f"self-loop at {u}")
+        e2 = (min(u, v), max(u, v))
+        if e2 in edges:
+            raise Invalid(f"duplicate edge {e2}")
+        edges.add(e2)
+    adj = [0] * n
+    for u, v in edges:
+        adj[u] |= 1 << v
+        adj[v] |= 1 << u
+    deg = [bin(a).count("1") for a in adj]
+    for u, v in inst.get("required_edges", []):
+        if (min(u, v), max(u, v)) not in edges:
+            raise Invalid(f"required edge ({u},{v}) is missing")
+    c = inst.get
+    if c("min_degree") is not None and min(deg) < c("min_degree"):
+        raise Invalid(f"minimum degree {min(deg)} < {c('min_degree')}")
+    if c("max_degree") is not None and max(deg) > c("max_degree"):
+        raise Invalid(f"maximum degree {max(deg)} > {c('max_degree')}")
+    if c("min_edges") is not None and len(edges) < c("min_edges"):
+        raise Invalid(f"{len(edges)} edges < {c('min_edges')}")
+    if c("max_edges") is not None and len(edges) > c("max_edges"):
+        raise Invalid(f"{len(edges)} edges > {c('max_edges')}")
+    if c("max_clique") is not None and _has_clique(adj, n, c("max_clique") + 1):
+        raise Invalid(f"graph contains a clique of size {c('max_clique') + 1}")
+    if c("max_independent_set") is not None:
+        full = (1 << n) - 1
+        comp = [full & ~adj[v] & ~(1 << v) for v in range(n)]
+        if _has_clique(comp, n, c("max_independent_set") + 1):
+            raise Invalid(f"graph contains an independent set of size {c('max_independent_set') + 1}")
+    if c("min_girth") is not None:
+        g = _girth(adj, n)
+        if g is not None and g < c("min_girth"):
+            raise Invalid(f"graph has a cycle of length {g} < {c('min_girth')}")
+    if c("min_connectivity") is not None:
+        k = c("min_connectivity")
+        if n <= k:
+            raise Invalid("too few vertices for this connectivity")
+        for s in range(k):
+            for I in itertools.combinations(range(n), s):
+                if not _connected(adj, n, sum(1 << v for v in I)):
+                    raise Invalid(f"deleting vertices {list(I)} disconnects the graph")
+    if c("mtf"):
+        for u, v in itertools.combinations(range(n), 2):
+            if adj[u] >> v & 1:
+                if adj[u] & adj[v]:
+                    raise Invalid(f"triangle on edge ({u},{v})")
+            elif not adj[u] & adj[v]:
+                raise Invalid(f"non-adjacent {u}, {v} have no common neighbour (edge could be added)")
+    if c("max_chromatic_number") is not None:
+        k = c("max_chromatic_number")
+        col = int_list(ans.get("coloring"), n, 0, k - 1, "coloring")
+        for u, v in edges:
+            if col[u] == col[v]:
+                raise Invalid(f"coloring: edge ({u},{v}) is monochromatic")
+    if c("min_chromatic_number") is not None and c("min_chromatic_number") >= 2:
+        m = c("min_chromatic_number")
+        if not edges or _colourable(adj, n, m - 1):
+            raise Invalid(f"graph is {m - 1}-colourable, chromatic number must be at least {m}")
+    return None, {"num_edges": len(edges)}
diff --git a/tasks/construct-mc-pysms-girth-degree-l1-s0/tests/instance.json b/tasks/construct-mc-pysms-girth-degree-l1-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..9ea9f7bd58978c6bed7d38a1fd2a8f4dd21b6519
--- /dev/null
+++ b/tasks/construct-mc-pysms-girth-degree-l1-s0/tests/instance.json
@@ -0,0 +1 @@
+{"vertices": 11, "required_edges": [], "max_degree": 5, "min_degree": 2, "min_girth": 6, "family": "mc_pysms_girth_degree", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-girth-degree-l1-s0/tests/run.py b/tasks/construct-mc-pysms-girth-degree-l1-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-pysms-girth-degree-l1-s0/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-pysms-girth-degree-l1-s0/tests/test.sh b/tasks/construct-mc-pysms-girth-degree-l1-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-pysms-girth-degree-l1-s0/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-pysms-graph-builder-l1-s1/environment/Dockerfile b/tasks/construct-mc-pysms-graph-builder-l1-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-pysms-graph-builder-l1-s1/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-pysms-graph-builder-l1-s1/environment/instance.json b/tasks/construct-mc-pysms-graph-builder-l1-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0c0747827e8194e82494ab02e351b15fa8d725aa
--- /dev/null
+++ b/tasks/construct-mc-pysms-graph-builder-l1-s1/environment/instance.json
@@ -0,0 +1 @@
+{"vertices": 12, "required_edges": [[5, 7], [4, 6]], "max_degree": 2, "min_degree": 1, "max_chromatic_number": 2, "min_chromatic_number": 2, "min_edges": 9, "max_edges": 11, "family": "mc_pysms_graph_builder", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-graph-builder-l1-s1/solution/solve.sh b/tasks/construct-mc-pysms-graph-builder-l1-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..5b359ec7e229559b24e008584ecd5d29bed5b443
--- /dev/null
+++ b/tasks/construct-mc-pysms-graph-builder-l1-s1/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"edges": [[0, 11], [1, 9], [1, 10], [2, 8], [2, 10], [3, 7], [3, 9], [4, 6], [4, 8], [5, 6], [5, 7]], "coloring": [0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-pysms-graph-builder-l1-s1/tests/check.py b/tasks/construct-mc-pysms-graph-builder-l1-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..4ab0b12508cf59b733f007a9d1f50831aec9efdf
--- /dev/null
+++ b/tasks/construct-mc-pysms-graph-builder-l1-s1/tests/check.py
@@ -0,0 +1,151 @@
+"""Graph-existence constraints (MathConstraint pysms_* problem types, CC-BY-4.0). The answer is a graph on
+the stated vertex set, plus a proper colouring when a chromatic-number upper bound is asked for. Every
+constraint present in the instance is checked exactly (no stored answer is compared).
+
+Upstream note: upstream verifies by fixing the edge variables in the pysms CNF and calling CaDiCaL; the
+min-chromatic-number bound of pysms_graph_builder is only a raw smsg solver parameter, not a clause, so the
+upstream verifier ignored it. It is enforced here. Edge endpoints are also type-checked (upstream accepted
+floats/bools that compare in range)."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def _has_clique(adj, n, k):
+    def rec(cand, size):
+        if size == k:
+            return True
+        while cand:
+            if size + bin(cand).count("1") < k:
+                return False
+            v = cand.bit_length() - 1
+            cand &= ~(1 << v)
+            if rec(cand & adj[v], size + 1):
+                return True
+        return False
+    return rec((1 << n) - 1, 0)
+
+
+def _connected(adj, n, removed):
+    alive = [v for v in range(n) if not removed >> v & 1]
+    if len(alive) <= 1:
+        return True
+    mask = sum(1 << v for v in alive)
+    seen = 1 << alive[0]
+    stack = [alive[0]]
+    while stack:
+        v = stack.pop()
+        new = adj[v] & mask & ~seen
+        seen |= new
+        while new:
+            u = new.bit_length() - 1
+            new &= ~(1 << u)
+            stack.append(u)
+    return seen == mask
+
+
+def _girth(adj, n):
+    best = None
+    for r in range(n):
+        dist, par, q = {r: 0}, {r: -1}, [r]
+        for v in q:
+            nb = adj[v]
+            while nb:
+                u = nb.bit_length() - 1
+                nb &= ~(1 << u)
+                if u not in dist:
+                    dist[u], par[u] = dist[v] + 1, v
+                    q.append(u)
+                elif par[v] != u:
+                    c = dist[u] + dist[v] + 1
+                    best = c if best is None else min(best, c)
+    return best  # None = acyclic
+
+
+def _colourable(adj, n, k):
+    col = [-1] * n
+    order = sorted(range(n), key=lambda v: -bin(adj[v]).count("1"))
+
+    def rec(i):
+        if i == n:
+            return True
+        v = order[i]
+        used = {col[u] for u in range(n) if adj[v] >> u & 1 and col[u] >= 0}
+        for c in range(k):
+            if c not in used:
+                col[v] = c
+                if rec(i + 1):
+                    return True
+        col[v] = -1
+        return False
+    return rec(0)
+
+
+def check(inst, ans):
+    n = inst["vertices"]
+    E = ans.get("edges")
+    if not isinstance(E, list):
+        raise Invalid("edges must be a list of [u, v] pairs")
+    edges = set()
+    for e in E:
+        u, v = int_list(e, 2, 0, n - 1, "edge")
+        if u == v:
+            raise Invalid(f"self-loop at {u}")
+        e2 = (min(u, v), max(u, v))
+        if e2 in edges:
+            raise Invalid(f"duplicate edge {e2}")
+        edges.add(e2)
+    adj = [0] * n
+    for u, v in edges:
+        adj[u] |= 1 << v
+        adj[v] |= 1 << u
+    deg = [bin(a).count("1") for a in adj]
+    for u, v in inst.get("required_edges", []):
+        if (min(u, v), max(u, v)) not in edges:
+            raise Invalid(f"required edge ({u},{v}) is missing")
+    c = inst.get
+    if c("min_degree") is not None and min(deg) < c("min_degree"):
+        raise Invalid(f"minimum degree {min(deg)} < {c('min_degree')}")
+    if c("max_degree") is not None and max(deg) > c("max_degree"):
+        raise Invalid(f"maximum degree {max(deg)} > {c('max_degree')}")
+    if c("min_edges") is not None and len(edges) < c("min_edges"):
+        raise Invalid(f"{len(edges)} edges < {c('min_edges')}")
+    if c("max_edges") is not None and len(edges) > c("max_edges"):
+        raise Invalid(f"{len(edges)} edges > {c('max_edges')}")
+    if c("max_clique") is not None and _has_clique(adj, n, c("max_clique") + 1):
+        raise Invalid(f"graph contains a clique of size {c('max_clique') + 1}")
+    if c("max_independent_set") is not None:
+        full = (1 << n) - 1
+        comp = [full & ~adj[v] & ~(1 << v) for v in range(n)]
+        if _has_clique(comp, n, c("max_independent_set") + 1):
+            raise Invalid(f"graph contains an independent set of size {c('max_independent_set') + 1}")
+    if c("min_girth") is not None:
+        g = _girth(adj, n)
+        if g is not None and g < c("min_girth"):
+            raise Invalid(f"graph has a cycle of length {g} < {c('min_girth')}")
+    if c("min_connectivity") is not None:
+        k = c("min_connectivity")
+        if n <= k:
+            raise Invalid("too few vertices for this connectivity")
+        for s in range(k):
+            for I in itertools.combinations(range(n), s):
+                if not _connected(adj, n, sum(1 << v for v in I)):
+                    raise Invalid(f"deleting vertices {list(I)} disconnects the graph")
+    if c("mtf"):
+        for u, v in itertools.combinations(range(n), 2):
+            if adj[u] >> v & 1:
+                if adj[u] & adj[v]:
+                    raise Invalid(f"triangle on edge ({u},{v})")
+            elif not adj[u] & adj[v]:
+                raise Invalid(f"non-adjacent {u}, {v} have no common neighbour (edge could be added)")
+    if c("max_chromatic_number") is not None:
+        k = c("max_chromatic_number")
+        col = int_list(ans.get("coloring"), n, 0, k - 1, "coloring")
+        for u, v in edges:
+            if col[u] == col[v]:
+                raise Invalid(f"coloring: edge ({u},{v}) is monochromatic")
+    if c("min_chromatic_number") is not None and c("min_chromatic_number") >= 2:
+        m = c("min_chromatic_number")
+        if not edges or _colourable(adj, n, m - 1):
+            raise Invalid(f"graph is {m - 1}-colourable, chromatic number must be at least {m}")
+    return None, {"num_edges": len(edges)}
diff --git a/tasks/construct-mc-pysms-graph-builder-l1-s1/tests/instance.json b/tasks/construct-mc-pysms-graph-builder-l1-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0c0747827e8194e82494ab02e351b15fa8d725aa
--- /dev/null
+++ b/tasks/construct-mc-pysms-graph-builder-l1-s1/tests/instance.json
@@ -0,0 +1 @@
+{"vertices": 12, "required_edges": [[5, 7], [4, 6]], "max_degree": 2, "min_degree": 1, "max_chromatic_number": 2, "min_chromatic_number": 2, "min_edges": 9, "max_edges": 11, "family": "mc_pysms_graph_builder", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-graph-builder-l1-s1/tests/run.py b/tasks/construct-mc-pysms-graph-builder-l1-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-pysms-graph-builder-l1-s1/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-pysms-graph-builder-l1-s1/tests/test.sh b/tasks/construct-mc-pysms-graph-builder-l1-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-pysms-graph-builder-l1-s1/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-pysms-min-connectivity-l2-s1/environment/Dockerfile b/tasks/construct-mc-pysms-min-connectivity-l2-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-connectivity-l2-s1/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-pysms-min-connectivity-l2-s1/environment/instance.json b/tasks/construct-mc-pysms-min-connectivity-l2-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e516c318c3a3ef90ff0924978dcb7d74a3e28336
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-connectivity-l2-s1/environment/instance.json
@@ -0,0 +1 @@
+{"vertices": 15, "required_edges": [], "min_connectivity": 1, "family": "mc_pysms_min_connectivity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-min-connectivity-l2-s1/solution/solve.sh b/tasks/construct-mc-pysms-min-connectivity-l2-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..202381af71f4ba7b8da0d1a923832175f9607239
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-connectivity-l2-s1/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"edges": [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [0, 6], [0, 7], [0, 8], [0, 9], [0, 10], [0, 11], [0, 12], [0, 13], [0, 14], [1, 2], [1, 3], [1, 4], [1, 5], [1, 6], [1, 7], [1, 8], [1, 9], [1, 10], [1, 11], [1, 12], [1, 13], [1, 14], [2, 3], [2, 4], [2, 5], [2, 6], [2, 7], [2, 8], [2, 9], [2, 10], [2, 11], [2, 12], [2, 13], [2, 14], [3, 4], [3, 5], [3, 6], [3, 7], [3, 8], [3, 9], [3, 10], [3, 11], [3, 12], [3, 13], [3, 14], [4, 5], [4, 6], [4, 7], [4, 8], [4, 9], [4, 10], [4, 11], [4, 12], [4, 13], [4, 14], [5, 6], [5, 7], [5, 8], [5, 9], [5, 10], [5, 11], [5, 12], [5, 13], [5, 14], [6, 7], [6, 8], [6, 9], [6, 10], [6, 11], [6, 12], [6, 13], [6, 14], [7, 8], [7, 9], [7, 10], [7, 11], [7, 12], [7, 13], [7, 14], [8, 9], [8, 10], [8, 11], [8, 12], [8, 13], [8, 14], [9, 10], [9, 11], [9, 12], [9, 13], [9, 14], [10, 11], [10, 12], [10, 13], [10, 14], [11, 12], [11, 13], [11, 14], [12, 13], [12, 14], [13, 14]]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-pysms-min-connectivity-l2-s1/tests/check.py b/tasks/construct-mc-pysms-min-connectivity-l2-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..4ab0b12508cf59b733f007a9d1f50831aec9efdf
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-connectivity-l2-s1/tests/check.py
@@ -0,0 +1,151 @@
+"""Graph-existence constraints (MathConstraint pysms_* problem types, CC-BY-4.0). The answer is a graph on
+the stated vertex set, plus a proper colouring when a chromatic-number upper bound is asked for. Every
+constraint present in the instance is checked exactly (no stored answer is compared).
+
+Upstream note: upstream verifies by fixing the edge variables in the pysms CNF and calling CaDiCaL; the
+min-chromatic-number bound of pysms_graph_builder is only a raw smsg solver parameter, not a clause, so the
+upstream verifier ignored it. It is enforced here. Edge endpoints are also type-checked (upstream accepted
+floats/bools that compare in range)."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def _has_clique(adj, n, k):
+    def rec(cand, size):
+        if size == k:
+            return True
+        while cand:
+            if size + bin(cand).count("1") < k:
+                return False
+            v = cand.bit_length() - 1
+            cand &= ~(1 << v)
+            if rec(cand & adj[v], size + 1):
+                return True
+        return False
+    return rec((1 << n) - 1, 0)
+
+
+def _connected(adj, n, removed):
+    alive = [v for v in range(n) if not removed >> v & 1]
+    if len(alive) <= 1:
+        return True
+    mask = sum(1 << v for v in alive)
+    seen = 1 << alive[0]
+    stack = [alive[0]]
+    while stack:
+        v = stack.pop()
+        new = adj[v] & mask & ~seen
+        seen |= new
+        while new:
+            u = new.bit_length() - 1
+            new &= ~(1 << u)
+            stack.append(u)
+    return seen == mask
+
+
+def _girth(adj, n):
+    best = None
+    for r in range(n):
+        dist, par, q = {r: 0}, {r: -1}, [r]
+        for v in q:
+            nb = adj[v]
+            while nb:
+                u = nb.bit_length() - 1
+                nb &= ~(1 << u)
+                if u not in dist:
+                    dist[u], par[u] = dist[v] + 1, v
+                    q.append(u)
+                elif par[v] != u:
+                    c = dist[u] + dist[v] + 1
+                    best = c if best is None else min(best, c)
+    return best  # None = acyclic
+
+
+def _colourable(adj, n, k):
+    col = [-1] * n
+    order = sorted(range(n), key=lambda v: -bin(adj[v]).count("1"))
+
+    def rec(i):
+        if i == n:
+            return True
+        v = order[i]
+        used = {col[u] for u in range(n) if adj[v] >> u & 1 and col[u] >= 0}
+        for c in range(k):
+            if c not in used:
+                col[v] = c
+                if rec(i + 1):
+                    return True
+        col[v] = -1
+        return False
+    return rec(0)
+
+
+def check(inst, ans):
+    n = inst["vertices"]
+    E = ans.get("edges")
+    if not isinstance(E, list):
+        raise Invalid("edges must be a list of [u, v] pairs")
+    edges = set()
+    for e in E:
+        u, v = int_list(e, 2, 0, n - 1, "edge")
+        if u == v:
+            raise Invalid(f"self-loop at {u}")
+        e2 = (min(u, v), max(u, v))
+        if e2 in edges:
+            raise Invalid(f"duplicate edge {e2}")
+        edges.add(e2)
+    adj = [0] * n
+    for u, v in edges:
+        adj[u] |= 1 << v
+        adj[v] |= 1 << u
+    deg = [bin(a).count("1") for a in adj]
+    for u, v in inst.get("required_edges", []):
+        if (min(u, v), max(u, v)) not in edges:
+            raise Invalid(f"required edge ({u},{v}) is missing")
+    c = inst.get
+    if c("min_degree") is not None and min(deg) < c("min_degree"):
+        raise Invalid(f"minimum degree {min(deg)} < {c('min_degree')}")
+    if c("max_degree") is not None and max(deg) > c("max_degree"):
+        raise Invalid(f"maximum degree {max(deg)} > {c('max_degree')}")
+    if c("min_edges") is not None and len(edges) < c("min_edges"):
+        raise Invalid(f"{len(edges)} edges < {c('min_edges')}")
+    if c("max_edges") is not None and len(edges) > c("max_edges"):
+        raise Invalid(f"{len(edges)} edges > {c('max_edges')}")
+    if c("max_clique") is not None and _has_clique(adj, n, c("max_clique") + 1):
+        raise Invalid(f"graph contains a clique of size {c('max_clique') + 1}")
+    if c("max_independent_set") is not None:
+        full = (1 << n) - 1
+        comp = [full & ~adj[v] & ~(1 << v) for v in range(n)]
+        if _has_clique(comp, n, c("max_independent_set") + 1):
+            raise Invalid(f"graph contains an independent set of size {c('max_independent_set') + 1}")
+    if c("min_girth") is not None:
+        g = _girth(adj, n)
+        if g is not None and g < c("min_girth"):
+            raise Invalid(f"graph has a cycle of length {g} < {c('min_girth')}")
+    if c("min_connectivity") is not None:
+        k = c("min_connectivity")
+        if n <= k:
+            raise Invalid("too few vertices for this connectivity")
+        for s in range(k):
+            for I in itertools.combinations(range(n), s):
+                if not _connected(adj, n, sum(1 << v for v in I)):
+                    raise Invalid(f"deleting vertices {list(I)} disconnects the graph")
+    if c("mtf"):
+        for u, v in itertools.combinations(range(n), 2):
+            if adj[u] >> v & 1:
+                if adj[u] & adj[v]:
+                    raise Invalid(f"triangle on edge ({u},{v})")
+            elif not adj[u] & adj[v]:
+                raise Invalid(f"non-adjacent {u}, {v} have no common neighbour (edge could be added)")
+    if c("max_chromatic_number") is not None:
+        k = c("max_chromatic_number")
+        col = int_list(ans.get("coloring"), n, 0, k - 1, "coloring")
+        for u, v in edges:
+            if col[u] == col[v]:
+                raise Invalid(f"coloring: edge ({u},{v}) is monochromatic")
+    if c("min_chromatic_number") is not None and c("min_chromatic_number") >= 2:
+        m = c("min_chromatic_number")
+        if not edges or _colourable(adj, n, m - 1):
+            raise Invalid(f"graph is {m - 1}-colourable, chromatic number must be at least {m}")
+    return None, {"num_edges": len(edges)}
diff --git a/tasks/construct-mc-pysms-min-connectivity-l2-s1/tests/instance.json b/tasks/construct-mc-pysms-min-connectivity-l2-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e516c318c3a3ef90ff0924978dcb7d74a3e28336
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-connectivity-l2-s1/tests/instance.json
@@ -0,0 +1 @@
+{"vertices": 15, "required_edges": [], "min_connectivity": 1, "family": "mc_pysms_min_connectivity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-min-connectivity-l2-s1/tests/run.py b/tasks/construct-mc-pysms-min-connectivity-l2-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-connectivity-l2-s1/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-pysms-min-connectivity-l2-s1/tests/test.sh b/tasks/construct-mc-pysms-min-connectivity-l2-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-connectivity-l2-s1/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-pysms-min-degree-l5-s2/environment/Dockerfile b/tasks/construct-mc-pysms-min-degree-l5-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-degree-l5-s2/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-pysms-min-degree-l5-s2/environment/instance.json b/tasks/construct-mc-pysms-min-degree-l5-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..7bf0156776325e8d8a69758e2dd3700b1281e3bb
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-degree-l5-s2/environment/instance.json
@@ -0,0 +1 @@
+{"vertices": 16, "required_edges": [], "min_degree": 1, "family": "mc_pysms_min_degree", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-min-degree-l5-s2/solution/solve.sh b/tasks/construct-mc-pysms-min-degree-l5-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..4b8141dfb1b77c83fdaab714902ed360207c54d0
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-degree-l5-s2/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"edges": [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [0, 6], [0, 7], [0, 8], [0, 9], [0, 10], [0, 11], [0, 12], [0, 13], [0, 14], [0, 15], [1, 2], [1, 3], [1, 4], [1, 5], [1, 6], [1, 7], [1, 8], [1, 9], [1, 10], [1, 11], [1, 12], [1, 13], [1, 14], [1, 15], [2, 3], [2, 4], [2, 5], [2, 6], [2, 7], [2, 8], [2, 9], [2, 10], [2, 11], [2, 12], [2, 13], [2, 14], [2, 15], [3, 4], [3, 5], [3, 6], [3, 7], [3, 8], [3, 9], [3, 10], [3, 11], [3, 12], [3, 13], [3, 14], [3, 15], [4, 5], [4, 6], [4, 7], [4, 8], [4, 9], [4, 10], [4, 11], [4, 12], [4, 13], [4, 14], [4, 15], [5, 6], [5, 7], [5, 8], [5, 9], [5, 10], [5, 11], [5, 12], [5, 13], [5, 14], [5, 15], [6, 7], [6, 8], [6, 9], [6, 10], [6, 11], [6, 12], [6, 13], [6, 14], [6, 15], [7, 8], [7, 9], [7, 10], [7, 11], [7, 12], [7, 13], [7, 14], [7, 15], [8, 9], [8, 10], [8, 11], [8, 12], [8, 13], [8, 14], [8, 15], [9, 10], [9, 11], [9, 12], [9, 13], [9, 14], [9, 15], [10, 11], [10, 12], [10, 13], [10, 14], [10, 15], [11, 12], [11, 13], [11, 14], [11, 15], [12, 13], [12, 14], [12, 15], [13, 14], [13, 15], [14, 15]]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-pysms-min-degree-l5-s2/tests/check.py b/tasks/construct-mc-pysms-min-degree-l5-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..4ab0b12508cf59b733f007a9d1f50831aec9efdf
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-degree-l5-s2/tests/check.py
@@ -0,0 +1,151 @@
+"""Graph-existence constraints (MathConstraint pysms_* problem types, CC-BY-4.0). The answer is a graph on
+the stated vertex set, plus a proper colouring when a chromatic-number upper bound is asked for. Every
+constraint present in the instance is checked exactly (no stored answer is compared).
+
+Upstream note: upstream verifies by fixing the edge variables in the pysms CNF and calling CaDiCaL; the
+min-chromatic-number bound of pysms_graph_builder is only a raw smsg solver parameter, not a clause, so the
+upstream verifier ignored it. It is enforced here. Edge endpoints are also type-checked (upstream accepted
+floats/bools that compare in range)."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def _has_clique(adj, n, k):
+    def rec(cand, size):
+        if size == k:
+            return True
+        while cand:
+            if size + bin(cand).count("1") < k:
+                return False
+            v = cand.bit_length() - 1
+            cand &= ~(1 << v)
+            if rec(cand & adj[v], size + 1):
+                return True
+        return False
+    return rec((1 << n) - 1, 0)
+
+
+def _connected(adj, n, removed):
+    alive = [v for v in range(n) if not removed >> v & 1]
+    if len(alive) <= 1:
+        return True
+    mask = sum(1 << v for v in alive)
+    seen = 1 << alive[0]
+    stack = [alive[0]]
+    while stack:
+        v = stack.pop()
+        new = adj[v] & mask & ~seen
+        seen |= new
+        while new:
+            u = new.bit_length() - 1
+            new &= ~(1 << u)
+            stack.append(u)
+    return seen == mask
+
+
+def _girth(adj, n):
+    best = None
+    for r in range(n):
+        dist, par, q = {r: 0}, {r: -1}, [r]
+        for v in q:
+            nb = adj[v]
+            while nb:
+                u = nb.bit_length() - 1
+                nb &= ~(1 << u)
+                if u not in dist:
+                    dist[u], par[u] = dist[v] + 1, v
+                    q.append(u)
+                elif par[v] != u:
+                    c = dist[u] + dist[v] + 1
+                    best = c if best is None else min(best, c)
+    return best  # None = acyclic
+
+
+def _colourable(adj, n, k):
+    col = [-1] * n
+    order = sorted(range(n), key=lambda v: -bin(adj[v]).count("1"))
+
+    def rec(i):
+        if i == n:
+            return True
+        v = order[i]
+        used = {col[u] for u in range(n) if adj[v] >> u & 1 and col[u] >= 0}
+        for c in range(k):
+            if c not in used:
+                col[v] = c
+                if rec(i + 1):
+                    return True
+        col[v] = -1
+        return False
+    return rec(0)
+
+
+def check(inst, ans):
+    n = inst["vertices"]
+    E = ans.get("edges")
+    if not isinstance(E, list):
+        raise Invalid("edges must be a list of [u, v] pairs")
+    edges = set()
+    for e in E:
+        u, v = int_list(e, 2, 0, n - 1, "edge")
+        if u == v:
+            raise Invalid(f"self-loop at {u}")
+        e2 = (min(u, v), max(u, v))
+        if e2 in edges:
+            raise Invalid(f"duplicate edge {e2}")
+        edges.add(e2)
+    adj = [0] * n
+    for u, v in edges:
+        adj[u] |= 1 << v
+        adj[v] |= 1 << u
+    deg = [bin(a).count("1") for a in adj]
+    for u, v in inst.get("required_edges", []):
+        if (min(u, v), max(u, v)) not in edges:
+            raise Invalid(f"required edge ({u},{v}) is missing")
+    c = inst.get
+    if c("min_degree") is not None and min(deg) < c("min_degree"):
+        raise Invalid(f"minimum degree {min(deg)} < {c('min_degree')}")
+    if c("max_degree") is not None and max(deg) > c("max_degree"):
+        raise Invalid(f"maximum degree {max(deg)} > {c('max_degree')}")
+    if c("min_edges") is not None and len(edges) < c("min_edges"):
+        raise Invalid(f"{len(edges)} edges < {c('min_edges')}")
+    if c("max_edges") is not None and len(edges) > c("max_edges"):
+        raise Invalid(f"{len(edges)} edges > {c('max_edges')}")
+    if c("max_clique") is not None and _has_clique(adj, n, c("max_clique") + 1):
+        raise Invalid(f"graph contains a clique of size {c('max_clique') + 1}")
+    if c("max_independent_set") is not None:
+        full = (1 << n) - 1
+        comp = [full & ~adj[v] & ~(1 << v) for v in range(n)]
+        if _has_clique(comp, n, c("max_independent_set") + 1):
+            raise Invalid(f"graph contains an independent set of size {c('max_independent_set') + 1}")
+    if c("min_girth") is not None:
+        g = _girth(adj, n)
+        if g is not None and g < c("min_girth"):
+            raise Invalid(f"graph has a cycle of length {g} < {c('min_girth')}")
+    if c("min_connectivity") is not None:
+        k = c("min_connectivity")
+        if n <= k:
+            raise Invalid("too few vertices for this connectivity")
+        for s in range(k):
+            for I in itertools.combinations(range(n), s):
+                if not _connected(adj, n, sum(1 << v for v in I)):
+                    raise Invalid(f"deleting vertices {list(I)} disconnects the graph")
+    if c("mtf"):
+        for u, v in itertools.combinations(range(n), 2):
+            if adj[u] >> v & 1:
+                if adj[u] & adj[v]:
+                    raise Invalid(f"triangle on edge ({u},{v})")
+            elif not adj[u] & adj[v]:
+                raise Invalid(f"non-adjacent {u}, {v} have no common neighbour (edge could be added)")
+    if c("max_chromatic_number") is not None:
+        k = c("max_chromatic_number")
+        col = int_list(ans.get("coloring"), n, 0, k - 1, "coloring")
+        for u, v in edges:
+            if col[u] == col[v]:
+                raise Invalid(f"coloring: edge ({u},{v}) is monochromatic")
+    if c("min_chromatic_number") is not None and c("min_chromatic_number") >= 2:
+        m = c("min_chromatic_number")
+        if not edges or _colourable(adj, n, m - 1):
+            raise Invalid(f"graph is {m - 1}-colourable, chromatic number must be at least {m}")
+    return None, {"num_edges": len(edges)}
diff --git a/tasks/construct-mc-pysms-min-degree-l5-s2/tests/instance.json b/tasks/construct-mc-pysms-min-degree-l5-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..7bf0156776325e8d8a69758e2dd3700b1281e3bb
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-degree-l5-s2/tests/instance.json
@@ -0,0 +1 @@
+{"vertices": 16, "required_edges": [], "min_degree": 1, "family": "mc_pysms_min_degree", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-min-degree-l5-s2/tests/run.py b/tasks/construct-mc-pysms-min-degree-l5-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-degree-l5-s2/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-pysms-min-degree-l5-s2/tests/test.sh b/tasks/construct-mc-pysms-min-degree-l5-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-degree-l5-s2/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-social-golfers-l5-s3/environment/Dockerfile b/tasks/construct-mc-social-golfers-l5-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-social-golfers-l5-s3/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-social-golfers-l5-s3/environment/instance.json b/tasks/construct-mc-social-golfers-l5-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..61ed08fe2e6c8237c4fae68198e0465b94bf1082
--- /dev/null
+++ b/tasks/construct-mc-social-golfers-l5-s3/environment/instance.json
@@ -0,0 +1 @@
+{"n_groups": 7, "group_size": 5, "n_weeks": 5, "family": "mc_social_golfers", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-social-golfers-l5-s3/solution/solve.sh b/tasks/construct-mc-social-golfers-l5-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..fe5a74c86f2410b9e56a6322639d08463ee4b2d9
--- /dev/null
+++ b/tasks/construct-mc-social-golfers-l5-s3/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"schedule": [[0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6], [5, 6, 4, 0, 1, 0, 6, 5, 4, 1, 2, 5, 6, 4, 3, 1, 2, 0, 6, 3, 3, 0, 5, 2, 1, 3, 6, 2, 5, 4, 4, 1, 0, 2, 3], [4, 0, 3, 1, 2, 2, 1, 3, 4, 0, 1, 6, 2, 5, 0, 3, 2, 0, 6, 5, 1, 4, 2, 6, 5, 6, 3, 4, 5, 0, 6, 1, 3, 5, 4], [0, 1, 2, 3, 4, 0, 2, 1, 3, 5, 0, 2, 3, 1, 6, 6, 1, 4, 5, 0, 5, 2, 6, 4, 3, 3, 4, 6, 5, 0, 6, 1, 5, 2, 4], [0, 1, 2, 3, 4, 2, 0, 6, 1, 3, 4, 5, 6, 0, 2, 0, 5, 6, 2, 1, 6, 4, 3, 1, 5, 0, 3, 6, 4, 5, 4, 2, 1, 3, 5]]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-social-golfers-l5-s3/tests/check.py b/tasks/construct-mc-social-golfers-l5-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..73977815b7b6094a8491075b6dc2ae1293e44930
--- /dev/null
+++ b/tasks/construct-mc-social-golfers-l5-s3/tests/check.py
@@ -0,0 +1,25 @@
+"""Social golfer problem (MathConstraint social_golfers, CSPLib prob010, CC-BY-4.0): each week partition
+the g*s players into g groups of size s; no two players share a group in more than one week."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    g, s, w = inst["n_groups"], inst["group_size"], inst["n_weeks"]
+    P = g * s
+    sched = ans.get("schedule")
+    if not isinstance(sched, list) or len(sched) != w:
+        raise Invalid(f"schedule must be a list of {w} weeks")
+    met = set()
+    for wk, row in enumerate(sched):
+        row = int_list(row, P, 0, g - 1, f"week {wk}")
+        groups = [[p for p in range(P) if row[p] == grp] for grp in range(g)]
+        for grp, mem in enumerate(groups):
+            if len(mem) != s:
+                raise Invalid(f"week {wk}: group {grp} has {len(mem)} players, need {s}")
+            for i in range(s):
+                for j in range(i + 1, s):
+                    pr = (mem[i], mem[j])
+                    if pr in met:
+                        raise Invalid(f"players {pr} play together in more than one week")
+                    met.add(pr)
+    return None, {}
diff --git a/tasks/construct-mc-social-golfers-l5-s3/tests/instance.json b/tasks/construct-mc-social-golfers-l5-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..61ed08fe2e6c8237c4fae68198e0465b94bf1082
--- /dev/null
+++ b/tasks/construct-mc-social-golfers-l5-s3/tests/instance.json
@@ -0,0 +1 @@
+{"n_groups": 7, "group_size": 5, "n_weeks": 5, "family": "mc_social_golfers", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-social-golfers-l5-s3/tests/run.py b/tasks/construct-mc-social-golfers-l5-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-social-golfers-l5-s3/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-social-golfers-l5-s3/tests/test.sh b/tasks/construct-mc-social-golfers-l5-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-social-golfers-l5-s3/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-van-der-waerden-l4-s2/environment/Dockerfile b/tasks/construct-mc-van-der-waerden-l4-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l4-s2/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-van-der-waerden-l4-s2/environment/instance.json b/tasks/construct-mc-van-der-waerden-l4-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0ed04a3ce06fc8d1a05503e91a9a765ce0b12e33
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l4-s2/environment/instance.json
@@ -0,0 +1 @@
+{"k": 2, "L": 5, "n": 100, "family": "mc_van_der_waerden", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-van-der-waerden-l4-s2/solution/solve.sh b/tasks/construct-mc-van-der-waerden-l4-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..9aaa0ca50246d1fcf3903b109b910ab484ccbb4c
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l4-s2/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"colors": [0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-van-der-waerden-l4-s2/tests/check.py b/tasks/construct-mc-van-der-waerden-l4-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..9853b089b6ac1773d80eac8e9e02ceb4019ad41f
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l4-s2/tests/check.py
@@ -0,0 +1,15 @@
+"""k-colouring of {1..n} with no monochromatic L-term arithmetic progression (MathConstraint
+van_der_waerden, CC-BY-4.0)."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k, L = inst["n"], inst["k"], inst["L"]
+    c = int_list(ans.get("colors"), n, 0, k - 1, "colors")
+    for a in range(n):
+        for d in range(1, n):
+            if a + (L - 1) * d >= n:
+                break
+            if all(c[a + j * d] == c[a] for j in range(1, L)):
+                raise Invalid(f"monochromatic progression {[a + 1 + j * d for j in range(L)]}")
+    return None, {}
diff --git a/tasks/construct-mc-van-der-waerden-l4-s2/tests/instance.json b/tasks/construct-mc-van-der-waerden-l4-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0ed04a3ce06fc8d1a05503e91a9a765ce0b12e33
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l4-s2/tests/instance.json
@@ -0,0 +1 @@
+{"k": 2, "L": 5, "n": 100, "family": "mc_van_der_waerden", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-van-der-waerden-l4-s2/tests/run.py b/tasks/construct-mc-van-der-waerden-l4-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l4-s2/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-van-der-waerden-l4-s2/tests/test.sh b/tasks/construct-mc-van-der-waerden-l4-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l4-s2/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-van-der-waerden-l4-s5/environment/Dockerfile b/tasks/construct-mc-van-der-waerden-l4-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l4-s5/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-van-der-waerden-l4-s5/environment/instance.json b/tasks/construct-mc-van-der-waerden-l4-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1acf1b43c03ee834627136d92195585fb8b47918
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l4-s5/environment/instance.json
@@ -0,0 +1 @@
+{"k": 4, "L": 3, "n": 40, "family": "mc_van_der_waerden", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-van-der-waerden-l4-s5/solution/solve.sh b/tasks/construct-mc-van-der-waerden-l4-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..517488452660147dc84726cfa9ecb3fa98aa93f2
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l4-s5/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"colors": [0, 2, 0, 2, 1, 1, 0, 2, 0, 3, 3, 0, 3, 3, 1, 2, 1, 2, 3, 3, 2, 3, 3, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 2, 2, 3, 3, 2, 3]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-van-der-waerden-l4-s5/tests/check.py b/tasks/construct-mc-van-der-waerden-l4-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..9853b089b6ac1773d80eac8e9e02ceb4019ad41f
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l4-s5/tests/check.py
@@ -0,0 +1,15 @@
+"""k-colouring of {1..n} with no monochromatic L-term arithmetic progression (MathConstraint
+van_der_waerden, CC-BY-4.0)."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k, L = inst["n"], inst["k"], inst["L"]
+    c = int_list(ans.get("colors"), n, 0, k - 1, "colors")
+    for a in range(n):
+        for d in range(1, n):
+            if a + (L - 1) * d >= n:
+                break
+            if all(c[a + j * d] == c[a] for j in range(1, L)):
+                raise Invalid(f"monochromatic progression {[a + 1 + j * d for j in range(L)]}")
+    return None, {}
diff --git a/tasks/construct-mc-van-der-waerden-l4-s5/tests/instance.json b/tasks/construct-mc-van-der-waerden-l4-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1acf1b43c03ee834627136d92195585fb8b47918
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l4-s5/tests/instance.json
@@ -0,0 +1 @@
+{"k": 4, "L": 3, "n": 40, "family": "mc_van_der_waerden", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-van-der-waerden-l4-s5/tests/run.py b/tasks/construct-mc-van-der-waerden-l4-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l4-s5/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-van-der-waerden-l4-s5/tests/test.sh b/tasks/construct-mc-van-der-waerden-l4-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l4-s5/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-vertex-cover-l1-s3/environment/Dockerfile b/tasks/construct-mc-vertex-cover-l1-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-l1-s3/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-vertex-cover-l1-s3/environment/instance.json b/tasks/construct-mc-vertex-cover-l1-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d57195868cfb7346d544a3ec1a869fc8d3088d46
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-l1-s3/environment/instance.json
@@ -0,0 +1 @@
+{"n": 20, "k": 11, "edges": [[0, 8], [0, 19], [1, 6], [1, 9], [1, 17], [1, 18], [2, 3], [3, 7], [3, 13], [3, 18], [5, 12], [5, 14], [5, 17], [6, 11], [6, 14], [8, 10], [8, 12], [8, 15], [9, 15], [9, 18], [12, 18], [12, 19], [13, 15], [14, 15], [15, 16], [15, 17], [15, 19], [16, 18], [17, 18]], "family": "mc_vertex_cover", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-vertex-cover-l1-s3/solution/solve.sh b/tasks/construct-mc-vertex-cover-l1-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..e0db8e5b7171528ceedca84cef2d46eef700dac8
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-l1-s3/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"cover": [1, 3, 5, 6, 8, 10, 11, 15, 17, 18, 19]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-vertex-cover-l1-s3/tests/check.py b/tasks/construct-mc-vertex-cover-l1-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..e3accb3da1c773fd2126b8b5b0a97b099e29c83b
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-l1-s3/tests/check.py
@@ -0,0 +1,16 @@
+"""Vertex cover of size <= k in a given graph (MathConstraint vertex_cover, CC-BY-4.0)."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    S = int_list(ans.get("cover"), None, 0, n - 1, "cover")
+    if len(set(S)) != len(S):
+        raise Invalid("repeated vertex")
+    if len(S) > k:
+        raise Invalid(f"cover has {len(S)} vertices, at most {k} allowed")
+    C = set(S)
+    for u, v in inst["edges"]:
+        if u not in C and v not in C:
+            raise Invalid(f"edge ({u},{v}) is not covered")
+    return None, {"size": len(S)}
diff --git a/tasks/construct-mc-vertex-cover-l1-s3/tests/instance.json b/tasks/construct-mc-vertex-cover-l1-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d57195868cfb7346d544a3ec1a869fc8d3088d46
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-l1-s3/tests/instance.json
@@ -0,0 +1 @@
+{"n": 20, "k": 11, "edges": [[0, 8], [0, 19], [1, 6], [1, 9], [1, 17], [1, 18], [2, 3], [3, 7], [3, 13], [3, 18], [5, 12], [5, 14], [5, 17], [6, 11], [6, 14], [8, 10], [8, 12], [8, 15], [9, 15], [9, 18], [12, 18], [12, 19], [13, 15], [14, 15], [15, 16], [15, 17], [15, 19], [16, 18], [17, 18]], "family": "mc_vertex_cover", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-vertex-cover-l1-s3/tests/run.py b/tasks/construct-mc-vertex-cover-l1-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-l1-s3/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-vertex-cover-l1-s3/tests/test.sh b/tasks/construct-mc-vertex-cover-l1-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-l1-s3/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-vertex-cover-l1-s4/environment/Dockerfile b/tasks/construct-mc-vertex-cover-l1-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-l1-s4/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-vertex-cover-l1-s4/environment/instance.json b/tasks/construct-mc-vertex-cover-l1-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d112d154f31d15a5d8090b334f50908fcad6de42
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-l1-s4/environment/instance.json
@@ -0,0 +1 @@
+{"n": 20, "k": 11, "edges": [[0, 10], [0, 17], [0, 18], [1, 8], [1, 19], [2, 9], [2, 16], [3, 10], [3, 14], [3, 16], [3, 17], [4, 9], [4, 14], [4, 17], [5, 18], [6, 9], [7, 18], [8, 9], [8, 12], [8, 14], [10, 12], [10, 14], [10, 18], [12, 14]], "family": "mc_vertex_cover", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-vertex-cover-l1-s4/solution/solve.sh b/tasks/construct-mc-vertex-cover-l1-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..a4021e847990ac56a2fe0e2531361f039703eb9d
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-l1-s4/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"cover": [0, 5, 8, 9, 10, 11, 14, 16, 17, 18, 19]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-vertex-cover-l1-s4/tests/check.py b/tasks/construct-mc-vertex-cover-l1-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..e3accb3da1c773fd2126b8b5b0a97b099e29c83b
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-l1-s4/tests/check.py
@@ -0,0 +1,16 @@
+"""Vertex cover of size <= k in a given graph (MathConstraint vertex_cover, CC-BY-4.0)."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    S = int_list(ans.get("cover"), None, 0, n - 1, "cover")
+    if len(set(S)) != len(S):
+        raise Invalid("repeated vertex")
+    if len(S) > k:
+        raise Invalid(f"cover has {len(S)} vertices, at most {k} allowed")
+    C = set(S)
+    for u, v in inst["edges"]:
+        if u not in C and v not in C:
+            raise Invalid(f"edge ({u},{v}) is not covered")
+    return None, {"size": len(S)}
diff --git a/tasks/construct-mc-vertex-cover-l1-s4/tests/instance.json b/tasks/construct-mc-vertex-cover-l1-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d112d154f31d15a5d8090b334f50908fcad6de42
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-l1-s4/tests/instance.json
@@ -0,0 +1 @@
+{"n": 20, "k": 11, "edges": [[0, 10], [0, 17], [0, 18], [1, 8], [1, 19], [2, 9], [2, 16], [3, 10], [3, 14], [3, 16], [3, 17], [4, 9], [4, 14], [4, 17], [5, 18], [6, 9], [7, 18], [8, 9], [8, 12], [8, 14], [10, 12], [10, 14], [10, 18], [12, 14]], "family": "mc_vertex_cover", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-vertex-cover-l1-s4/tests/run.py b/tasks/construct-mc-vertex-cover-l1-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-l1-s4/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-vertex-cover-l1-s4/tests/test.sh b/tasks/construct-mc-vertex-cover-l1-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-l1-s4/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-nppc-3col-l1-s4/tests/run.py b/tasks/construct-nppc-3col-l1-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-3col-l1-s4/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-nppc-3col-l5-s2/environment/Dockerfile b/tasks/construct-nppc-3col-l5-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-3col-l5-s2/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-nppc-3col-l5-s2/environment/instance.json b/tasks/construct-nppc-3col-l5-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..fdea6a40aedf9d8a92b2477feac4d1c2bcc93803
--- /dev/null
+++ b/tasks/construct-nppc-3col-l5-s2/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 60, "edges": [[0, 4], [0, 44], [0, 58], [1, 3], [1, 6], [1, 9], [1, 26], [1, 55], [2, 11], [2, 29], [2, 37], [2, 47], [2, 50], [2, 55], [3, 28], [3, 41], [3, 42], [3, 47], [3, 50], [3, 53], [4, 7], [4, 19], [4, 20], [4, 25], [4, 26], [4, 31], [4, 56], [5, 9], [5, 13], [5, 26], [5, 34], [5, 37], [5, 57], [5, 59], [6, 16], [6, 28], [6, 31], [6, 48], [6, 51], [7, 8], [7, 24], [7, 29], [7, 32], [7, 35], [7, 50], [8, 40], [8, 47], [8, 56], [9, 19], [9, 31], [9, 36], [9, 43], [9, 44], [9, 46], [9, 59], [10, 11], [10, 18], [10, 27], [10, 31], [10, 34], [10, 39], [10, 41], [10, 44], [10, 54], [11, 24], [11, 32], [11, 46], [11, 47], [11, 54], [12, 13], [12, 23], [12, 30], [12, 43], [12, 44], [12, 48], [13, 22], [13, 26], [13, 30], [13, 31], [13, 34], [14, 19], [14, 29], [14, 39], [14, 43], [14, 44], [14, 54], [14, 57], [15, 35], [15, 48], [15, 49], [15, 52], [15, 57], [16, 22], [16, 27], [16, 28], [16, 42], [16, 59], [17, 33], [17, 41], [17, 52], [18, 28], [18, 34], [18, 37], [18, 45], [18, 47], [19, 22], [19, 24], [20, 42], [20, 48], [20, 50], [21, 26], [21, 47], [21, 50], [22, 31], [22, 52], [22, 59], [23, 32], [23, 51], [24, 31], [24, 37], [24, 41], [25, 28], [25, 39], [25, 47], [25, 48], [26, 28], [26, 42], [26, 53], [26, 54], [26, 58], [27, 28], [27, 30], [27, 39], [27, 41], [27, 48], [27, 54], [28, 36], [28, 41], [28, 52], [28, 55], [28, 57], [29, 39], [29, 46], [29, 58], [30, 35], [30, 38], [30, 50], [30, 59], [31, 34], [31, 35], [31, 37], [31, 49], [31, 53], [31, 59], [32, 37], [33, 36], [33, 43], [33, 54], [34, 38], [34, 39], [34, 46], [34, 48], [34, 50], [35, 38], [35, 39], [35, 44], [35, 53], [36, 47], [37, 44], [37, 45], [37, 51], [38, 49], [39, 40], [39, 53], [40, 44], [40, 56], [41, 45], [41, 50], [41, 55], [42, 49], [42, 54], [43, 47], [43, 49], [43, 51], [43, 53], [43, 59], [44, 45], [44, 51], [44, 53], [44, 55], [44, 57], [45, 48], [45, 49], [47, 48], [47, 52], [47, 54], [49, 58], [53, 57], [54, 57], [57, 59]], "family": "nppc_three_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-3col-l5-s2/solution/solve.sh b/tasks/construct-nppc-3col-l5-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..730360e64d3dc04e373e1866609e7d25838afa40
--- /dev/null
+++ b/tasks/construct-nppc-3col-l5-s2/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"coloring": [1, 2, 2, 1, 0, 2, 1, 2, 1, 1, 0, 1, 1, 0, 0, 0, 2, 0, 2, 2, 1, 2, 1, 2, 0, 1, 1, 1, 0, 1, 2, 2, 0, 1, 1, 1, 2, 1, 0, 2, 0, 2, 0, 2, 2, 0, 0, 0, 2, 1, 0, 0, 2, 0, 2, 1, 2, 1, 0, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-3col-l5-s2/tests/check.py b/tasks/construct-nppc-3col-l5-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..72362ed9180250b6c53ce0c71867ca476ac2af20
--- /dev/null
+++ b/tasks/construct-nppc-3col-l5-s2/tests/check.py
@@ -0,0 +1,12 @@
+"""Proper 3-colouring of a given graph. Accepts any valid colouring."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n = inst["num_nodes"]
+    # upstream NPPC verifier skips the range/length check, so an all-distinct colouring passed it
+    c = int_list(ans.get("coloring"), n, 0, 2, "coloring")
+    for u, v in inst["edges"]:
+        if c[u] == c[v]:
+            raise Invalid(f"edge ({u},{v}) is monochromatic")
+    return None, {}
diff --git a/tasks/construct-nppc-3col-l5-s2/tests/instance.json b/tasks/construct-nppc-3col-l5-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..fdea6a40aedf9d8a92b2477feac4d1c2bcc93803
--- /dev/null
+++ b/tasks/construct-nppc-3col-l5-s2/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 60, "edges": [[0, 4], [0, 44], [0, 58], [1, 3], [1, 6], [1, 9], [1, 26], [1, 55], [2, 11], [2, 29], [2, 37], [2, 47], [2, 50], [2, 55], [3, 28], [3, 41], [3, 42], [3, 47], [3, 50], [3, 53], [4, 7], [4, 19], [4, 20], [4, 25], [4, 26], [4, 31], [4, 56], [5, 9], [5, 13], [5, 26], [5, 34], [5, 37], [5, 57], [5, 59], [6, 16], [6, 28], [6, 31], [6, 48], [6, 51], [7, 8], [7, 24], [7, 29], [7, 32], [7, 35], [7, 50], [8, 40], [8, 47], [8, 56], [9, 19], [9, 31], [9, 36], [9, 43], [9, 44], [9, 46], [9, 59], [10, 11], [10, 18], [10, 27], [10, 31], [10, 34], [10, 39], [10, 41], [10, 44], [10, 54], [11, 24], [11, 32], [11, 46], [11, 47], [11, 54], [12, 13], [12, 23], [12, 30], [12, 43], [12, 44], [12, 48], [13, 22], [13, 26], [13, 30], [13, 31], [13, 34], [14, 19], [14, 29], [14, 39], [14, 43], [14, 44], [14, 54], [14, 57], [15, 35], [15, 48], [15, 49], [15, 52], [15, 57], [16, 22], [16, 27], [16, 28], [16, 42], [16, 59], [17, 33], [17, 41], [17, 52], [18, 28], [18, 34], [18, 37], [18, 45], [18, 47], [19, 22], [19, 24], [20, 42], [20, 48], [20, 50], [21, 26], [21, 47], [21, 50], [22, 31], [22, 52], [22, 59], [23, 32], [23, 51], [24, 31], [24, 37], [24, 41], [25, 28], [25, 39], [25, 47], [25, 48], [26, 28], [26, 42], [26, 53], [26, 54], [26, 58], [27, 28], [27, 30], [27, 39], [27, 41], [27, 48], [27, 54], [28, 36], [28, 41], [28, 52], [28, 55], [28, 57], [29, 39], [29, 46], [29, 58], [30, 35], [30, 38], [30, 50], [30, 59], [31, 34], [31, 35], [31, 37], [31, 49], [31, 53], [31, 59], [32, 37], [33, 36], [33, 43], [33, 54], [34, 38], [34, 39], [34, 46], [34, 48], [34, 50], [35, 38], [35, 39], [35, 44], [35, 53], [36, 47], [37, 44], [37, 45], [37, 51], [38, 49], [39, 40], [39, 53], [40, 44], [40, 56], [41, 45], [41, 50], [41, 55], [42, 49], [42, 54], [43, 47], [43, 49], [43, 51], [43, 53], [43, 59], [44, 45], [44, 51], [44, 53], [44, 55], [44, 57], [45, 48], [45, 49], [47, 48], [47, 52], [47, 54], [49, 58], [53, 57], [54, 57], [57, 59]], "family": "nppc_three_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-3col-l5-s2/tests/run.py b/tasks/construct-nppc-3col-l5-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-3col-l5-s2/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-nppc-3col-l5-s2/tests/test.sh b/tasks/construct-nppc-3col-l5-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-3col-l5-s2/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-nppc-3col-l5-s5/environment/Dockerfile b/tasks/construct-nppc-3col-l5-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-3col-l5-s5/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-nppc-3col-l5-s5/environment/instance.json b/tasks/construct-nppc-3col-l5-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..aa7c97207f8648ad94eac4275bdf3a9d88c6bfa9
--- /dev/null
+++ b/tasks/construct-nppc-3col-l5-s5/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 60, "edges": [[0, 2], [0, 34], [0, 39], [0, 45], [1, 13], [1, 32], [2, 3], [2, 13], [2, 15], [2, 24], [2, 41], [3, 10], [3, 18], [3, 37], [3, 42], [3, 44], [3, 47], [3, 53], [4, 8], [4, 17], [4, 22], [4, 23], [4, 31], [4, 35], [4, 42], [4, 48], [4, 53], [5, 27], [5, 32], [5, 37], [5, 56], [6, 28], [6, 42], [6, 45], [6, 47], [6, 48], [6, 57], [7, 8], [7, 51], [7, 53], [7, 59], [8, 22], [8, 27], [8, 35], [8, 39], [8, 49], [8, 58], [9, 31], [9, 44], [9, 47], [9, 51], [9, 54], [10, 22], [10, 27], [10, 34], [10, 38], [10, 39], [10, 47], [10, 50], [10, 55], [11, 14], [11, 16], [11, 20], [11, 23], [11, 29], [11, 49], [11, 58], [12, 26], [12, 41], [12, 43], [12, 46], [12, 47], [12, 52], [13, 18], [13, 20], [13, 32], [13, 37], [13, 44], [13, 48], [13, 55], [14, 16], [14, 28], [14, 33], [14, 42], [15, 28], [15, 43], [15, 48], [15, 49], [15, 52], [15, 55], [16, 21], [16, 25], [16, 35], [16, 43], [16, 46], [16, 47], [16, 54], [16, 55], [17, 20], [17, 22], [17, 39], [17, 42], [17, 47], [18, 26], [18, 27], [18, 35], [18, 52], [18, 53], [18, 55], [19, 34], [20, 22], [20, 34], [20, 35], [20, 41], [20, 47], [20, 53], [20, 57], [21, 24], [21, 30], [21, 58], [22, 31], [22, 36], [22, 43], [22, 45], [22, 59], [23, 35], [23, 40], [23, 42], [23, 49], [24, 32], [24, 34], [24, 45], [24, 56], [25, 29], [25, 33], [25, 39], [25, 42], [25, 46], [26, 30], [26, 43], [26, 48], [26, 50], [26, 55], [27, 31], [27, 32], [27, 43], [27, 45], [27, 55], [27, 59], [28, 39], [28, 43], [29, 38], [29, 53], [29, 55], [30, 32], [30, 36], [30, 38], [30, 44], [30, 46], [30, 54], [31, 34], [31, 39], [31, 55], [32, 58], [33, 50], [34, 37], [34, 43], [35, 36], [35, 37], [35, 40], [35, 49], [35, 58], [36, 49], [36, 54], [37, 50], [37, 55], [37, 58], [38, 48], [38, 50], [39, 43], [39, 44], [40, 41], [42, 44], [42, 59], [43, 54], [45, 49], [45, 50], [45, 54], [45, 58], [46, 49], [46, 50], [46, 52], [47, 55], [47, 59], [48, 52], [48, 53], [49, 51], [51, 59], [55, 57], [57, 58]], "family": "nppc_three_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-3col-l5-s5/solution/solve.sh b/tasks/construct-nppc-3col-l5-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..7386ad054c6f2d62bdb42f1aca4bf0950c2238e7
--- /dev/null
+++ b/tasks/construct-nppc-3col-l5-s5/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"coloring": [0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 2, 1, 2, 0, 2, 1, 1, 1, 1, 0, 2, 2, 1, 0, 0, 2, 2, 1, 1, 0, 1, 1, 1, 2, 2, 1, 1, 2, 2, 0, 2, 2, 0, 1, 1, 2, 2, 1, 0, 0, 2, 0, 2, 2, 0, 2, 2, 0, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-3col-l5-s5/tests/check.py b/tasks/construct-nppc-3col-l5-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..72362ed9180250b6c53ce0c71867ca476ac2af20
--- /dev/null
+++ b/tasks/construct-nppc-3col-l5-s5/tests/check.py
@@ -0,0 +1,12 @@
+"""Proper 3-colouring of a given graph. Accepts any valid colouring."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n = inst["num_nodes"]
+    # upstream NPPC verifier skips the range/length check, so an all-distinct colouring passed it
+    c = int_list(ans.get("coloring"), n, 0, 2, "coloring")
+    for u, v in inst["edges"]:
+        if c[u] == c[v]:
+            raise Invalid(f"edge ({u},{v}) is monochromatic")
+    return None, {}
diff --git a/tasks/construct-nppc-3col-l5-s5/tests/instance.json b/tasks/construct-nppc-3col-l5-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..aa7c97207f8648ad94eac4275bdf3a9d88c6bfa9
--- /dev/null
+++ b/tasks/construct-nppc-3col-l5-s5/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 60, "edges": [[0, 2], [0, 34], [0, 39], [0, 45], [1, 13], [1, 32], [2, 3], [2, 13], [2, 15], [2, 24], [2, 41], [3, 10], [3, 18], [3, 37], [3, 42], [3, 44], [3, 47], [3, 53], [4, 8], [4, 17], [4, 22], [4, 23], [4, 31], [4, 35], [4, 42], [4, 48], [4, 53], [5, 27], [5, 32], [5, 37], [5, 56], [6, 28], [6, 42], [6, 45], [6, 47], [6, 48], [6, 57], [7, 8], [7, 51], [7, 53], [7, 59], [8, 22], [8, 27], [8, 35], [8, 39], [8, 49], [8, 58], [9, 31], [9, 44], [9, 47], [9, 51], [9, 54], [10, 22], [10, 27], [10, 34], [10, 38], [10, 39], [10, 47], [10, 50], [10, 55], [11, 14], [11, 16], [11, 20], [11, 23], [11, 29], [11, 49], [11, 58], [12, 26], [12, 41], [12, 43], [12, 46], [12, 47], [12, 52], [13, 18], [13, 20], [13, 32], [13, 37], [13, 44], [13, 48], [13, 55], [14, 16], [14, 28], [14, 33], [14, 42], [15, 28], [15, 43], [15, 48], [15, 49], [15, 52], [15, 55], [16, 21], [16, 25], [16, 35], [16, 43], [16, 46], [16, 47], [16, 54], [16, 55], [17, 20], [17, 22], [17, 39], [17, 42], [17, 47], [18, 26], [18, 27], [18, 35], [18, 52], [18, 53], [18, 55], [19, 34], [20, 22], [20, 34], [20, 35], [20, 41], [20, 47], [20, 53], [20, 57], [21, 24], [21, 30], [21, 58], [22, 31], [22, 36], [22, 43], [22, 45], [22, 59], [23, 35], [23, 40], [23, 42], [23, 49], [24, 32], [24, 34], [24, 45], [24, 56], [25, 29], [25, 33], [25, 39], [25, 42], [25, 46], [26, 30], [26, 43], [26, 48], [26, 50], [26, 55], [27, 31], [27, 32], [27, 43], [27, 45], [27, 55], [27, 59], [28, 39], [28, 43], [29, 38], [29, 53], [29, 55], [30, 32], [30, 36], [30, 38], [30, 44], [30, 46], [30, 54], [31, 34], [31, 39], [31, 55], [32, 58], [33, 50], [34, 37], [34, 43], [35, 36], [35, 37], [35, 40], [35, 49], [35, 58], [36, 49], [36, 54], [37, 50], [37, 55], [37, 58], [38, 48], [38, 50], [39, 43], [39, 44], [40, 41], [42, 44], [42, 59], [43, 54], [45, 49], [45, 50], [45, 54], [45, 58], [46, 49], [46, 50], [46, 52], [47, 55], [47, 59], [48, 52], [48, 53], [49, 51], [51, 59], [55, 57], [57, 58]], "family": "nppc_three_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-3col-l5-s5/tests/run.py b/tasks/construct-nppc-3col-l5-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-3col-l5-s5/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-nppc-3col-l5-s5/tests/test.sh b/tasks/construct-nppc-3col-l5-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-3col-l5-s5/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-nppc-bandwidth-l3-s0/environment/Dockerfile b/tasks/construct-nppc-bandwidth-l3-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l3-s0/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-nppc-bandwidth-l3-s0/environment/instance.json b/tasks/construct-nppc-bandwidth-l3-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..df3ef7cdf16774fd2adc3b000e1c7b0789c484fe
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l3-s0/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 8, "bandwidth": 2, "edges": [[0, 2], [1, 3], [2, 4], [2, 5], [3, 6], [4, 5], [5, 6], [6, 7]], "family": "nppc_bandwidth", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-bandwidth-l3-s0/solution/solve.sh b/tasks/construct-nppc-bandwidth-l3-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..a5a6c04d171cfd1a76d8b2affd256cc26d9bf893
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l3-s0/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"layout": [0, 4, 1, 6, 2, 3, 5, 7]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-bandwidth-l3-s0/tests/check.py b/tasks/construct-nppc-bandwidth-l3-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..0f847cded257e6c404063dfafde33fa44163afa2
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l3-s0/tests/check.py
@@ -0,0 +1,14 @@
+"""Bandwidth <= b: layout must be a permutation of 0..n-1 with |p[u]-p[v]| <= b on every edge."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, b = inst["num_nodes"], inst["bandwidth"]
+    # upstream NPPC verifier compares sorted(layout) to range(n), which also accepts floats/bools
+    p = int_list(ans.get("layout"), n, 0, n - 1, "layout")
+    if len(set(p)) != n:
+        raise Invalid("layout is not a permutation of 0..n-1")
+    for u, v in inst["edges"]:
+        if abs(p[u] - p[v]) > b:
+            raise Invalid(f"edge ({u},{v}) has stretch {abs(p[u] - p[v])} > {b}")
+    return None, {}
diff --git a/tasks/construct-nppc-bandwidth-l3-s0/tests/instance.json b/tasks/construct-nppc-bandwidth-l3-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..df3ef7cdf16774fd2adc3b000e1c7b0789c484fe
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l3-s0/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 8, "bandwidth": 2, "edges": [[0, 2], [1, 3], [2, 4], [2, 5], [3, 6], [4, 5], [5, 6], [6, 7]], "family": "nppc_bandwidth", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-bandwidth-l3-s0/tests/run.py b/tasks/construct-nppc-bandwidth-l3-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l3-s0/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-nppc-bandwidth-l3-s0/tests/test.sh b/tasks/construct-nppc-bandwidth-l3-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l3-s0/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-nppc-bandwidth-l3-s6/environment/Dockerfile b/tasks/construct-nppc-bandwidth-l3-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l3-s6/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-nppc-bandwidth-l3-s6/environment/instance.json b/tasks/construct-nppc-bandwidth-l3-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f7b94dc0852dd337811ea4ba9a65fd7afac94789
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l3-s6/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 8, "bandwidth": 2, "edges": [[0, 3], [1, 2], [1, 5], [1, 6], [2, 5], [3, 4], [4, 6], [5, 7]], "family": "nppc_bandwidth", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-bandwidth-l3-s6/solution/solve.sh b/tasks/construct-nppc-bandwidth-l3-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..71f604e40892239947b14b989914a57710fd6342
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l3-s6/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"layout": [6, 3, 2, 7, 5, 1, 4, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-bandwidth-l3-s6/tests/check.py b/tasks/construct-nppc-bandwidth-l3-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..0f847cded257e6c404063dfafde33fa44163afa2
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l3-s6/tests/check.py
@@ -0,0 +1,14 @@
+"""Bandwidth <= b: layout must be a permutation of 0..n-1 with |p[u]-p[v]| <= b on every edge."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, b = inst["num_nodes"], inst["bandwidth"]
+    # upstream NPPC verifier compares sorted(layout) to range(n), which also accepts floats/bools
+    p = int_list(ans.get("layout"), n, 0, n - 1, "layout")
+    if len(set(p)) != n:
+        raise Invalid("layout is not a permutation of 0..n-1")
+    for u, v in inst["edges"]:
+        if abs(p[u] - p[v]) > b:
+            raise Invalid(f"edge ({u},{v}) has stretch {abs(p[u] - p[v])} > {b}")
+    return None, {}
diff --git a/tasks/construct-nppc-bandwidth-l3-s6/tests/instance.json b/tasks/construct-nppc-bandwidth-l3-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f7b94dc0852dd337811ea4ba9a65fd7afac94789
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l3-s6/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 8, "bandwidth": 2, "edges": [[0, 3], [1, 2], [1, 5], [1, 6], [2, 5], [3, 4], [4, 6], [5, 7]], "family": "nppc_bandwidth", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-bandwidth-l3-s6/tests/run.py b/tasks/construct-nppc-bandwidth-l3-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l3-s6/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-nppc-bandwidth-l3-s6/tests/test.sh b/tasks/construct-nppc-bandwidth-l3-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l3-s6/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-nppc-clique-l2-s9/environment/Dockerfile b/tasks/construct-nppc-clique-l2-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-clique-l2-s9/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-nppc-clique-l2-s9/environment/instance.json b/tasks/construct-nppc-clique-l2-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0c1fa7e741036060bb2742a69d4a34974bcf0dba
--- /dev/null
+++ b/tasks/construct-nppc-clique-l2-s9/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 14, "k": 7, "edges": [[0, 3], [0, 5], [0, 9], [1, 2], [1, 6], [1, 10], [1, 11], [1, 13], [2, 3], [2, 6], [2, 7], [2, 8], [2, 9], [2, 10], [2, 11], [2, 12], [2, 13], [3, 8], [3, 9], [3, 12], [3, 13], [4, 6], [4, 8], [4, 9], [4, 10], [4, 13], [5, 9], [5, 11], [6, 9], [6, 10], [6, 11], [6, 12], [6, 13], [7, 8], [7, 10], [7, 11], [7, 13], [9, 10], [9, 11], [9, 12], [9, 13], [10, 11], [10, 12], [10, 13], [11, 12], [11, 13], [12, 13]], "family": "nppc_clique", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-clique-l2-s9/solution/solve.sh b/tasks/construct-nppc-clique-l2-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..0e7fa13e9bc351ac49a0488619b76fd5be414ebb
--- /dev/null
+++ b/tasks/construct-nppc-clique-l2-s9/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"clique": [2, 6, 9, 10, 11, 12, 13]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-clique-l2-s9/tests/check.py b/tasks/construct-nppc-clique-l2-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..98eb4fc9097fcfa31988cf0a8b79280cec2b7980
--- /dev/null
+++ b/tasks/construct-nppc-clique-l2-s9/tests/check.py
@@ -0,0 +1,19 @@
+"""Clique of size >= k. Accepts any valid clique."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["num_nodes"], inst["k"]
+    c = int_list(ans.get("clique"), None, 0, n - 1, "clique")
+    # upstream NPPC verifier checks len(list) >= k but only tests pairs u < v, so [v, v, ..., v] passed
+    if len(set(c)) != len(c):
+        raise Invalid("clique contains duplicates")
+    if len(c) < k:
+        raise Invalid(f"clique has {len(c)} < {k} vertices")
+    es = {(min(u, v), max(u, v)) for u, v in inst["edges"]}
+    for u, v in itertools.combinations(sorted(c), 2):
+        if (u, v) not in es:
+            raise Invalid(f"no edge between {u} and {v}")
+    return None, {"size": len(c)}
diff --git a/tasks/construct-nppc-clique-l2-s9/tests/instance.json b/tasks/construct-nppc-clique-l2-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0c1fa7e741036060bb2742a69d4a34974bcf0dba
--- /dev/null
+++ b/tasks/construct-nppc-clique-l2-s9/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 14, "k": 7, "edges": [[0, 3], [0, 5], [0, 9], [1, 2], [1, 6], [1, 10], [1, 11], [1, 13], [2, 3], [2, 6], [2, 7], [2, 8], [2, 9], [2, 10], [2, 11], [2, 12], [2, 13], [3, 8], [3, 9], [3, 12], [3, 13], [4, 6], [4, 8], [4, 9], [4, 10], [4, 13], [5, 9], [5, 11], [6, 9], [6, 10], [6, 11], [6, 12], [6, 13], [7, 8], [7, 10], [7, 11], [7, 13], [9, 10], [9, 11], [9, 12], [9, 13], [10, 11], [10, 12], [10, 13], [11, 12], [11, 13], [12, 13]], "family": "nppc_clique", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-clique-l2-s9/tests/run.py b/tasks/construct-nppc-clique-l2-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-clique-l2-s9/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-nppc-clique-l2-s9/tests/test.sh b/tasks/construct-nppc-clique-l2-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-clique-l2-s9/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-nppc-clique-l6-s0/environment/Dockerfile b/tasks/construct-nppc-clique-l6-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-clique-l6-s0/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-nppc-clique-l6-s0/environment/instance.json b/tasks/construct-nppc-clique-l6-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..899f8cfef93e0611a290224571984d582bd758af
--- /dev/null
+++ b/tasks/construct-nppc-clique-l6-s0/environment/instance.json
@@ -0,0 +1 @@
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90], [89, 93], [89, 97], [89, 99], [90, 91], [90, 93], [90, 94], [90, 95], [90, 98], [91, 94], [91, 97], [91, 98], [92, 93], [92, 94], [92, 96], [92, 97], [92, 99], [93, 94], [94, 96], [95, 96], [95, 97], [95, 99], [96, 97]], "family": "nppc_clique", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-clique-l6-s0/solution/solve.sh b/tasks/construct-nppc-clique-l6-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..e918533f95dc86727c516dec9ec07e47b524cbab
--- /dev/null
+++ b/tasks/construct-nppc-clique-l6-s0/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"clique": [5, 17, 27, 35, 40, 43, 46, 47, 48, 50, 51, 52, 55, 56, 69, 70, 74, 79, 87, 91]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-clique-l6-s0/tests/check.py b/tasks/construct-nppc-clique-l6-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..98eb4fc9097fcfa31988cf0a8b79280cec2b7980
--- /dev/null
+++ b/tasks/construct-nppc-clique-l6-s0/tests/check.py
@@ -0,0 +1,19 @@
+"""Clique of size >= k. Accepts any valid clique."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["num_nodes"], inst["k"]
+    c = int_list(ans.get("clique"), None, 0, n - 1, "clique")
+    # upstream NPPC verifier checks len(list) >= k but only tests pairs u < v, so [v, v, ..., v] passed
+    if len(set(c)) != len(c):
+        raise Invalid("clique contains duplicates")
+    if len(c) < k:
+        raise Invalid(f"clique has {len(c)} < {k} vertices")
+    es = {(min(u, v), max(u, v)) for u, v in inst["edges"]}
+    for u, v in itertools.combinations(sorted(c), 2):
+        if (u, v) not in es:
+            raise Invalid(f"no edge between {u} and {v}")
+    return None, {"size": len(c)}
diff --git a/tasks/construct-nppc-clique-l6-s0/tests/instance.json b/tasks/construct-nppc-clique-l6-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..899f8cfef93e0611a290224571984d582bd758af
--- /dev/null
+++ b/tasks/construct-nppc-clique-l6-s0/tests/instance.json
@@ -0,0 +1 @@
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90], [89, 93], [89, 97], [89, 99], [90, 91], [90, 93], [90, 94], [90, 95], [90, 98], [91, 94], [91, 97], [91, 98], [92, 93], [92, 94], [92, 96], [92, 97], [92, 99], [93, 94], [94, 96], [95, 96], [95, 97], [95, 99], [96, 97]], "family": "nppc_clique", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-clique-l6-s0/tests/run.py b/tasks/construct-nppc-clique-l6-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-clique-l6-s0/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-nppc-clique-l6-s0/tests/test.sh b/tasks/construct-nppc-clique-l6-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-clique-l6-s0/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-nppc-clique-l6-s1/environment/instance.json b/tasks/construct-nppc-clique-l6-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b5f4f0de4b57cc9bca00330661bf2cbae5d63981
--- /dev/null
+++ b/tasks/construct-nppc-clique-l6-s1/environment/instance.json
@@ -0,0 +1 @@
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\ No newline at end of file
diff --git a/tasks/construct-nppc-clique-l6-s6/environment/Dockerfile b/tasks/construct-nppc-clique-l6-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-clique-l6-s6/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-nppc-clique-l6-s6/environment/instance.json b/tasks/construct-nppc-clique-l6-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..9239898365a0ad82199ddf02d2a023017ccf5caf
--- /dev/null
+++ b/tasks/construct-nppc-clique-l6-s6/environment/instance.json
@@ -0,0 +1 @@
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\ No newline at end of file
diff --git a/tasks/construct-nppc-clique-l6-s6/solution/solve.sh b/tasks/construct-nppc-clique-l6-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..e03256b08bd0a8bac2b9c28f3f85b0241eb1f4c2
--- /dev/null
+++ b/tasks/construct-nppc-clique-l6-s6/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"clique": [0, 1, 3, 6, 12, 13, 14, 16, 19, 21, 25, 28, 36, 37, 49, 50, 66, 71, 86, 92]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-clique-l6-s6/tests/check.py b/tasks/construct-nppc-clique-l6-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..98eb4fc9097fcfa31988cf0a8b79280cec2b7980
--- /dev/null
+++ b/tasks/construct-nppc-clique-l6-s6/tests/check.py
@@ -0,0 +1,19 @@
+"""Clique of size >= k. Accepts any valid clique."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["num_nodes"], inst["k"]
+    c = int_list(ans.get("clique"), None, 0, n - 1, "clique")
+    # upstream NPPC verifier checks len(list) >= k but only tests pairs u < v, so [v, v, ..., v] passed
+    if len(set(c)) != len(c):
+        raise Invalid("clique contains duplicates")
+    if len(c) < k:
+        raise Invalid(f"clique has {len(c)} < {k} vertices")
+    es = {(min(u, v), max(u, v)) for u, v in inst["edges"]}
+    for u, v in itertools.combinations(sorted(c), 2):
+        if (u, v) not in es:
+            raise Invalid(f"no edge between {u} and {v}")
+    return None, {"size": len(c)}
diff --git a/tasks/construct-nppc-clique-l6-s6/tests/instance.json b/tasks/construct-nppc-clique-l6-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..9239898365a0ad82199ddf02d2a023017ccf5caf
--- /dev/null
+++ b/tasks/construct-nppc-clique-l6-s6/tests/instance.json
@@ -0,0 +1 @@
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\ No newline at end of file
diff --git a/tasks/construct-nppc-clique-l6-s6/tests/run.py b/tasks/construct-nppc-clique-l6-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-clique-l6-s6/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-nppc-clique-l6-s6/tests/test.sh b/tasks/construct-nppc-clique-l6-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-clique-l6-s6/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-nppc-dominating-set-l2-s0/environment/Dockerfile b/tasks/construct-nppc-dominating-set-l2-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l2-s0/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-nppc-dominating-set-l2-s0/environment/instance.json b/tasks/construct-nppc-dominating-set-l2-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..5dd3b7c38c6591040c8bfc63beb7a73ba95b43b0
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l2-s0/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 30, "k": 15, "edges": [[0, 3], [0, 5], [0, 6], [0, 11], [0, 15], [0, 17], [0, 19], [0, 21], [0, 23], [0, 26], [1, 12], [1, 15], [1, 16], [1, 17], [1, 18], [1, 19], [1, 22], [1, 23], [2, 3], [2, 6], [2, 7], [2, 9], [2, 15], [2, 16], [2, 25], [2, 26], [2, 28], [2, 29], [3, 7], [3, 12], [3, 17], [3, 20], [3, 22], [3, 23], [3, 29], [4, 5], [4, 6], [4, 8], [4, 9], [4, 10], [4, 11], [4, 16], [4, 18], [4, 20], [4, 22], [4, 23], [4, 24], [4, 25], [5, 20], [5, 25], [5, 26], [5, 28], [5, 29], [6, 9], [6, 11], [6, 13], [6, 14], [6, 16], [6, 20], [6, 21], [6, 24], [7, 8], [7, 13], [7, 17], [7, 18], [7, 21], [7, 24], [7, 28], [8, 11], [8, 13], [8, 14], [8, 18], [8, 20], [8, 21], [8, 22], [8, 23], [8, 25], [8, 28], [9, 11], [9, 12], [9, 13], [9, 16], [9, 22], [9, 23], [9, 24], [9, 25], [10, 13], [10, 15], [10, 17], [10, 26], [10, 29], [11, 12], [11, 14], [11, 21], [11, 23], [11, 28], [12, 13], [12, 14], [12, 15], [12, 18], [12, 19], [12, 20], [12, 21], [12, 22], [12, 26], [12, 28], [13, 22], [13, 24], [13, 27], [14, 15], [14, 18], [14, 24], [14, 26], [15, 16], [15, 19], [15, 22], [15, 23], [15, 27], [15, 29], [16, 19], [16, 27], [17, 21], [17, 23], [17, 28], [18, 19], [18, 21], [18, 22], [18, 24], [18, 25], [19, 23], [19, 24], [19, 26], [19, 29], [20, 23], [20, 27], [21, 24], [21, 26], [21, 29], [22, 23], [22, 28], [22, 29], [23, 24], [24, 27], [24, 28], [25, 27], [25, 28], [25, 29], [26, 28], [27, 28], [27, 29]], "family": "nppc_dominating_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-dominating-set-l2-s0/solution/solve.sh b/tasks/construct-nppc-dominating-set-l2-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..05e58ed94ee93a28f8c35a6b91a5acef5f28b0fb
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l2-s0/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"dominating_set": [6, 7, 9, 11, 12, 13, 15, 16, 19, 20, 21, 23, 24, 25, 28]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-dominating-set-l2-s0/tests/check.py b/tasks/construct-nppc-dominating-set-l2-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..0677a7bcbccb02297d022ffa4b5dd15da6371bff
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l2-s0/tests/check.py
@@ -0,0 +1,23 @@
+"""Dominating set of size <= k. Accepts any valid dominating set."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["num_nodes"], inst["k"]
+    # upstream NPPC verifier has no type/range checks (out-of-range labels and floats are silently used)
+    d = int_list(ans.get("dominating_set"), None, 0, n - 1, "dominating_set")
+    if len(set(d)) != len(d):
+        raise Invalid("dominating_set contains duplicates")
+    if not d or len(d) > k:
+        raise Invalid(f"dominating_set must have 1..{k} vertices, got {len(d)}")
+    dom = set(d)
+    s = set(d)
+    for u, v in inst["edges"]:
+        if u in s:
+            dom.add(v)
+        if v in s:
+            dom.add(u)
+    if len(dom) != n:
+        miss = min(set(range(n)) - dom)
+        raise Invalid(f"vertex {miss} is not dominated")
+    return None, {"size": len(d)}
diff --git a/tasks/construct-nppc-dominating-set-l2-s0/tests/instance.json b/tasks/construct-nppc-dominating-set-l2-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..5dd3b7c38c6591040c8bfc63beb7a73ba95b43b0
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l2-s0/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 30, "k": 15, "edges": [[0, 3], [0, 5], [0, 6], [0, 11], [0, 15], [0, 17], [0, 19], [0, 21], [0, 23], [0, 26], [1, 12], [1, 15], [1, 16], [1, 17], [1, 18], [1, 19], [1, 22], [1, 23], [2, 3], [2, 6], [2, 7], [2, 9], [2, 15], [2, 16], [2, 25], [2, 26], [2, 28], [2, 29], [3, 7], [3, 12], [3, 17], [3, 20], [3, 22], [3, 23], [3, 29], [4, 5], [4, 6], [4, 8], [4, 9], [4, 10], [4, 11], [4, 16], [4, 18], [4, 20], [4, 22], [4, 23], [4, 24], [4, 25], [5, 20], [5, 25], [5, 26], [5, 28], [5, 29], [6, 9], [6, 11], [6, 13], [6, 14], [6, 16], [6, 20], [6, 21], [6, 24], [7, 8], [7, 13], [7, 17], [7, 18], [7, 21], [7, 24], [7, 28], [8, 11], [8, 13], [8, 14], [8, 18], [8, 20], [8, 21], [8, 22], [8, 23], [8, 25], [8, 28], [9, 11], [9, 12], [9, 13], [9, 16], [9, 22], [9, 23], [9, 24], [9, 25], [10, 13], [10, 15], [10, 17], [10, 26], [10, 29], [11, 12], [11, 14], [11, 21], [11, 23], [11, 28], [12, 13], [12, 14], [12, 15], [12, 18], [12, 19], [12, 20], [12, 21], [12, 22], [12, 26], [12, 28], [13, 22], [13, 24], [13, 27], [14, 15], [14, 18], [14, 24], [14, 26], [15, 16], [15, 19], [15, 22], [15, 23], [15, 27], [15, 29], [16, 19], [16, 27], [17, 21], [17, 23], [17, 28], [18, 19], [18, 21], [18, 22], [18, 24], [18, 25], [19, 23], [19, 24], [19, 26], [19, 29], [20, 23], [20, 27], [21, 24], [21, 26], [21, 29], [22, 23], [22, 28], [22, 29], [23, 24], [24, 27], [24, 28], [25, 27], [25, 28], [25, 29], [26, 28], [27, 28], [27, 29]], "family": "nppc_dominating_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-dominating-set-l2-s0/tests/run.py b/tasks/construct-nppc-dominating-set-l2-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l2-s0/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-nppc-dominating-set-l2-s0/tests/test.sh b/tasks/construct-nppc-dominating-set-l2-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l2-s0/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-nppc-dominating-set-l2-s7/environment/Dockerfile b/tasks/construct-nppc-dominating-set-l2-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l2-s7/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-nppc-dominating-set-l2-s7/environment/instance.json b/tasks/construct-nppc-dominating-set-l2-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1032527a02f28dbd0d5f7e1e5302d7de5946cf8e
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l2-s7/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 30, "k": 15, "edges": [[0, 4], [0, 11], [0, 16], [0, 20], [0, 28], [1, 2], [1, 3], [1, 8], [1, 9], [1, 10], [1, 16], [1, 21], [1, 22], [2, 7], [2, 12], [2, 15], [2, 27], [3, 4], [3, 6], [3, 10], [3, 15], [3, 17], [3, 19], [3, 22], [4, 6], [4, 10], [4, 19], [4, 20], [4, 23], [4, 25], [4, 27], [4, 28], [4, 29], [5, 6], [5, 7], [5, 11], [5, 12], [5, 17], [5, 19], [5, 21], [5, 22], [5, 23], [5, 25], [5, 28], [6, 10], [6, 11], [6, 19], [6, 21], [6, 25], [6, 27], [6, 28], [7, 17], [7, 18], [7, 20], [7, 22], [7, 27], [7, 28], [7, 29], [8, 11], [8, 21], [8, 22], [8, 25], [8, 26], [8, 27], [8, 29], [9, 10], [9, 12], [9, 15], [9, 17], [9, 18], [9, 24], [9, 27], [9, 28], [10, 15], [11, 17], [11, 18], [11, 25], [11, 26], [11, 28], [12, 14], [12, 15], [12, 19], [12, 20], [12, 22], [12, 24], [12, 25], [12, 27], [12, 28], [13, 14], [13, 16], [13, 17], [13, 18], [13, 21], [13, 22], [13, 23], [13, 27], [13, 28], [14, 16], [14, 19], [14, 21], [14, 22], [14, 23], [14, 24], [15, 16], [15, 19], [15, 22], [15, 25], [15, 29], [16, 17], [16, 18], [16, 19], [16, 22], [16, 26], [17, 22], [17, 25], [17, 26], [17, 27], [17, 28], [17, 29], [18, 21], [18, 25], [18, 29], [19, 21], [19, 28], [20, 21], [20, 25], [20, 29], [21, 23], [21, 24], [21, 26], [21, 29], [22, 23], [22, 24], [22, 25], [22, 26], [23, 24], [24, 25], [24, 29], [25, 27], [25, 29], [27, 28], [27, 29]], "family": "nppc_dominating_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-dominating-set-l2-s9/environment/Dockerfile b/tasks/construct-nppc-dominating-set-l2-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l2-s9/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-nppc-dominating-set-l2-s9/environment/instance.json b/tasks/construct-nppc-dominating-set-l2-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..395e611647365b77cb457d2575480796961735da
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l2-s9/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 30, "k": 15, "edges": [[0, 2], [0, 10], [0, 16], [0, 21], [0, 22], [0, 23], [0, 28], [1, 2], [1, 3], [1, 4], [1, 5], [1, 7], [1, 10], [1, 11], [1, 12], [1, 19], [1, 20], [1, 21], [1, 22], [1, 23], [1, 26], [1, 27], [2, 4], [2, 7], [2, 10], [2, 11], [2, 17], [2, 18], [2, 24], [2, 25], [2, 29], [3, 5], [3, 7], [3, 9], [3, 10], [3, 11], [3, 12], [3, 14], [3, 15], [3, 16], [3, 25], [3, 28], [4, 7], [4, 10], [4, 12], [4, 15], [4, 22], [4, 25], [5, 6], [5, 9], [5, 13], [5, 17], [5, 18], [5, 23], [5, 24], [5, 26], [5, 29], [6, 12], [6, 13], [6, 15], [6, 16], [6, 20], [6, 21], [7, 8], [7, 9], [7, 10], [7, 11], [7, 17], [7, 19], [7, 21], [7, 22], [8, 11], [8, 18], [8, 20], [8, 21], [8, 22], [8, 25], [8, 27], [9, 16], [9, 17], [9, 29], [10, 11], [10, 17], [10, 18], [10, 24], [10, 26], [10, 27], [10, 29], [11, 12], [11, 14], [11, 18], [11, 19], [11, 23], [11, 25], [11, 26], [12, 22], [12, 23], [13, 28], [14, 15], [14, 19], [14, 21], [14, 26], [14, 29], [15, 16], [15, 17], [15, 18], [15, 21], [16, 20], [17, 23], [17, 24], [17, 26], [18, 19], [18, 20], [18, 22], [18, 23], [18, 27], [19, 20], [19, 21], [19, 26], [20, 21], [20, 28], [21, 24], [21, 26], [21, 27], [21, 28], [22, 24], [22, 26], [22, 27], [23, 28], [24, 27], [24, 28], [26, 27], [26, 28]], "family": "nppc_dominating_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-dominating-set-l2-s9/solution/solve.sh b/tasks/construct-nppc-dominating-set-l2-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..5224db672c573b935173325b418b768ae2e6c613
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l2-s9/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"dominating_set": [0, 1, 5, 7, 8, 10, 11, 15, 19, 20, 21, 22, 23, 24, 25]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-dominating-set-l2-s9/tests/check.py b/tasks/construct-nppc-dominating-set-l2-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..0677a7bcbccb02297d022ffa4b5dd15da6371bff
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l2-s9/tests/check.py
@@ -0,0 +1,23 @@
+"""Dominating set of size <= k. Accepts any valid dominating set."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["num_nodes"], inst["k"]
+    # upstream NPPC verifier has no type/range checks (out-of-range labels and floats are silently used)
+    d = int_list(ans.get("dominating_set"), None, 0, n - 1, "dominating_set")
+    if len(set(d)) != len(d):
+        raise Invalid("dominating_set contains duplicates")
+    if not d or len(d) > k:
+        raise Invalid(f"dominating_set must have 1..{k} vertices, got {len(d)}")
+    dom = set(d)
+    s = set(d)
+    for u, v in inst["edges"]:
+        if u in s:
+            dom.add(v)
+        if v in s:
+            dom.add(u)
+    if len(dom) != n:
+        miss = min(set(range(n)) - dom)
+        raise Invalid(f"vertex {miss} is not dominated")
+    return None, {"size": len(d)}
diff --git a/tasks/construct-nppc-dominating-set-l2-s9/tests/instance.json b/tasks/construct-nppc-dominating-set-l2-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..395e611647365b77cb457d2575480796961735da
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l2-s9/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 30, "k": 15, "edges": [[0, 2], [0, 10], [0, 16], [0, 21], [0, 22], [0, 23], [0, 28], [1, 2], [1, 3], [1, 4], [1, 5], [1, 7], [1, 10], [1, 11], [1, 12], [1, 19], [1, 20], [1, 21], [1, 22], [1, 23], [1, 26], [1, 27], [2, 4], [2, 7], [2, 10], [2, 11], [2, 17], [2, 18], [2, 24], [2, 25], [2, 29], [3, 5], [3, 7], [3, 9], [3, 10], [3, 11], [3, 12], [3, 14], [3, 15], [3, 16], [3, 25], [3, 28], [4, 7], [4, 10], [4, 12], [4, 15], [4, 22], [4, 25], [5, 6], [5, 9], [5, 13], [5, 17], [5, 18], [5, 23], [5, 24], [5, 26], [5, 29], [6, 12], [6, 13], [6, 15], [6, 16], [6, 20], [6, 21], [7, 8], [7, 9], [7, 10], [7, 11], [7, 17], [7, 19], [7, 21], [7, 22], [8, 11], [8, 18], [8, 20], [8, 21], [8, 22], [8, 25], [8, 27], [9, 16], [9, 17], [9, 29], [10, 11], [10, 17], [10, 18], [10, 24], [10, 26], [10, 27], [10, 29], [11, 12], [11, 14], [11, 18], [11, 19], [11, 23], [11, 25], [11, 26], [12, 22], [12, 23], [13, 28], [14, 15], [14, 19], [14, 21], [14, 26], [14, 29], [15, 16], [15, 17], [15, 18], [15, 21], [16, 20], [17, 23], [17, 24], [17, 26], [18, 19], [18, 20], [18, 22], [18, 23], [18, 27], [19, 20], [19, 21], [19, 26], [20, 21], [20, 28], [21, 24], [21, 26], [21, 27], [21, 28], [22, 24], [22, 26], [22, 27], [23, 28], [24, 27], [24, 28], [26, 27], [26, 28]], "family": "nppc_dominating_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-dominating-set-l2-s9/tests/run.py b/tasks/construct-nppc-dominating-set-l2-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l2-s9/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-nppc-dominating-set-l2-s9/tests/test.sh b/tasks/construct-nppc-dominating-set-l2-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l2-s9/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-nppc-dominating-set-l6-s1/environment/Dockerfile b/tasks/construct-nppc-dominating-set-l6-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l6-s1/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-nppc-dominating-set-l6-s1/environment/instance.json b/tasks/construct-nppc-dominating-set-l6-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..838b0cb40ae67a0058094f355409752995d80d5a
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l6-s1/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 300, "k": 20, "edges": [[0, 12], [0, 29], [0, 36], [0, 41], [0, 52], [0, 93], [0, 111], [0, 149], [0, 153], [0, 158], [0, 195], [0, 266], [0, 277], [0, 281], [1, 7], [1, 19], [1, 30], [1, 56], [1, 60], [1, 69], [1, 97], [1, 148], [1, 153], [1, 172], [1, 181], [1, 231], [1, 240], [1, 259], [1, 269], [1, 271], [1, 275], [1, 278], [2, 17], [2, 31], [2, 33], [2, 76], [2, 83], [2, 90], [2, 110], [2, 132], [2, 135], [2, 138], [2, 159], [2, 160], [2, 177], [2, 189], [2, 199], [2, 210], [2, 212], [2, 225], [2, 226], [2, 230], [2, 231], [2, 244], [2, 272], [2, 294], [3, 11], [3, 112], [3, 124], [3, 130], [3, 144], [3, 148], [3, 149], [3, 156], [3, 160], [3, 170], [3, 191], [3, 220], [3, 222], [3, 226], [3, 259], [3, 286], [3, 291], [3, 293], [4, 9], [4, 14], [4, 56], [4, 76], [4, 81], [4, 102], [4, 109], [4, 110], [4, 199], [4, 289], [4, 294], [5, 9], [5, 14], [5, 22], [5, 32], [5, 36], [5, 77], [5, 84], [5, 106], [5, 135], [5, 175], [5, 186], [5, 190], [5, 199], [5, 236], [5, 246], [5, 273], [5, 296], [5, 299], [6, 9], [6, 26], [6, 41], [6, 47], [6, 66], [6, 68], [6, 90], [6, 100], [6, 126], [6, 146], [6, 169], [6, 182], [6, 185], [6, 189], [6, 210], [6, 214], [6, 216], [6, 224], [6, 232], [6, 235], [6, 248], [6, 251], [6, 266], [6, 284], [7, 8], [7, 35], [7, 39], [7, 61], [7, 65], [7, 73], [7, 123], [7, 137], [7, 220], [7, 228], [7, 229], [7, 231], [7, 267], [7, 292], [8, 13], [8, 15], [8, 38], [8, 47], [8, 48], [8, 56], [8, 68], [8, 79], [8, 106], [8, 140], [8, 171], [8, 183], [8, 220], [8, 223], [8, 249], [8, 251], [8, 255], [8, 282], [8, 299], [9, 16], [9, 31], [9, 40], [9, 72], [9, 73], [9, 81], [9, 97], [9, 106], [9, 113], [9, 141], [9, 147], [9, 163], [9, 164], [9, 167], [9, 183], [9, 184], [9, 212], [9, 220], [9, 239], [9, 244], [9, 251], [9, 252], [9, 265], [9, 273], [10, 38], [10, 58], [10, 74], [10, 93], [10, 96], [10, 101], [10, 104], [10, 109], [10, 132], [10, 133], [10, 136], [10, 176], [10, 177], [10, 181], [10, 182], [10, 185], [10, 196], [10, 200], [10, 212], [10, 214], [10, 219], [10, 235], [10, 240], [10, 264], [10, 276], [11, 63], [11, 72], [11, 80], [11, 121], [11, 141], [11, 157], [11, 166], [11, 167], [11, 192], [11, 206], [11, 230], [11, 255], [11, 256], [11, 263], [11, 276], [11, 293], [12, 18], [12, 26], [12, 48], [12, 65], [12, 84], [12, 125], [12, 141], [12, 144], [12, 170], [12, 177], [12, 186], [12, 193], [12, 195], [12, 206], [12, 210], [12, 211], [12, 259], [12, 270], [12, 280], [12, 287], [12, 299], [13, 35], [13, 38], [13, 39], [13, 65], [13, 68], [13, 78], [13, 115], [13, 132], [13, 151], [13, 171], [13, 187], [13, 204], [13, 239], [13, 246], [13, 268], [14, 45], [14, 48], [14, 67], [14, 78], [14, 84], [14, 111], [14, 114], [14, 122], [14, 153], [14, 193], [14, 236], [14, 258], [14, 261], [14, 272], [14, 287], [15, 16], [15, 25], [15, 28], [15, 33], [15, 40], [15, 56], [15, 98], [15, 112], [15, 123], [15, 128], [15, 141], [15, 160], [15, 163], [15, 204], [15, 226], [15, 229], [15, 252], [15, 257], [15, 266], [15, 268], [15, 270], [15, 290], [16, 50], [16, 51], [16, 57], [16, 64], [16, 68], [16, 70], [16, 72], [16, 92], [16, 108], [16, 111], [16, 112], [16, 128], [16, 133], [16, 136], [16, 140], [16, 149], [16, 169], [16, 184], [16, 203], [16, 234], [16, 241], [16, 246], [16, 252], [16, 256], [16, 260], [16, 268], [16, 270], [16, 273], [16, 287], [16, 290], [17, 58], [17, 71], [17, 72], [17, 93], [17, 100], [17, 126], [17, 149], [17, 153], [17, 193], [17, 228], [17, 253], [17, 274], [17, 296], [18, 29], [18, 31], [18, 42], [18, 43], [18, 68], [18, 85], [18, 104], [18, 107], [18, 111], [18, 127], [18, 130], [18, 132], [18, 139], [18, 181], [18, 202], [18, 206], [18, 207], [18, 221], [18, 226], [18, 234], [18, 251], [19, 22], [19, 37], [19, 54], [19, 55], [19, 88], [19, 102], [19, 144], [19, 173], [19, 183], [19, 246], [19, 254], [19, 276], [20, 21], [20, 37], [20, 38], [20, 66], [20, 70], [20, 80], [20, 94], [20, 114], [20, 135], [20, 146], [20, 156], [20, 169], [20, 185], [20, 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296], [224, 240], [224, 243], [224, 262], [224, 269], [224, 271], [225, 239], [225, 247], [225, 251], [225, 266], [225, 285], [225, 286], [225, 296], [226, 231], [226, 243], [226, 254], [226, 256], [226, 268], [226, 284], [227, 231], [227, 235], [228, 235], [228, 250], [228, 253], [228, 263], [228, 267], [228, 276], [228, 287], [228, 296], [229, 243], [229, 287], [230, 234], [230, 244], [230, 253], [230, 295], [231, 248], [231, 253], [231, 281], [231, 286], [231, 288], [232, 260], [232, 261], [232, 272], [233, 279], [233, 286], [234, 243], [234, 249], [234, 268], [234, 278], [234, 295], [235, 237], [235, 238], [235, 269], [235, 287], [236, 256], [236, 266], [236, 277], [236, 279], [236, 288], [237, 257], [237, 271], [237, 272], [238, 271], [238, 273], [239, 243], [239, 256], [239, 261], [239, 275], [239, 278], [239, 282], [239, 285], [240, 251], [240, 267], [240, 293], [241, 263], [241, 268], [241, 275], [241, 276], [241, 277], [241, 289], [242, 279], [242, 281], [243, 269], [244, 268], [244, 279], [244, 281], [244, 287], [245, 274], [245, 277], [245, 288], [245, 298], [246, 266], [246, 270], [246, 290], [247, 253], [247, 277], [247, 283], [247, 285], [248, 271], [248, 293], [249, 264], [249, 288], [250, 254], [250, 256], [250, 263], [250, 281], [250, 288], [250, 291], [251, 262], [251, 267], [251, 268], [251, 276], [252, 271], [252, 272], [252, 291], [253, 272], [253, 278], [254, 259], [254, 274], [254, 279], [254, 290], [255, 266], [256, 258], [256, 278], [256, 299], [257, 265], [257, 274], [257, 278], [257, 279], [258, 264], [258, 271], [258, 294], [259, 283], [260, 279], [260, 297], [262, 267], [262, 274], [262, 294], [263, 271], [264, 273], [264, 285], [266, 267], [266, 298], [267, 296], [268, 294], [269, 294], [270, 279], [271, 276], [271, 277], [271, 298], [272, 277], [272, 284], [273, 283], [273, 287], [273, 288], [273, 295], [274, 279], [274, 281], [274, 282], [275, 283], [275, 290], [275, 293], [275, 294], [276, 291], [277, 283], [278, 292], [280, 283], [282, 285], [284, 297], [284, 298], [285, 286], [285, 289], [286, 291], [288, 295], [294, 298], [294, 299]], "family": "nppc_dominating_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-dominating-set-l6-s1/solution/solve.sh b/tasks/construct-nppc-dominating-set-l6-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..f46547847a0d579130a3cdb70cc4739dd7f54d7a
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l6-s1/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"dominating_set": [10, 16, 25, 30, 36, 56, 68, 81, 98, 102, 121, 132, 141, 156, 175, 193, 199, 220, 228, 269]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-dominating-set-l6-s1/tests/check.py b/tasks/construct-nppc-dominating-set-l6-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..0677a7bcbccb02297d022ffa4b5dd15da6371bff
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l6-s1/tests/check.py
@@ -0,0 +1,23 @@
+"""Dominating set of size <= k. Accepts any valid dominating set."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["num_nodes"], inst["k"]
+    # upstream NPPC verifier has no type/range checks (out-of-range labels and floats are silently used)
+    d = int_list(ans.get("dominating_set"), None, 0, n - 1, "dominating_set")
+    if len(set(d)) != len(d):
+        raise Invalid("dominating_set contains duplicates")
+    if not d or len(d) > k:
+        raise Invalid(f"dominating_set must have 1..{k} vertices, got {len(d)}")
+    dom = set(d)
+    s = set(d)
+    for u, v in inst["edges"]:
+        if u in s:
+            dom.add(v)
+        if v in s:
+            dom.add(u)
+    if len(dom) != n:
+        miss = min(set(range(n)) - dom)
+        raise Invalid(f"vertex {miss} is not dominated")
+    return None, {"size": len(d)}
diff --git a/tasks/construct-nppc-dominating-set-l6-s1/tests/instance.json b/tasks/construct-nppc-dominating-set-l6-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..838b0cb40ae67a0058094f355409752995d80d5a
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l6-s1/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 300, "k": 20, "edges": [[0, 12], [0, 29], [0, 36], [0, 41], [0, 52], [0, 93], [0, 111], [0, 149], [0, 153], [0, 158], [0, 195], [0, 266], [0, 277], [0, 281], [1, 7], [1, 19], [1, 30], [1, 56], [1, 60], [1, 69], [1, 97], [1, 148], [1, 153], [1, 172], [1, 181], [1, 231], [1, 240], [1, 259], [1, 269], [1, 271], [1, 275], [1, 278], [2, 17], [2, 31], [2, 33], [2, 76], [2, 83], [2, 90], [2, 110], [2, 132], [2, 135], [2, 138], [2, 159], [2, 160], [2, 177], [2, 189], [2, 199], [2, 210], [2, 212], [2, 225], [2, 226], [2, 230], [2, 231], [2, 244], [2, 272], [2, 294], [3, 11], [3, 112], [3, 124], [3, 130], [3, 144], [3, 148], [3, 149], [3, 156], [3, 160], [3, 170], [3, 191], [3, 220], [3, 222], [3, 226], [3, 259], [3, 286], [3, 291], [3, 293], [4, 9], [4, 14], [4, 56], [4, 76], [4, 81], [4, 102], [4, 109], [4, 110], [4, 199], [4, 289], [4, 294], [5, 9], [5, 14], [5, 22], [5, 32], [5, 36], [5, 77], [5, 84], [5, 106], [5, 135], [5, 175], [5, 186], [5, 190], [5, 199], [5, 236], [5, 246], [5, 273], [5, 296], [5, 299], [6, 9], [6, 26], [6, 41], [6, 47], [6, 66], [6, 68], [6, 90], [6, 100], [6, 126], [6, 146], [6, 169], [6, 182], [6, 185], [6, 189], [6, 210], [6, 214], [6, 216], [6, 224], [6, 232], [6, 235], [6, 248], [6, 251], [6, 266], [6, 284], [7, 8], [7, 35], [7, 39], [7, 61], [7, 65], [7, 73], [7, 123], [7, 137], [7, 220], [7, 228], [7, 229], [7, 231], [7, 267], [7, 292], [8, 13], [8, 15], [8, 38], [8, 47], [8, 48], [8, 56], [8, 68], [8, 79], [8, 106], [8, 140], [8, 171], [8, 183], [8, 220], [8, 223], [8, 249], [8, 251], [8, 255], [8, 282], [8, 299], [9, 16], [9, 31], [9, 40], [9, 72], [9, 73], [9, 81], [9, 97], [9, 106], [9, 113], [9, 141], [9, 147], [9, 163], [9, 164], [9, 167], [9, 183], [9, 184], [9, 212], [9, 220], [9, 239], [9, 244], [9, 251], [9, 252], [9, 265], [9, 273], [10, 38], [10, 58], [10, 74], [10, 93], [10, 96], [10, 101], [10, 104], [10, 109], [10, 132], [10, 133], [10, 136], [10, 176], [10, 177], [10, 181], [10, 182], [10, 185], [10, 196], 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268], [244, 279], [244, 281], [244, 287], [245, 274], [245, 277], [245, 288], [245, 298], [246, 266], [246, 270], [246, 290], [247, 253], [247, 277], [247, 283], [247, 285], [248, 271], [248, 293], [249, 264], [249, 288], [250, 254], [250, 256], [250, 263], [250, 281], [250, 288], [250, 291], [251, 262], [251, 267], [251, 268], [251, 276], [252, 271], [252, 272], [252, 291], [253, 272], [253, 278], [254, 259], [254, 274], [254, 279], [254, 290], [255, 266], [256, 258], [256, 278], [256, 299], [257, 265], [257, 274], [257, 278], [257, 279], [258, 264], [258, 271], [258, 294], [259, 283], [260, 279], [260, 297], [262, 267], [262, 274], [262, 294], [263, 271], [264, 273], [264, 285], [266, 267], [266, 298], [267, 296], [268, 294], [269, 294], [270, 279], [271, 276], [271, 277], [271, 298], [272, 277], [272, 284], [273, 283], [273, 287], [273, 288], [273, 295], [274, 279], [274, 281], [274, 282], [275, 283], [275, 290], [275, 293], [275, 294], [276, 291], [277, 283], [278, 292], [280, 283], [282, 285], [284, 297], [284, 298], [285, 286], [285, 289], [286, 291], [288, 295], [294, 298], [294, 299]], "family": "nppc_dominating_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-dominating-set-l6-s1/tests/run.py b/tasks/construct-nppc-dominating-set-l6-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l6-s1/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-nppc-dominating-set-l6-s1/tests/test.sh b/tasks/construct-nppc-dominating-set-l6-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l6-s1/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-nppc-dominating-set-l6-s6/environment/Dockerfile b/tasks/construct-nppc-dominating-set-l6-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l6-s6/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-nppc-dominating-set-l6-s6/environment/instance.json b/tasks/construct-nppc-dominating-set-l6-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..5645bca713ea75b9d23571da650ead44b168eee2
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l6-s6/environment/instance.json
@@ -0,0 +1 @@
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[281, 286], [281, 291], [281, 298], [281, 299], [282, 288], [283, 288], [289, 299], [291, 299]], "family": "nppc_dominating_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-dominating-set-l6-s6/solution/solve.sh b/tasks/construct-nppc-dominating-set-l6-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..48e9ddc07dc101380a150cd6fcfb21eee85df774
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l6-s6/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"dominating_set": [34, 76, 82, 85, 86, 103, 114, 116, 125, 129, 202, 216, 220, 230, 249, 255, 260, 281, 288, 289]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-dominating-set-l6-s6/tests/check.py b/tasks/construct-nppc-dominating-set-l6-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..0677a7bcbccb02297d022ffa4b5dd15da6371bff
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l6-s6/tests/check.py
@@ -0,0 +1,23 @@
+"""Dominating set of size <= k. Accepts any valid dominating set."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["num_nodes"], inst["k"]
+    # upstream NPPC verifier has no type/range checks (out-of-range labels and floats are silently used)
+    d = int_list(ans.get("dominating_set"), None, 0, n - 1, "dominating_set")
+    if len(set(d)) != len(d):
+        raise Invalid("dominating_set contains duplicates")
+    if not d or len(d) > k:
+        raise Invalid(f"dominating_set must have 1..{k} vertices, got {len(d)}")
+    dom = set(d)
+    s = set(d)
+    for u, v in inst["edges"]:
+        if u in s:
+            dom.add(v)
+        if v in s:
+            dom.add(u)
+    if len(dom) != n:
+        miss = min(set(range(n)) - dom)
+        raise Invalid(f"vertex {miss} is not dominated")
+    return None, {"size": len(d)}
diff --git a/tasks/construct-nppc-dominating-set-l6-s6/tests/instance.json b/tasks/construct-nppc-dominating-set-l6-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..5645bca713ea75b9d23571da650ead44b168eee2
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l6-s6/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 300, "k": 20, "edges": [[0, 10], [0, 33], [0, 56], [0, 78], [0, 79], [0, 86], [0, 92], [0, 94], [0, 96], [0, 109], [0, 132], [0, 168], [0, 169], [0, 175], [0, 189], [0, 218], [0, 231], [0, 240], [0, 286], [0, 292], [1, 46], [1, 70], [1, 77], [1, 95], [1, 113], [1, 133], [1, 156], [1, 162], [1, 230], [1, 240], [1, 246], [1, 268], [1, 297], [2, 33], [2, 47], [2, 53], [2, 55], [2, 64], [2, 82], [2, 86], [2, 94], [2, 144], [2, 149], [2, 155], [2, 157], [2, 188], [2, 190], [2, 191], [2, 203], [2, 225], [2, 248], [2, 259], [2, 290], [3, 19], [3, 66], [3, 74], [3, 81], [3, 85], [3, 88], [3, 146], [3, 156], [3, 165], [3, 173], [3, 187], [3, 195], [3, 202], [3, 221], [3, 224], [3, 227], [3, 231], [3, 241], [3, 246], [3, 291], [3, 293], [4, 14], [4, 29], [4, 48], [4, 55], [4, 64], [4, 70], [4, 80], [4, 86], [4, 95], [4, 126], [4, 143], [4, 161], [4, 162], [4, 218], [4, 225], [4, 234], [4, 245], [4, 253], [4, 259], [4, 266], [4, 281], [5, 37], [5, 69], [5, 88], [5, 103], [5, 119], 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[244, 273], [244, 274], [244, 283], [244, 292], [244, 295], [245, 250], [245, 264], [245, 279], [245, 288], [245, 289], [246, 257], [246, 269], [246, 272], [247, 253], [247, 275], [247, 287], [247, 290], [247, 296], [248, 252], [248, 288], [249, 287], [249, 290], [249, 296], [250, 255], [251, 258], [251, 285], [251, 286], [251, 291], [252, 255], [254, 261], [254, 288], [254, 297], [255, 265], [255, 266], [255, 273], [255, 278], [256, 257], [256, 260], [257, 275], [257, 286], [257, 293], [258, 289], [259, 289], [260, 261], [260, 276], [260, 286], [261, 295], [262, 266], [262, 280], [262, 281], [262, 297], [263, 275], [264, 268], [264, 288], [265, 268], [265, 287], [265, 289], [265, 298], [266, 267], [266, 269], [266, 294], [267, 275], [267, 281], [268, 276], [269, 294], [269, 299], [270, 271], [270, 283], [270, 298], [271, 275], [271, 280], [271, 289], [273, 286], [273, 290], [273, 298], [274, 279], [274, 287], [275, 280], [275, 298], [277, 280], [277, 292], [279, 294], [280, 293], [281, 286], [281, 291], [281, 298], [281, 299], [282, 288], [283, 288], [289, 299], [291, 299]], "family": "nppc_dominating_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-dominating-set-l6-s6/tests/run.py b/tasks/construct-nppc-dominating-set-l6-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l6-s6/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-nppc-dominating-set-l6-s6/tests/test.sh b/tasks/construct-nppc-dominating-set-l6-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l6-s6/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-nppc-quad-diophantine-l2-s8/environment/Dockerfile b/tasks/construct-nppc-quad-diophantine-l2-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-l2-s8/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-nppc-quad-diophantine-l2-s8/environment/instance.json b/tasks/construct-nppc-quad-diophantine-l2-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d60ae11d118784fc4d3bbe2995710d61a0f44e97
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-l2-s8/environment/instance.json
@@ -0,0 +1 @@
+{"a": 74, "b": 830, "c": 217514, "family": "nppc_quad_diophantine", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-quad-diophantine-l2-s8/solution/solve.sh b/tasks/construct-nppc-quad-diophantine-l2-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..6f312cc4fbe188cc4e489a406fa2e6d1844c2666
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-l2-s8/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": 34, "y": 159}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-quad-diophantine-l2-s8/tests/check.py b/tasks/construct-nppc-quad-diophantine-l2-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..88989d2a8839b089c9b07dd2d3c26c21bfdca58c
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-l2-s8/tests/check.py
@@ -0,0 +1,11 @@
+"""a*x^2 + b*y = c in positive integers. Accepts any solution (exact integer arithmetic)."""
+from run import Invalid, as_int
+
+
+def check(inst, ans):
+    # upstream NPPC verifier only checks x > 0 and y > 0, so non-integer (float) x, y were accepted
+    x = as_int(ans.get("x"), "x", lo=1)
+    y = as_int(ans.get("y"), "y", lo=1)
+    if inst["a"] * x * x + inst["b"] * y != inst["c"]:
+        raise Invalid("a*x^2 + b*y != c")
+    return None, {}
diff --git a/tasks/construct-nppc-quad-diophantine-l2-s8/tests/instance.json b/tasks/construct-nppc-quad-diophantine-l2-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d60ae11d118784fc4d3bbe2995710d61a0f44e97
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-l2-s8/tests/instance.json
@@ -0,0 +1 @@
+{"a": 74, "b": 830, "c": 217514, "family": "nppc_quad_diophantine", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-quad-diophantine-l2-s8/tests/run.py b/tasks/construct-nppc-quad-diophantine-l2-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-l2-s8/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-nppc-quad-diophantine-l2-s8/tests/test.sh b/tasks/construct-nppc-quad-diophantine-l2-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-l2-s8/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-nppc-quad-diophantine-l6-s0/environment/Dockerfile b/tasks/construct-nppc-quad-diophantine-l6-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-l6-s0/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-nppc-quad-diophantine-l6-s0/environment/instance.json b/tasks/construct-nppc-quad-diophantine-l6-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..299a4da68cb2a2836a8fa51c2ae59683eecffd13
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-l6-s0/environment/instance.json
@@ -0,0 +1 @@
+{"a": 5877870180, "b": 3987024243, "c": 166435860223745549608143329040, "family": "nppc_quad_diophantine", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-quad-diophantine-l6-s0/solution/solve.sh b/tasks/construct-nppc-quad-diophantine-l6-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..733c08e47472a67ce15e562bb6f948b4e133b432
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-l6-s0/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": 5321247480, "y": 5743183280}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-quad-diophantine-l6-s0/tests/check.py b/tasks/construct-nppc-quad-diophantine-l6-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..88989d2a8839b089c9b07dd2d3c26c21bfdca58c
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-l6-s0/tests/check.py
@@ -0,0 +1,11 @@
+"""a*x^2 + b*y = c in positive integers. Accepts any solution (exact integer arithmetic)."""
+from run import Invalid, as_int
+
+
+def check(inst, ans):
+    # upstream NPPC verifier only checks x > 0 and y > 0, so non-integer (float) x, y were accepted
+    x = as_int(ans.get("x"), "x", lo=1)
+    y = as_int(ans.get("y"), "y", lo=1)
+    if inst["a"] * x * x + inst["b"] * y != inst["c"]:
+        raise Invalid("a*x^2 + b*y != c")
+    return None, {}
diff --git a/tasks/construct-nppc-quad-diophantine-l6-s0/tests/instance.json b/tasks/construct-nppc-quad-diophantine-l6-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..299a4da68cb2a2836a8fa51c2ae59683eecffd13
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-l6-s0/tests/instance.json
@@ -0,0 +1 @@
+{"a": 5877870180, "b": 3987024243, "c": 166435860223745549608143329040, "family": "nppc_quad_diophantine", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-quad-diophantine-l6-s0/tests/run.py b/tasks/construct-nppc-quad-diophantine-l6-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-l6-s0/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-nppc-quad-diophantine-l6-s0/tests/test.sh b/tasks/construct-nppc-quad-diophantine-l6-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-l6-s0/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-nppc-quad-diophantine-l6-s7/environment/Dockerfile b/tasks/construct-nppc-quad-diophantine-l6-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-l6-s7/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-nppc-quad-diophantine-l6-s7/environment/instance.json b/tasks/construct-nppc-quad-diophantine-l6-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4786dd578f86efcec235bd9c039abf3517db6704
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-l6-s7/environment/instance.json
@@ -0,0 +1 @@
+{"a": 8499667983, "b": 7980065863, "c": 148468647729573796485120421324, "family": "nppc_quad_diophantine", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-quad-diophantine-l6-s7/solution/solve.sh b/tasks/construct-nppc-quad-diophantine-l6-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..eceffa5f82fd26212e4c071f816b7b9186d58ffe
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-l6-s7/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": 4179423648, "y": 1670742484}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-quad-diophantine-l6-s7/tests/check.py b/tasks/construct-nppc-quad-diophantine-l6-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..88989d2a8839b089c9b07dd2d3c26c21bfdca58c
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-l6-s7/tests/check.py
@@ -0,0 +1,11 @@
+"""a*x^2 + b*y = c in positive integers. Accepts any solution (exact integer arithmetic)."""
+from run import Invalid, as_int
+
+
+def check(inst, ans):
+    # upstream NPPC verifier only checks x > 0 and y > 0, so non-integer (float) x, y were accepted
+    x = as_int(ans.get("x"), "x", lo=1)
+    y = as_int(ans.get("y"), "y", lo=1)
+    if inst["a"] * x * x + inst["b"] * y != inst["c"]:
+        raise Invalid("a*x^2 + b*y != c")
+    return None, {}
diff --git a/tasks/construct-nppc-quad-diophantine-l6-s7/tests/instance.json b/tasks/construct-nppc-quad-diophantine-l6-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4786dd578f86efcec235bd9c039abf3517db6704
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-l6-s7/tests/instance.json
@@ -0,0 +1 @@
+{"a": 8499667983, "b": 7980065863, "c": 148468647729573796485120421324, "family": "nppc_quad_diophantine", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-quad-diophantine-l6-s7/tests/run.py b/tasks/construct-nppc-quad-diophantine-l6-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-l6-s7/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-nppc-quad-diophantine-l6-s7/tests/test.sh b/tasks/construct-nppc-quad-diophantine-l6-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-l6-s7/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-nppc-quadratic-congruence-l2-s8/environment/Dockerfile b/tasks/construct-nppc-quadratic-congruence-l2-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-l2-s8/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-nppc-quadratic-congruence-l2-s8/environment/instance.json b/tasks/construct-nppc-quadratic-congruence-l2-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..7ffd3e1a94bbb5c6540c5a5c5c47cbef251050f6
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-l2-s8/environment/instance.json
@@ -0,0 +1 @@
+{"a": 2586, "b": 5709, "c": 11811, "family": "nppc_quadratic_congruence", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-quadratic-congruence-l2-s8/solution/solve.sh b/tasks/construct-nppc-quadratic-congruence-l2-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..4a5c753a07b57ffddfc9a0f2ecb1707876970efe
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-l2-s8/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": 9063}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-quadratic-congruence-l2-s8/tests/check.py b/tasks/construct-nppc-quadratic-congruence-l2-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..1a751c071a276ca0aae7fb05bfdcd7b6e0c6d3a6
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-l2-s8/tests/check.py
@@ -0,0 +1,11 @@
+"""Quadratic congruence: integer x with 1 <= x < c and x^2 = a (mod b). Accepts any such x."""
+from run import Invalid, as_int
+
+
+def check(inst, ans):
+    # upstream NPPC verifier did int(x) (accepting "12", 12.9 -> 12) and only checked x < c, so negative x
+    # (e.g. -r for any root r) and x = 0 passed and the bound c was meaningless
+    x = as_int(ans.get("x"), "x", lo=1, hi=inst["c"] - 1)
+    if x * x % inst["b"] != inst["a"]:
+        raise Invalid("x^2 mod b != a")
+    return None, {}
diff --git a/tasks/construct-nppc-quadratic-congruence-l2-s8/tests/instance.json b/tasks/construct-nppc-quadratic-congruence-l2-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..7ffd3e1a94bbb5c6540c5a5c5c47cbef251050f6
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-l2-s8/tests/instance.json
@@ -0,0 +1 @@
+{"a": 2586, "b": 5709, "c": 11811, "family": "nppc_quadratic_congruence", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-quadratic-congruence-l2-s8/tests/run.py b/tasks/construct-nppc-quadratic-congruence-l2-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-l2-s8/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-nppc-quadratic-congruence-l2-s8/tests/test.sh b/tasks/construct-nppc-quadratic-congruence-l2-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-l2-s8/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-nppc-quadratic-congruence-l6-s0/environment/Dockerfile b/tasks/construct-nppc-quadratic-congruence-l6-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-l6-s0/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-nppc-quadratic-congruence-l6-s0/environment/instance.json b/tasks/construct-nppc-quadratic-congruence-l6-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2df5df8c1e0e144cff8c779aac73c429741d1b46
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-l6-s0/environment/instance.json
@@ -0,0 +1 @@
+{"a": 69949447710857463966058429949540960708984393755, "b": 116752723035413833583551890218610134433309379219, "c": 170690396901712565523618602570108501781620541, "family": "nppc_quadratic_congruence", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-quadratic-congruence-l6-s0/solution/solve.sh b/tasks/construct-nppc-quadratic-congruence-l6-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..990d3f52b6f333f2fd78dc47d928a399c2724e47
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-l6-s0/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": 167764679954075549326197778087457773127852696}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-quadratic-congruence-l6-s0/tests/check.py b/tasks/construct-nppc-quadratic-congruence-l6-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..1a751c071a276ca0aae7fb05bfdcd7b6e0c6d3a6
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-l6-s0/tests/check.py
@@ -0,0 +1,11 @@
+"""Quadratic congruence: integer x with 1 <= x < c and x^2 = a (mod b). Accepts any such x."""
+from run import Invalid, as_int
+
+
+def check(inst, ans):
+    # upstream NPPC verifier did int(x) (accepting "12", 12.9 -> 12) and only checked x < c, so negative x
+    # (e.g. -r for any root r) and x = 0 passed and the bound c was meaningless
+    x = as_int(ans.get("x"), "x", lo=1, hi=inst["c"] - 1)
+    if x * x % inst["b"] != inst["a"]:
+        raise Invalid("x^2 mod b != a")
+    return None, {}
diff --git a/tasks/construct-nppc-quadratic-congruence-l6-s0/tests/instance.json b/tasks/construct-nppc-quadratic-congruence-l6-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2df5df8c1e0e144cff8c779aac73c429741d1b46
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-l6-s0/tests/instance.json
@@ -0,0 +1 @@
+{"a": 69949447710857463966058429949540960708984393755, "b": 116752723035413833583551890218610134433309379219, "c": 170690396901712565523618602570108501781620541, "family": "nppc_quadratic_congruence", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-quadratic-congruence-l6-s0/tests/run.py b/tasks/construct-nppc-quadratic-congruence-l6-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-l6-s0/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-nppc-quadratic-congruence-l6-s0/tests/test.sh b/tasks/construct-nppc-quadratic-congruence-l6-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-l6-s0/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-nppc-quadratic-congruence-l6-s7/environment/Dockerfile b/tasks/construct-nppc-quadratic-congruence-l6-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-l6-s7/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-nppc-quadratic-congruence-l6-s7/environment/instance.json b/tasks/construct-nppc-quadratic-congruence-l6-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..57fb7196eecbcada1a045ad9cee399212feb49a0
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-l6-s7/environment/instance.json
@@ -0,0 +1 @@
+{"a": 153879964348175119621192249586863784264926265037, "b": 258583211762801849157885436569255510813072484469, "c": 33159055645726876205987857463064569187576621730, "family": "nppc_quadratic_congruence", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-quadratic-congruence-l6-s7/solution/solve.sh b/tasks/construct-nppc-quadratic-congruence-l6-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..095390abee6fe2d9fef7e07ef533c72bb73aa014
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-l6-s7/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": 33146452745072971592474474663902396844682688447}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-quadratic-congruence-l6-s7/tests/check.py b/tasks/construct-nppc-quadratic-congruence-l6-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..1a751c071a276ca0aae7fb05bfdcd7b6e0c6d3a6
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-l6-s7/tests/check.py
@@ -0,0 +1,11 @@
+"""Quadratic congruence: integer x with 1 <= x < c and x^2 = a (mod b). Accepts any such x."""
+from run import Invalid, as_int
+
+
+def check(inst, ans):
+    # upstream NPPC verifier did int(x) (accepting "12", 12.9 -> 12) and only checked x < c, so negative x
+    # (e.g. -r for any root r) and x = 0 passed and the bound c was meaningless
+    x = as_int(ans.get("x"), "x", lo=1, hi=inst["c"] - 1)
+    if x * x % inst["b"] != inst["a"]:
+        raise Invalid("x^2 mod b != a")
+    return None, {}
diff --git a/tasks/construct-nppc-quadratic-congruence-l6-s7/tests/instance.json b/tasks/construct-nppc-quadratic-congruence-l6-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..57fb7196eecbcada1a045ad9cee399212feb49a0
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-l6-s7/tests/instance.json
@@ -0,0 +1 @@
+{"a": 153879964348175119621192249586863784264926265037, "b": 258583211762801849157885436569255510813072484469, "c": 33159055645726876205987857463064569187576621730, "family": "nppc_quadratic_congruence", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-quadratic-congruence-l6-s7/tests/run.py b/tasks/construct-nppc-quadratic-congruence-l6-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-l6-s7/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-nppc-quadratic-congruence-l6-s7/tests/test.sh b/tasks/construct-nppc-quadratic-congruence-l6-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-l6-s7/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rg-graph-color-l3-s6/environment/Dockerfile b/tasks/construct-rg-graph-color-l3-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l3-s6/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rg-graph-color-l3-s6/environment/instance.json b/tasks/construct-rg-graph-color-l3-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4bfff175071a22b1f46968402a8a7dace7ea73bd
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l3-s6/environment/instance.json
@@ -0,0 +1 @@
+{"num_vertices": 10, "num_colors": 3, "edges": [[0, 5], [1, 2], [3, 5], [3, 7], [4, 5], [4, 7], [4, 8], [4, 9], [5, 6], [6, 8]], "family": "rg_graph_color", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-graph-color-l3-s6/solution/solve.sh b/tasks/construct-rg-graph-color-l3-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..1e0375008d262cab2f7448ec7f245e8d53239187
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l3-s6/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"coloring": [1, 1, 2, 1, 1, 2, 1, 2, 2, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-rg-graph-color-l3-s6/tests/check.py b/tasks/construct-rg-graph-color-l3-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..e9328912405d1c894970c90415be62bf9556ed36
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l3-s6/tests/check.py
@@ -0,0 +1,13 @@
+"""Proper colouring with colours 1..k (Reasoning Gym graph_color, Apache-2.0). Any valid colouring.
+Upstream note: its verifier checks `color in allowed_colors`, so True (== 1) and 1.0 were accepted as
+colours, and extra keys were ignored. Here colours must be ints in 1..k, one per vertex."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["num_vertices"], inst["num_colors"]
+    c = int_list(ans.get("coloring"), n, 1, k, "coloring")
+    for u, v in inst["edges"]:
+        if c[u] == c[v]:
+            raise Invalid(f"adjacent vertices {u} and {v} share colour {c[u]}")
+    return None, {}
diff --git a/tasks/construct-rg-graph-color-l3-s6/tests/instance.json b/tasks/construct-rg-graph-color-l3-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4bfff175071a22b1f46968402a8a7dace7ea73bd
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l3-s6/tests/instance.json
@@ -0,0 +1 @@
+{"num_vertices": 10, "num_colors": 3, "edges": [[0, 5], [1, 2], [3, 5], [3, 7], [4, 5], [4, 7], [4, 8], [4, 9], [5, 6], [6, 8]], "family": "rg_graph_color", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-graph-color-l3-s6/tests/run.py b/tasks/construct-rg-graph-color-l3-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l3-s6/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rg-graph-color-l3-s6/tests/test.sh b/tasks/construct-rg-graph-color-l3-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l3-s6/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rg-graph-color-l3-s8/environment/Dockerfile b/tasks/construct-rg-graph-color-l3-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l3-s8/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rg-graph-color-l3-s8/environment/instance.json b/tasks/construct-rg-graph-color-l3-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..dddc85cea787e8eeb5fe33e6c951d887f8cc1a33
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l3-s8/environment/instance.json
@@ -0,0 +1 @@
+{"num_vertices": 10, "num_colors": 3, "edges": [[1, 9], [2, 6], [2, 7], [3, 5], [3, 6], [4, 7], [8, 9]], "family": "rg_graph_color", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-graph-color-l3-s8/solution/solve.sh b/tasks/construct-rg-graph-color-l3-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..0543710a078c670d4828dc4e236fdbb5c49c996a
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l3-s8/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"coloring": [1, 1, 1, 1, 1, 2, 2, 2, 1, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-rg-graph-color-l3-s8/tests/check.py b/tasks/construct-rg-graph-color-l3-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..e9328912405d1c894970c90415be62bf9556ed36
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l3-s8/tests/check.py
@@ -0,0 +1,13 @@
+"""Proper colouring with colours 1..k (Reasoning Gym graph_color, Apache-2.0). Any valid colouring.
+Upstream note: its verifier checks `color in allowed_colors`, so True (== 1) and 1.0 were accepted as
+colours, and extra keys were ignored. Here colours must be ints in 1..k, one per vertex."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["num_vertices"], inst["num_colors"]
+    c = int_list(ans.get("coloring"), n, 1, k, "coloring")
+    for u, v in inst["edges"]:
+        if c[u] == c[v]:
+            raise Invalid(f"adjacent vertices {u} and {v} share colour {c[u]}")
+    return None, {}
diff --git a/tasks/construct-rg-graph-color-l3-s8/tests/instance.json b/tasks/construct-rg-graph-color-l3-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..dddc85cea787e8eeb5fe33e6c951d887f8cc1a33
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l3-s8/tests/instance.json
@@ -0,0 +1 @@
+{"num_vertices": 10, "num_colors": 3, "edges": [[1, 9], [2, 6], [2, 7], [3, 5], [3, 6], [4, 7], [8, 9]], "family": "rg_graph_color", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-graph-color-l3-s8/tests/run.py b/tasks/construct-rg-graph-color-l3-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l3-s8/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rg-graph-color-l3-s8/tests/test.sh b/tasks/construct-rg-graph-color-l3-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l3-s8/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rg-intermediate-integration-l1-s2/environment/Dockerfile b/tasks/construct-rg-intermediate-integration-l1-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-l1-s2/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rg-intermediate-integration-l1-s2/environment/instance.json b/tasks/construct-rg-intermediate-integration-l1-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..935117b7d36d2d751221a9512fe9e3eb47ff6630
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-l1-s2/environment/instance.json
@@ -0,0 +1 @@
+{"variable": "x", "integrand": "(9*x + 8)**3", "problem_type": "linear", "family": "rg_intermediate_integration", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-intermediate-integration-l1-s2/solution/solve.sh b/tasks/construct-rg-intermediate-integration-l1-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..2ab2a9617d134d437f12cc30369705f5591cb3b2
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-l1-s2/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"antiderivative": "729*x**4/4 + 648*x**3 + 864*x**2 + 512*x + C"}
+RLMATH_EOF
diff --git a/tasks/construct-rg-intermediate-integration-l1-s2/tests/check.py b/tasks/construct-rg-intermediate-integration-l1-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..a9e364a8ede6c8215a16a04ee6b0d3529adc647d
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-l1-s2/tests/check.py
@@ -0,0 +1,158 @@
+"""Antiderivative check (Reasoning Gym simple_integration / intermediate_integration, Apache-2.0).
+
+The answer string is parsed with a whitelist AST walker (no eval), differentiated with SymPy and compared
+with the integrand numerically (30-digit evaluation) at fixed points inside the integrand's real domain.
+Any additive constant (e.g. "+ C") is allowed.
+
+Upstream notes: upstream scores with sympy.parse_expr, which calls eval() on the model's string (arbitrary
+code execution), and it accepts an unevaluated "Integral(f, x)" since its derivative is f. Both are
+rejected here: only arithmetic, whitelisted elementary functions, the variable, C, E and pi are allowed."""
+import ast
+import math
+import signal
+
+import sympy
+
+from run import Invalid
+
+MAX_LEN = 3000
+MAX_NODES = 600
+MAX_POW_NEST = 4  # nested powers with a variable exponent (x**x**x**...) blow up differentiation
+TIME_LIMIT = 40  # seconds; a slower answer is rejected
+FUNCS = {"sin": sympy.sin, "cos": sympy.cos, "tan": sympy.tan, "cot": sympy.cot, "sec": sympy.sec,
+         "csc": sympy.csc, "asin": sympy.asin, "acos": sympy.acos, "atan": sympy.atan, "acot": sympy.acot,
+         "arcsin": sympy.asin, "arccos": sympy.acos, "arctan": sympy.atan, "sinh": sympy.sinh,
+         "cosh": sympy.cosh, "tanh": sympy.tanh, "asinh": sympy.asinh, "acosh": sympy.acosh,
+         "atanh": sympy.atanh, "exp": sympy.exp, "log": sympy.log, "ln": sympy.log, "sqrt": sympy.sqrt,
+         "Abs": sympy.Abs, "abs": sympy.Abs}
+POINTS = [0.3137, 0.5521, 0.8093, 1.1171, 1.4519, 1.9033, 2.3471, 2.8123, 3.3319, 3.9107, 4.6231,
+          -0.3547, -0.6529, -0.9511, -1.3027, -1.7519, -2.2013, -2.7131, -3.1397, -3.8203, -4.4111,
+          0.1231, -0.1523, 0.2129, -0.2617, 0.0517, -0.0719, 5.4127, -5.3311, 6.3019, -6.2203, 7.2213,
+          -7.1117, 8.9021, -9.3107, 10.7129, -10.6337, 12.9011, -12.8231, 15.3113, -15.2719]
+MIN_POINTS = 5
+
+
+def parse(s, var):
+    if not isinstance(s, str) or not s.strip():
+        raise Invalid("antiderivative must be a non-empty string")
+    if len(s) > MAX_LEN:
+        raise Invalid("expression too long")
+    s = s.strip()
+    try:
+        tree = ast.parse(s, mode="eval")
+    except SyntaxError:
+        raise Invalid("expression does not parse (use ** for powers and * for products)")
+    if sum(1 for _ in ast.walk(tree)) > MAX_NODES:
+        raise Invalid("expression too large")
+
+    def pow_nest(nd):
+        kids = [pow_nest(c) for c in ast.iter_child_nodes(nd)]
+        own = isinstance(nd, ast.BinOp) and isinstance(nd.op, ast.Pow) and any(
+            isinstance(c, ast.Name) and c.id == var for c in ast.walk(nd.right))
+        return max(kids, default=0) + (1 if own else 0)
+    if pow_nest(tree) > MAX_POW_NEST:
+        raise Invalid("too many nested powers with a variable exponent")
+    x = sympy.Symbol(var, real=True)
+    names = {var: x, "C": sympy.Symbol("C"), "E": sympy.E, "pi": sympy.pi}
+
+    def conv(nd):
+        if isinstance(nd, ast.Expression):
+            return conv(nd.body)
+        if isinstance(nd, ast.Constant):
+            v = nd.value
+            if isinstance(v, bool) or not isinstance(v, (int, float)):
+                raise Invalid(f"constant {v!r} not allowed")
+            if isinstance(v, int):
+                if abs(v) > 10 ** 18:
+                    raise Invalid("integer constant too large")
+                return sympy.Integer(v)
+            if not math.isfinite(v):
+                raise Invalid("non-finite constant")
+            return sympy.Float(v, 30)
+        if isinstance(nd, ast.Name):
+            if nd.id not in names:
+                raise Invalid(f"unknown name {nd.id!r} (the variable is {var})")
+            return names[nd.id]
+        if isinstance(nd, ast.UnaryOp) and isinstance(nd.op, (ast.USub, ast.UAdd)):
+            a = conv(nd.operand)
+            return -a if isinstance(nd.op, ast.USub) else a
+        if isinstance(nd, ast.BinOp):
+            a, b = conv(nd.left), conv(nd.right)
+            if isinstance(nd.op, ast.Add):
+                return a + b
+            if isinstance(nd.op, ast.Sub):
+                return a - b
+            if isinstance(nd.op, ast.Mult):
+                return a * b
+            if isinstance(nd.op, ast.Div):
+                if b == 0:
+                    raise Invalid("division by zero")
+                return a / b
+            if isinstance(nd.op, ast.Pow):
+                if not b.free_symbols:
+                    bv = complex(b.evalf(20))
+                    if bv.imag != 0 or abs(bv.real) > 100:
+                        raise Invalid("exponent too large")
+                    if not a.free_symbols and abs(complex(a.evalf(20))) > 1e6:
+                        raise Invalid("numeric power too large")
+                return a ** b
+            raise Invalid("only + - * / ** are allowed (use ** for powers)")
+        if isinstance(nd, ast.Call):
+            if not isinstance(nd.func, ast.Name) or nd.func.id not in FUNCS:
+                raise Invalid(f"function {ast.unparse(nd.func)!r} not allowed")
+            if nd.keywords or len(nd.args) != 1:
+                raise Invalid(f"{nd.func.id} takes exactly one argument")
+            return FUNCS[nd.func.id](conv(nd.args[0]))
+        raise Invalid(f"syntax element {type(nd).__name__} not allowed")
+
+    return conv(tree), x
+
+
+def value(expr, x, p):
+    v = expr.evalf(30, subs={x: sympy.Float(p, 30)})
+    try:
+        z = complex(v)
+    except (TypeError, ValueError):
+        return None
+    if not (math.isfinite(z.real) and math.isfinite(z.imag)):
+        return None
+    return z
+
+
+def _timeout(signum, frame):
+    raise Invalid(f"answer took more than {TIME_LIMIT} s to check")
+
+
+def check(inst, ans):
+    if hasattr(signal, "SIGALRM"):
+        old = signal.signal(signal.SIGALRM, _timeout)
+        signal.alarm(TIME_LIMIT)
+        try:
+            return _check(inst, ans)
+        finally:
+            signal.alarm(0)
+            signal.signal(signal.SIGALRM, old)
+    return _check(inst, ans)
+
+
+def _check(inst, ans):
+    var = inst["variable"]
+    F, x = parse(ans.get("antiderivative"), var)
+    f, x2 = parse(inst["integrand"], var)
+    if not F.has(x):
+        raise Invalid("antiderivative does not depend on the variable")
+    dF = sympy.diff(F, x)
+    used = 0
+    for p in POINTS:
+        fv = value(f, x2, p)
+        if fv is None or abs(fv.imag) > 1e-20 * (1 + abs(fv.real)):
+            continue  # outside the integrand's real domain
+        gv = value(dF, x, p)
+        if gv is None:
+            raise Invalid(f"derivative of the answer cannot be evaluated at {var} = {p}")
+        if abs(gv - fv) > 1e-9 * max(1.0, abs(fv)):
+            raise Invalid(f"derivative of the answer differs from the integrand at {var} = {p}")
+        used += 1
+    if used < MIN_POINTS:
+        raise Invalid("too few evaluation points in the integrand's domain")
+    return None, {"points": used}
diff --git a/tasks/construct-rg-intermediate-integration-l1-s2/tests/instance.json b/tasks/construct-rg-intermediate-integration-l1-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..935117b7d36d2d751221a9512fe9e3eb47ff6630
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-l1-s2/tests/instance.json
@@ -0,0 +1 @@
+{"variable": "x", "integrand": "(9*x + 8)**3", "problem_type": "linear", "family": "rg_intermediate_integration", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-intermediate-integration-l1-s2/tests/run.py b/tasks/construct-rg-intermediate-integration-l1-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-l1-s2/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rg-intermediate-integration-l1-s2/tests/test.sh b/tasks/construct-rg-intermediate-integration-l1-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-l1-s2/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rg-intermediate-integration-l1-s3/environment/Dockerfile b/tasks/construct-rg-intermediate-integration-l1-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-l1-s3/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rg-intermediate-integration-l1-s3/environment/instance.json b/tasks/construct-rg-intermediate-integration-l1-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..136ec487519adac57dd07e3279f0e7f711b9d24c
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-l1-s3/environment/instance.json
@@ -0,0 +1 @@
+{"variable": "x", "integrand": "(4*x - 10)**4", "problem_type": "linear", "family": "rg_intermediate_integration", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-intermediate-integration-l1-s3/solution/solve.sh b/tasks/construct-rg-intermediate-integration-l1-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..5c8d34111442e38204f486bbbf8055848d1cfd04
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-l1-s3/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"antiderivative": "256*x**5/5 - 640*x**4 + 3200*x**3 - 8000*x**2 + 10000*x + C"}
+RLMATH_EOF
diff --git a/tasks/construct-rg-intermediate-integration-l1-s3/tests/check.py b/tasks/construct-rg-intermediate-integration-l1-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..a9e364a8ede6c8215a16a04ee6b0d3529adc647d
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-l1-s3/tests/check.py
@@ -0,0 +1,158 @@
+"""Antiderivative check (Reasoning Gym simple_integration / intermediate_integration, Apache-2.0).
+
+The answer string is parsed with a whitelist AST walker (no eval), differentiated with SymPy and compared
+with the integrand numerically (30-digit evaluation) at fixed points inside the integrand's real domain.
+Any additive constant (e.g. "+ C") is allowed.
+
+Upstream notes: upstream scores with sympy.parse_expr, which calls eval() on the model's string (arbitrary
+code execution), and it accepts an unevaluated "Integral(f, x)" since its derivative is f. Both are
+rejected here: only arithmetic, whitelisted elementary functions, the variable, C, E and pi are allowed."""
+import ast
+import math
+import signal
+
+import sympy
+
+from run import Invalid
+
+MAX_LEN = 3000
+MAX_NODES = 600
+MAX_POW_NEST = 4  # nested powers with a variable exponent (x**x**x**...) blow up differentiation
+TIME_LIMIT = 40  # seconds; a slower answer is rejected
+FUNCS = {"sin": sympy.sin, "cos": sympy.cos, "tan": sympy.tan, "cot": sympy.cot, "sec": sympy.sec,
+         "csc": sympy.csc, "asin": sympy.asin, "acos": sympy.acos, "atan": sympy.atan, "acot": sympy.acot,
+         "arcsin": sympy.asin, "arccos": sympy.acos, "arctan": sympy.atan, "sinh": sympy.sinh,
+         "cosh": sympy.cosh, "tanh": sympy.tanh, "asinh": sympy.asinh, "acosh": sympy.acosh,
+         "atanh": sympy.atanh, "exp": sympy.exp, "log": sympy.log, "ln": sympy.log, "sqrt": sympy.sqrt,
+         "Abs": sympy.Abs, "abs": sympy.Abs}
+POINTS = [0.3137, 0.5521, 0.8093, 1.1171, 1.4519, 1.9033, 2.3471, 2.8123, 3.3319, 3.9107, 4.6231,
+          -0.3547, -0.6529, -0.9511, -1.3027, -1.7519, -2.2013, -2.7131, -3.1397, -3.8203, -4.4111,
+          0.1231, -0.1523, 0.2129, -0.2617, 0.0517, -0.0719, 5.4127, -5.3311, 6.3019, -6.2203, 7.2213,
+          -7.1117, 8.9021, -9.3107, 10.7129, -10.6337, 12.9011, -12.8231, 15.3113, -15.2719]
+MIN_POINTS = 5
+
+
+def parse(s, var):
+    if not isinstance(s, str) or not s.strip():
+        raise Invalid("antiderivative must be a non-empty string")
+    if len(s) > MAX_LEN:
+        raise Invalid("expression too long")
+    s = s.strip()
+    try:
+        tree = ast.parse(s, mode="eval")
+    except SyntaxError:
+        raise Invalid("expression does not parse (use ** for powers and * for products)")
+    if sum(1 for _ in ast.walk(tree)) > MAX_NODES:
+        raise Invalid("expression too large")
+
+    def pow_nest(nd):
+        kids = [pow_nest(c) for c in ast.iter_child_nodes(nd)]
+        own = isinstance(nd, ast.BinOp) and isinstance(nd.op, ast.Pow) and any(
+            isinstance(c, ast.Name) and c.id == var for c in ast.walk(nd.right))
+        return max(kids, default=0) + (1 if own else 0)
+    if pow_nest(tree) > MAX_POW_NEST:
+        raise Invalid("too many nested powers with a variable exponent")
+    x = sympy.Symbol(var, real=True)
+    names = {var: x, "C": sympy.Symbol("C"), "E": sympy.E, "pi": sympy.pi}
+
+    def conv(nd):
+        if isinstance(nd, ast.Expression):
+            return conv(nd.body)
+        if isinstance(nd, ast.Constant):
+            v = nd.value
+            if isinstance(v, bool) or not isinstance(v, (int, float)):
+                raise Invalid(f"constant {v!r} not allowed")
+            if isinstance(v, int):
+                if abs(v) > 10 ** 18:
+                    raise Invalid("integer constant too large")
+                return sympy.Integer(v)
+            if not math.isfinite(v):
+                raise Invalid("non-finite constant")
+            return sympy.Float(v, 30)
+        if isinstance(nd, ast.Name):
+            if nd.id not in names:
+                raise Invalid(f"unknown name {nd.id!r} (the variable is {var})")
+            return names[nd.id]
+        if isinstance(nd, ast.UnaryOp) and isinstance(nd.op, (ast.USub, ast.UAdd)):
+            a = conv(nd.operand)
+            return -a if isinstance(nd.op, ast.USub) else a
+        if isinstance(nd, ast.BinOp):
+            a, b = conv(nd.left), conv(nd.right)
+            if isinstance(nd.op, ast.Add):
+                return a + b
+            if isinstance(nd.op, ast.Sub):
+                return a - b
+            if isinstance(nd.op, ast.Mult):
+                return a * b
+            if isinstance(nd.op, ast.Div):
+                if b == 0:
+                    raise Invalid("division by zero")
+                return a / b
+            if isinstance(nd.op, ast.Pow):
+                if not b.free_symbols:
+                    bv = complex(b.evalf(20))
+                    if bv.imag != 0 or abs(bv.real) > 100:
+                        raise Invalid("exponent too large")
+                    if not a.free_symbols and abs(complex(a.evalf(20))) > 1e6:
+                        raise Invalid("numeric power too large")
+                return a ** b
+            raise Invalid("only + - * / ** are allowed (use ** for powers)")
+        if isinstance(nd, ast.Call):
+            if not isinstance(nd.func, ast.Name) or nd.func.id not in FUNCS:
+                raise Invalid(f"function {ast.unparse(nd.func)!r} not allowed")
+            if nd.keywords or len(nd.args) != 1:
+                raise Invalid(f"{nd.func.id} takes exactly one argument")
+            return FUNCS[nd.func.id](conv(nd.args[0]))
+        raise Invalid(f"syntax element {type(nd).__name__} not allowed")
+
+    return conv(tree), x
+
+
+def value(expr, x, p):
+    v = expr.evalf(30, subs={x: sympy.Float(p, 30)})
+    try:
+        z = complex(v)
+    except (TypeError, ValueError):
+        return None
+    if not (math.isfinite(z.real) and math.isfinite(z.imag)):
+        return None
+    return z
+
+
+def _timeout(signum, frame):
+    raise Invalid(f"answer took more than {TIME_LIMIT} s to check")
+
+
+def check(inst, ans):
+    if hasattr(signal, "SIGALRM"):
+        old = signal.signal(signal.SIGALRM, _timeout)
+        signal.alarm(TIME_LIMIT)
+        try:
+            return _check(inst, ans)
+        finally:
+            signal.alarm(0)
+            signal.signal(signal.SIGALRM, old)
+    return _check(inst, ans)
+
+
+def _check(inst, ans):
+    var = inst["variable"]
+    F, x = parse(ans.get("antiderivative"), var)
+    f, x2 = parse(inst["integrand"], var)
+    if not F.has(x):
+        raise Invalid("antiderivative does not depend on the variable")
+    dF = sympy.diff(F, x)
+    used = 0
+    for p in POINTS:
+        fv = value(f, x2, p)
+        if fv is None or abs(fv.imag) > 1e-20 * (1 + abs(fv.real)):
+            continue  # outside the integrand's real domain
+        gv = value(dF, x, p)
+        if gv is None:
+            raise Invalid(f"derivative of the answer cannot be evaluated at {var} = {p}")
+        if abs(gv - fv) > 1e-9 * max(1.0, abs(fv)):
+            raise Invalid(f"derivative of the answer differs from the integrand at {var} = {p}")
+        used += 1
+    if used < MIN_POINTS:
+        raise Invalid("too few evaluation points in the integrand's domain")
+    return None, {"points": used}
diff --git a/tasks/construct-rg-intermediate-integration-l1-s3/tests/instance.json b/tasks/construct-rg-intermediate-integration-l1-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..136ec487519adac57dd07e3279f0e7f711b9d24c
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-l1-s3/tests/instance.json
@@ -0,0 +1 @@
+{"variable": "x", "integrand": "(4*x - 10)**4", "problem_type": "linear", "family": "rg_intermediate_integration", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-intermediate-integration-l1-s3/tests/run.py b/tasks/construct-rg-intermediate-integration-l1-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-l1-s3/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rg-intermediate-integration-l1-s3/tests/test.sh b/tasks/construct-rg-intermediate-integration-l1-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-l1-s3/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rg-simple-integration-l3-s0/environment/Dockerfile b/tasks/construct-rg-simple-integration-l3-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l3-s0/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rg-simple-integration-l3-s0/environment/instance.json b/tasks/construct-rg-simple-integration-l3-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1015265fb1941c53ca82d7620efee84e484d512c
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l3-s0/environment/instance.json
@@ -0,0 +1 @@
+{"variable": "x", "integrand": "-2*x**4 - 4*x**3 + 3*x**2", "family": "rg_simple_integration", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-simple-integration-l3-s0/solution/solve.sh b/tasks/construct-rg-simple-integration-l3-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..549441c16c21063ec7b4e418aae2f7ad793074f9
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l3-s0/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"antiderivative": "-2*x**5/5 - x**4 + x**3 + C"}
+RLMATH_EOF
diff --git a/tasks/construct-rg-simple-integration-l3-s0/tests/check.py b/tasks/construct-rg-simple-integration-l3-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..a9e364a8ede6c8215a16a04ee6b0d3529adc647d
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l3-s0/tests/check.py
@@ -0,0 +1,158 @@
+"""Antiderivative check (Reasoning Gym simple_integration / intermediate_integration, Apache-2.0).
+
+The answer string is parsed with a whitelist AST walker (no eval), differentiated with SymPy and compared
+with the integrand numerically (30-digit evaluation) at fixed points inside the integrand's real domain.
+Any additive constant (e.g. "+ C") is allowed.
+
+Upstream notes: upstream scores with sympy.parse_expr, which calls eval() on the model's string (arbitrary
+code execution), and it accepts an unevaluated "Integral(f, x)" since its derivative is f. Both are
+rejected here: only arithmetic, whitelisted elementary functions, the variable, C, E and pi are allowed."""
+import ast
+import math
+import signal
+
+import sympy
+
+from run import Invalid
+
+MAX_LEN = 3000
+MAX_NODES = 600
+MAX_POW_NEST = 4  # nested powers with a variable exponent (x**x**x**...) blow up differentiation
+TIME_LIMIT = 40  # seconds; a slower answer is rejected
+FUNCS = {"sin": sympy.sin, "cos": sympy.cos, "tan": sympy.tan, "cot": sympy.cot, "sec": sympy.sec,
+         "csc": sympy.csc, "asin": sympy.asin, "acos": sympy.acos, "atan": sympy.atan, "acot": sympy.acot,
+         "arcsin": sympy.asin, "arccos": sympy.acos, "arctan": sympy.atan, "sinh": sympy.sinh,
+         "cosh": sympy.cosh, "tanh": sympy.tanh, "asinh": sympy.asinh, "acosh": sympy.acosh,
+         "atanh": sympy.atanh, "exp": sympy.exp, "log": sympy.log, "ln": sympy.log, "sqrt": sympy.sqrt,
+         "Abs": sympy.Abs, "abs": sympy.Abs}
+POINTS = [0.3137, 0.5521, 0.8093, 1.1171, 1.4519, 1.9033, 2.3471, 2.8123, 3.3319, 3.9107, 4.6231,
+          -0.3547, -0.6529, -0.9511, -1.3027, -1.7519, -2.2013, -2.7131, -3.1397, -3.8203, -4.4111,
+          0.1231, -0.1523, 0.2129, -0.2617, 0.0517, -0.0719, 5.4127, -5.3311, 6.3019, -6.2203, 7.2213,
+          -7.1117, 8.9021, -9.3107, 10.7129, -10.6337, 12.9011, -12.8231, 15.3113, -15.2719]
+MIN_POINTS = 5
+
+
+def parse(s, var):
+    if not isinstance(s, str) or not s.strip():
+        raise Invalid("antiderivative must be a non-empty string")
+    if len(s) > MAX_LEN:
+        raise Invalid("expression too long")
+    s = s.strip()
+    try:
+        tree = ast.parse(s, mode="eval")
+    except SyntaxError:
+        raise Invalid("expression does not parse (use ** for powers and * for products)")
+    if sum(1 for _ in ast.walk(tree)) > MAX_NODES:
+        raise Invalid("expression too large")
+
+    def pow_nest(nd):
+        kids = [pow_nest(c) for c in ast.iter_child_nodes(nd)]
+        own = isinstance(nd, ast.BinOp) and isinstance(nd.op, ast.Pow) and any(
+            isinstance(c, ast.Name) and c.id == var for c in ast.walk(nd.right))
+        return max(kids, default=0) + (1 if own else 0)
+    if pow_nest(tree) > MAX_POW_NEST:
+        raise Invalid("too many nested powers with a variable exponent")
+    x = sympy.Symbol(var, real=True)
+    names = {var: x, "C": sympy.Symbol("C"), "E": sympy.E, "pi": sympy.pi}
+
+    def conv(nd):
+        if isinstance(nd, ast.Expression):
+            return conv(nd.body)
+        if isinstance(nd, ast.Constant):
+            v = nd.value
+            if isinstance(v, bool) or not isinstance(v, (int, float)):
+                raise Invalid(f"constant {v!r} not allowed")
+            if isinstance(v, int):
+                if abs(v) > 10 ** 18:
+                    raise Invalid("integer constant too large")
+                return sympy.Integer(v)
+            if not math.isfinite(v):
+                raise Invalid("non-finite constant")
+            return sympy.Float(v, 30)
+        if isinstance(nd, ast.Name):
+            if nd.id not in names:
+                raise Invalid(f"unknown name {nd.id!r} (the variable is {var})")
+            return names[nd.id]
+        if isinstance(nd, ast.UnaryOp) and isinstance(nd.op, (ast.USub, ast.UAdd)):
+            a = conv(nd.operand)
+            return -a if isinstance(nd.op, ast.USub) else a
+        if isinstance(nd, ast.BinOp):
+            a, b = conv(nd.left), conv(nd.right)
+            if isinstance(nd.op, ast.Add):
+                return a + b
+            if isinstance(nd.op, ast.Sub):
+                return a - b
+            if isinstance(nd.op, ast.Mult):
+                return a * b
+            if isinstance(nd.op, ast.Div):
+                if b == 0:
+                    raise Invalid("division by zero")
+                return a / b
+            if isinstance(nd.op, ast.Pow):
+                if not b.free_symbols:
+                    bv = complex(b.evalf(20))
+                    if bv.imag != 0 or abs(bv.real) > 100:
+                        raise Invalid("exponent too large")
+                    if not a.free_symbols and abs(complex(a.evalf(20))) > 1e6:
+                        raise Invalid("numeric power too large")
+                return a ** b
+            raise Invalid("only + - * / ** are allowed (use ** for powers)")
+        if isinstance(nd, ast.Call):
+            if not isinstance(nd.func, ast.Name) or nd.func.id not in FUNCS:
+                raise Invalid(f"function {ast.unparse(nd.func)!r} not allowed")
+            if nd.keywords or len(nd.args) != 1:
+                raise Invalid(f"{nd.func.id} takes exactly one argument")
+            return FUNCS[nd.func.id](conv(nd.args[0]))
+        raise Invalid(f"syntax element {type(nd).__name__} not allowed")
+
+    return conv(tree), x
+
+
+def value(expr, x, p):
+    v = expr.evalf(30, subs={x: sympy.Float(p, 30)})
+    try:
+        z = complex(v)
+    except (TypeError, ValueError):
+        return None
+    if not (math.isfinite(z.real) and math.isfinite(z.imag)):
+        return None
+    return z
+
+
+def _timeout(signum, frame):
+    raise Invalid(f"answer took more than {TIME_LIMIT} s to check")
+
+
+def check(inst, ans):
+    if hasattr(signal, "SIGALRM"):
+        old = signal.signal(signal.SIGALRM, _timeout)
+        signal.alarm(TIME_LIMIT)
+        try:
+            return _check(inst, ans)
+        finally:
+            signal.alarm(0)
+            signal.signal(signal.SIGALRM, old)
+    return _check(inst, ans)
+
+
+def _check(inst, ans):
+    var = inst["variable"]
+    F, x = parse(ans.get("antiderivative"), var)
+    f, x2 = parse(inst["integrand"], var)
+    if not F.has(x):
+        raise Invalid("antiderivative does not depend on the variable")
+    dF = sympy.diff(F, x)
+    used = 0
+    for p in POINTS:
+        fv = value(f, x2, p)
+        if fv is None or abs(fv.imag) > 1e-20 * (1 + abs(fv.real)):
+            continue  # outside the integrand's real domain
+        gv = value(dF, x, p)
+        if gv is None:
+            raise Invalid(f"derivative of the answer cannot be evaluated at {var} = {p}")
+        if abs(gv - fv) > 1e-9 * max(1.0, abs(fv)):
+            raise Invalid(f"derivative of the answer differs from the integrand at {var} = {p}")
+        used += 1
+    if used < MIN_POINTS:
+        raise Invalid("too few evaluation points in the integrand's domain")
+    return None, {"points": used}
diff --git a/tasks/construct-rg-simple-integration-l3-s0/tests/instance.json b/tasks/construct-rg-simple-integration-l3-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1015265fb1941c53ca82d7620efee84e484d512c
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l3-s0/tests/instance.json
@@ -0,0 +1 @@
+{"variable": "x", "integrand": "-2*x**4 - 4*x**3 + 3*x**2", "family": "rg_simple_integration", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-simple-integration-l3-s0/tests/run.py b/tasks/construct-rg-simple-integration-l3-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l3-s0/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rg-simple-integration-l3-s0/tests/test.sh b/tasks/construct-rg-simple-integration-l3-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l3-s0/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rg-simple-integration-l3-s9/environment/Dockerfile b/tasks/construct-rg-simple-integration-l3-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l3-s9/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rg-simple-integration-l3-s9/environment/instance.json b/tasks/construct-rg-simple-integration-l3-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f5b2c7da29b4c5f166b69da2ed59b43e20e48af5
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l3-s9/environment/instance.json
@@ -0,0 +1 @@
+{"variable": "x", "integrand": "9*x**8 - 35*x**6/4 + 45*x**4/8 + x**2/2", "family": "rg_simple_integration", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-simple-integration-l3-s9/solution/solve.sh b/tasks/construct-rg-simple-integration-l3-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..505474bd6a27b97ffad170de382b67bda06b74e5
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l3-s9/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"antiderivative": "x**9 - 5*x**7/4 + 9*x**5/8 + x**3/6 + C"}
+RLMATH_EOF
diff --git a/tasks/construct-rg-simple-integration-l3-s9/tests/check.py b/tasks/construct-rg-simple-integration-l3-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..a9e364a8ede6c8215a16a04ee6b0d3529adc647d
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l3-s9/tests/check.py
@@ -0,0 +1,158 @@
+"""Antiderivative check (Reasoning Gym simple_integration / intermediate_integration, Apache-2.0).
+
+The answer string is parsed with a whitelist AST walker (no eval), differentiated with SymPy and compared
+with the integrand numerically (30-digit evaluation) at fixed points inside the integrand's real domain.
+Any additive constant (e.g. "+ C") is allowed.
+
+Upstream notes: upstream scores with sympy.parse_expr, which calls eval() on the model's string (arbitrary
+code execution), and it accepts an unevaluated "Integral(f, x)" since its derivative is f. Both are
+rejected here: only arithmetic, whitelisted elementary functions, the variable, C, E and pi are allowed."""
+import ast
+import math
+import signal
+
+import sympy
+
+from run import Invalid
+
+MAX_LEN = 3000
+MAX_NODES = 600
+MAX_POW_NEST = 4  # nested powers with a variable exponent (x**x**x**...) blow up differentiation
+TIME_LIMIT = 40  # seconds; a slower answer is rejected
+FUNCS = {"sin": sympy.sin, "cos": sympy.cos, "tan": sympy.tan, "cot": sympy.cot, "sec": sympy.sec,
+         "csc": sympy.csc, "asin": sympy.asin, "acos": sympy.acos, "atan": sympy.atan, "acot": sympy.acot,
+         "arcsin": sympy.asin, "arccos": sympy.acos, "arctan": sympy.atan, "sinh": sympy.sinh,
+         "cosh": sympy.cosh, "tanh": sympy.tanh, "asinh": sympy.asinh, "acosh": sympy.acosh,
+         "atanh": sympy.atanh, "exp": sympy.exp, "log": sympy.log, "ln": sympy.log, "sqrt": sympy.sqrt,
+         "Abs": sympy.Abs, "abs": sympy.Abs}
+POINTS = [0.3137, 0.5521, 0.8093, 1.1171, 1.4519, 1.9033, 2.3471, 2.8123, 3.3319, 3.9107, 4.6231,
+          -0.3547, -0.6529, -0.9511, -1.3027, -1.7519, -2.2013, -2.7131, -3.1397, -3.8203, -4.4111,
+          0.1231, -0.1523, 0.2129, -0.2617, 0.0517, -0.0719, 5.4127, -5.3311, 6.3019, -6.2203, 7.2213,
+          -7.1117, 8.9021, -9.3107, 10.7129, -10.6337, 12.9011, -12.8231, 15.3113, -15.2719]
+MIN_POINTS = 5
+
+
+def parse(s, var):
+    if not isinstance(s, str) or not s.strip():
+        raise Invalid("antiderivative must be a non-empty string")
+    if len(s) > MAX_LEN:
+        raise Invalid("expression too long")
+    s = s.strip()
+    try:
+        tree = ast.parse(s, mode="eval")
+    except SyntaxError:
+        raise Invalid("expression does not parse (use ** for powers and * for products)")
+    if sum(1 for _ in ast.walk(tree)) > MAX_NODES:
+        raise Invalid("expression too large")
+
+    def pow_nest(nd):
+        kids = [pow_nest(c) for c in ast.iter_child_nodes(nd)]
+        own = isinstance(nd, ast.BinOp) and isinstance(nd.op, ast.Pow) and any(
+            isinstance(c, ast.Name) and c.id == var for c in ast.walk(nd.right))
+        return max(kids, default=0) + (1 if own else 0)
+    if pow_nest(tree) > MAX_POW_NEST:
+        raise Invalid("too many nested powers with a variable exponent")
+    x = sympy.Symbol(var, real=True)
+    names = {var: x, "C": sympy.Symbol("C"), "E": sympy.E, "pi": sympy.pi}
+
+    def conv(nd):
+        if isinstance(nd, ast.Expression):
+            return conv(nd.body)
+        if isinstance(nd, ast.Constant):
+            v = nd.value
+            if isinstance(v, bool) or not isinstance(v, (int, float)):
+                raise Invalid(f"constant {v!r} not allowed")
+            if isinstance(v, int):
+                if abs(v) > 10 ** 18:
+                    raise Invalid("integer constant too large")
+                return sympy.Integer(v)
+            if not math.isfinite(v):
+                raise Invalid("non-finite constant")
+            return sympy.Float(v, 30)
+        if isinstance(nd, ast.Name):
+            if nd.id not in names:
+                raise Invalid(f"unknown name {nd.id!r} (the variable is {var})")
+            return names[nd.id]
+        if isinstance(nd, ast.UnaryOp) and isinstance(nd.op, (ast.USub, ast.UAdd)):
+            a = conv(nd.operand)
+            return -a if isinstance(nd.op, ast.USub) else a
+        if isinstance(nd, ast.BinOp):
+            a, b = conv(nd.left), conv(nd.right)
+            if isinstance(nd.op, ast.Add):
+                return a + b
+            if isinstance(nd.op, ast.Sub):
+                return a - b
+            if isinstance(nd.op, ast.Mult):
+                return a * b
+            if isinstance(nd.op, ast.Div):
+                if b == 0:
+                    raise Invalid("division by zero")
+                return a / b
+            if isinstance(nd.op, ast.Pow):
+                if not b.free_symbols:
+                    bv = complex(b.evalf(20))
+                    if bv.imag != 0 or abs(bv.real) > 100:
+                        raise Invalid("exponent too large")
+                    if not a.free_symbols and abs(complex(a.evalf(20))) > 1e6:
+                        raise Invalid("numeric power too large")
+                return a ** b
+            raise Invalid("only + - * / ** are allowed (use ** for powers)")
+        if isinstance(nd, ast.Call):
+            if not isinstance(nd.func, ast.Name) or nd.func.id not in FUNCS:
+                raise Invalid(f"function {ast.unparse(nd.func)!r} not allowed")
+            if nd.keywords or len(nd.args) != 1:
+                raise Invalid(f"{nd.func.id} takes exactly one argument")
+            return FUNCS[nd.func.id](conv(nd.args[0]))
+        raise Invalid(f"syntax element {type(nd).__name__} not allowed")
+
+    return conv(tree), x
+
+
+def value(expr, x, p):
+    v = expr.evalf(30, subs={x: sympy.Float(p, 30)})
+    try:
+        z = complex(v)
+    except (TypeError, ValueError):
+        return None
+    if not (math.isfinite(z.real) and math.isfinite(z.imag)):
+        return None
+    return z
+
+
+def _timeout(signum, frame):
+    raise Invalid(f"answer took more than {TIME_LIMIT} s to check")
+
+
+def check(inst, ans):
+    if hasattr(signal, "SIGALRM"):
+        old = signal.signal(signal.SIGALRM, _timeout)
+        signal.alarm(TIME_LIMIT)
+        try:
+            return _check(inst, ans)
+        finally:
+            signal.alarm(0)
+            signal.signal(signal.SIGALRM, old)
+    return _check(inst, ans)
+
+
+def _check(inst, ans):
+    var = inst["variable"]
+    F, x = parse(ans.get("antiderivative"), var)
+    f, x2 = parse(inst["integrand"], var)
+    if not F.has(x):
+        raise Invalid("antiderivative does not depend on the variable")
+    dF = sympy.diff(F, x)
+    used = 0
+    for p in POINTS:
+        fv = value(f, x2, p)
+        if fv is None or abs(fv.imag) > 1e-20 * (1 + abs(fv.real)):
+            continue  # outside the integrand's real domain
+        gv = value(dF, x, p)
+        if gv is None:
+            raise Invalid(f"derivative of the answer cannot be evaluated at {var} = {p}")
+        if abs(gv - fv) > 1e-9 * max(1.0, abs(fv)):
+            raise Invalid(f"derivative of the answer differs from the integrand at {var} = {p}")
+        used += 1
+    if used < MIN_POINTS:
+        raise Invalid("too few evaluation points in the integrand's domain")
+    return None, {"points": used}
diff --git a/tasks/construct-rg-simple-integration-l3-s9/tests/instance.json b/tasks/construct-rg-simple-integration-l3-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f5b2c7da29b4c5f166b69da2ed59b43e20e48af5
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l3-s9/tests/instance.json
@@ -0,0 +1 @@
+{"variable": "x", "integrand": "9*x**8 - 35*x**6/4 + 45*x**4/8 + x**2/2", "family": "rg_simple_integration", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-simple-integration-l3-s9/tests/run.py b/tasks/construct-rg-simple-integration-l3-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l3-s9/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rg-simple-integration-l3-s9/tests/test.sh b/tasks/construct-rg-simple-integration-l3-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l3-s9/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-antiderivative-l2-s0/environment/Dockerfile b/tasks/construct-rlve-antiderivative-l2-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l2-s0/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-antiderivative-l2-s0/environment/instance.json b/tasks/construct-rlve-antiderivative-l2-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8b0cac3c72254d29a9a1a7967b1e09b4fef5522b
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l2-s0/environment/instance.json
@@ -0,0 +1 @@
+{"f_prime": "7/(12*x**(5/12))", "max_ops": 28, "max_len": 580, "family": "rlve_antiderivative", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-antiderivative-l2-s0/solution/solve.sh b/tasks/construct-rlve-antiderivative-l2-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..43e107a098f5c8e5014dba684e737f2192cb0d8b
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l2-s0/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"F": "x**(7/12)"}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-antiderivative-l2-s0/tests/check.py b/tasks/construct-rlve-antiderivative-l2-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..c6b30b93bbbb7795b54d973e43f25b23c82cb7fc
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l2-s0/tests/check.py
@@ -0,0 +1,107 @@
+"""Antiderivative check. The answer string is tokenised against a whitelist (x, integers, + - * / ** ( ) and a few
+elementary functions) before sympy parses it, so no Python eval of untrusted text and no `integrate`/`Integral`."""
+import re
+import signal
+
+import sympy
+from sympy.parsing.sympy_parser import auto_number, parse_expr
+
+from run import Invalid
+
+x = sympy.Symbol("x")
+FUNCS = {"sin": sympy.sin, "cos": sympy.cos, "tan": sympy.tan, "exp": sympy.exp, "log": sympy.log,
+         "sqrt": sympy.sqrt, "asin": sympy.asin, "acos": sympy.acos, "atan": sympy.atan, "sinh": sympy.sinh,
+         "cosh": sympy.cosh, "tanh": sympy.tanh, "E": sympy.E, "pi": sympy.pi, "x": x}
+TOK = re.compile(r"\s*(?:(\d+)|([A-Za-z]+)|(\*\*|[-+*/()]))")
+NPTS = 40
+
+
+def safe_parse(s, max_len):
+    if not isinstance(s, str) or not s.strip():
+        raise Invalid("F must be a non-empty string")
+    if len(s) > max_len:
+        raise Invalid("expression too long")
+    pos = 0
+    while pos < len(s):
+        m = TOK.match(s, pos)
+        if not m or m.end() == pos:
+            if s[pos:].strip() == "":
+                break
+            raise Invalid(f"illegal character near position {pos}")
+        if m.group(1) and len(m.group(1)) > 12:
+            raise Invalid("integer literal too long")
+        if m.group(2) and m.group(2) not in FUNCS:
+            raise Invalid(f"unknown name {m.group(2)!r}")
+        pos = m.end()
+    try:
+        e = parse_expr(s, local_dict=dict(FUNCS), global_dict={"__builtins__": {}, "Integer": sympy.Integer},
+                       transformations=(auto_number,))
+    except Invalid:
+        raise
+    except Exception as ex:
+        raise Invalid(f"cannot parse F: {type(ex).__name__}")
+    if not isinstance(e, sympy.Expr):
+        raise Invalid("F is not an expression")
+    if e.free_symbols - {x}:
+        raise Invalid("F may only contain the variable x")
+    return e
+
+
+def _val(e, pt):
+    v = e.evalf(40, subs={x: pt})
+    if not v.is_number or v.has(sympy.zoo, sympy.nan, sympy.oo, -sympy.oo):
+        return None
+    return v
+
+
+class _Timeout(Exception):
+    pass
+
+
+def _alarm(*_):
+    raise _Timeout()
+
+
+def check(inst, ans):
+    old = None
+    try:
+        old = signal.signal(signal.SIGALRM, _alarm)
+        signal.alarm(40)
+    except ValueError:  # not in main thread
+        old = None
+    try:
+        return _check(inst, ans)
+    except _Timeout:
+        raise Invalid("evaluation timed out")
+    except RecursionError:
+        raise Invalid("expression too deep")
+    finally:
+        if old is not None:
+            signal.alarm(0)
+            signal.signal(signal.SIGALRM, old)
+
+
+def _check(inst, ans):
+    F = safe_parse(ans.get("F"), inst["max_len"])
+    if sympy.count_ops(F) > inst["max_ops"]:
+        raise Invalid("expression too complex")
+    if any(isinstance(a, sympy.Rational) and max(abs(a.p), abs(a.q)) > 10 ** 12 for a in F.atoms(sympy.Rational)):
+        raise Invalid("numeric constants too large")
+    D = sympy.diff(F, x)
+    f = sympy.sympify(inst["f_prime"])  # trusted instance data
+    ok = 0
+    for k in range(NPTS):
+        pt = sympy.Rational(2 * k + 1, NPTS // 2 * 2) * 2 - 2 + sympy.Rational(1, 997)
+        vf = _val(f, pt)
+        if vf is None:
+            continue
+        vd = _val(D, pt)
+        if vd is None:
+            continue
+        err = sympy.Abs(vd - vf).evalf(40)
+        if err > sympy.Float("1e-20", 40) * (1 + sympy.Abs(vf).evalf(40)):
+            raise Invalid(f"F'(x) != f(x) at x = {float(pt):.4f}")
+        ok += 1
+    if ok < NPTS // 4:
+        raise Invalid("F' could not be compared with f on enough test points")
+    return None, {"points_checked": ok}
diff --git a/tasks/construct-rlve-antiderivative-l2-s0/tests/instance.json b/tasks/construct-rlve-antiderivative-l2-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8b0cac3c72254d29a9a1a7967b1e09b4fef5522b
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l2-s0/tests/instance.json
@@ -0,0 +1 @@
+{"f_prime": "7/(12*x**(5/12))", "max_ops": 28, "max_len": 580, "family": "rlve_antiderivative", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-antiderivative-l2-s0/tests/run.py b/tasks/construct-rlve-antiderivative-l2-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l2-s0/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-antiderivative-l2-s0/tests/test.sh b/tasks/construct-rlve-antiderivative-l2-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l2-s0/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-antiderivative-l2-s7/environment/Dockerfile b/tasks/construct-rlve-antiderivative-l2-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l2-s7/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-antiderivative-l2-s7/environment/instance.json b/tasks/construct-rlve-antiderivative-l2-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..cff6ab6bf14b64746e927df54b6e342bdb406192
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l2-s7/environment/instance.json
@@ -0,0 +1 @@
+{"f_prime": "x*(2*x + 4) + (x + 2)**2", "max_ops": 32, "max_len": 640, "family": "rlve_antiderivative", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-antiderivative-l2-s7/solution/solve.sh b/tasks/construct-rlve-antiderivative-l2-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..c9d8282707a63f2c9e60c8ca9dff4be4d9bce225
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l2-s7/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"F": "x*(x + 2)**2"}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-antiderivative-l2-s7/tests/check.py b/tasks/construct-rlve-antiderivative-l2-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..c6b30b93bbbb7795b54d973e43f25b23c82cb7fc
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l2-s7/tests/check.py
@@ -0,0 +1,107 @@
+"""Antiderivative check. The answer string is tokenised against a whitelist (x, integers, + - * / ** ( ) and a few
+elementary functions) before sympy parses it, so no Python eval of untrusted text and no `integrate`/`Integral`."""
+import re
+import signal
+
+import sympy
+from sympy.parsing.sympy_parser import auto_number, parse_expr
+
+from run import Invalid
+
+x = sympy.Symbol("x")
+FUNCS = {"sin": sympy.sin, "cos": sympy.cos, "tan": sympy.tan, "exp": sympy.exp, "log": sympy.log,
+         "sqrt": sympy.sqrt, "asin": sympy.asin, "acos": sympy.acos, "atan": sympy.atan, "sinh": sympy.sinh,
+         "cosh": sympy.cosh, "tanh": sympy.tanh, "E": sympy.E, "pi": sympy.pi, "x": x}
+TOK = re.compile(r"\s*(?:(\d+)|([A-Za-z]+)|(\*\*|[-+*/()]))")
+NPTS = 40
+
+
+def safe_parse(s, max_len):
+    if not isinstance(s, str) or not s.strip():
+        raise Invalid("F must be a non-empty string")
+    if len(s) > max_len:
+        raise Invalid("expression too long")
+    pos = 0
+    while pos < len(s):
+        m = TOK.match(s, pos)
+        if not m or m.end() == pos:
+            if s[pos:].strip() == "":
+                break
+            raise Invalid(f"illegal character near position {pos}")
+        if m.group(1) and len(m.group(1)) > 12:
+            raise Invalid("integer literal too long")
+        if m.group(2) and m.group(2) not in FUNCS:
+            raise Invalid(f"unknown name {m.group(2)!r}")
+        pos = m.end()
+    try:
+        e = parse_expr(s, local_dict=dict(FUNCS), global_dict={"__builtins__": {}, "Integer": sympy.Integer},
+                       transformations=(auto_number,))
+    except Invalid:
+        raise
+    except Exception as ex:
+        raise Invalid(f"cannot parse F: {type(ex).__name__}")
+    if not isinstance(e, sympy.Expr):
+        raise Invalid("F is not an expression")
+    if e.free_symbols - {x}:
+        raise Invalid("F may only contain the variable x")
+    return e
+
+
+def _val(e, pt):
+    v = e.evalf(40, subs={x: pt})
+    if not v.is_number or v.has(sympy.zoo, sympy.nan, sympy.oo, -sympy.oo):
+        return None
+    return v
+
+
+class _Timeout(Exception):
+    pass
+
+
+def _alarm(*_):
+    raise _Timeout()
+
+
+def check(inst, ans):
+    old = None
+    try:
+        old = signal.signal(signal.SIGALRM, _alarm)
+        signal.alarm(40)
+    except ValueError:  # not in main thread
+        old = None
+    try:
+        return _check(inst, ans)
+    except _Timeout:
+        raise Invalid("evaluation timed out")
+    except RecursionError:
+        raise Invalid("expression too deep")
+    finally:
+        if old is not None:
+            signal.alarm(0)
+            signal.signal(signal.SIGALRM, old)
+
+
+def _check(inst, ans):
+    F = safe_parse(ans.get("F"), inst["max_len"])
+    if sympy.count_ops(F) > inst["max_ops"]:
+        raise Invalid("expression too complex")
+    if any(isinstance(a, sympy.Rational) and max(abs(a.p), abs(a.q)) > 10 ** 12 for a in F.atoms(sympy.Rational)):
+        raise Invalid("numeric constants too large")
+    D = sympy.diff(F, x)
+    f = sympy.sympify(inst["f_prime"])  # trusted instance data
+    ok = 0
+    for k in range(NPTS):
+        pt = sympy.Rational(2 * k + 1, NPTS // 2 * 2) * 2 - 2 + sympy.Rational(1, 997)
+        vf = _val(f, pt)
+        if vf is None:
+            continue
+        vd = _val(D, pt)
+        if vd is None:
+            continue
+        err = sympy.Abs(vd - vf).evalf(40)
+        if err > sympy.Float("1e-20", 40) * (1 + sympy.Abs(vf).evalf(40)):
+            raise Invalid(f"F'(x) != f(x) at x = {float(pt):.4f}")
+        ok += 1
+    if ok < NPTS // 4:
+        raise Invalid("F' could not be compared with f on enough test points")
+    return None, {"points_checked": ok}
diff --git a/tasks/construct-rlve-antiderivative-l2-s7/tests/instance.json b/tasks/construct-rlve-antiderivative-l2-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..cff6ab6bf14b64746e927df54b6e342bdb406192
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l2-s7/tests/instance.json
@@ -0,0 +1 @@
+{"f_prime": "x*(2*x + 4) + (x + 2)**2", "max_ops": 32, "max_len": 640, "family": "rlve_antiderivative", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-antiderivative-l2-s7/tests/run.py b/tasks/construct-rlve-antiderivative-l2-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l2-s7/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-antiderivative-l2-s7/tests/test.sh b/tasks/construct-rlve-antiderivative-l2-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l2-s7/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-antiderivative-l6-s6/environment/Dockerfile b/tasks/construct-rlve-antiderivative-l6-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l6-s6/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-antiderivative-l6-s6/environment/instance.json b/tasks/construct-rlve-antiderivative-l6-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a3125a42ad865dab381b1e940e745155e174b29e
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l6-s6/environment/instance.json
@@ -0,0 +1 @@
+{"f_prime": "-2*sin(x - 1/3)", "max_ops": 40, "max_len": 760, "family": "rlve_antiderivative", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-antiderivative-l6-s6/solution/solve.sh b/tasks/construct-rlve-antiderivative-l6-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..c79ba1dcc0056c7bbac6d03d9fe6ad9bd759799c
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l6-s6/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"F": "2*cos(x - 1/3) - 1"}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-antiderivative-l6-s6/tests/check.py b/tasks/construct-rlve-antiderivative-l6-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..c6b30b93bbbb7795b54d973e43f25b23c82cb7fc
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l6-s6/tests/check.py
@@ -0,0 +1,107 @@
+"""Antiderivative check. The answer string is tokenised against a whitelist (x, integers, + - * / ** ( ) and a few
+elementary functions) before sympy parses it, so no Python eval of untrusted text and no `integrate`/`Integral`."""
+import re
+import signal
+
+import sympy
+from sympy.parsing.sympy_parser import auto_number, parse_expr
+
+from run import Invalid
+
+x = sympy.Symbol("x")
+FUNCS = {"sin": sympy.sin, "cos": sympy.cos, "tan": sympy.tan, "exp": sympy.exp, "log": sympy.log,
+         "sqrt": sympy.sqrt, "asin": sympy.asin, "acos": sympy.acos, "atan": sympy.atan, "sinh": sympy.sinh,
+         "cosh": sympy.cosh, "tanh": sympy.tanh, "E": sympy.E, "pi": sympy.pi, "x": x}
+TOK = re.compile(r"\s*(?:(\d+)|([A-Za-z]+)|(\*\*|[-+*/()]))")
+NPTS = 40
+
+
+def safe_parse(s, max_len):
+    if not isinstance(s, str) or not s.strip():
+        raise Invalid("F must be a non-empty string")
+    if len(s) > max_len:
+        raise Invalid("expression too long")
+    pos = 0
+    while pos < len(s):
+        m = TOK.match(s, pos)
+        if not m or m.end() == pos:
+            if s[pos:].strip() == "":
+                break
+            raise Invalid(f"illegal character near position {pos}")
+        if m.group(1) and len(m.group(1)) > 12:
+            raise Invalid("integer literal too long")
+        if m.group(2) and m.group(2) not in FUNCS:
+            raise Invalid(f"unknown name {m.group(2)!r}")
+        pos = m.end()
+    try:
+        e = parse_expr(s, local_dict=dict(FUNCS), global_dict={"__builtins__": {}, "Integer": sympy.Integer},
+                       transformations=(auto_number,))
+    except Invalid:
+        raise
+    except Exception as ex:
+        raise Invalid(f"cannot parse F: {type(ex).__name__}")
+    if not isinstance(e, sympy.Expr):
+        raise Invalid("F is not an expression")
+    if e.free_symbols - {x}:
+        raise Invalid("F may only contain the variable x")
+    return e
+
+
+def _val(e, pt):
+    v = e.evalf(40, subs={x: pt})
+    if not v.is_number or v.has(sympy.zoo, sympy.nan, sympy.oo, -sympy.oo):
+        return None
+    return v
+
+
+class _Timeout(Exception):
+    pass
+
+
+def _alarm(*_):
+    raise _Timeout()
+
+
+def check(inst, ans):
+    old = None
+    try:
+        old = signal.signal(signal.SIGALRM, _alarm)
+        signal.alarm(40)
+    except ValueError:  # not in main thread
+        old = None
+    try:
+        return _check(inst, ans)
+    except _Timeout:
+        raise Invalid("evaluation timed out")
+    except RecursionError:
+        raise Invalid("expression too deep")
+    finally:
+        if old is not None:
+            signal.alarm(0)
+            signal.signal(signal.SIGALRM, old)
+
+
+def _check(inst, ans):
+    F = safe_parse(ans.get("F"), inst["max_len"])
+    if sympy.count_ops(F) > inst["max_ops"]:
+        raise Invalid("expression too complex")
+    if any(isinstance(a, sympy.Rational) and max(abs(a.p), abs(a.q)) > 10 ** 12 for a in F.atoms(sympy.Rational)):
+        raise Invalid("numeric constants too large")
+    D = sympy.diff(F, x)
+    f = sympy.sympify(inst["f_prime"])  # trusted instance data
+    ok = 0
+    for k in range(NPTS):
+        pt = sympy.Rational(2 * k + 1, NPTS // 2 * 2) * 2 - 2 + sympy.Rational(1, 997)
+        vf = _val(f, pt)
+        if vf is None:
+            continue
+        vd = _val(D, pt)
+        if vd is None:
+            continue
+        err = sympy.Abs(vd - vf).evalf(40)
+        if err > sympy.Float("1e-20", 40) * (1 + sympy.Abs(vf).evalf(40)):
+            raise Invalid(f"F'(x) != f(x) at x = {float(pt):.4f}")
+        ok += 1
+    if ok < NPTS // 4:
+        raise Invalid("F' could not be compared with f on enough test points")
+    return None, {"points_checked": ok}
diff --git a/tasks/construct-rlve-antiderivative-l6-s6/tests/instance.json b/tasks/construct-rlve-antiderivative-l6-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a3125a42ad865dab381b1e940e745155e174b29e
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l6-s6/tests/instance.json
@@ -0,0 +1 @@
+{"f_prime": "-2*sin(x - 1/3)", "max_ops": 40, "max_len": 760, "family": "rlve_antiderivative", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-antiderivative-l6-s6/tests/run.py b/tasks/construct-rlve-antiderivative-l6-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l6-s6/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-antiderivative-l6-s6/tests/test.sh b/tasks/construct-rlve-antiderivative-l6-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l6-s6/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-antiderivative-l6-s8/environment/Dockerfile b/tasks/construct-rlve-antiderivative-l6-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l6-s8/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-antiderivative-l6-s8/environment/instance.json b/tasks/construct-rlve-antiderivative-l6-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1da81140f9428053c5203305ecba9f4bb2aebb73
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l6-s8/environment/instance.json
@@ -0,0 +1 @@
+{"f_prime": "2*cos(2*x)/3 - 2", "max_ops": 48, "max_len": 860, "family": "rlve_antiderivative", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-antiderivative-l6-s8/solution/solve.sh b/tasks/construct-rlve-antiderivative-l6-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..86c51654d30ae70f19ef2096dc53bbd7963d58ac
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l6-s8/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"F": "-2*x + sin(2*x)/3 - 4/3"}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-antiderivative-l6-s8/tests/check.py b/tasks/construct-rlve-antiderivative-l6-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..c6b30b93bbbb7795b54d973e43f25b23c82cb7fc
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l6-s8/tests/check.py
@@ -0,0 +1,107 @@
+"""Antiderivative check. The answer string is tokenised against a whitelist (x, integers, + - * / ** ( ) and a few
+elementary functions) before sympy parses it, so no Python eval of untrusted text and no `integrate`/`Integral`."""
+import re
+import signal
+
+import sympy
+from sympy.parsing.sympy_parser import auto_number, parse_expr
+
+from run import Invalid
+
+x = sympy.Symbol("x")
+FUNCS = {"sin": sympy.sin, "cos": sympy.cos, "tan": sympy.tan, "exp": sympy.exp, "log": sympy.log,
+         "sqrt": sympy.sqrt, "asin": sympy.asin, "acos": sympy.acos, "atan": sympy.atan, "sinh": sympy.sinh,
+         "cosh": sympy.cosh, "tanh": sympy.tanh, "E": sympy.E, "pi": sympy.pi, "x": x}
+TOK = re.compile(r"\s*(?:(\d+)|([A-Za-z]+)|(\*\*|[-+*/()]))")
+NPTS = 40
+
+
+def safe_parse(s, max_len):
+    if not isinstance(s, str) or not s.strip():
+        raise Invalid("F must be a non-empty string")
+    if len(s) > max_len:
+        raise Invalid("expression too long")
+    pos = 0
+    while pos < len(s):
+        m = TOK.match(s, pos)
+        if not m or m.end() == pos:
+            if s[pos:].strip() == "":
+                break
+            raise Invalid(f"illegal character near position {pos}")
+        if m.group(1) and len(m.group(1)) > 12:
+            raise Invalid("integer literal too long")
+        if m.group(2) and m.group(2) not in FUNCS:
+            raise Invalid(f"unknown name {m.group(2)!r}")
+        pos = m.end()
+    try:
+        e = parse_expr(s, local_dict=dict(FUNCS), global_dict={"__builtins__": {}, "Integer": sympy.Integer},
+                       transformations=(auto_number,))
+    except Invalid:
+        raise
+    except Exception as ex:
+        raise Invalid(f"cannot parse F: {type(ex).__name__}")
+    if not isinstance(e, sympy.Expr):
+        raise Invalid("F is not an expression")
+    if e.free_symbols - {x}:
+        raise Invalid("F may only contain the variable x")
+    return e
+
+
+def _val(e, pt):
+    v = e.evalf(40, subs={x: pt})
+    if not v.is_number or v.has(sympy.zoo, sympy.nan, sympy.oo, -sympy.oo):
+        return None
+    return v
+
+
+class _Timeout(Exception):
+    pass
+
+
+def _alarm(*_):
+    raise _Timeout()
+
+
+def check(inst, ans):
+    old = None
+    try:
+        old = signal.signal(signal.SIGALRM, _alarm)
+        signal.alarm(40)
+    except ValueError:  # not in main thread
+        old = None
+    try:
+        return _check(inst, ans)
+    except _Timeout:
+        raise Invalid("evaluation timed out")
+    except RecursionError:
+        raise Invalid("expression too deep")
+    finally:
+        if old is not None:
+            signal.alarm(0)
+            signal.signal(signal.SIGALRM, old)
+
+
+def _check(inst, ans):
+    F = safe_parse(ans.get("F"), inst["max_len"])
+    if sympy.count_ops(F) > inst["max_ops"]:
+        raise Invalid("expression too complex")
+    if any(isinstance(a, sympy.Rational) and max(abs(a.p), abs(a.q)) > 10 ** 12 for a in F.atoms(sympy.Rational)):
+        raise Invalid("numeric constants too large")
+    D = sympy.diff(F, x)
+    f = sympy.sympify(inst["f_prime"])  # trusted instance data
+    ok = 0
+    for k in range(NPTS):
+        pt = sympy.Rational(2 * k + 1, NPTS // 2 * 2) * 2 - 2 + sympy.Rational(1, 997)
+        vf = _val(f, pt)
+        if vf is None:
+            continue
+        vd = _val(D, pt)
+        if vd is None:
+            continue
+        err = sympy.Abs(vd - vf).evalf(40)
+        if err > sympy.Float("1e-20", 40) * (1 + sympy.Abs(vf).evalf(40)):
+            raise Invalid(f"F'(x) != f(x) at x = {float(pt):.4f}")
+        ok += 1
+    if ok < NPTS // 4:
+        raise Invalid("F' could not be compared with f on enough test points")
+    return None, {"points_checked": ok}
diff --git a/tasks/construct-rlve-antiderivative-l6-s8/tests/instance.json b/tasks/construct-rlve-antiderivative-l6-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1da81140f9428053c5203305ecba9f4bb2aebb73
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l6-s8/tests/instance.json
@@ -0,0 +1 @@
+{"f_prime": "2*cos(2*x)/3 - 2", "max_ops": 48, "max_len": 860, "family": "rlve_antiderivative", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-antiderivative-l6-s8/tests/run.py b/tasks/construct-rlve-antiderivative-l6-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l6-s8/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-antiderivative-l6-s8/tests/test.sh b/tasks/construct-rlve-antiderivative-l6-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l6-s8/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-bezout-identity-l3-s3/environment/Dockerfile b/tasks/construct-rlve-bezout-identity-l3-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l3-s3/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-bezout-identity-l3-s3/environment/instance.json b/tasks/construct-rlve-bezout-identity-l3-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8bd22d5af28e8529df15ae4ecb2c7601dd614d64
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l3-s3/environment/instance.json
@@ -0,0 +1 @@
+{"A": [346, 118, -2, -428, 222, -418], "target": 2, "family": "rlve_bezout_identity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l3-s3/solution/solve.sh b/tasks/construct-rlve-bezout-identity-l3-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..38579d9004352bf477b24c9d3f83f942ab792cea
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l3-s3/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"X": [0, 0, -1, 0, 0, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-bezout-identity-l3-s3/tests/check.py b/tasks/construct-rlve-bezout-identity-l3-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..b4cb860ed36974a12b71adc9a0fa3f1ad85c814a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l3-s3/tests/check.py
@@ -0,0 +1,10 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    A = inst["A"]
+    X = int_list(ans.get("X"), len(A), name="X")
+    S = sum(a * x for a, x in zip(A, X))
+    if S != inst["target"]:
+        raise Invalid(f"S = {S}, required {inst['target']}")
+    return None, {}
diff --git a/tasks/construct-rlve-bezout-identity-l3-s3/tests/instance.json b/tasks/construct-rlve-bezout-identity-l3-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8bd22d5af28e8529df15ae4ecb2c7601dd614d64
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l3-s3/tests/instance.json
@@ -0,0 +1 @@
+{"A": [346, 118, -2, -428, 222, -418], "target": 2, "family": "rlve_bezout_identity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l3-s3/tests/run.py b/tasks/construct-rlve-bezout-identity-l3-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l3-s3/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-bezout-identity-l3-s3/tests/test.sh b/tasks/construct-rlve-bezout-identity-l3-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l3-s3/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-construct-hack-interval-l1-s8/environment/Dockerfile b/tasks/construct-rlve-construct-hack-interval-l1-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l1-s8/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-construct-hack-interval-l1-s8/environment/instance.json b/tasks/construct-rlve-construct-hack-interval-l1-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..6e60e369800888ac7eeae1c6af547cdef65ffba6
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l1-s8/environment/instance.json
@@ -0,0 +1 @@
+{"mod": 66, "family": "rlve_construct_hack_interval", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-construct-hack-interval-l1-s8/solution/solve.sh b/tasks/construct-rlve-construct-hack-interval-l1-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..4a4eca1049d223e4267545a6d62eb053b79e1216
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l1-s8/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"L": 1, "R": 15}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-construct-hack-interval-l1-s8/tests/check.py b/tasks/construct-rlve-construct-hack-interval-l1-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..34b46e9758e3a50ed16c9c2a29cfc0abe663926b
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l1-s8/tests/check.py
@@ -0,0 +1,30 @@
+from run import Invalid, as_int
+
+
+def digit_sum_prefix(n):
+    """sum of digit sums of all integers in [0, n] (exact, digit DP)."""
+    if n <= 0:
+        return 0
+    s = str(n)
+    total, prefix_sum = 0, 0
+    k = len(s)
+    for i, ch in enumerate(s):
+        d = int(ch)
+        rest = k - i - 1
+        p = 10 ** rest
+        # numbers with this digit < d, free suffix of `rest` digits
+        total += d * prefix_sum * p + (d * (d - 1) // 2) * p + d * (45 * rest * p // 10 if rest else 0)
+        prefix_sum += d
+    return total + prefix_sum  # include n itself
+
+
+def check(inst, ans):
+    mod = inst["mod"]
+    L = as_int(ans.get("L"), "L", lo=1)
+    R = as_int(ans.get("R"), "R", lo=1, hi=10 * mod)
+    if L > R:
+        raise Invalid("need L <= R")
+    total = digit_sum_prefix(R) - digit_sum_prefix(L - 1)
+    if total % mod:
+        raise Invalid(f"sum of digit sums over [L, R] is {total}, not divisible by {mod}")
+    return None, {}
diff --git a/tasks/construct-rlve-construct-hack-interval-l1-s8/tests/instance.json b/tasks/construct-rlve-construct-hack-interval-l1-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..6e60e369800888ac7eeae1c6af547cdef65ffba6
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l1-s8/tests/instance.json
@@ -0,0 +1 @@
+{"mod": 66, "family": "rlve_construct_hack_interval", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-construct-hack-interval-l1-s8/tests/run.py b/tasks/construct-rlve-construct-hack-interval-l1-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l1-s8/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-construct-hack-interval-l1-s8/tests/test.sh b/tasks/construct-rlve-construct-hack-interval-l1-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l1-s8/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-construct-hack-interval-l1-s9/environment/Dockerfile b/tasks/construct-rlve-construct-hack-interval-l1-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l1-s9/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-construct-hack-interval-l1-s9/environment/instance.json b/tasks/construct-rlve-construct-hack-interval-l1-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4def0362c154dbe60d31813aebd468b48760a05d
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l1-s9/environment/instance.json
@@ -0,0 +1 @@
+{"mod": 63, "family": "rlve_construct_hack_interval", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-construct-hack-interval-l1-s9/solution/solve.sh b/tasks/construct-rlve-construct-hack-interval-l1-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..53ba94071658d50cfe93e6fbee5cd25dc4b79847
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l1-s9/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"L": 3, "R": 15}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-construct-hack-interval-l1-s9/tests/check.py b/tasks/construct-rlve-construct-hack-interval-l1-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..34b46e9758e3a50ed16c9c2a29cfc0abe663926b
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l1-s9/tests/check.py
@@ -0,0 +1,30 @@
+from run import Invalid, as_int
+
+
+def digit_sum_prefix(n):
+    """sum of digit sums of all integers in [0, n] (exact, digit DP)."""
+    if n <= 0:
+        return 0
+    s = str(n)
+    total, prefix_sum = 0, 0
+    k = len(s)
+    for i, ch in enumerate(s):
+        d = int(ch)
+        rest = k - i - 1
+        p = 10 ** rest
+        # numbers with this digit < d, free suffix of `rest` digits
+        total += d * prefix_sum * p + (d * (d - 1) // 2) * p + d * (45 * rest * p // 10 if rest else 0)
+        prefix_sum += d
+    return total + prefix_sum  # include n itself
+
+
+def check(inst, ans):
+    mod = inst["mod"]
+    L = as_int(ans.get("L"), "L", lo=1)
+    R = as_int(ans.get("R"), "R", lo=1, hi=10 * mod)
+    if L > R:
+        raise Invalid("need L <= R")
+    total = digit_sum_prefix(R) - digit_sum_prefix(L - 1)
+    if total % mod:
+        raise Invalid(f"sum of digit sums over [L, R] is {total}, not divisible by {mod}")
+    return None, {}
diff --git a/tasks/construct-rlve-construct-hack-interval-l1-s9/tests/instance.json b/tasks/construct-rlve-construct-hack-interval-l1-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4def0362c154dbe60d31813aebd468b48760a05d
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l1-s9/tests/instance.json
@@ -0,0 +1 @@
+{"mod": 63, "family": "rlve_construct_hack_interval", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-construct-hack-interval-l1-s9/tests/run.py b/tasks/construct-rlve-construct-hack-interval-l1-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l1-s9/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-construct-hack-interval-l1-s9/tests/test.sh b/tasks/construct-rlve-construct-hack-interval-l1-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l1-s9/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-construct-hack-interval-l5-s6/environment/Dockerfile b/tasks/construct-rlve-construct-hack-interval-l5-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l5-s6/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-construct-hack-interval-l5-s6/environment/instance.json b/tasks/construct-rlve-construct-hack-interval-l5-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f51ef9c1eb97a3ca7257c833ab726216a0a56380
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l5-s6/environment/instance.json
@@ -0,0 +1 @@
+{"mod": 294208032536424, "family": "rlve_construct_hack_interval", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-construct-hack-interval-l5-s6/solution/solve.sh b/tasks/construct-rlve-construct-hack-interval-l5-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..9ca2d4dd0e36fb665015708bcd492cef2f5fdc0f
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l5-s6/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"L": 3416065072848, "R": 13416065072847}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-construct-hack-interval-l5-s6/tests/check.py b/tasks/construct-rlve-construct-hack-interval-l5-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..34b46e9758e3a50ed16c9c2a29cfc0abe663926b
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l5-s6/tests/check.py
@@ -0,0 +1,30 @@
+from run import Invalid, as_int
+
+
+def digit_sum_prefix(n):
+    """sum of digit sums of all integers in [0, n] (exact, digit DP)."""
+    if n <= 0:
+        return 0
+    s = str(n)
+    total, prefix_sum = 0, 0
+    k = len(s)
+    for i, ch in enumerate(s):
+        d = int(ch)
+        rest = k - i - 1
+        p = 10 ** rest
+        # numbers with this digit < d, free suffix of `rest` digits
+        total += d * prefix_sum * p + (d * (d - 1) // 2) * p + d * (45 * rest * p // 10 if rest else 0)
+        prefix_sum += d
+    return total + prefix_sum  # include n itself
+
+
+def check(inst, ans):
+    mod = inst["mod"]
+    L = as_int(ans.get("L"), "L", lo=1)
+    R = as_int(ans.get("R"), "R", lo=1, hi=10 * mod)
+    if L > R:
+        raise Invalid("need L <= R")
+    total = digit_sum_prefix(R) - digit_sum_prefix(L - 1)
+    if total % mod:
+        raise Invalid(f"sum of digit sums over [L, R] is {total}, not divisible by {mod}")
+    return None, {}
diff --git a/tasks/construct-rlve-construct-hack-interval-l5-s6/tests/instance.json b/tasks/construct-rlve-construct-hack-interval-l5-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f51ef9c1eb97a3ca7257c833ab726216a0a56380
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l5-s6/tests/instance.json
@@ -0,0 +1 @@
+{"mod": 294208032536424, "family": "rlve_construct_hack_interval", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-construct-hack-interval-l5-s6/tests/run.py b/tasks/construct-rlve-construct-hack-interval-l5-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l5-s6/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-construct-hack-interval-l5-s6/tests/test.sh b/tasks/construct-rlve-construct-hack-interval-l5-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l5-s6/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-crt-l4-s7/environment/Dockerfile b/tasks/construct-rlve-crt-l4-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-crt-l4-s7/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-crt-l4-s7/environment/instance.json b/tasks/construct-rlve-crt-l4-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8baf8bde566fdab27664a4789f866912675601b9
--- /dev/null
+++ b/tasks/construct-rlve-crt-l4-s7/environment/instance.json
@@ -0,0 +1 @@
+{"moduli": [121952, 240234, 257629, 526334, 346469], "residues": [66026, 73366, 38576, 27500, 207365], "family": "rlve_crt", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-crt-l4-s7/solution/solve.sh b/tasks/construct-rlve-crt-l4-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..6b483504fbaca27abee1bc646c9009840479e8de
--- /dev/null
+++ b/tasks/construct-rlve-crt-l4-s7/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": 553834}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-crt-l4-s7/tests/check.py b/tasks/construct-rlve-crt-l4-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..886d95606ef325930b24794c0959c2ba7b254ebd
--- /dev/null
+++ b/tasks/construct-rlve-crt-l4-s7/tests/check.py
@@ -0,0 +1,11 @@
+from run import Invalid, as_int
+
+
+def check(inst, ans):
+    x = as_int(ans.get("x"), "x", lo=0)
+    if x.bit_length() > 20000:
+        raise Invalid("x is unreasonably large")
+    for a, b in zip(inst["residues"], inst["moduli"]):
+        if x % b != a:
+            raise Invalid(f"x mod {b} = {x % b}, need {a}")
+    return None, {}
diff --git a/tasks/construct-rlve-crt-l4-s7/tests/instance.json b/tasks/construct-rlve-crt-l4-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8baf8bde566fdab27664a4789f866912675601b9
--- /dev/null
+++ b/tasks/construct-rlve-crt-l4-s7/tests/instance.json
@@ -0,0 +1 @@
+{"moduli": [121952, 240234, 257629, 526334, 346469], "residues": [66026, 73366, 38576, 27500, 207365], "family": "rlve_crt", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-crt-l4-s7/tests/run.py b/tasks/construct-rlve-crt-l4-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-crt-l4-s7/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-crt-l4-s7/tests/test.sh b/tasks/construct-rlve-crt-l4-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-crt-l4-s7/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-crt-l4-s9/environment/Dockerfile b/tasks/construct-rlve-crt-l4-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-crt-l4-s9/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-crt-l4-s9/environment/instance.json b/tasks/construct-rlve-crt-l4-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a2bbd8651f99bd42751e8e52290b776990060a30
--- /dev/null
+++ b/tasks/construct-rlve-crt-l4-s9/environment/instance.json
@@ -0,0 +1 @@
+{"moduli": [17945, 55175, 118462, 51193, 24055], "residues": [5691, 20956, 12844, 28920, 11031], "family": "rlve_crt", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-crt-l4-s9/solution/solve.sh b/tasks/construct-rlve-crt-l4-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..3f57d8702d3e7f35ecaabc3becfd26f76b28e94c
--- /dev/null
+++ b/tasks/construct-rlve-crt-l4-s9/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": 131306}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-crt-l4-s9/tests/check.py b/tasks/construct-rlve-crt-l4-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..886d95606ef325930b24794c0959c2ba7b254ebd
--- /dev/null
+++ b/tasks/construct-rlve-crt-l4-s9/tests/check.py
@@ -0,0 +1,11 @@
+from run import Invalid, as_int
+
+
+def check(inst, ans):
+    x = as_int(ans.get("x"), "x", lo=0)
+    if x.bit_length() > 20000:
+        raise Invalid("x is unreasonably large")
+    for a, b in zip(inst["residues"], inst["moduli"]):
+        if x % b != a:
+            raise Invalid(f"x mod {b} = {x % b}, need {a}")
+    return None, {}
diff --git a/tasks/construct-rlve-crt-l4-s9/tests/instance.json b/tasks/construct-rlve-crt-l4-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a2bbd8651f99bd42751e8e52290b776990060a30
--- /dev/null
+++ b/tasks/construct-rlve-crt-l4-s9/tests/instance.json
@@ -0,0 +1 @@
+{"moduli": [17945, 55175, 118462, 51193, 24055], "residues": [5691, 20956, 12844, 28920, 11031], "family": "rlve_crt", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-crt-l4-s9/tests/run.py b/tasks/construct-rlve-crt-l4-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-crt-l4-s9/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-crt-l4-s9/tests/test.sh b/tasks/construct-rlve-crt-l4-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-crt-l4-s9/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s1/environment/Dockerfile b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s1/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s1/environment/instance.json b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..37a95f08c02b7aa2a2046292d9e50c4f4d90ed97
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s1/environment/instance.json
@@ -0,0 +1 @@
+{"A": [4, 7, 6, 2], "family": "rlve_distinct_subset_sum_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s1/solution/solve.sh b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..160a8afb6e6d2f9c672059c0659942ce401d98b7
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s1/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"B": [6, 2, 7, 4]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s1/tests/check.py b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..06bbbcf872c2fca05785f40fdfdcf85a5e5a0e7c
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s1/tests/check.py
@@ -0,0 +1,22 @@
+from collections import Counter
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    A = inst["A"]
+    n = len(A)
+    B = int_list(ans.get("B"), n, name="B")
+    if Counter(B) != Counter(A):
+        raise Invalid("B is not a permutation of A")
+    d = [a - b for a, b in zip(A, B)]
+    # count subsets (including empty and full) with sum 0; the empty and full sets always qualify
+    ways = Counter({0: 1})
+    for x in d:
+        nxt = Counter(ways)
+        for s, c in ways.items():
+            nxt[s + x] += c
+        ways = nxt
+    if ways[0] != 2:
+        raise Invalid(f"{ways[0] - 2} non-empty proper subsets have equal sums in A and B")
+    return None, {}
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s1/tests/instance.json b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..37a95f08c02b7aa2a2046292d9e50c4f4d90ed97
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s1/tests/instance.json
@@ -0,0 +1 @@
+{"A": [4, 7, 6, 2], "family": "rlve_distinct_subset_sum_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s1/tests/run.py b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s1/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s1/tests/test.sh b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s1/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s6/environment/Dockerfile b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s6/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s6/environment/instance.json b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2be6dee00b62905cb9a2833acbdd069d4fb51095
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s6/environment/instance.json
@@ -0,0 +1 @@
+{"A": [2, 7, 4, 1], "family": "rlve_distinct_subset_sum_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s6/solution/solve.sh b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..da3f0996776111c0f186ac55a847fa6e284091fd
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s6/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"B": [4, 1, 7, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s6/tests/check.py b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..06bbbcf872c2fca05785f40fdfdcf85a5e5a0e7c
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s6/tests/check.py
@@ -0,0 +1,22 @@
+from collections import Counter
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    A = inst["A"]
+    n = len(A)
+    B = int_list(ans.get("B"), n, name="B")
+    if Counter(B) != Counter(A):
+        raise Invalid("B is not a permutation of A")
+    d = [a - b for a, b in zip(A, B)]
+    # count subsets (including empty and full) with sum 0; the empty and full sets always qualify
+    ways = Counter({0: 1})
+    for x in d:
+        nxt = Counter(ways)
+        for s, c in ways.items():
+            nxt[s + x] += c
+        ways = nxt
+    if ways[0] != 2:
+        raise Invalid(f"{ways[0] - 2} non-empty proper subsets have equal sums in A and B")
+    return None, {}
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s6/tests/instance.json b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2be6dee00b62905cb9a2833acbdd069d4fb51095
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s6/tests/instance.json
@@ -0,0 +1 @@
+{"A": [2, 7, 4, 1], "family": "rlve_distinct_subset_sum_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s6/tests/run.py b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s6/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s6/tests/test.sh b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l1-s6/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s0/environment/Dockerfile b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s0/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s0/environment/instance.json b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8d4678ca7c3ed216ea9536083d28f600cb23be7e
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s0/environment/instance.json
@@ -0,0 +1 @@
+{"A": [19, 1, 26, 25, 33, 0, 20, 34, 11, 32, 31, 9, 35, 2, 30, 8, 16, 27], "family": "rlve_distinct_subset_sum_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s0/solution/solve.sh b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..69a124d4fce9fe269d5f76eae66c5a87698033f5
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s0/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"B": [20, 2, 27, 26, 34, 1, 25, 35, 16, 33, 32, 11, 0, 8, 31, 9, 19, 30]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s0/tests/check.py b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..06bbbcf872c2fca05785f40fdfdcf85a5e5a0e7c
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s0/tests/check.py
@@ -0,0 +1,22 @@
+from collections import Counter
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    A = inst["A"]
+    n = len(A)
+    B = int_list(ans.get("B"), n, name="B")
+    if Counter(B) != Counter(A):
+        raise Invalid("B is not a permutation of A")
+    d = [a - b for a, b in zip(A, B)]
+    # count subsets (including empty and full) with sum 0; the empty and full sets always qualify
+    ways = Counter({0: 1})
+    for x in d:
+        nxt = Counter(ways)
+        for s, c in ways.items():
+            nxt[s + x] += c
+        ways = nxt
+    if ways[0] != 2:
+        raise Invalid(f"{ways[0] - 2} non-empty proper subsets have equal sums in A and B")
+    return None, {}
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s0/tests/instance.json b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8d4678ca7c3ed216ea9536083d28f600cb23be7e
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s0/tests/instance.json
@@ -0,0 +1 @@
+{"A": [19, 1, 26, 25, 33, 0, 20, 34, 11, 32, 31, 9, 35, 2, 30, 8, 16, 27], "family": "rlve_distinct_subset_sum_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s0/tests/run.py b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s0/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s0/tests/test.sh b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s0/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s9/environment/Dockerfile b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s9/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s9/environment/instance.json b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8451b649ce1646e0f17aa6d4cf410fda2b7c12e8
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s9/environment/instance.json
@@ -0,0 +1 @@
+{"A": [5, 22, 26, 17, 24, 35, 3, 6, 7, 28, 33, 13, 27, 2, 30, 18, 34, 31], "family": "rlve_distinct_subset_sum_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s9/solution/solve.sh b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..db185ec93fa04a92f698ab2e81ce960a96e34dee
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s9/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"B": [6, 24, 27, 18, 26, 2, 5, 7, 13, 30, 34, 17, 28, 3, 31, 22, 35, 33]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s9/tests/check.py b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..06bbbcf872c2fca05785f40fdfdcf85a5e5a0e7c
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s9/tests/check.py
@@ -0,0 +1,22 @@
+from collections import Counter
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    A = inst["A"]
+    n = len(A)
+    B = int_list(ans.get("B"), n, name="B")
+    if Counter(B) != Counter(A):
+        raise Invalid("B is not a permutation of A")
+    d = [a - b for a, b in zip(A, B)]
+    # count subsets (including empty and full) with sum 0; the empty and full sets always qualify
+    ways = Counter({0: 1})
+    for x in d:
+        nxt = Counter(ways)
+        for s, c in ways.items():
+            nxt[s + x] += c
+        ways = nxt
+    if ways[0] != 2:
+        raise Invalid(f"{ways[0] - 2} non-empty proper subsets have equal sums in A and B")
+    return None, {}
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s9/tests/instance.json b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8451b649ce1646e0f17aa6d4cf410fda2b7c12e8
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s9/tests/instance.json
@@ -0,0 +1 @@
+{"A": [5, 22, 26, 17, 24, 35, 3, 6, 7, 28, 33, 13, 27, 2, 30, 18, 34, 31], "family": "rlve_distinct_subset_sum_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s9/tests/run.py b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s9/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s9/tests/test.sh b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l5-s9/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-even-degree-partition-l4-s0/environment/Dockerfile b/tasks/construct-rlve-even-degree-partition-l4-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l4-s0/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-even-degree-partition-l4-s0/environment/instance.json b/tasks/construct-rlve-even-degree-partition-l4-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..fb2c533d23a8bea62de09d9b1500fdd49252022b
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l4-s0/environment/instance.json
@@ -0,0 +1 @@
+{"n": 18, "edges": [[0, 3], [0, 6], [0, 9], [0, 15], [0, 16], [0, 17], [1, 2], [1, 3], [1, 6], [1, 8], [1, 12], [1, 16], [1, 17], [2, 3], [2, 5], [2, 8], [2, 9], [2, 13], [2, 16], [2, 17], [3, 4], [3, 6], [3, 7], [3, 8], [3, 9], [3, 11], [3, 12], [3, 13], [3, 15], [3, 16], [3, 17], [4, 6], [4, 8], [4, 9], [4, 13], [4, 16], [4, 17], [5, 12], [5, 13], [6, 9], [6, 10], [7, 14], [7, 15], [7, 17], [8, 9], [8, 10], [8, 11], [8, 14], [8, 16], [8, 17], [9, 12], [9, 15], [9, 16], [10, 13], [10, 14], [10, 15], [12, 13], [13, 15], [13, 16], [14, 17], [15, 16]], "family": "rlve_even_degree_partition", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-even-degree-partition-l4-s0/solution/solve.sh b/tasks/construct-rlve-even-degree-partition-l4-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..134268712b1d33c162223e8312fb2f5126fa9ca1
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l4-s0/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"groups": [2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-even-degree-partition-l4-s0/tests/check.py b/tasks/construct-rlve-even-degree-partition-l4-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..9c7f0f71ea6d10264ececcc933b97a4a0e6eaf8e
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l4-s0/tests/check.py
@@ -0,0 +1,15 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n = inst["n"]
+    g = int_list(ans.get("groups"), n, 1, 2, "groups")
+    deg = [0] * n
+    for u, v in inst["edges"]:
+        if g[u] == g[v]:
+            deg[u] += 1
+            deg[v] += 1
+    for v in range(n):
+        if deg[v] % 2:
+            raise Invalid(f"vertex {v} has {deg[v]} same-group neighbours")
+    return None, {}
diff --git a/tasks/construct-rlve-even-degree-partition-l4-s0/tests/instance.json b/tasks/construct-rlve-even-degree-partition-l4-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..fb2c533d23a8bea62de09d9b1500fdd49252022b
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l4-s0/tests/instance.json
@@ -0,0 +1 @@
+{"n": 18, "edges": [[0, 3], [0, 6], [0, 9], [0, 15], [0, 16], [0, 17], [1, 2], [1, 3], [1, 6], [1, 8], [1, 12], [1, 16], [1, 17], [2, 3], [2, 5], [2, 8], [2, 9], [2, 13], [2, 16], [2, 17], [3, 4], [3, 6], [3, 7], [3, 8], [3, 9], [3, 11], [3, 12], [3, 13], [3, 15], [3, 16], [3, 17], [4, 6], [4, 8], [4, 9], [4, 13], [4, 16], [4, 17], [5, 12], [5, 13], [6, 9], [6, 10], [7, 14], [7, 15], [7, 17], [8, 9], [8, 10], [8, 11], [8, 14], [8, 16], [8, 17], [9, 12], [9, 15], [9, 16], [10, 13], [10, 14], [10, 15], [12, 13], [13, 15], [13, 16], [14, 17], [15, 16]], "family": "rlve_even_degree_partition", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-even-degree-partition-l4-s0/tests/run.py b/tasks/construct-rlve-even-degree-partition-l4-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l4-s0/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-even-degree-partition-l4-s0/tests/test.sh b/tasks/construct-rlve-even-degree-partition-l4-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l4-s0/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-even-degree-partition-l4-s7/environment/Dockerfile b/tasks/construct-rlve-even-degree-partition-l4-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l4-s7/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-even-degree-partition-l4-s7/environment/instance.json b/tasks/construct-rlve-even-degree-partition-l4-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..adb82f5707f57993572e413e214da5de85614583
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l4-s7/environment/instance.json
@@ -0,0 +1 @@
+{"n": 18, "edges": [[0, 1], [0, 3], [0, 9], [0, 10], [0, 13], [0, 14], [0, 15], [1, 3], [1, 4], [1, 5], [1, 6], [1, 10], [1, 11], [1, 14], [1, 15], [1, 16], [1, 17], [2, 4], [2, 9], [2, 10], [2, 12], [2, 15], [3, 7], [3, 8], [3, 9], [3, 10], [3, 11], [3, 12], [3, 13], [3, 14], [4, 5], [4, 6], [4, 8], [4, 10], [4, 13], [4, 14], [5, 8], [5, 10], [5, 11], [5, 12], [5, 13], [5, 15], [5, 16], [5, 17], [6, 9], [6, 10], [6, 14], [8, 10], [8, 12], [8, 13], [8, 17], [9, 10], [9, 11], [9, 12], [10, 11], [10, 13], [10, 14], [10, 15], [10, 17], [11, 12], [11, 13], [11, 15], [11, 16], [12, 14], [13, 14], [13, 15], [13, 16], [13, 17], [14, 17], [15, 17], [16, 17]], "family": "rlve_even_degree_partition", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-even-degree-partition-l4-s7/solution/solve.sh b/tasks/construct-rlve-even-degree-partition-l4-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..4512dd73001720af5181faa7888ce7f4bd385890
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l4-s7/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"groups": [2, 2, 2, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-even-degree-partition-l4-s7/tests/check.py b/tasks/construct-rlve-even-degree-partition-l4-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..9c7f0f71ea6d10264ececcc933b97a4a0e6eaf8e
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l4-s7/tests/check.py
@@ -0,0 +1,15 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n = inst["n"]
+    g = int_list(ans.get("groups"), n, 1, 2, "groups")
+    deg = [0] * n
+    for u, v in inst["edges"]:
+        if g[u] == g[v]:
+            deg[u] += 1
+            deg[v] += 1
+    for v in range(n):
+        if deg[v] % 2:
+            raise Invalid(f"vertex {v} has {deg[v]} same-group neighbours")
+    return None, {}
diff --git a/tasks/construct-rlve-even-degree-partition-l4-s7/tests/instance.json b/tasks/construct-rlve-even-degree-partition-l4-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..adb82f5707f57993572e413e214da5de85614583
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l4-s7/tests/instance.json
@@ -0,0 +1 @@
+{"n": 18, "edges": [[0, 1], [0, 3], [0, 9], [0, 10], [0, 13], [0, 14], [0, 15], [1, 3], [1, 4], [1, 5], [1, 6], [1, 10], [1, 11], [1, 14], [1, 15], [1, 16], [1, 17], [2, 4], [2, 9], [2, 10], [2, 12], [2, 15], [3, 7], [3, 8], [3, 9], [3, 10], [3, 11], [3, 12], [3, 13], [3, 14], [4, 5], [4, 6], [4, 8], [4, 10], [4, 13], [4, 14], [5, 8], [5, 10], [5, 11], [5, 12], [5, 13], [5, 15], [5, 16], [5, 17], [6, 9], [6, 10], [6, 14], [8, 10], [8, 12], [8, 13], [8, 17], [9, 10], [9, 11], [9, 12], [10, 11], [10, 13], [10, 14], [10, 15], [10, 17], [11, 12], [11, 13], [11, 15], [11, 16], [12, 14], [13, 14], [13, 15], [13, 16], [13, 17], [14, 17], [15, 17], [16, 17]], "family": "rlve_even_degree_partition", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-even-degree-partition-l4-s7/tests/run.py b/tasks/construct-rlve-even-degree-partition-l4-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l4-s7/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-even-degree-partition-l4-s7/tests/test.sh b/tasks/construct-rlve-even-degree-partition-l4-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l4-s7/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-graph-isomorphism-l2-s3/environment/Dockerfile b/tasks/construct-rlve-graph-isomorphism-l2-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l2-s3/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-graph-isomorphism-l2-s3/environment/instance.json b/tasks/construct-rlve-graph-isomorphism-l2-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c939948a2a0cef7daca1dda1124b97d50cd28fdb
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l2-s3/environment/instance.json
@@ -0,0 +1 @@
+{"n": 8, "edges1": [[1, 4], [1, 3], [3, 7], [0, 5], [4, 6], [3, 6], [2, 3], [4, 7], [1, 5], [4, 5], [3, 4], [0, 3]], "edges2": [[5, 7], [5, 6], [6, 7], [1, 5], [3, 4], [3, 5], [1, 7], [2, 7], [4, 7], [3, 6], [0, 7], [2, 5]], "family": "rlve_graph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-graph-isomorphism-l2-s3/solution/solve.sh b/tasks/construct-rlve-graph-isomorphism-l2-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..3cc71a1132996235efc9a119ba7fb69257ed7d84
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l2-s3/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"p": [4, 6, 0, 7, 5, 3, 2, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-graph-isomorphism-l2-s3/tests/check.py b/tasks/construct-rlve-graph-isomorphism-l2-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..64b755f5ab12e7270811ccfc2f02d104715517b7
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l2-s3/tests/check.py
@@ -0,0 +1,13 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n = inst["n"]
+    p = int_list(ans.get("p"), n, 0, n - 1, "p")
+    if len(set(p)) != n:
+        raise Invalid("p is not a permutation")
+    e2 = {(min(u, v), max(u, v)) for u, v in inst["edges2"]}
+    img = {(min(p[u], p[v]), max(p[u], p[v])) for u, v in inst["edges1"]}
+    if img != e2:
+        raise Invalid(f"{len(img - e2)} edges of G1 are not mapped onto edges of G2")
+    return None, {}
diff --git a/tasks/construct-rlve-graph-isomorphism-l2-s3/tests/instance.json b/tasks/construct-rlve-graph-isomorphism-l2-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c939948a2a0cef7daca1dda1124b97d50cd28fdb
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l2-s3/tests/instance.json
@@ -0,0 +1 @@
+{"n": 8, "edges1": [[1, 4], [1, 3], [3, 7], [0, 5], [4, 6], [3, 6], [2, 3], [4, 7], [1, 5], [4, 5], [3, 4], [0, 3]], "edges2": [[5, 7], [5, 6], [6, 7], [1, 5], [3, 4], [3, 5], [1, 7], [2, 7], [4, 7], [3, 6], [0, 7], [2, 5]], "family": "rlve_graph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-graph-isomorphism-l2-s3/tests/run.py b/tasks/construct-rlve-graph-isomorphism-l2-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l2-s3/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-graph-isomorphism-l2-s3/tests/test.sh b/tasks/construct-rlve-graph-isomorphism-l2-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l2-s3/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-graph-isomorphism-l6-s2/environment/Dockerfile b/tasks/construct-rlve-graph-isomorphism-l6-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l6-s2/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-graph-isomorphism-l6-s2/environment/instance.json b/tasks/construct-rlve-graph-isomorphism-l6-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..bfc9e08c342ccc1f086b033205a70c85a1719f6f
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l6-s2/environment/instance.json
@@ -0,0 +1 @@
+{"n": 40, "edges1": [[4, 36], [30, 34], [0, 6], [27, 32], [19, 26], [22, 31], [26, 29], [7, 34], [10, 12], [8, 38], [2, 16], [20, 33], [15, 23], [3, 10], [19, 35], [1, 26], [3, 39], [4, 16], [17, 36], [0, 24], [18, 38], [21, 39], [15, 17], [7, 26], [4, 29], [12, 14], [32, 39], [13, 32], [14, 34], [20, 29], [11, 24], [10, 29], [28, 29], [33, 36], [17, 22], [4, 30], [2, 4], [2, 13], [0, 16], [16, 37], [5, 22], [35, 39], [14, 25], [2, 10], [30, 35], [5, 16], [7, 31], [6, 37], [9, 20], [35, 38], [15, 34], [8, 32], [1, 33], [5, 12], [26, 32], [3, 25], [21, 31], [10, 28], [11, 29], [16, 21], [4, 9], [24, 27], [2, 20], [6, 14], [25, 28], [3, 30], [6, 17], [2, 8], [37, 39], [1, 22], [10, 20], [17, 33], [13, 25], [11, 14], [19, 21], [21, 36], [16, 28], [12, 21], [2, 33], [8, 15], [12, 27], [24, 33], [1, 16], [14, 33], [28, 33], [1, 32], [7, 38], [22, 23], [3, 16], [13, 23], [14, 22], [0, 10], [6, 8], [9, 12], [3, 6], [7, 21], [27, 35], [2, 28], [3, 18], [26, 28], [6, 12], [30, 32], [12, 16], 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[15, 35], [21, 37], [11, 21], [33, 39], [11, 27], [10, 39], [26, 33], [33, 34], [10, 15], [4, 12], [2, 3], [29, 38], [12, 26], [4, 10], [18, 35], [19, 29], [12, 30], [20, 31], [9, 27], [16, 39], [25, 34], [9, 10], [6, 36], [12, 33], [12, 34], [1, 8], [1, 12], [6, 24], [30, 33], [1, 21], [18, 20], [21, 29], [9, 15], [15, 36], [21, 24], [5, 14], [8, 33], [19, 27], [9, 30], [8, 23], [21, 25], [5, 34], [37, 38], [15, 24], [15, 27], [25, 39], [11, 19], [11, 36], [20, 28], [6, 26], [1, 23], [5, 7], [0, 35], [20, 30], [21, 34], [24, 35], [0, 21], [34, 37], [11, 20], [3, 8], [5, 13], [11, 18], [26, 39], [18, 24], [10, 21], [21, 38], [13, 33], [10, 19], [11, 12], [29, 39], [10, 30], [7, 19], [1, 24], [3, 5], [11, 33], [0, 7], [14, 21], [8, 24], [0, 15], [1, 34], [5, 9], [8, 35], [19, 24], [32, 38], [6, 9], [9, 16], [32, 37], [21, 22], [0, 9], [12, 20], [8, 17], [29, 34], [4, 32], [8, 12], [31, 32], [19, 38], [2, 35], [0, 31], [9, 29], [12, 28], [0, 29], [0, 38], [8, 16], [2, 14], [6, 39], [21, 30], [29, 35], [38, 39], [8, 30], [6, 33], [28, 36], [1, 18], [29, 30], [9, 23], [12, 24]], "edges2": [[25, 27], [1, 32], [8, 11], [21, 34], [26, 32], [28, 31], [12, 24], [2, 21], [2, 20], [6, 7], [32, 34], [20, 22], [20, 32], [9, 12], [11, 24], [0, 30], [0, 4], [26, 28], [24, 38], [26, 37], [10, 19], [17, 18], [18, 27], [6, 39], [16, 30], [20, 37], [4, 31], [12, 26], [15, 38], [24, 26], [1, 31], [6, 11], [0, 12], [4, 17], [5, 8], [21, 22], [12, 22], [13, 31], [28, 30], [11, 39], [20, 39], [26, 36], [3, 8], [6, 17], [16, 31], [3, 23], [25, 38], [12, 31], [7, 21], [4, 16], [19, 33], [13, 19], [0, 13], [32, 37], [10, 15], [6, 12], [15, 36], [5, 22], [7, 11], [7, 18], [14, 39], [9, 34], [1, 26], [1, 14], [15, 16], [23, 26], [5, 28], [7, 14], [16, 33], [10, 12], [19, 23], [13, 18], [2, 34], [11, 37], [10, 18], [4, 12], [5, 11], [14, 20], [7, 38], [10, 39], [8, 32], [0, 37], [14, 17], [35, 38], [2, 32], [1, 8], [2, 25], [19, 37], [27, 31], [9, 39], [0, 10], [17, 20], [3, 25], [15, 31], [33, 36], [4, 14], [8, 37], [2, 30], [1, 11], [30, 32], [12, 35], [9, 10], [8, 29], [21, 24], [35, 39], [3, 32], [16, 20], [5, 24], [10, 38], [23, 28], [22, 33], [24, 35], [18, 39], [9, 28], [3, 12], [18, 31], [8, 9], [11, 27], [16, 19], [11, 34], [10, 16], [15, 17], [2, 24], [7, 23], [22, 26], [11, 36], [3, 6], [1, 36], [19, 36], [7, 27], [31, 34], [28, 39], [14, 34], [1, 7], [7, 31], [15, 25], [4, 30], [1, 27], [14, 36], [30, 39], [5, 12], [13, 14], [28, 35], [3, 28], [29, 37], [11, 17], [11, 28], [37, 39], [21, 31], [26, 29], [12, 16], [27, 37], [5, 13], [11, 20], [8, 26], [22, 24], [31, 33], [2, 3], [8, 31], [8, 13], [0, 6], [3, 11], [15, 39], [1, 18], [17, 30], [14, 37], [19, 24], [19, 29], [29, 35], [21, 38], [1, 6], [29, 32], [27, 39], [27, 29], [29, 38], [15, 24], [23, 32], [4, 11], [2, 29], [8, 28], [20, 27], [14, 28], [6, 37], [22, 36], [26, 35], [21, 39], [26, 27], [10, 33], [0, 22], [12, 34], [21, 29], [6, 16], [10, 34], [9, 24], [4, 19], [1, 15], [28, 33], [4, 18], [7, 15], [3, 35], [10, 24], [3, 9], [23, 35], [17, 25], [15, 34], [6, 28], [6, 30], [9, 19], [31, 37], [14, 18], [0, 24], [1, 20], [24, 36], [5, 25], [22, 23], [36, 37], [8, 15], [2, 18], [16, 36], [26, 38], [0, 11], [17, 22], [6, 10], [15, 26], [4, 15], [1, 29], [11, 30], [22, 38], [14, 33], [31, 39], [27, 34], [2, 14], [23, 37], [16, 27], [13, 36], [24, 39], [27, 28], [9, 33], [31, 36], [3, 22], [30, 34], [13, 33], [3, 31], [24, 27], [13, 39], [8, 36], [13, 34], [13, 22], [8, 27], [21, 30], [15, 33], [11, 35], [8, 24], [16, 24], [4, 27], [29, 34], [8, 16], [2, 33], [33, 35], [13, 21], [0, 36], [12, 30], [9, 17], [34, 36], [5, 15], [2, 11], [0, 8], [13, 24], [0, 23], [10, 14], [30, 38], [19, 32], [1, 33], [13, 37], [4, 8], [11, 23], [9, 13], [6, 29], [19, 35], [5, 16], [5, 21], [4, 26], [9, 27], [33, 39], [19, 21], [0, 34], [10, 37], [8, 39], [24, 29], [9, 23], [2, 17], [12, 29], [4, 5], [15, 23], [15, 18], [7, 12], [11, 26], [3, 36], [13, 29], [3, 27], [10, 31], [7, 30], [4, 32], [30, 31], [5, 17], [20, 34], [9, 31], [0, 26], [16, 35], [23, 30], [15, 37], [16, 22], [16, 17], [5, 27], [6, 15], [15, 22], [9, 14], [5, 23], [9, 22], [12, 20], [12, 18], [0, 25], [5, 7], [18, 25], [22, 28], [19, 30], [11, 16], [11, 38], [14, 31], [3, 19], [9, 16], [13, 16], [9, 26], [9, 21], [28, 38], [0, 31], [22, 30], [1, 16], [9, 30], [4, 25], [4, 33], [0, 39], [24, 37], [17, 23], [8, 17], [1, 4], [20, 33], [8, 33], [13, 25], [24, 32], [3, 38], [25, 31], [21, 25], [1, 10], [11, 12], [19, 31], [26, 33], [8, 34], [19, 22], [29, 39], [30, 36], [34, 35], [18, 38], [6, 21], [23, 39], [29, 30], [9, 36], [4, 10], [4, 28], [12, 25], [12, 32], [14, 35], [3, 20], [1, 35], [25, 30], [6, 35], [7, 36], [8, 19], [30, 33], [7, 10], [9, 11], [25, 33], [20, 29], [7, 20], [28, 36], [0, 15], [23, 29], [8, 14], [3, 18], [18, 23], [17, 34], [16, 23], [5, 33], [6, 8], [27, 30], [11, 22], [3, 30], [1, 19], [2, 19], [12, 27], [9, 20], [1, 34], [18, 21], [1, 37], [15, 27], [6, 9], [11, 32], [1, 21], [2, 22], [3, 34], [12, 38], [13, 15], [2, 9], [29, 36], [7, 24], [13, 30], [17, 29], [0, 19], [12, 14], [0, 35], [0, 5], [0, 14], [7, 37], [10, 25], [14, 21], [2, 26], [1, 22], [12, 33], [11, 19]], "family": "rlve_graph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-graph-isomorphism-l6-s2/solution/solve.sh b/tasks/construct-rlve-graph-isomorphism-l6-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..3e14d3c5ad3eb8ceebe0f864ab0baa80bc6adbe9
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l6-s2/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"p": [30, 27, 10, 37, 33, 29, 0, 32, 31, 13, 19, 17, 8, 38, 6, 21, 39, 25, 20, 2, 16, 11, 12, 18, 34, 7, 26, 14, 24, 22, 36, 4, 28, 15, 1, 9, 5, 35, 3, 23]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-graph-isomorphism-l6-s2/tests/check.py b/tasks/construct-rlve-graph-isomorphism-l6-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..64b755f5ab12e7270811ccfc2f02d104715517b7
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l6-s2/tests/check.py
@@ -0,0 +1,13 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n = inst["n"]
+    p = int_list(ans.get("p"), n, 0, n - 1, "p")
+    if len(set(p)) != n:
+        raise Invalid("p is not a permutation")
+    e2 = {(min(u, v), max(u, v)) for u, v in inst["edges2"]}
+    img = {(min(p[u], p[v]), max(p[u], p[v])) for u, v in inst["edges1"]}
+    if img != e2:
+        raise Invalid(f"{len(img - e2)} edges of G1 are not mapped onto edges of G2")
+    return None, {}
diff --git a/tasks/construct-rlve-graph-isomorphism-l6-s2/tests/instance.json b/tasks/construct-rlve-graph-isomorphism-l6-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..bfc9e08c342ccc1f086b033205a70c85a1719f6f
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l6-s2/tests/instance.json
@@ -0,0 +1 @@
+{"n": 40, "edges1": [[4, 36], [30, 34], [0, 6], [27, 32], [19, 26], [22, 31], [26, 29], [7, 34], [10, 12], [8, 38], [2, 16], [20, 33], [15, 23], [3, 10], [19, 35], [1, 26], [3, 39], [4, 16], [17, 36], [0, 24], [18, 38], [21, 39], [15, 17], [7, 26], [4, 29], [12, 14], [32, 39], [13, 32], [14, 34], [20, 29], [11, 24], [10, 29], [28, 29], [33, 36], [17, 22], [4, 30], [2, 4], [2, 13], [0, 16], [16, 37], [5, 22], [35, 39], [14, 25], [2, 10], [30, 35], [5, 16], [7, 31], [6, 37], [9, 20], [35, 38], [15, 34], [8, 32], [1, 33], [5, 12], [26, 32], [3, 25], [21, 31], [10, 28], [11, 29], [16, 21], [4, 9], [24, 27], [2, 20], [6, 14], [25, 28], [3, 30], [6, 17], [2, 8], [37, 39], [1, 22], [10, 20], [17, 33], [13, 25], [11, 14], [19, 21], [21, 36], [16, 28], [12, 21], [2, 33], [8, 15], [12, 27], [24, 33], [1, 16], [14, 33], [28, 33], [1, 32], [7, 38], [22, 23], [3, 16], [13, 23], [14, 22], [0, 10], [6, 8], [9, 12], [3, 6], [7, 21], [27, 35], [2, 28], [3, 18], [26, 28], [6, 12], [30, 32], [12, 16], 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\ No newline at end of file
diff --git a/tasks/construct-rlve-graph-isomorphism-l6-s2/tests/run.py b/tasks/construct-rlve-graph-isomorphism-l6-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l6-s2/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-graph-isomorphism-l6-s2/tests/test.sh b/tasks/construct-rlve-graph-isomorphism-l6-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l6-s2/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-graph-isomorphism-l6-s5/environment/Dockerfile b/tasks/construct-rlve-graph-isomorphism-l6-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l6-s5/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-graph-isomorphism-l6-s5/environment/instance.json b/tasks/construct-rlve-graph-isomorphism-l6-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..233ca16173d9601a72a7d42ad7e64e5979237697
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l6-s5/environment/instance.json
@@ -0,0 +1 @@
+{"n": 40, "edges1": [[13, 37], [14, 22], [8, 12], [33, 39], [4, 34], [18, 36], [12, 14], [15, 24], [9, 39], [31, 35], [15, 39], [13, 34], [2, 20], [24, 39], [9, 19], [5, 25], [19, 28], [10, 15], [12, 30], [8, 38], [1, 24], [29, 39], [15, 21], [16, 37], [25, 38], [3, 29], [25, 35], [8, 36], [9, 11], [27, 36], [3, 27], [2, 4], [5, 19], [20, 24], [13, 32], [12, 18], [3, 24], [9, 25], [17, 22], [30, 32], [25, 37], [10, 11], [29, 30], [0, 30], [10, 39], [27, 33], [14, 31], [13, 16], [1, 34], [4, 21], [5, 38], [30, 34], [6, 23], [5, 12], [15, 27], [21, 38], [3, 36], [13, 20], [3, 13], [7, 39], [14, 17], [14, 37], [16, 21], [35, 38], [0, 37], [18, 38], [6, 35], [4, 27], [26, 28], [20, 21], [3, 5], [25, 39], [12, 13], [6, 26], [1, 10], [13, 18], [13, 24], [3, 4], [4, 29], [17, 37], [17, 25], [28, 38], [7, 20], [9, 16], [6, 36], [0, 10], [23, 24], [16, 32], [0, 19], [4, 8], [3, 37], [4, 16], [19, 20], [20, 27], [0, 9], [25, 28], [12, 23], [29, 37], [7, 21], [19, 36], [26, 32], [15, 31], [8, 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39], [11, 37], [15, 16], [31, 35], [0, 34], [22, 35], [7, 29], [19, 22], [6, 32], [20, 37], [5, 24], [3, 14], [9, 30], [0, 15], [20, 26], [2, 18], [9, 14], [30, 37], [17, 38], [13, 18], [9, 12], [13, 37]], "family": "rlve_graph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-graph-isomorphism-l6-s5/solution/solve.sh b/tasks/construct-rlve-graph-isomorphism-l6-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..3fe11644c15dca61c9159b479283b4f5fd26d98b
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l6-s5/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"p": [21, 0, 36, 9, 39, 37, 32, 2, 14, 23, 30, 8, 35, 1, 25, 38, 28, 6, 27, 24, 31, 11, 19, 33, 20, 29, 7, 26, 12, 4, 34, 10, 22, 18, 15, 17, 3, 5, 13, 16]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-graph-isomorphism-l6-s5/tests/check.py b/tasks/construct-rlve-graph-isomorphism-l6-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..64b755f5ab12e7270811ccfc2f02d104715517b7
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l6-s5/tests/check.py
@@ -0,0 +1,13 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n = inst["n"]
+    p = int_list(ans.get("p"), n, 0, n - 1, "p")
+    if len(set(p)) != n:
+        raise Invalid("p is not a permutation")
+    e2 = {(min(u, v), max(u, v)) for u, v in inst["edges2"]}
+    img = {(min(p[u], p[v]), max(p[u], p[v])) for u, v in inst["edges1"]}
+    if img != e2:
+        raise Invalid(f"{len(img - e2)} edges of G1 are not mapped onto edges of G2")
+    return None, {}
diff --git a/tasks/construct-rlve-graph-isomorphism-l6-s5/tests/instance.json b/tasks/construct-rlve-graph-isomorphism-l6-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..233ca16173d9601a72a7d42ad7e64e5979237697
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l6-s5/tests/instance.json
@@ -0,0 +1 @@
+{"n": 40, "edges1": [[13, 37], [14, 22], [8, 12], [33, 39], [4, 34], [18, 36], [12, 14], [15, 24], [9, 39], [31, 35], [15, 39], [13, 34], [2, 20], [24, 39], [9, 19], [5, 25], [19, 28], [10, 15], [12, 30], [8, 38], [1, 24], [29, 39], [15, 21], [16, 37], [25, 38], [3, 29], [25, 35], [8, 36], [9, 11], [27, 36], [3, 27], [2, 4], [5, 19], [20, 24], [13, 32], [12, 18], [3, 24], [9, 25], [17, 22], [30, 32], [25, 37], [10, 11], [29, 30], [0, 30], [10, 39], [27, 33], [14, 31], [13, 16], [1, 34], [4, 21], [5, 38], [30, 34], [6, 23], [5, 12], [15, 27], [21, 38], [3, 36], [13, 20], [3, 13], [7, 39], [14, 17], [14, 37], [16, 21], [35, 38], [0, 37], [18, 38], [6, 35], [4, 27], [26, 28], [20, 21], [3, 5], [25, 39], [12, 13], [6, 26], [1, 10], [13, 18], [13, 24], [3, 4], [4, 29], [17, 37], [17, 25], [28, 38], [7, 20], [9, 16], [6, 36], [0, 10], [23, 24], [16, 32], [0, 19], [4, 8], [3, 37], [4, 16], [19, 20], [20, 27], [0, 9], [25, 28], [12, 23], [29, 37], [7, 21], [19, 36], [26, 32], [15, 31], [8, 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[16, 39], [13, 34], [36, 37], [14, 15], [1, 23], [10, 28], [30, 35], [31, 38], [26, 36], [4, 15], [3, 20], [21, 22], [2, 31], [8, 18], [29, 36], [6, 7], [35, 39], [21, 24], [1, 5], [15, 19], [4, 39], [31, 36], [0, 35], [19, 20], [2, 39], [7, 22], [17, 31], [18, 36], [21, 34], [12, 20], [10, 32], [29, 37], [35, 37], [1, 20], [20, 33], [8, 27], [22, 36], [18, 28], [1, 18], [19, 33], [10, 31], [6, 17], [24, 27], [13, 22], [32, 33], [15, 24], [21, 30], [5, 30], [16, 18], [8, 29], [11, 38], [24, 35], [12, 27], [17, 24], [27, 28], [8, 10], [28, 39], [34, 35], [0, 16], [15, 35], [18, 26], [26, 27], [3, 39], [8, 13], [17, 36], [7, 27], [15, 38], [14, 32], [20, 25], [25, 31], [31, 33], [3, 25], [10, 13], [26, 29], [14, 39], [3, 26], [10, 36], [11, 29], [14, 33], [6, 11], [12, 32], [3, 22], [9, 19], [4, 34], [12, 35], [37, 38], [9, 38], [9, 37], [27, 34], [11, 30], [21, 31], [9, 29], [9, 34], [6, 36], [12, 15], [17, 27], [38, 39], [19, 34], [0, 17], [8, 30], [5, 18], [10, 33], [9, 32], [24, 32], [5, 10], [27, 30], [15, 21], [32, 36], [4, 31], [1, 25], [11, 39], [13, 14], [7, 28], [2, 23], [1, 16], [3, 24], [6, 13], [35, 38], [4, 24], [14, 18], [10, 14], [0, 20], [7, 9], [24, 39], [19, 25], [6, 25], [3, 34], [1, 35], [8, 21], [13, 17], [10, 25], [0, 26], [7, 12], [18, 31], [10, 38], [2, 37], [6, 20], [16, 29], [3, 36], [12, 13], [24, 37], [25, 35], [33, 35], [9, 21], [8, 38], [11, 12], [6, 29], [8, 23], [4, 16], [13, 28], [1, 27], [8, 12], [20, 38], [12, 38], [13, 16], [5, 23], [7, 31], [5, 28], [23, 29], [17, 35], [5, 35], [12, 24], [4, 28], [7, 8], [35, 36], [14, 35], [12, 34], [2, 11], [1, 32], [23, 31], [16, 17], [9, 23], [15, 39], [1, 2], [25, 34], [12, 29], [3, 10], [0, 36], [0, 8], [6, 10], [26, 34], [5, 38], [9, 26], [7, 23], [17, 34], [25, 27], [13, 20], [6, 16], [23, 32], [5, 15], [1, 10], [2, 20], [6, 34], [23, 28], [24, 36], [30, 34], [4, 25], [17, 39], [5, 6], [11, 31], [2, 12], [18, 39], [20, 27], [2, 16], [15, 33], [14, 22], [8, 20], [36, 38], [1, 26], [16, 21], [25, 32], [1, 39], [21, 35], [15, 34], [2, 3], [17, 25], [13, 24], [12, 16], [1, 31], [32, 38], [13, 23], [15, 25], [16, 36], [9, 20], [22, 27], [2, 17], [9, 39], [6, 31], [18, 21], [1, 11], [19, 29], [27, 31], [13, 35], [20, 34], [4, 12], [11, 14], [19, 35], [3, 28], [5, 13], [11, 28], [23, 27], [5, 32], [13, 33], [6, 38], [16, 35], [4, 26], [14, 28], [25, 36], [2, 36], [5, 29], [18, 19], [4, 36], [24, 31], [26, 28], [26, 38], [0, 7], [8, 39], [1, 15], [31, 37], [23, 38], [16, 25], [31, 39], [20, 23], [13, 27], [5, 19], [12, 28], [3, 15], [3, 23], [14, 26], [0, 3], [20, 24], [30, 31], [9, 16], [33, 36], [19, 28], [16, 23], [22, 28], [3, 32], [23, 24], [1, 6], [16, 32], [5, 31], [0, 29], [10, 27], [3, 35], [8, 22], [4, 8], [10, 17], [16, 30], [27, 39], [17, 29], [12, 37], [16, 20], [24, 26], [2, 38], [0, 10], [0, 6], [20, 29], [18, 29], [12, 21], [16, 38], [4, 5], [13, 15], [9, 35], [22, 31], [2, 24], [14, 16], [1, 28], [16, 19], [20, 31], [10, 39], [3, 29], [13, 29], [7, 32], [21, 39], [11, 37], [15, 16], [31, 35], [0, 34], [22, 35], [7, 29], [19, 22], [6, 32], [20, 37], [5, 24], [3, 14], [9, 30], [0, 15], [20, 26], [2, 18], [9, 14], [30, 37], [17, 38], [13, 18], [9, 12], [13, 37]], "family": "rlve_graph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-graph-isomorphism-l6-s5/tests/run.py b/tasks/construct-rlve-graph-isomorphism-l6-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l6-s5/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-graph-isomorphism-l6-s5/tests/test.sh b/tasks/construct-rlve-graph-isomorphism-l6-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l6-s5/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-grid-parity-construction-l2-s4/environment/Dockerfile b/tasks/construct-rlve-grid-parity-construction-l2-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l2-s4/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-grid-parity-construction-l2-s4/environment/instance.json b/tasks/construct-rlve-grid-parity-construction-l2-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..7d54b8cb5f4719015d0b912ad573c1a63d11a112
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l2-s4/environment/instance.json
@@ -0,0 +1 @@
+{"parity": ["00011", "01011", "11000", "11110"], "family": "rlve_grid_parity_construction", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-grid-parity-construction-l2-s4/solution/solve.sh b/tasks/construct-rlve-grid-parity-construction-l2-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..921e124d8725ee2877900163db49534fbc6c508c
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l2-s4/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"grid": ["00000", "00011", "01111", "01101"]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-grid-parity-construction-l2-s4/tests/check.py b/tasks/construct-rlve-grid-parity-construction-l2-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..89d5d286593348d0000ac10db74e1c6242be6c97
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l2-s4/tests/check.py
@@ -0,0 +1,22 @@
+from run import Invalid
+
+
+def check(inst, ans):
+    P = inst["parity"]
+    N, M = len(P), len(P[0])
+    G = ans.get("grid")
+    if not isinstance(G, list) or len(G) != N:
+        raise Invalid(f"grid must be a list of {N} strings")
+    for r in G:
+        if not isinstance(r, str) or len(r) != M or set(r) - {"0", "1"}:
+            raise Invalid(f"each row must be a string of {M} characters '0'/'1'")
+    for i in range(N):
+        for j in range(M):
+            v = int(G[i][j])
+            for di, dj in ((-1, 0), (1, 0), (0, -1), (0, 1)):
+                a, b = i + di, j + dj
+                if 0 <= a < N and 0 <= b < M:
+                    v ^= int(G[a][b])
+            if v != int(P[i][j]):
+                raise Invalid(f"parity mismatch at ({i}, {j})")
+    return None, {}
diff --git a/tasks/construct-rlve-grid-parity-construction-l2-s4/tests/instance.json b/tasks/construct-rlve-grid-parity-construction-l2-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..7d54b8cb5f4719015d0b912ad573c1a63d11a112
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l2-s4/tests/instance.json
@@ -0,0 +1 @@
+{"parity": ["00011", "01011", "11000", "11110"], "family": "rlve_grid_parity_construction", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-grid-parity-construction-l2-s4/tests/run.py b/tasks/construct-rlve-grid-parity-construction-l2-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l2-s4/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-grid-parity-construction-l2-s4/tests/test.sh b/tasks/construct-rlve-grid-parity-construction-l2-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l2-s4/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-grid-parity-construction-l6-s2/environment/Dockerfile b/tasks/construct-rlve-grid-parity-construction-l6-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l6-s2/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-grid-parity-construction-l6-s2/environment/instance.json b/tasks/construct-rlve-grid-parity-construction-l6-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e74bf2320c026479cbf12a70285ebd0c470d0f66
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l6-s2/environment/instance.json
@@ -0,0 +1 @@
+{"parity": ["1101110101100000001", "1000010100111111101", "0100011100001110011", "1000110110111110110", "0001101001001101001", "1111010100101001001"], "family": "rlve_grid_parity_construction", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-grid-parity-construction-l6-s2/solution/solve.sh b/tasks/construct-rlve-grid-parity-construction-l6-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..124b4f6a01d98e4070727b112e228bdbbd230c18
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l6-s2/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"grid": ["1111011110111111111", "1011111001111111111", "1110111100111111100", "1011111110101110110", "1111110110100101011", "1101110111011110111"]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-grid-parity-construction-l6-s2/tests/check.py b/tasks/construct-rlve-grid-parity-construction-l6-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..89d5d286593348d0000ac10db74e1c6242be6c97
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l6-s2/tests/check.py
@@ -0,0 +1,22 @@
+from run import Invalid
+
+
+def check(inst, ans):
+    P = inst["parity"]
+    N, M = len(P), len(P[0])
+    G = ans.get("grid")
+    if not isinstance(G, list) or len(G) != N:
+        raise Invalid(f"grid must be a list of {N} strings")
+    for r in G:
+        if not isinstance(r, str) or len(r) != M or set(r) - {"0", "1"}:
+            raise Invalid(f"each row must be a string of {M} characters '0'/'1'")
+    for i in range(N):
+        for j in range(M):
+            v = int(G[i][j])
+            for di, dj in ((-1, 0), (1, 0), (0, -1), (0, 1)):
+                a, b = i + di, j + dj
+                if 0 <= a < N and 0 <= b < M:
+                    v ^= int(G[a][b])
+            if v != int(P[i][j]):
+                raise Invalid(f"parity mismatch at ({i}, {j})")
+    return None, {}
diff --git a/tasks/construct-rlve-grid-parity-construction-l6-s2/tests/instance.json b/tasks/construct-rlve-grid-parity-construction-l6-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e74bf2320c026479cbf12a70285ebd0c470d0f66
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l6-s2/tests/instance.json
@@ -0,0 +1 @@
+{"parity": ["1101110101100000001", "1000010100111111101", "0100011100001110011", "1000110110111110110", "0001101001001101001", "1111010100101001001"], "family": "rlve_grid_parity_construction", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-grid-parity-construction-l6-s2/tests/run.py b/tasks/construct-rlve-grid-parity-construction-l6-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l6-s2/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-grid-parity-construction-l6-s2/tests/test.sh b/tasks/construct-rlve-grid-parity-construction-l6-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l6-s2/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-grid-parity-construction-l6-s5/environment/Dockerfile b/tasks/construct-rlve-grid-parity-construction-l6-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l6-s5/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-grid-parity-construction-l6-s5/environment/instance.json b/tasks/construct-rlve-grid-parity-construction-l6-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c4859f0e5c23b9e6555fdacedad2e6453dc27dad
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l6-s5/environment/instance.json
@@ -0,0 +1 @@
+{"parity": ["000000", "000000", "000000", "000000", "000000", "000000", "000000", "000000", "000000", "000000", "000000", "000000", "100000", "110000", "100000", "000000", "010000", "101100", "001110", "111000", "101000", "000000"], "family": "rlve_grid_parity_construction", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-grid-parity-construction-l6-s5/solution/solve.sh b/tasks/construct-rlve-grid-parity-construction-l6-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..86542e768ddefe9d780e8af602b6e8f78b6396bf
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l6-s5/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"grid": ["000000", "000000", "000000", "000000", "000000", "000000", "000000", "000000", "000000", "000000", "000000", "000000", "000000", "100000", "000000", "000000", "000000", "010000", "010100", "101000", "000000", "000000"]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-grid-parity-construction-l6-s5/tests/check.py b/tasks/construct-rlve-grid-parity-construction-l6-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..89d5d286593348d0000ac10db74e1c6242be6c97
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l6-s5/tests/check.py
@@ -0,0 +1,22 @@
+from run import Invalid
+
+
+def check(inst, ans):
+    P = inst["parity"]
+    N, M = len(P), len(P[0])
+    G = ans.get("grid")
+    if not isinstance(G, list) or len(G) != N:
+        raise Invalid(f"grid must be a list of {N} strings")
+    for r in G:
+        if not isinstance(r, str) or len(r) != M or set(r) - {"0", "1"}:
+            raise Invalid(f"each row must be a string of {M} characters '0'/'1'")
+    for i in range(N):
+        for j in range(M):
+            v = int(G[i][j])
+            for di, dj in ((-1, 0), (1, 0), (0, -1), (0, 1)):
+                a, b = i + di, j + dj
+                if 0 <= a < N and 0 <= b < M:
+                    v ^= int(G[a][b])
+            if v != int(P[i][j]):
+                raise Invalid(f"parity mismatch at ({i}, {j})")
+    return None, {}
diff --git a/tasks/construct-rlve-grid-parity-construction-l6-s5/tests/instance.json b/tasks/construct-rlve-grid-parity-construction-l6-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c4859f0e5c23b9e6555fdacedad2e6453dc27dad
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l6-s5/tests/instance.json
@@ -0,0 +1 @@
+{"parity": ["000000", "000000", "000000", "000000", "000000", "000000", "000000", "000000", "000000", "000000", "000000", "000000", "100000", "110000", "100000", "000000", "010000", "101100", "001110", "111000", "101000", "000000"], "family": "rlve_grid_parity_construction", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-grid-parity-construction-l6-s5/tests/run.py b/tasks/construct-rlve-grid-parity-construction-l6-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l6-s5/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-grid-parity-construction-l6-s5/tests/test.sh b/tasks/construct-rlve-grid-parity-construction-l6-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l6-s5/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-imp-party-l4-s4/environment/Dockerfile b/tasks/construct-rlve-imp-party-l4-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-l4-s4/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-imp-party-l4-s4/environment/instance.json b/tasks/construct-rlve-imp-party-l4-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3f24dac74a54af9a0f3712a2fc7dcb61c1858437
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-l4-s4/environment/instance.json
@@ -0,0 +1 @@
+{"n": 12, "edges": [[0, 1], [0, 3], [0, 4], [0, 5], [0, 6], [0, 7], [0, 8], [0, 10], [0, 11], [0, 12], [0, 13], [0, 14], [0, 15], [0, 16], [0, 17], [0, 18], [0, 19], [0, 20], [0, 21], [0, 23], [0, 25], [0, 27], [0, 28], [0, 29], [0, 30], [0, 31], [0, 32], [0, 33], [1, 2], [1, 3], [1, 4], [1, 5], [1, 6], [1, 7], [1, 8], [1, 10], [1, 11], [1, 12], [1, 13], [1, 14], [1, 15], [1, 16], [1, 17], [1, 18], [1, 19], [1, 21], [1, 23], [1, 25], [1, 27], [1, 28], [1, 29], [1, 30], [1, 31], [1, 32], [1, 33], [1, 34], [2, 3], [2, 7], [2, 11], [2, 15], [2, 18], [2, 25], [2, 27], [2, 28], [2, 31], [2, 32], [2, 33], [3, 5], [3, 6], [3, 7], [3, 8], [3, 11], [3, 12], [3, 14], [3, 15], [3, 16], [3, 17], [3, 18], [3, 19], [3, 20], [3, 21], [3, 22], [3, 23], [3, 24], [3, 25], [3, 26], [3, 27], [3, 28], [3, 29], [3, 31], [3, 32], [3, 33], [3, 35], [4, 7], [4, 8], [4, 11], [4, 16], [4, 19], [4, 23], [4, 25], [4, 27], [4, 29], [4, 32], [5, 6], [5, 7], [5, 8], [5, 9], [5, 11], [5, 12], [5, 14], [5, 15], [5, 16], [5, 17], [5, 18], [5, 19], [5, 21], [5, 22], [5, 23], [5, 24], [5, 25], [5, 27], [5, 28], [5, 29], [5, 31], [5, 32], [5, 33], [5, 35], [6, 7], [6, 8], [6, 10], [6, 11], [6, 12], [6, 13], [6, 14], [6, 15], [6, 16], [6, 17], [6, 18], [6, 19], [6, 21], [6, 23], [6, 25], [6, 26], [6, 27], [6, 28], [6, 29], [6, 31], [6, 32], [6, 33], [6, 34], [7, 8], [7, 11], [7, 12], [7, 14], [7, 15], [7, 16], [7, 17], [7, 18], [7, 19], [7, 21], [7, 23], [7, 24], [7, 25], [7, 26], [7, 27], [7, 28], [7, 29], [7, 30], [7, 31], [7, 32], [7, 33], [8, 11], [8, 12], [8, 13], [8, 14], [8, 15], [8, 16], [8, 17], [8, 18], [8, 19], [8, 20], [8, 21], [8, 22], [8, 23], [8, 24], [8, 25], [8, 26], [8, 27], [8, 28], [8, 29], [8, 31], [8, 32], [8, 33], [8, 34], [8, 35], [9, 12], [9, 14], [9, 15], [9, 16], [9, 23], [9, 25], [9, 28], [9, 31], [10, 11], [10, 15], [10, 16], [10, 21], [10, 23], [10, 25], [10, 28], [10, 32], [11, 12], [11, 14], [11, 15], [11, 16], [11, 17], [11, 18], [11, 19], [11, 20], [11, 21], [11, 23], [11, 24], [11, 25], [11, 26], [11, 27], [11, 28], [11, 29], [11, 31], [11, 32], [11, 33], [11, 34], [11, 35], [12, 13], [12, 14], [12, 15], [12, 16], [12, 17], [12, 18], [12, 19], [12, 20], [12, 21], [12, 22], [12, 23], [12, 25], [12, 27], [12, 28], [12, 29], [12, 30], [12, 31], [12, 32], [12, 33], [12, 35], [13, 14], [13, 15], [13, 18], [13, 19], [13, 25], [13, 29], [13, 32], [13, 33], [14, 15], [14, 16], [14, 17], [14, 18], [14, 19], [14, 21], [14, 22], [14, 23], [14, 25], [14, 27], [14, 28], [14, 29], [14, 31], [14, 32], [14, 33], [14, 34], [14, 35], [15, 16], [15, 17], [15, 18], [15, 19], [15, 21], [15, 23], [15, 24], [15, 25], [15, 27], [15, 28], [15, 29], [15, 30], [15, 31], [15, 32], [15, 33], [15, 34], [15, 35], [16, 17], [16, 18], [16, 19], [16, 20], [16, 21], [16, 22], [16, 23], [16, 24], [16, 25], [16, 27], [16, 28], [16, 29], [16, 31], [16, 32], [16, 33], [16, 35], [17, 18], [17, 19], [17, 21], [17, 22], [17, 23], [17, 25], [17, 26], [17, 27], [17, 28], [17, 29], [17, 31], [17, 32], [17, 33], [17, 35], [18, 19], [18, 21], [18, 23], [18, 25], [18, 26], [18, 27], [18, 28], [18, 29], [18, 31], [18, 32], [18, 33], [18, 35], [19, 21], [19, 23], [19, 25], [19, 26], [19, 27], [19, 28], [19, 29], [19, 31], [19, 32], [19, 33], [19, 34], [20, 23], [20, 25], [20, 28], [20, 32], [21, 23], [21, 24], [21, 25], [21, 26], [21, 27], [21, 28], [21, 29], [21, 31], [21, 32], [21, 33], [21, 34], [21, 35], [22, 23], [22, 25], [22, 29], [22, 32], [22, 33], [23, 24], [23, 25], [23, 27], [23, 28], [23, 29], [23, 30], [23, 31], [23, 32], [23, 33], [23, 34], [24, 25], [24, 28], [24, 29], [24, 32], [24, 33], [25, 27], [25, 28], [25, 29], [25, 31], [25, 32], [25, 33], [25, 34], [25, 35], [26, 32], [26, 33], [27, 28], [27, 29], [27, 30], [27, 31], [27, 32], [27, 33], [27, 35], [28, 29], [28, 31], [28, 32], [28, 33], [28, 34], [29, 30], [29, 31], [29, 32], [29, 33], [29, 34], [30, 32], [31, 32], [31, 33], [31, 34], [31, 35], [32, 33], [32, 34], [33, 34]], "family": "rlve_imp_party", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-imp-party-l4-s4/solution/solve.sh b/tasks/construct-rlve-imp-party-l4-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..c4ce92e7a067b5142981b989f073c943ecaea041
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-l4-s4/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"clique": [6, 7, 11, 14, 16, 17, 19, 23, 25, 27, 31, 33]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-imp-party-l4-s4/tests/check.py b/tasks/construct-rlve-imp-party-l4-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..97b5a037a2ba8bf4495cf87cc074d1c60b11311c
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-l4-s4/tests/check.py
@@ -0,0 +1,14 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n = inst["n"]
+    c = int_list(ans.get("clique"), n, 0, 3 * n - 1, "clique")
+    if len(set(c)) != n:
+        raise Invalid("vertices must be distinct")
+    es = {(min(u, v), max(u, v)) for u, v in inst["edges"]}
+    for i in range(n):
+        for j in range(i + 1, n):
+            if (min(c[i], c[j]), max(c[i], c[j])) not in es:
+                raise Invalid(f"{c[i]} and {c[j]} are not adjacent")
+    return None, {}
diff --git a/tasks/construct-rlve-imp-party-l4-s4/tests/instance.json b/tasks/construct-rlve-imp-party-l4-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3f24dac74a54af9a0f3712a2fc7dcb61c1858437
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-l4-s4/tests/instance.json
@@ -0,0 +1 @@
+{"n": 12, "edges": [[0, 1], [0, 3], [0, 4], [0, 5], [0, 6], [0, 7], [0, 8], [0, 10], [0, 11], [0, 12], [0, 13], [0, 14], [0, 15], [0, 16], [0, 17], [0, 18], [0, 19], [0, 20], [0, 21], [0, 23], [0, 25], [0, 27], [0, 28], [0, 29], [0, 30], [0, 31], [0, 32], [0, 33], [1, 2], [1, 3], [1, 4], [1, 5], [1, 6], [1, 7], [1, 8], [1, 10], [1, 11], [1, 12], [1, 13], [1, 14], [1, 15], [1, 16], [1, 17], [1, 18], [1, 19], [1, 21], [1, 23], [1, 25], [1, 27], [1, 28], [1, 29], [1, 30], [1, 31], [1, 32], [1, 33], [1, 34], [2, 3], [2, 7], [2, 11], [2, 15], [2, 18], [2, 25], [2, 27], [2, 28], [2, 31], [2, 32], [2, 33], [3, 5], [3, 6], [3, 7], [3, 8], [3, 11], [3, 12], [3, 14], [3, 15], [3, 16], [3, 17], [3, 18], [3, 19], [3, 20], [3, 21], [3, 22], [3, 23], [3, 24], [3, 25], [3, 26], [3, 27], [3, 28], [3, 29], [3, 31], [3, 32], [3, 33], [3, 35], [4, 7], [4, 8], [4, 11], [4, 16], [4, 19], [4, 23], [4, 25], [4, 27], [4, 29], [4, 32], [5, 6], [5, 7], [5, 8], [5, 9], [5, 11], [5, 12], [5, 14], [5, 15], [5, 16], [5, 17], [5, 18], [5, 19], [5, 21], [5, 22], [5, 23], [5, 24], [5, 25], [5, 27], [5, 28], [5, 29], [5, 31], [5, 32], [5, 33], [5, 35], [6, 7], [6, 8], [6, 10], [6, 11], [6, 12], [6, 13], [6, 14], [6, 15], [6, 16], [6, 17], [6, 18], [6, 19], [6, 21], [6, 23], [6, 25], [6, 26], [6, 27], [6, 28], [6, 29], [6, 31], [6, 32], [6, 33], [6, 34], [7, 8], [7, 11], [7, 12], [7, 14], [7, 15], [7, 16], [7, 17], [7, 18], [7, 19], [7, 21], [7, 23], [7, 24], [7, 25], [7, 26], [7, 27], [7, 28], [7, 29], [7, 30], [7, 31], [7, 32], [7, 33], [8, 11], [8, 12], [8, 13], [8, 14], [8, 15], [8, 16], [8, 17], [8, 18], [8, 19], [8, 20], [8, 21], [8, 22], [8, 23], [8, 24], [8, 25], [8, 26], [8, 27], [8, 28], [8, 29], [8, 31], [8, 32], [8, 33], [8, 34], [8, 35], [9, 12], [9, 14], [9, 15], [9, 16], [9, 23], [9, 25], [9, 28], [9, 31], [10, 11], [10, 15], [10, 16], [10, 21], [10, 23], [10, 25], [10, 28], [10, 32], [11, 12], [11, 14], [11, 15], [11, 16], [11, 17], [11, 18], [11, 19], [11, 20], [11, 21], [11, 23], [11, 24], [11, 25], [11, 26], [11, 27], [11, 28], [11, 29], [11, 31], [11, 32], [11, 33], [11, 34], [11, 35], [12, 13], [12, 14], [12, 15], [12, 16], [12, 17], [12, 18], [12, 19], [12, 20], [12, 21], [12, 22], [12, 23], [12, 25], [12, 27], [12, 28], [12, 29], [12, 30], [12, 31], [12, 32], [12, 33], [12, 35], [13, 14], [13, 15], [13, 18], [13, 19], [13, 25], [13, 29], [13, 32], [13, 33], [14, 15], [14, 16], [14, 17], [14, 18], [14, 19], [14, 21], [14, 22], [14, 23], [14, 25], [14, 27], [14, 28], [14, 29], [14, 31], [14, 32], [14, 33], [14, 34], [14, 35], [15, 16], [15, 17], [15, 18], [15, 19], [15, 21], [15, 23], [15, 24], [15, 25], [15, 27], [15, 28], [15, 29], [15, 30], [15, 31], [15, 32], [15, 33], [15, 34], [15, 35], [16, 17], [16, 18], [16, 19], [16, 20], [16, 21], [16, 22], [16, 23], [16, 24], [16, 25], [16, 27], [16, 28], [16, 29], [16, 31], [16, 32], [16, 33], [16, 35], [17, 18], [17, 19], [17, 21], [17, 22], [17, 23], [17, 25], [17, 26], [17, 27], [17, 28], [17, 29], [17, 31], [17, 32], [17, 33], [17, 35], [18, 19], [18, 21], [18, 23], [18, 25], [18, 26], [18, 27], [18, 28], [18, 29], [18, 31], [18, 32], [18, 33], [18, 35], [19, 21], [19, 23], [19, 25], [19, 26], [19, 27], [19, 28], [19, 29], [19, 31], [19, 32], [19, 33], [19, 34], [20, 23], [20, 25], [20, 28], [20, 32], [21, 23], [21, 24], [21, 25], [21, 26], [21, 27], [21, 28], [21, 29], [21, 31], [21, 32], [21, 33], [21, 34], [21, 35], [22, 23], [22, 25], [22, 29], [22, 32], [22, 33], [23, 24], [23, 25], [23, 27], [23, 28], [23, 29], [23, 30], [23, 31], [23, 32], [23, 33], [23, 34], [24, 25], [24, 28], [24, 29], [24, 32], [24, 33], [25, 27], [25, 28], [25, 29], [25, 31], [25, 32], [25, 33], [25, 34], [25, 35], [26, 32], [26, 33], [27, 28], [27, 29], [27, 30], [27, 31], [27, 32], [27, 33], [27, 35], [28, 29], [28, 31], [28, 32], [28, 33], [28, 34], [29, 30], [29, 31], [29, 32], [29, 33], [29, 34], [30, 32], [31, 32], [31, 33], [31, 34], [31, 35], [32, 33], [32, 34], [33, 34]], "family": "rlve_imp_party", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-imp-party-l4-s4/tests/run.py b/tasks/construct-rlve-imp-party-l4-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-l4-s4/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-imp-party-l4-s4/tests/test.sh b/tasks/construct-rlve-imp-party-l4-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-l4-s4/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-integer-linear-system-l3-s4/environment/Dockerfile b/tasks/construct-rlve-integer-linear-system-l3-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-l3-s4/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-integer-linear-system-l3-s4/environment/instance.json b/tasks/construct-rlve-integer-linear-system-l3-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..34f72d98ce9c2542457de393dcfe1616f857d3b7
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-l3-s4/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [[1, 1, 1, 0, 0, 1, 0, 0], [1, 0, 0, 0, 0, 1, 0, 0], [0, 0, 1, 1, 0, 1, 0, 0], [0, 0, 1, 0, 1, 1, 0, 0], [1, 0, 1, 0, 1, 0, 0, 1], [1, 0, 0, 1, 0, 0, 1, 1], [0, 1, 0, 0, 0, 0, 0, 1], [0, 1, 1, 1, 0, 0, 1, 1], [0, 1, 1, 0, 0, 0, 1, 0], [1, 1, 0, 0, 1, 1, 1, 1], [1, 0, 1, 0, 1, 0, 1, 0], [1, 0, 1, 1, 1, 0, 1, 1], [1, 1, 0, 0, 1, 1, 1, 0], [0, 1, 0, 0, 0, 1, 1, 1], [1, 0, 1, 1, 0, 1, 0, 1], [1, 1, 0, 1, 1, 1, 0, 0], [1, 1, 0, 0, 1, 1, 1, 1], [1, 0, 0, 1, 0, 0, 1, 1], [0, 0, 0, 1, 1, 1, 0, 0], [0, 1, 1, 0, 1, 1, 1, 1]], "rhs": [22, 13, 14, 13, 16, 15, 6, 17, 12, 24, 17, 22, 22, 15, 23, 22, 24, 15, 11, 22], "family": "rlve_integer_linear_system", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-integer-linear-system-l3-s4/solution/solve.sh b/tasks/construct-rlve-integer-linear-system-l3-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..3879132ae991028cad99f7d5afa5b751dc001c04
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-l3-s4/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": [7, 4, 5, 3, 2, 6, 3, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-integer-linear-system-l3-s4/tests/check.py b/tasks/construct-rlve-integer-linear-system-l3-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..ff19002b3fddc9597f84ee31ef5ffbee5d3fac32
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-l3-s4/tests/check.py
@@ -0,0 +1,10 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n = len(inst["coeffs"][0])
+    x = int_list(ans.get("x"), n, name="x")
+    for k, (e, r) in enumerate(zip(inst["coeffs"], inst["rhs"])):
+        if sum(c * v for c, v in zip(e, x)) != r:
+            raise Invalid(f"equation {k} is violated")
+    return None, {}
diff --git a/tasks/construct-rlve-integer-linear-system-l3-s4/tests/instance.json b/tasks/construct-rlve-integer-linear-system-l3-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..34f72d98ce9c2542457de393dcfe1616f857d3b7
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-l3-s4/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [[1, 1, 1, 0, 0, 1, 0, 0], [1, 0, 0, 0, 0, 1, 0, 0], [0, 0, 1, 1, 0, 1, 0, 0], [0, 0, 1, 0, 1, 1, 0, 0], [1, 0, 1, 0, 1, 0, 0, 1], [1, 0, 0, 1, 0, 0, 1, 1], [0, 1, 0, 0, 0, 0, 0, 1], [0, 1, 1, 1, 0, 0, 1, 1], [0, 1, 1, 0, 0, 0, 1, 0], [1, 1, 0, 0, 1, 1, 1, 1], [1, 0, 1, 0, 1, 0, 1, 0], [1, 0, 1, 1, 1, 0, 1, 1], [1, 1, 0, 0, 1, 1, 1, 0], [0, 1, 0, 0, 0, 1, 1, 1], [1, 0, 1, 1, 0, 1, 0, 1], [1, 1, 0, 1, 1, 1, 0, 0], [1, 1, 0, 0, 1, 1, 1, 1], [1, 0, 0, 1, 0, 0, 1, 1], [0, 0, 0, 1, 1, 1, 0, 0], [0, 1, 1, 0, 1, 1, 1, 1]], "rhs": [22, 13, 14, 13, 16, 15, 6, 17, 12, 24, 17, 22, 22, 15, 23, 22, 24, 15, 11, 22], "family": "rlve_integer_linear_system", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-integer-linear-system-l3-s4/tests/run.py b/tasks/construct-rlve-integer-linear-system-l3-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-l3-s4/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-integer-linear-system-l3-s4/tests/test.sh b/tasks/construct-rlve-integer-linear-system-l3-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-l3-s4/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-largest-convex-polygon-l4-s2/environment/Dockerfile b/tasks/construct-rlve-largest-convex-polygon-l4-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l4-s2/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-largest-convex-polygon-l4-s2/environment/instance.json b/tasks/construct-rlve-largest-convex-polygon-l4-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4355fef1dc97fab595ebc1324cf4922b25b65c3d
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l4-s2/environment/instance.json
@@ -0,0 +1 @@
+{"points": [[2, 14], [12, 11], [1, 2], [5, 12], [2, 1], [11, 16], [18, 7], [12, 8], [1, 9], [9, 12], [13, 10], [18, 16], [11, 15], [4, 8], [0, 18], [14, 2], [4, 17], [13, 11]], "k": 9, "family": "rlve_largest_convex_polygon", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-convex-polygon-l4-s2/solution/solve.sh b/tasks/construct-rlve-largest-convex-polygon-l4-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..319811a7ce1900cb9b42e3f040a6b2cd3f5811b9
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l4-s2/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"indices": [2, 8, 0, 16, 12, 17, 10, 7, 4]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-largest-convex-polygon-l4-s2/tests/check.py b/tasks/construct-rlve-largest-convex-polygon-l4-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3a09cc858f41ce5745e4e491188cd42360c65dae
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l4-s2/tests/check.py
@@ -0,0 +1,27 @@
+from run import Invalid, int_list
+
+
+def _cross(o, a, b):
+    return (a[0] - o[0]) * (b[1] - o[1]) - (a[1] - o[1]) * (b[0] - o[0])
+
+
+def check(inst, ans):
+    P, k = inst["points"], inst["k"]
+    idx = int_list(ans.get("indices"), None, 0, len(P) - 1, "indices")
+    if len(set(idx)) != len(idx):
+        raise Invalid("indices must be distinct")
+    if len(idx) < k:
+        raise Invalid(f"{len(idx)} points given, need {k}")
+    pts = sorted(tuple(P[i]) for i in idx)
+    lower, upper = [], []
+    for p in pts:  # monotone chain, strict (collinear points dropped)
+        while len(lower) >= 2 and _cross(lower[-2], lower[-1], p) <= 0:
+            lower.pop()
+        lower.append(p)
+    for p in reversed(pts):
+        while len(upper) >= 2 and _cross(upper[-2], upper[-1], p) <= 0:
+            upper.pop()
+        upper.append(p)
+    if len(lower) + len(upper) - 2 != len(pts):
+        raise Invalid("the chosen points are not in convex position")
+    return None, {}
diff --git a/tasks/construct-rlve-largest-convex-polygon-l4-s2/tests/instance.json b/tasks/construct-rlve-largest-convex-polygon-l4-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4355fef1dc97fab595ebc1324cf4922b25b65c3d
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l4-s2/tests/instance.json
@@ -0,0 +1 @@
+{"points": [[2, 14], [12, 11], [1, 2], [5, 12], [2, 1], [11, 16], [18, 7], [12, 8], [1, 9], [9, 12], [13, 10], [18, 16], [11, 15], [4, 8], [0, 18], [14, 2], [4, 17], [13, 11]], "k": 9, "family": "rlve_largest_convex_polygon", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-convex-polygon-l4-s2/tests/run.py b/tasks/construct-rlve-largest-convex-polygon-l4-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l4-s2/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-largest-convex-polygon-l4-s2/tests/test.sh b/tasks/construct-rlve-largest-convex-polygon-l4-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l4-s2/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-largest-convex-polygon-l4-s5/environment/Dockerfile b/tasks/construct-rlve-largest-convex-polygon-l4-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l4-s5/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-largest-convex-polygon-l4-s5/environment/instance.json b/tasks/construct-rlve-largest-convex-polygon-l4-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..258d435bd33219e3d9ff2bf41716b5e17ddf8206
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l4-s5/environment/instance.json
@@ -0,0 +1 @@
+{"points": [[4, 4], [9, 10], [6, 2], [10, 5], [6, 15], [2, 7], [0, 0], [11, 16], [17, 0], [4, 6], [10, 7], [0, 18], [3, 13], [5, 6], [15, 9], [11, 17], [3, 15], [2, 8]], "k": 9, "family": "rlve_largest_convex_polygon", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-convex-polygon-l4-s5/solution/solve.sh b/tasks/construct-rlve-largest-convex-polygon-l4-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..86c985c2eb669268114889294ac0cfbc5d1a98e7
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l4-s5/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"indices": [5, 17, 12, 4, 15, 7, 1, 13, 9]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-largest-convex-polygon-l4-s5/tests/check.py b/tasks/construct-rlve-largest-convex-polygon-l4-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3a09cc858f41ce5745e4e491188cd42360c65dae
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l4-s5/tests/check.py
@@ -0,0 +1,27 @@
+from run import Invalid, int_list
+
+
+def _cross(o, a, b):
+    return (a[0] - o[0]) * (b[1] - o[1]) - (a[1] - o[1]) * (b[0] - o[0])
+
+
+def check(inst, ans):
+    P, k = inst["points"], inst["k"]
+    idx = int_list(ans.get("indices"), None, 0, len(P) - 1, "indices")
+    if len(set(idx)) != len(idx):
+        raise Invalid("indices must be distinct")
+    if len(idx) < k:
+        raise Invalid(f"{len(idx)} points given, need {k}")
+    pts = sorted(tuple(P[i]) for i in idx)
+    lower, upper = [], []
+    for p in pts:  # monotone chain, strict (collinear points dropped)
+        while len(lower) >= 2 and _cross(lower[-2], lower[-1], p) <= 0:
+            lower.pop()
+        lower.append(p)
+    for p in reversed(pts):
+        while len(upper) >= 2 and _cross(upper[-2], upper[-1], p) <= 0:
+            upper.pop()
+        upper.append(p)
+    if len(lower) + len(upper) - 2 != len(pts):
+        raise Invalid("the chosen points are not in convex position")
+    return None, {}
diff --git a/tasks/construct-rlve-largest-convex-polygon-l4-s5/tests/instance.json b/tasks/construct-rlve-largest-convex-polygon-l4-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..258d435bd33219e3d9ff2bf41716b5e17ddf8206
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l4-s5/tests/instance.json
@@ -0,0 +1 @@
+{"points": [[4, 4], [9, 10], [6, 2], [10, 5], [6, 15], [2, 7], [0, 0], [11, 16], [17, 0], [4, 6], [10, 7], [0, 18], [3, 13], [5, 6], [15, 9], [11, 17], [3, 15], [2, 8]], "k": 9, "family": "rlve_largest_convex_polygon", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-convex-polygon-l4-s5/tests/run.py b/tasks/construct-rlve-largest-convex-polygon-l4-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l4-s5/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-largest-convex-polygon-l4-s5/tests/test.sh b/tasks/construct-rlve-largest-convex-polygon-l4-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l4-s5/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-largest-rectangle-l6-s2/environment/Dockerfile b/tasks/construct-rlve-largest-rectangle-l6-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l6-s2/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-largest-rectangle-l6-s2/environment/instance.json b/tasks/construct-rlve-largest-rectangle-l6-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..046a680d987c159749394c6ed95800a04c3e0fda
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l6-s2/environment/instance.json
@@ -0,0 +1 @@
+{"points": [[-42, -3], [-6, -7], [4, -20], [7, 3], [55, -53], [-49, 1], [-31, 58], [-13, 43], [50, -59], [-46, 22], [17, -10], [-40, 5], [-46, 35], [49, 50], [31, -18], [52, -17], [-16, -3], [57, 20], [42, -22], [-49, 49], [-14, -4], [31, -4], [22, 25], [23, -54], [7, -45], [-41, -3], [-48, 58], [-33, -47], [60, 11], [-33, 0], [16, 3], [4, -41], [18, 49], [45, 24], [-33, 52], [59, -25], [40, 5], [-54, 34], [0, 48], [-20, 6], [-34, -23], [-12, 28], [47, -52], [22, 14], [38, -10], [36, -20], [2, 27], [4, 38], [23, -11], [-30, -13], [-60, -15], [-56, -44], [25, -54], [-57, 34], [-6, 32], [-1, 36], [-51, 52], [59, -52], [-44, -22], [-48, 20]], "area": 269, "family": "rlve_largest_rectangle", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-rectangle-l6-s2/solution/solve.sh b/tasks/construct-rlve-largest-rectangle-l6-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..33673bef2f42d9610e99874bc933d17b99a88802
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l6-s2/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"indices": [1, 2, 10, 3]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-largest-rectangle-l6-s2/tests/check.py b/tasks/construct-rlve-largest-rectangle-l6-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2251b2af23a481e598fa1e48d0a8249f70b6cc44
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l6-s2/tests/check.py
@@ -0,0 +1,24 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    P = inst["points"]
+    idx = int_list(ans.get("indices"), 4, 0, len(P) - 1, "indices")
+    if len(set(idx)) != 4:
+        raise Invalid("indices must be distinct")
+    A, *rest = [tuple(P[i]) for i in idx]
+    if len(set([A] + rest)) != 4:
+        raise Invalid("points must be distinct")
+    # the vertex farthest from A is opposite; the other two are A's neighbours
+    rest.sort(key=lambda X: (X[0] - A[0]) ** 2 + (X[1] - A[1]) ** 2)
+    B, D, C = rest
+    u = (B[0] - A[0], B[1] - A[1])
+    v = (D[0] - A[0], D[1] - A[1])
+    if u[0] * v[0] + u[1] * v[1] != 0:
+        raise Invalid("not a rectangle (no right angle at the first vertex)")
+    if (B[0] + D[0] - A[0], B[1] + D[1] - A[1]) != C:
+        raise Invalid("not a rectangle")
+    area = abs(u[0] * v[1] - u[1] * v[0])
+    if area < inst["area"]:
+        raise Invalid(f"area {area} < {inst['area']}")
+    return None, {"area": area}
diff --git a/tasks/construct-rlve-largest-rectangle-l6-s2/tests/instance.json b/tasks/construct-rlve-largest-rectangle-l6-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..046a680d987c159749394c6ed95800a04c3e0fda
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l6-s2/tests/instance.json
@@ -0,0 +1 @@
+{"points": [[-42, -3], [-6, -7], [4, -20], [7, 3], [55, -53], [-49, 1], [-31, 58], [-13, 43], [50, -59], [-46, 22], [17, -10], [-40, 5], [-46, 35], [49, 50], [31, -18], [52, -17], [-16, -3], [57, 20], [42, -22], [-49, 49], [-14, -4], [31, -4], [22, 25], [23, -54], [7, -45], [-41, -3], [-48, 58], [-33, -47], [60, 11], [-33, 0], [16, 3], [4, -41], [18, 49], [45, 24], [-33, 52], [59, -25], [40, 5], [-54, 34], [0, 48], [-20, 6], [-34, -23], [-12, 28], [47, -52], [22, 14], [38, -10], [36, -20], [2, 27], [4, 38], [23, -11], [-30, -13], [-60, -15], [-56, -44], [25, -54], [-57, 34], [-6, 32], [-1, 36], [-51, 52], [59, -52], [-44, -22], [-48, 20]], "area": 269, "family": "rlve_largest_rectangle", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-rectangle-l6-s2/tests/run.py b/tasks/construct-rlve-largest-rectangle-l6-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l6-s2/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-largest-rectangle-l6-s2/tests/test.sh b/tasks/construct-rlve-largest-rectangle-l6-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l6-s2/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-largest-rectangle-l6-s5/environment/Dockerfile b/tasks/construct-rlve-largest-rectangle-l6-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l6-s5/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-largest-rectangle-l6-s5/environment/instance.json b/tasks/construct-rlve-largest-rectangle-l6-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3666d822e5a0fe6483047ac35b425ac4d780d989
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l6-s5/environment/instance.json
@@ -0,0 +1 @@
+{"points": [[12, 42], [47, 43], [3, -57], [10, -10], [35, 55], [-4, 5], [-18, 32], [-38, 3], [-21, -18], [-17, 57], [27, -53], [-26, -20], [20, -26], [-6, -40], [-18, -18], [-20, 17], [54, 53], [60, 12], [-43, -28], [14, 5], [47, -53], [-50, 7], [-52, 35], [4, 26], [-30, -21], [-27, -3], [10, -46], [29, 1], [3, 34], [31, 18], [30, -45], [45, -23], [8, 47], [26, 11], [2, 36], [34, 19], [51, -14], [-19, -9], [30, 4], [40, -45], [-27, -23], [18, 19], [12, 52], [35, 47], [39, -24], [-2, -39], [-12, -60], [-24, -24], [-27, 18], [8, 27], [6, -24], [-12, 28], [-9, -16], [36, 30], [0, -15], [-27, -15], [-21, 59], [-11, -38], [49, -38], [-34, 45]], "area": 45, "family": "rlve_largest_rectangle", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-rectangle-l6-s5/solution/solve.sh b/tasks/construct-rlve-largest-rectangle-l6-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..f51e625ba57fd33e44f3ae09f455a2630327d107
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l6-s5/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"indices": [8, 47, 24, 55]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-largest-rectangle-l6-s5/tests/check.py b/tasks/construct-rlve-largest-rectangle-l6-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2251b2af23a481e598fa1e48d0a8249f70b6cc44
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l6-s5/tests/check.py
@@ -0,0 +1,24 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    P = inst["points"]
+    idx = int_list(ans.get("indices"), 4, 0, len(P) - 1, "indices")
+    if len(set(idx)) != 4:
+        raise Invalid("indices must be distinct")
+    A, *rest = [tuple(P[i]) for i in idx]
+    if len(set([A] + rest)) != 4:
+        raise Invalid("points must be distinct")
+    # the vertex farthest from A is opposite; the other two are A's neighbours
+    rest.sort(key=lambda X: (X[0] - A[0]) ** 2 + (X[1] - A[1]) ** 2)
+    B, D, C = rest
+    u = (B[0] - A[0], B[1] - A[1])
+    v = (D[0] - A[0], D[1] - A[1])
+    if u[0] * v[0] + u[1] * v[1] != 0:
+        raise Invalid("not a rectangle (no right angle at the first vertex)")
+    if (B[0] + D[0] - A[0], B[1] + D[1] - A[1]) != C:
+        raise Invalid("not a rectangle")
+    area = abs(u[0] * v[1] - u[1] * v[0])
+    if area < inst["area"]:
+        raise Invalid(f"area {area} < {inst['area']}")
+    return None, {"area": area}
diff --git a/tasks/construct-rlve-largest-rectangle-l6-s5/tests/instance.json b/tasks/construct-rlve-largest-rectangle-l6-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3666d822e5a0fe6483047ac35b425ac4d780d989
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l6-s5/tests/instance.json
@@ -0,0 +1 @@
+{"points": [[12, 42], [47, 43], [3, -57], [10, -10], [35, 55], [-4, 5], [-18, 32], [-38, 3], [-21, -18], [-17, 57], [27, -53], [-26, -20], [20, -26], [-6, -40], [-18, -18], [-20, 17], [54, 53], [60, 12], [-43, -28], [14, 5], [47, -53], [-50, 7], [-52, 35], [4, 26], [-30, -21], [-27, -3], [10, -46], [29, 1], [3, 34], [31, 18], [30, -45], [45, -23], [8, 47], [26, 11], [2, 36], [34, 19], [51, -14], [-19, -9], [30, 4], [40, -45], [-27, -23], [18, 19], [12, 52], [35, 47], [39, -24], [-2, -39], [-12, -60], [-24, -24], [-27, 18], [8, 27], [6, -24], [-12, 28], [-9, -16], [36, 30], [0, -15], [-27, -15], [-21, 59], [-11, -38], [49, -38], [-34, 45]], "area": 45, "family": "rlve_largest_rectangle", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-rectangle-l6-s5/tests/run.py b/tasks/construct-rlve-largest-rectangle-l6-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l6-s5/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-largest-rectangle-l6-s5/tests/test.sh b/tasks/construct-rlve-largest-rectangle-l6-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l6-s5/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l1-s8/environment/Dockerfile b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s8/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l1-s8/environment/instance.json b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..153e2ff6a4f0a6f084f4c1f78093640291e2454e
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s8/environment/instance.json
@@ -0,0 +1 @@
+{"A": ["100", "000"], "B": ["000", "010"], "family": "rlve_matrix_permutation_equivalence", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l1-s8/solution/solve.sh b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..5f8d5de1bac69ca909d5dda6ec14992c8a171c45
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s8/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"a": [1, 0], "b": [1, 0, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l1-s8/tests/check.py b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..310a1d2c8b6afd7a82a35ba2a4c802c1ce289d03
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s8/tests/check.py
@@ -0,0 +1,15 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    A, B = inst["A"], inst["B"]
+    N, M = len(A), len(A[0])
+    a = int_list(ans.get("a"), N, 0, N - 1, "a")
+    b = int_list(ans.get("b"), M, 0, M - 1, "b")
+    if len(set(a)) != N or len(set(b)) != M:
+        raise Invalid("a and b must be permutations")
+    for i in range(N):
+        for j in range(M):
+            if A[a[i]][b[j]] != B[i][j]:
+                raise Invalid(f"mismatch at ({i}, {j})")
+    return None, {}
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l1-s8/tests/instance.json b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..153e2ff6a4f0a6f084f4c1f78093640291e2454e
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s8/tests/instance.json
@@ -0,0 +1 @@
+{"A": ["100", "000"], "B": ["000", "010"], "family": "rlve_matrix_permutation_equivalence", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l1-s8/tests/run.py b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s8/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l1-s8/tests/test.sh b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s8/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l1-s9/environment/Dockerfile b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s9/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l1-s9/environment/instance.json b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b85bf75664e65de3a9898313218e03b9c6c231bb
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s9/environment/instance.json
@@ -0,0 +1 @@
+{"A": ["111", "101", "110"], "B": ["111", "011", "110"], "family": "rlve_matrix_permutation_equivalence", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l1-s9/solution/solve.sh b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..cd6beae962631594a973bdd060fdd5fe4c024c70
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s9/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"a": [0, 2, 1], "b": [2, 0, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l1-s9/tests/check.py b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..310a1d2c8b6afd7a82a35ba2a4c802c1ce289d03
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s9/tests/check.py
@@ -0,0 +1,15 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    A, B = inst["A"], inst["B"]
+    N, M = len(A), len(A[0])
+    a = int_list(ans.get("a"), N, 0, N - 1, "a")
+    b = int_list(ans.get("b"), M, 0, M - 1, "b")
+    if len(set(a)) != N or len(set(b)) != M:
+        raise Invalid("a and b must be permutations")
+    for i in range(N):
+        for j in range(M):
+            if A[a[i]][b[j]] != B[i][j]:
+                raise Invalid(f"mismatch at ({i}, {j})")
+    return None, {}
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l1-s9/tests/instance.json b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b85bf75664e65de3a9898313218e03b9c6c231bb
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s9/tests/instance.json
@@ -0,0 +1 @@
+{"A": ["111", "101", "110"], "B": ["111", "011", "110"], "family": "rlve_matrix_permutation_equivalence", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l1-s9/tests/run.py b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s9/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l1-s9/tests/test.sh b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l1-s9/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l5-s6/environment/Dockerfile b/tasks/construct-rlve-matrix-permutation-equivalence-l5-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l5-s6/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l5-s6/environment/instance.json b/tasks/construct-rlve-matrix-permutation-equivalence-l5-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..ae08e3bb1c15a62673ad2cdfecc702df2deebdcb
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l5-s6/environment/instance.json
@@ -0,0 +1 @@
+{"A": ["111011101111", "101110111111"], "B": ["110011111111", "111111010111"], "family": "rlve_matrix_permutation_equivalence", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l5-s6/solution/solve.sh b/tasks/construct-rlve-matrix-permutation-equivalence-l5-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..5b5eb9e3b93d862baf1c1a7331eeb689ae8e0ee1
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l5-s6/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"a": [0, 1], "b": [10, 4, 7, 3, 8, 0, 1, 2, 5, 9, 6, 11]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l5-s6/tests/check.py b/tasks/construct-rlve-matrix-permutation-equivalence-l5-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..310a1d2c8b6afd7a82a35ba2a4c802c1ce289d03
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l5-s6/tests/check.py
@@ -0,0 +1,15 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    A, B = inst["A"], inst["B"]
+    N, M = len(A), len(A[0])
+    a = int_list(ans.get("a"), N, 0, N - 1, "a")
+    b = int_list(ans.get("b"), M, 0, M - 1, "b")
+    if len(set(a)) != N or len(set(b)) != M:
+        raise Invalid("a and b must be permutations")
+    for i in range(N):
+        for j in range(M):
+            if A[a[i]][b[j]] != B[i][j]:
+                raise Invalid(f"mismatch at ({i}, {j})")
+    return None, {}
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l5-s6/tests/instance.json b/tasks/construct-rlve-matrix-permutation-equivalence-l5-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..ae08e3bb1c15a62673ad2cdfecc702df2deebdcb
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l5-s6/tests/instance.json
@@ -0,0 +1 @@
+{"A": ["111011101111", "101110111111"], "B": ["110011111111", "111111010111"], "family": "rlve_matrix_permutation_equivalence", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l5-s6/tests/run.py b/tasks/construct-rlve-matrix-permutation-equivalence-l5-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l5-s6/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l5-s6/tests/test.sh b/tasks/construct-rlve-matrix-permutation-equivalence-l5-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l5-s6/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l4-s3/environment/Dockerfile b/tasks/construct-rlve-max-achromatic-coloring-l4-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l4-s3/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l4-s3/environment/instance.json b/tasks/construct-rlve-max-achromatic-coloring-l4-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..71c4c53a3a912f59392044e94ed937eed7e41a67
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l4-s3/environment/instance.json
@@ -0,0 +1 @@
+{"n": 12, "k": 6, "edges": [[0, 6], [0, 11], [1, 3], [1, 5], [1, 7], [1, 11], [2, 3], [2, 5], [2, 6], [2, 7], [3, 4], [3, 9], [3, 11], [4, 5], [4, 10], [4, 11], [5, 7], [5, 8], [6, 9], [6, 11], [7, 9], [7, 11], [8, 11]], "family": "rlve_max_achromatic_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l4-s3/solution/solve.sh b/tasks/construct-rlve-max-achromatic-coloring-l4-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..48faa7c57a32cb30f5cb1d69b48e83ced9917add
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l4-s3/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"colors": [0, 1, 0, 2, 0, 2, 1, 3, 4, 4, 4, 5]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l4-s3/tests/check.py b/tasks/construct-rlve-max-achromatic-coloring-l4-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..bb676a518264e5db4fb0964e27ccffccf6980807
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l4-s3/tests/check.py
@@ -0,0 +1,16 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    c = int_list(ans.get("colors"), n, 0, k - 1, "colors")
+    if len(set(c)) != k:
+        raise Invalid(f"uses {len(set(c))} colours, need all {k}")
+    pairs = set()
+    for u, v in inst["edges"]:
+        if c[u] == c[v]:
+            raise Invalid(f"edge ({u},{v}) is monochromatic")
+        pairs.add((min(c[u], c[v]), max(c[u], c[v])))
+    if len(pairs) != k * (k - 1) // 2:
+        raise Invalid("some pair of colours is not joined by an edge")
+    return None, {}
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l4-s3/tests/instance.json b/tasks/construct-rlve-max-achromatic-coloring-l4-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..71c4c53a3a912f59392044e94ed937eed7e41a67
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l4-s3/tests/instance.json
@@ -0,0 +1 @@
+{"n": 12, "k": 6, "edges": [[0, 6], [0, 11], [1, 3], [1, 5], [1, 7], [1, 11], [2, 3], [2, 5], [2, 6], [2, 7], [3, 4], [3, 9], [3, 11], [4, 5], [4, 10], [4, 11], [5, 7], [5, 8], [6, 9], [6, 11], [7, 9], [7, 11], [8, 11]], "family": "rlve_max_achromatic_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l4-s3/tests/run.py b/tasks/construct-rlve-max-achromatic-coloring-l4-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l4-s3/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l4-s3/tests/test.sh b/tasks/construct-rlve-max-achromatic-coloring-l4-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l4-s3/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l4-s4/environment/Dockerfile b/tasks/construct-rlve-max-achromatic-coloring-l4-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l4-s4/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l4-s4/environment/instance.json b/tasks/construct-rlve-max-achromatic-coloring-l4-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..bdbbea0054d343ad884948486a06aab54925a5ad
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l4-s4/environment/instance.json
@@ -0,0 +1 @@
+{"n": 12, "k": 6, "edges": [[0, 2], [0, 3], [0, 5], [0, 11], [1, 2], [1, 5], [1, 6], [1, 7], [1, 10], [2, 5], [2, 6], [2, 9], [2, 10], [3, 4], [3, 7], [3, 9], [3, 10], [3, 11], [4, 5], [5, 8], [6, 10], [6, 11], [7, 10]], "family": "rlve_max_achromatic_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l4-s4/solution/solve.sh b/tasks/construct-rlve-max-achromatic-coloring-l4-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..dfe0dbf8e8df0f513ccbba6328d89136d8a684a8
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l4-s4/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"colors": [0, 0, 1, 2, 0, 2, 3, 4, 3, 4, 5, 4]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l4-s4/tests/check.py b/tasks/construct-rlve-max-achromatic-coloring-l4-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..bb676a518264e5db4fb0964e27ccffccf6980807
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l4-s4/tests/check.py
@@ -0,0 +1,16 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    c = int_list(ans.get("colors"), n, 0, k - 1, "colors")
+    if len(set(c)) != k:
+        raise Invalid(f"uses {len(set(c))} colours, need all {k}")
+    pairs = set()
+    for u, v in inst["edges"]:
+        if c[u] == c[v]:
+            raise Invalid(f"edge ({u},{v}) is monochromatic")
+        pairs.add((min(c[u], c[v]), max(c[u], c[v])))
+    if len(pairs) != k * (k - 1) // 2:
+        raise Invalid("some pair of colours is not joined by an edge")
+    return None, {}
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l4-s4/tests/instance.json b/tasks/construct-rlve-max-achromatic-coloring-l4-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..bdbbea0054d343ad884948486a06aab54925a5ad
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l4-s4/tests/instance.json
@@ -0,0 +1 @@
+{"n": 12, "k": 6, "edges": [[0, 2], [0, 3], [0, 5], [0, 11], [1, 2], [1, 5], [1, 6], [1, 7], [1, 10], [2, 5], [2, 6], [2, 9], [2, 10], [3, 4], [3, 7], [3, 9], [3, 10], [3, 11], [4, 5], [5, 8], [6, 10], [6, 11], [7, 10]], "family": "rlve_max_achromatic_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l4-s4/tests/run.py b/tasks/construct-rlve-max-achromatic-coloring-l4-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l4-s4/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l4-s4/tests/test.sh b/tasks/construct-rlve-max-achromatic-coloring-l4-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l4-s4/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-maximum-clique-l6-s0/environment/Dockerfile b/tasks/construct-rlve-maximum-clique-l6-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-maximum-clique-l6-s0/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-maximum-clique-l6-s0/environment/instance.json b/tasks/construct-rlve-maximum-clique-l6-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1f818f2d1b0727dbe9288bb4753265cdcc683eb3
--- /dev/null
+++ b/tasks/construct-rlve-maximum-clique-l6-s0/environment/instance.json
@@ -0,0 +1 @@
+{"n": 56, "k": 10, "edges": [[0, 1], [0, 3], [0, 5], [0, 6], [0, 8], [0, 10], [0, 13], [0, 14], [0, 16], [0, 17], [0, 18], [0, 19], [0, 21], [0, 23], [0, 24], [0, 26], [0, 27], [0, 28], [0, 31], [0, 32], [0, 36], [0, 37], [0, 39], [0, 40], [0, 42], [0, 44], [0, 45], [0, 50], [0, 51], [0, 53], [0, 55], [1, 2], [1, 3], [1, 4], [1, 5], [1, 9], [1, 10], [1, 12], [1, 14], [1, 15], [1, 17], [1, 18], [1, 19], [1, 20], [1, 23], [1, 24], [1, 25], [1, 26], [1, 27], [1, 28], [1, 31], [1, 33], [1, 34], [1, 35], [1, 39], [1, 41], [1, 42], [1, 43], [1, 44], [1, 45], [1, 46], [1, 47], [1, 49], [1, 50], [1, 52], [1, 53], [2, 4], [2, 6], [2, 8], [2, 9], [2, 10], [2, 11], [2, 17], [2, 18], [2, 20], [2, 21], [2, 23], [2, 27], [2, 28], [2, 29], [2, 30], [2, 31], [2, 32], [2, 33], [2, 35], [2, 38], [2, 39], [2, 40], [2, 42], [2, 45], [2, 46], [2, 47], [2, 48], [2, 50], [2, 51], [2, 53], [2, 54], [2, 55], [3, 4], [3, 7], [3, 8], [3, 10], [3, 11], [3, 12], [3, 13], [3, 14], [3, 15], [3, 19], [3, 24], [3, 25], [3, 26], [3, 27], [3, 28], [3, 32], [3, 33], [3, 34], [3, 35], [3, 38], [3, 40], [3, 41], [3, 44], [3, 47], [3, 48], [3, 50], [3, 54], [3, 55], [4, 6], [4, 7], [4, 8], [4, 9], [4, 10], [4, 13], [4, 15], [4, 16], [4, 17], [4, 18], [4, 19], [4, 23], [4, 25], [4, 26], [4, 27], [4, 28], [4, 29], [4, 31], [4, 32], [4, 33], [4, 34], [4, 35], [4, 39], [4, 40], [4, 43], [4, 47], [4, 49], [4, 51], [5, 6], [5, 12], [5, 14], [5, 15], [5, 16], [5, 18], [5, 19], [5, 20], [5, 21], [5, 22], [5, 24], [5, 25], [5, 26], [5, 27], [5, 32], [5, 33], [5, 34], [5, 35], [5, 36], [5, 40], [5, 41], [5, 43], [5, 45], [5, 47], [5, 48], [5, 50], [5, 51], [5, 52], [5, 53], [5, 55], [6, 7], [6, 9], [6, 10], [6, 12], [6, 14], [6, 15], [6, 16], [6, 17], [6, 21], [6, 22], [6, 23], [6, 24], [6, 25], [6, 26], [6, 27], [6, 28], [6, 30], [6, 33], [6, 34], [6, 35], [6, 36], [6, 39], [6, 42], [6, 44], [6, 46], [6, 49], [6, 51], [6, 52], [6, 53], [6, 55], [7, 9], [7, 10], [7, 11], [7, 12], [7, 13], [7, 14], [7, 15], [7, 16], [7, 18], [7, 19], [7, 20], [7, 21], [7, 23], [7, 24], [7, 27], [7, 29], [7, 30], [7, 32], [7, 34], [7, 36], [7, 37], [7, 38], [7, 42], [7, 43], [7, 44], [7, 47], [7, 48], [7, 50], [7, 51], [7, 53], [7, 55], [8, 9], [8, 11], [8, 12], [8, 13], [8, 14], [8, 15], [8, 16], [8, 17], [8, 19], [8, 20], [8, 23], [8, 26], [8, 29], [8, 34], [8, 36], [8, 38], [8, 40], [8, 43], [8, 45], [8, 47], [8, 48], [8, 49], [8, 51], [8, 53], [8, 55], [9, 13], [9, 19], [9, 20], [9, 21], [9, 22], [9, 23], [9, 24], [9, 27], [9, 29], [9, 30], [9, 31], [9, 32], [9, 35], [9, 37], [9, 40], [9, 41], [9, 42], [9, 43], [9, 45], [9, 46], [9, 47], [9, 48], [9, 49], [9, 50], [9, 52], [9, 53], [9, 55], [10, 12], [10, 17], [10, 18], [10, 19], [10, 20], [10, 21], [10, 23], [10, 25], [10, 26], [10, 29], [10, 31], [10, 32], [10, 33], [10, 38], [10, 39], [10, 42], [10, 45], [10, 47], [10, 49], [10, 54], [10, 55], [11, 13], [11, 14], [11, 16], [11, 18], [11, 19], [11, 20], [11, 22], [11, 23], [11, 24], [11, 27], [11, 28], [11, 29], [11, 31], [11, 32], [11, 36], [11, 38], [11, 39], [11, 40], [11, 41], [11, 42], [11, 45], [11, 47], [11, 50], [11, 54], [11, 55], [12, 14], [12, 17], [12, 19], [12, 20], [12, 27], [12, 28], [12, 29], [12, 30], [12, 31], [12, 32], [12, 39], [12, 40], [12, 43], [12, 45], [12, 46], [12, 48], [12, 50], [12, 51], [12, 53], [12, 54], [13, 14], [13, 16], [13, 21], [13, 22], [13, 24], [13, 26], [13, 28], [13, 29], [13, 30], [13, 32], [13, 33], [13, 34], [13, 35], [13, 36], [13, 37], [13, 39], [13, 41], [13, 42], [13, 43], [13, 44], [13, 46], [13, 48], [13, 50], [13, 51], [13, 52], [13, 54], [13, 55], [14, 16], [14, 18], [14, 19], [14, 20], [14, 21], [14, 24], [14, 25], [14, 26], [14, 28], [14, 31], [14, 33], [14, 34], [14, 35], [14, 37], [14, 38], [14, 39], [14, 41], [14, 45], [14, 46], [14, 47], [14, 49], [14, 50], [14, 51], [14, 52], [14, 55], [15, 19], [15, 22], [15, 23], [15, 26], [15, 30], [15, 33], [15, 34], [15, 36], [15, 40], [15, 41], [15, 43], [15, 44], [15, 46], [15, 49], 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"construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-maximum-clique-l6-s0/solution/solve.sh b/tasks/construct-rlve-maximum-clique-l6-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..25e30955d6a0ccb6a7aa5fb0e390ffcb6df4097a
--- /dev/null
+++ b/tasks/construct-rlve-maximum-clique-l6-s0/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"clique": [1, 2, 9, 20, 23, 27, 42, 46, 50, 53]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-maximum-clique-l6-s0/tests/check.py b/tasks/construct-rlve-maximum-clique-l6-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..95e00c78f7c986ba3aecbdc8af3cf8ed221abd05
--- /dev/null
+++ b/tasks/construct-rlve-maximum-clique-l6-s0/tests/check.py
@@ -0,0 +1,16 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    c = int_list(ans.get("clique"), None, 0, n - 1, "clique")
+    if len(set(c)) != len(c):
+        raise Invalid("vertices must be distinct")
+    if len(c) < k:
+        raise Invalid(f"clique has {len(c)} vertices, need {k}")
+    es = {(min(u, v), max(u, v)) for u, v in inst["edges"]}
+    for i in range(len(c)):
+        for j in range(i + 1, len(c)):
+            if (min(c[i], c[j]), max(c[i], c[j])) not in es:
+                raise Invalid(f"{c[i]} and {c[j]} are not adjacent")
+    return None, {}
diff --git a/tasks/construct-rlve-maximum-clique-l6-s0/tests/instance.json b/tasks/construct-rlve-maximum-clique-l6-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1f818f2d1b0727dbe9288bb4753265cdcc683eb3
--- /dev/null
+++ b/tasks/construct-rlve-maximum-clique-l6-s0/tests/instance.json
@@ -0,0 +1 @@
+{"n": 56, "k": 10, "edges": [[0, 1], [0, 3], [0, 5], [0, 6], [0, 8], [0, 10], [0, 13], [0, 14], [0, 16], [0, 17], [0, 18], [0, 19], [0, 21], [0, 23], [0, 24], [0, 26], [0, 27], [0, 28], [0, 31], [0, 32], [0, 36], [0, 37], [0, 39], [0, 40], [0, 42], [0, 44], [0, 45], [0, 50], [0, 51], [0, 53], [0, 55], [1, 2], [1, 3], [1, 4], [1, 5], [1, 9], [1, 10], [1, 12], [1, 14], [1, 15], [1, 17], [1, 18], [1, 19], [1, 20], [1, 23], [1, 24], [1, 25], [1, 26], [1, 27], [1, 28], [1, 31], [1, 33], [1, 34], [1, 35], [1, 39], [1, 41], [1, 42], [1, 43], [1, 44], [1, 45], [1, 46], [1, 47], [1, 49], [1, 50], [1, 52], [1, 53], [2, 4], [2, 6], [2, 8], [2, 9], [2, 10], [2, 11], [2, 17], [2, 18], [2, 20], [2, 21], [2, 23], [2, 27], [2, 28], [2, 29], [2, 30], [2, 31], [2, 32], [2, 33], [2, 35], [2, 38], [2, 39], [2, 40], [2, 42], [2, 45], [2, 46], [2, 47], [2, 48], [2, 50], [2, 51], [2, 53], [2, 54], [2, 55], [3, 4], [3, 7], [3, 8], [3, 10], [3, 11], [3, 12], [3, 13], [3, 14], [3, 15], [3, 19], [3, 24], [3, 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"construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-maximum-clique-l6-s0/tests/run.py b/tasks/construct-rlve-maximum-clique-l6-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-maximum-clique-l6-s0/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-maximum-clique-l6-s0/tests/test.sh b/tasks/construct-rlve-maximum-clique-l6-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-maximum-clique-l6-s0/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-maximum-clique-l6-s1/environment/Dockerfile b/tasks/construct-rlve-maximum-clique-l6-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-maximum-clique-l6-s1/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-maximum-clique-l6-s1/environment/instance.json b/tasks/construct-rlve-maximum-clique-l6-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f24c226e302274ffd7db233c0156b1364cea2d5d
--- /dev/null
+++ b/tasks/construct-rlve-maximum-clique-l6-s1/environment/instance.json
@@ -0,0 +1 @@
+{"n": 56, "k": 9, "edges": [[0, 1], [0, 2], [0, 4], [0, 5], [0, 6], [0, 8], [0, 9], [0, 10], [0, 11], [0, 13], [0, 14], [0, 17], [0, 21], [0, 22], [0, 24], [0, 25], [0, 26], [0, 27], [0, 30], [0, 32], [0, 36], [0, 38], [0, 39], [0, 41], [0, 42], [0, 45], [0, 46], [0, 48], [0, 49], [0, 50], [0, 51], [0, 53], [0, 54], [0, 55], [1, 2], [1, 3], [1, 5], [1, 6], [1, 7], [1, 9], [1, 11], [1, 12], [1, 16], [1, 17], [1, 18], [1, 20], [1, 22], [1, 26], [1, 28], [1, 29], [1, 30], [1, 31], [1, 32], [1, 33], [1, 34], [1, 35], [1, 38], [1, 41], [1, 42], [1, 43], [1, 44], [1, 46], [1, 49], [1, 50], [1, 53], [1, 54], [2, 3], [2, 6], [2, 7], [2, 8], [2, 10], [2, 11], [2, 12], [2, 13], [2, 15], [2, 16], [2, 17], [2, 19], [2, 21], [2, 22], [2, 23], [2, 26], [2, 27], [2, 29], [2, 30], [2, 32], [2, 35], [2, 36], [2, 37], [2, 40], [2, 41], [2, 43], [2, 45], [2, 46], [2, 47], [2, 48], [2, 49], [2, 50], [2, 52], [2, 55], [3, 4], [3, 10], [3, 12], [3, 14], [3, 15], [3, 16], [3, 19], [3, 20], [3, 25], [3, 27], 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"construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-maximum-clique-l6-s1/solution/solve.sh b/tasks/construct-rlve-maximum-clique-l6-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..5dbba7b85786fbcc10c0874660f804b619238423
--- /dev/null
+++ b/tasks/construct-rlve-maximum-clique-l6-s1/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"clique": [2, 6, 10, 19, 26, 29, 35, 36, 45]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-maximum-clique-l6-s1/tests/check.py b/tasks/construct-rlve-maximum-clique-l6-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..95e00c78f7c986ba3aecbdc8af3cf8ed221abd05
--- /dev/null
+++ b/tasks/construct-rlve-maximum-clique-l6-s1/tests/check.py
@@ -0,0 +1,16 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    c = int_list(ans.get("clique"), None, 0, n - 1, "clique")
+    if len(set(c)) != len(c):
+        raise Invalid("vertices must be distinct")
+    if len(c) < k:
+        raise Invalid(f"clique has {len(c)} vertices, need {k}")
+    es = {(min(u, v), max(u, v)) for u, v in inst["edges"]}
+    for i in range(len(c)):
+        for j in range(i + 1, len(c)):
+            if (min(c[i], c[j]), max(c[i], c[j])) not in es:
+                raise Invalid(f"{c[i]} and {c[j]} are not adjacent")
+    return None, {}
diff --git a/tasks/construct-rlve-maximum-clique-l6-s1/tests/instance.json b/tasks/construct-rlve-maximum-clique-l6-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f24c226e302274ffd7db233c0156b1364cea2d5d
--- /dev/null
+++ b/tasks/construct-rlve-maximum-clique-l6-s1/tests/instance.json
@@ -0,0 +1 @@
+{"n": 56, "k": 9, "edges": [[0, 1], [0, 2], [0, 4], [0, 5], [0, 6], [0, 8], [0, 9], [0, 10], [0, 11], [0, 13], [0, 14], [0, 17], [0, 21], [0, 22], [0, 24], [0, 25], [0, 26], [0, 27], [0, 30], [0, 32], [0, 36], [0, 38], [0, 39], [0, 41], [0, 42], [0, 45], [0, 46], [0, 48], [0, 49], [0, 50], [0, 51], [0, 53], [0, 54], [0, 55], [1, 2], [1, 3], [1, 5], [1, 6], [1, 7], [1, 9], [1, 11], [1, 12], [1, 16], [1, 17], [1, 18], [1, 20], [1, 22], [1, 26], [1, 28], [1, 29], [1, 30], [1, 31], [1, 32], [1, 33], [1, 34], [1, 35], [1, 38], [1, 41], [1, 42], [1, 43], [1, 44], [1, 46], [1, 49], [1, 50], [1, 53], [1, 54], [2, 3], [2, 6], [2, 7], [2, 8], [2, 10], [2, 11], [2, 12], [2, 13], [2, 15], [2, 16], [2, 17], [2, 19], [2, 21], [2, 22], [2, 23], [2, 26], [2, 27], [2, 29], [2, 30], [2, 32], [2, 35], [2, 36], [2, 37], [2, 40], [2, 41], [2, 43], [2, 45], [2, 46], [2, 47], [2, 48], [2, 49], [2, 50], [2, 52], [2, 55], [3, 4], [3, 10], [3, 12], [3, 14], [3, 15], [3, 16], [3, 19], [3, 20], [3, 25], [3, 27], 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"construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-maximum-clique-l6-s1/tests/run.py b/tasks/construct-rlve-maximum-clique-l6-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-maximum-clique-l6-s1/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-maximum-clique-l6-s1/tests/test.sh b/tasks/construct-rlve-maximum-clique-l6-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-maximum-clique-l6-s1/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-min-chromatic-number-l1-s2/environment/Dockerfile b/tasks/construct-rlve-min-chromatic-number-l1-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l1-s2/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-min-chromatic-number-l1-s2/environment/instance.json b/tasks/construct-rlve-min-chromatic-number-l1-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e4a25726e757e9bcf8e9f5999b207d12677f860e
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l1-s2/environment/instance.json
@@ -0,0 +1 @@
+{"n": 6, "k": 3, "edges": [[0, 1], [0, 3], [0, 5], [2, 3], [3, 5]], "family": "rlve_min_chromatic_number", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-chromatic-number-l1-s2/solution/solve.sh b/tasks/construct-rlve-min-chromatic-number-l1-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..5568051aa557ee297f3f543d42b02b3f624bd2eb
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l1-s2/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"colors": [0, 1, 0, 1, 0, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-min-chromatic-number-l1-s2/tests/check.py b/tasks/construct-rlve-min-chromatic-number-l1-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..d2b77793fb15e0d97174b3c28e69a86e2a26996f
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l1-s2/tests/check.py
@@ -0,0 +1,10 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    c = int_list(ans.get("colors"), n, 0, k - 1, "colors")
+    for u, v in inst["edges"]:
+        if c[u] == c[v]:
+            raise Invalid(f"edge ({u},{v}) is monochromatic")
+    return None, {}
diff --git a/tasks/construct-rlve-min-chromatic-number-l1-s2/tests/instance.json b/tasks/construct-rlve-min-chromatic-number-l1-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e4a25726e757e9bcf8e9f5999b207d12677f860e
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l1-s2/tests/instance.json
@@ -0,0 +1 @@
+{"n": 6, "k": 3, "edges": [[0, 1], [0, 3], [0, 5], [2, 3], [3, 5]], "family": "rlve_min_chromatic_number", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-chromatic-number-l1-s2/tests/run.py b/tasks/construct-rlve-min-chromatic-number-l1-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l1-s2/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-min-chromatic-number-l1-s2/tests/test.sh b/tasks/construct-rlve-min-chromatic-number-l1-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l1-s2/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-min-chromatic-number-l1-s5/environment/Dockerfile b/tasks/construct-rlve-min-chromatic-number-l1-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l1-s5/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-min-chromatic-number-l1-s5/environment/instance.json b/tasks/construct-rlve-min-chromatic-number-l1-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a69b742999300ec78a02a549e9d324e7f81a9027
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l1-s5/environment/instance.json
@@ -0,0 +1 @@
+{"n": 6, "k": 3, "edges": [[3, 4], [3, 5], [4, 5]], "family": "rlve_min_chromatic_number", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-chromatic-number-l1-s5/solution/solve.sh b/tasks/construct-rlve-min-chromatic-number-l1-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..6513022ca9c1c71a657a7572e37008665ef3f7cf
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l1-s5/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"colors": [0, 0, 0, 0, 1, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-min-chromatic-number-l1-s5/tests/check.py b/tasks/construct-rlve-min-chromatic-number-l1-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..d2b77793fb15e0d97174b3c28e69a86e2a26996f
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l1-s5/tests/check.py
@@ -0,0 +1,10 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    c = int_list(ans.get("colors"), n, 0, k - 1, "colors")
+    for u, v in inst["edges"]:
+        if c[u] == c[v]:
+            raise Invalid(f"edge ({u},{v}) is monochromatic")
+    return None, {}
diff --git a/tasks/construct-rlve-min-chromatic-number-l1-s5/tests/instance.json b/tasks/construct-rlve-min-chromatic-number-l1-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a69b742999300ec78a02a549e9d324e7f81a9027
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l1-s5/tests/instance.json
@@ -0,0 +1 @@
+{"n": 6, "k": 3, "edges": [[3, 4], [3, 5], [4, 5]], "family": "rlve_min_chromatic_number", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-chromatic-number-l1-s5/tests/run.py b/tasks/construct-rlve-min-chromatic-number-l1-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l1-s5/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-min-chromatic-number-l1-s5/tests/test.sh b/tasks/construct-rlve-min-chromatic-number-l1-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l1-s5/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-min-chromatic-number-l5-s3/environment/Dockerfile b/tasks/construct-rlve-min-chromatic-number-l5-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l5-s3/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-min-chromatic-number-l5-s3/environment/instance.json b/tasks/construct-rlve-min-chromatic-number-l5-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8f40f666978f3995afc35d5af811b19425e9823b
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l5-s3/environment/instance.json
@@ -0,0 +1 @@
+{"n": 24, "k": 6, "edges": [[0, 1], [0, 7], [0, 8], [0, 10], [0, 14], [0, 16], [0, 20], [0, 21], [0, 22], [0, 23], [1, 3], [1, 5], [1, 9], [1, 10], [1, 13], [1, 14], [1, 15], [1, 22], [2, 4], [2, 5], [2, 9], [2, 10], [2, 14], [2, 17], [2, 18], [2, 21], [3, 9], [3, 10], [3, 11], [3, 12], [3, 13], [3, 16], [3, 17], [3, 21], [4, 6], [4, 7], [4, 8], [4, 10], [4, 11], [4, 14], [4, 15], [4, 18], [5, 6], [5, 8], [5, 9], [5, 11], [5, 12], [5, 13], [5, 14], [5, 18], [5, 19], [5, 20], [6, 9], [6, 16], [6, 17], [6, 18], [6, 20], [6, 21], [6, 23], [7, 8], [7, 9], [7, 10], [7, 14], [7, 17], [7, 19], [7, 20], [7, 21], [7, 22], [7, 23], [8, 9], [8, 11], [8, 12], [8, 17], [8, 18], [8, 19], [8, 20], [8, 21], [8, 22], [9, 12], [9, 15], [9, 16], [9, 17], [9, 20], [10, 13], [10, 15], [10, 17], [10, 23], [11, 12], [11, 14], [11, 15], [11, 16], [11, 17], [11, 18], [11, 20], [11, 22], [11, 23], [12, 13], [12, 17], [13, 14], [13, 15], [13, 16], [13, 17], [13, 20], [13, 21], [13, 23], [14, 15], [14, 16], [14, 19], [14, 20], [15, 17], [15, 18], [15, 19], [15, 21], [16, 20], [16, 22], [18, 19], [18, 20], [18, 22], [18, 23], [19, 22], [19, 23], [20, 21], [20, 23], [21, 23]], "family": "rlve_min_chromatic_number", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-chromatic-number-l5-s3/solution/solve.sh b/tasks/construct-rlve-min-chromatic-number-l5-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..0a1822f2397fedea20600d0a8d90247a1c75fd4a
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l5-s3/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"colors": [0, 1, 0, 0, 1, 2, 0, 3, 4, 5, 2, 5, 3, 5, 4, 0, 3, 1, 3, 1, 1, 2, 2, 4]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-min-chromatic-number-l5-s3/tests/check.py b/tasks/construct-rlve-min-chromatic-number-l5-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..d2b77793fb15e0d97174b3c28e69a86e2a26996f
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l5-s3/tests/check.py
@@ -0,0 +1,10 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    c = int_list(ans.get("colors"), n, 0, k - 1, "colors")
+    for u, v in inst["edges"]:
+        if c[u] == c[v]:
+            raise Invalid(f"edge ({u},{v}) is monochromatic")
+    return None, {}
diff --git a/tasks/construct-rlve-min-chromatic-number-l5-s3/tests/instance.json b/tasks/construct-rlve-min-chromatic-number-l5-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8f40f666978f3995afc35d5af811b19425e9823b
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l5-s3/tests/instance.json
@@ -0,0 +1 @@
+{"n": 24, "k": 6, "edges": [[0, 1], [0, 7], [0, 8], [0, 10], [0, 14], [0, 16], [0, 20], [0, 21], [0, 22], [0, 23], [1, 3], [1, 5], [1, 9], [1, 10], [1, 13], [1, 14], [1, 15], [1, 22], [2, 4], [2, 5], [2, 9], [2, 10], [2, 14], [2, 17], [2, 18], [2, 21], [3, 9], [3, 10], [3, 11], [3, 12], [3, 13], [3, 16], [3, 17], [3, 21], [4, 6], [4, 7], [4, 8], [4, 10], [4, 11], [4, 14], [4, 15], [4, 18], [5, 6], [5, 8], [5, 9], [5, 11], [5, 12], [5, 13], [5, 14], [5, 18], [5, 19], [5, 20], [6, 9], [6, 16], [6, 17], [6, 18], [6, 20], [6, 21], [6, 23], [7, 8], [7, 9], [7, 10], [7, 14], [7, 17], [7, 19], [7, 20], [7, 21], [7, 22], [7, 23], [8, 9], [8, 11], [8, 12], [8, 17], [8, 18], [8, 19], [8, 20], [8, 21], [8, 22], [9, 12], [9, 15], [9, 16], [9, 17], [9, 20], [10, 13], [10, 15], [10, 17], [10, 23], [11, 12], [11, 14], [11, 15], [11, 16], [11, 17], [11, 18], [11, 20], [11, 22], [11, 23], [12, 13], [12, 17], [13, 14], [13, 15], [13, 16], [13, 17], [13, 20], [13, 21], [13, 23], [14, 15], [14, 16], [14, 19], [14, 20], [15, 17], [15, 18], [15, 19], [15, 21], [16, 20], [16, 22], [18, 19], [18, 20], [18, 22], [18, 23], [19, 22], [19, 23], [20, 21], [20, 23], [21, 23]], "family": "rlve_min_chromatic_number", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-chromatic-number-l5-s3/tests/run.py b/tasks/construct-rlve-min-chromatic-number-l5-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l5-s3/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-min-chromatic-number-l5-s3/tests/test.sh b/tasks/construct-rlve-min-chromatic-number-l5-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l5-s3/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-min-chromatic-number-l5-s4/environment/Dockerfile b/tasks/construct-rlve-min-chromatic-number-l5-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l5-s4/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-min-chromatic-number-l5-s4/environment/instance.json b/tasks/construct-rlve-min-chromatic-number-l5-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..659380a0fc74dfe0c40ac986c2166b7e021dd882
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l5-s4/environment/instance.json
@@ -0,0 +1 @@
+{"n": 24, "k": 7, "edges": [[0, 5], [0, 6], [0, 8], [0, 9], [0, 10], [0, 11], [0, 14], [0, 16], [0, 17], [0, 20], [1, 2], [1, 3], [1, 9], [1, 10], [1, 11], [1, 14], [1, 16], [1, 19], [1, 21], [1, 22], [2, 6], [2, 7], [2, 8], [2, 9], [2, 10], [2, 13], [2, 14], [2, 16], [2, 17], [2, 18], [2, 22], [3, 4], [3, 5], [3, 6], [3, 8], [3, 16], [3, 18], [3, 21], [3, 22], [4, 5], [4, 7], [4, 9], [4, 10], [4, 14], [4, 15], [4, 16], [4, 17], [4, 18], [4, 20], [5, 8], [5, 9], [5, 10], [5, 12], [5, 13], [5, 14], [5, 15], [5, 16], [5, 17], [5, 18], [5, 19], [5, 20], [5, 21], [5, 22], [6, 7], [6, 8], [6, 10], [6, 16], [6, 18], [6, 21], [6, 23], [7, 8], [7, 11], [7, 12], [7, 14], [7, 15], [7, 16], [7, 20], [7, 21], [7, 22], [7, 23], [8, 9], [8, 11], [8, 12], [8, 13], [8, 15], [8, 19], [8, 21], [8, 22], [8, 23], [9, 11], [9, 13], [9, 14], [9, 15], [9, 16], [9, 17], [9, 19], [9, 20], [10, 11], [10, 12], [10, 13], [10, 16], [10, 17], [10, 18], [10, 19], [10, 20], [10, 21], [10, 22], [10, 23], [11, 12], [11, 13], [11, 15], [11, 16], [11, 17], [11, 18], [11, 19], [11, 20], [11, 22], [12, 13], [12, 14], [12, 17], [12, 20], [12, 22], [12, 23], [13, 14], [13, 18], [13, 19], [13, 22], [13, 23], [14, 17], [14, 18], [14, 20], [14, 21], [14, 22], [15, 16], [15, 19], [15, 20], [15, 21], [15, 23], [16, 18], [16, 19], [16, 21], [16, 23], [17, 18], [17, 19], [17, 22], [18, 19], [18, 22], [19, 20], [19, 21], [19, 22], [22, 23]], "family": "rlve_min_chromatic_number", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-chromatic-number-l5-s4/solution/solve.sh b/tasks/construct-rlve-min-chromatic-number-l5-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..468db11d224b6cc03e5356363998f3d660fb8327
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l5-s4/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"colors": [0, 0, 1, 1, 2, 3, 2, 0, 4, 5, 4, 3, 1, 6, 4, 1, 6, 6, 0, 2, 6, 5, 5, 3]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-min-chromatic-number-l5-s4/tests/check.py b/tasks/construct-rlve-min-chromatic-number-l5-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..d2b77793fb15e0d97174b3c28e69a86e2a26996f
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l5-s4/tests/check.py
@@ -0,0 +1,10 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    c = int_list(ans.get("colors"), n, 0, k - 1, "colors")
+    for u, v in inst["edges"]:
+        if c[u] == c[v]:
+            raise Invalid(f"edge ({u},{v}) is monochromatic")
+    return None, {}
diff --git a/tasks/construct-rlve-min-chromatic-number-l5-s4/tests/instance.json b/tasks/construct-rlve-min-chromatic-number-l5-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..659380a0fc74dfe0c40ac986c2166b7e021dd882
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l5-s4/tests/instance.json
@@ -0,0 +1 @@
+{"n": 24, "k": 7, "edges": [[0, 5], [0, 6], [0, 8], [0, 9], [0, 10], [0, 11], [0, 14], [0, 16], [0, 17], [0, 20], [1, 2], [1, 3], [1, 9], [1, 10], [1, 11], [1, 14], [1, 16], [1, 19], [1, 21], [1, 22], [2, 6], [2, 7], [2, 8], [2, 9], [2, 10], [2, 13], [2, 14], [2, 16], [2, 17], [2, 18], [2, 22], [3, 4], [3, 5], [3, 6], [3, 8], [3, 16], [3, 18], [3, 21], [3, 22], [4, 5], [4, 7], [4, 9], [4, 10], [4, 14], [4, 15], [4, 16], [4, 17], [4, 18], [4, 20], [5, 8], [5, 9], [5, 10], [5, 12], [5, 13], [5, 14], [5, 15], [5, 16], [5, 17], [5, 18], [5, 19], [5, 20], [5, 21], [5, 22], [6, 7], [6, 8], [6, 10], [6, 16], [6, 18], [6, 21], [6, 23], [7, 8], [7, 11], [7, 12], [7, 14], [7, 15], [7, 16], [7, 20], [7, 21], [7, 22], [7, 23], [8, 9], [8, 11], [8, 12], [8, 13], [8, 15], [8, 19], [8, 21], [8, 22], [8, 23], [9, 11], [9, 13], [9, 14], [9, 15], [9, 16], [9, 17], [9, 19], [9, 20], [10, 11], [10, 12], [10, 13], [10, 16], [10, 17], [10, 18], [10, 19], [10, 20], [10, 21], [10, 22], [10, 23], [11, 12], [11, 13], [11, 15], [11, 16], [11, 17], [11, 18], [11, 19], [11, 20], [11, 22], [12, 13], [12, 14], [12, 17], [12, 20], [12, 22], [12, 23], [13, 14], [13, 18], [13, 19], [13, 22], [13, 23], [14, 17], [14, 18], [14, 20], [14, 21], [14, 22], [15, 16], [15, 19], [15, 20], [15, 21], [15, 23], [16, 18], [16, 19], [16, 21], [16, 23], [17, 18], [17, 19], [17, 22], [18, 19], [18, 22], [19, 20], [19, 21], [19, 22], [22, 23]], "family": "rlve_min_chromatic_number", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-chromatic-number-l5-s4/tests/run.py b/tasks/construct-rlve-min-chromatic-number-l5-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l5-s4/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-min-chromatic-number-l5-s4/tests/test.sh b/tasks/construct-rlve-min-chromatic-number-l5-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l5-s4/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l1-s6/environment/Dockerfile b/tasks/construct-rlve-min-harmonious-coloring-l1-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l1-s6/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l1-s6/environment/instance.json b/tasks/construct-rlve-min-harmonious-coloring-l1-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..17399f941d16df271d45a03018594ffb335b1a0b
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l1-s6/environment/instance.json
@@ -0,0 +1 @@
+{"n": 6, "k": 3, "edges": [[0, 4], [0, 5]], "family": "rlve_min_harmonious_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l1-s6/solution/solve.sh b/tasks/construct-rlve-min-harmonious-coloring-l1-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..6513022ca9c1c71a657a7572e37008665ef3f7cf
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l1-s6/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"colors": [0, 0, 0, 0, 1, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l1-s6/tests/check.py b/tasks/construct-rlve-min-harmonious-coloring-l1-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..91282a31c765b6103e6575180adb47abddb6a506
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l1-s6/tests/check.py
@@ -0,0 +1,15 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    c = int_list(ans.get("colors"), n, 0, k - 1, "colors")
+    seen = set()
+    for u, v in inst["edges"]:
+        if c[u] == c[v]:
+            raise Invalid(f"edge ({u},{v}) is monochromatic")
+        pair = (min(c[u], c[v]), max(c[u], c[v]))
+        if pair in seen:
+            raise Invalid(f"colour pair {pair} appears on two edges")
+        seen.add(pair)
+    return None, {}
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l1-s6/tests/instance.json b/tasks/construct-rlve-min-harmonious-coloring-l1-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..17399f941d16df271d45a03018594ffb335b1a0b
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l1-s6/tests/instance.json
@@ -0,0 +1 @@
+{"n": 6, "k": 3, "edges": [[0, 4], [0, 5]], "family": "rlve_min_harmonious_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l1-s6/tests/run.py b/tasks/construct-rlve-min-harmonious-coloring-l1-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l1-s6/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l1-s6/tests/test.sh b/tasks/construct-rlve-min-harmonious-coloring-l1-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l1-s6/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l1-s8/environment/Dockerfile b/tasks/construct-rlve-min-harmonious-coloring-l1-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l1-s8/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l1-s8/environment/instance.json b/tasks/construct-rlve-min-harmonious-coloring-l1-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4db3fff456d16b93e0a90950b1354f8aae73f8a6
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l1-s8/environment/instance.json
@@ -0,0 +1 @@
+{"n": 6, "k": 3, "edges": [[0, 4], [2, 3], [3, 4]], "family": "rlve_min_harmonious_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l1-s8/solution/solve.sh b/tasks/construct-rlve-min-harmonious-coloring-l1-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..507750783edb4737b84f155ecabef4d2b36c3f75
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l1-s8/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"colors": [0, 0, 0, 1, 2, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l1-s8/tests/check.py b/tasks/construct-rlve-min-harmonious-coloring-l1-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..91282a31c765b6103e6575180adb47abddb6a506
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l1-s8/tests/check.py
@@ -0,0 +1,15 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    c = int_list(ans.get("colors"), n, 0, k - 1, "colors")
+    seen = set()
+    for u, v in inst["edges"]:
+        if c[u] == c[v]:
+            raise Invalid(f"edge ({u},{v}) is monochromatic")
+        pair = (min(c[u], c[v]), max(c[u], c[v]))
+        if pair in seen:
+            raise Invalid(f"colour pair {pair} appears on two edges")
+        seen.add(pair)
+    return None, {}
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l1-s8/tests/instance.json b/tasks/construct-rlve-min-harmonious-coloring-l1-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4db3fff456d16b93e0a90950b1354f8aae73f8a6
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l1-s8/tests/instance.json
@@ -0,0 +1 @@
+{"n": 6, "k": 3, "edges": [[0, 4], [2, 3], [3, 4]], "family": "rlve_min_harmonious_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l1-s8/tests/run.py b/tasks/construct-rlve-min-harmonious-coloring-l1-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l1-s8/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l1-s8/tests/test.sh b/tasks/construct-rlve-min-harmonious-coloring-l1-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l1-s8/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l5-s0/environment/Dockerfile b/tasks/construct-rlve-min-harmonious-coloring-l5-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l5-s0/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l5-s0/environment/instance.json b/tasks/construct-rlve-min-harmonious-coloring-l5-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4fb2fcd52994e24271acbb0f77e3990b37f20a21
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l5-s0/environment/instance.json
@@ -0,0 +1 @@
+{"n": 15, "k": 9, "edges": [[0, 4], [0, 11], [1, 4], [1, 9], [1, 13], [2, 3], [2, 5], [2, 14], [3, 5], [4, 6], [4, 8], [4, 9], [5, 8], [5, 10], [5, 11], [5, 12], [5, 14], [6, 9], [6, 13], [6, 14], [7, 11], [9, 11], [10, 11], [10, 12], [11, 14], [12, 13]], "family": "rlve_min_harmonious_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l5-s0/solution/solve.sh b/tasks/construct-rlve-min-harmonious-coloring-l5-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..7cc799234db678768fd8f39427433bb4cb7e4871
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l5-s0/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"colors": [0, 1, 0, 2, 3, 4, 2, 2, 5, 6, 3, 7, 8, 5, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l5-s0/tests/check.py b/tasks/construct-rlve-min-harmonious-coloring-l5-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..91282a31c765b6103e6575180adb47abddb6a506
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l5-s0/tests/check.py
@@ -0,0 +1,15 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    c = int_list(ans.get("colors"), n, 0, k - 1, "colors")
+    seen = set()
+    for u, v in inst["edges"]:
+        if c[u] == c[v]:
+            raise Invalid(f"edge ({u},{v}) is monochromatic")
+        pair = (min(c[u], c[v]), max(c[u], c[v]))
+        if pair in seen:
+            raise Invalid(f"colour pair {pair} appears on two edges")
+        seen.add(pair)
+    return None, {}
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l5-s0/tests/instance.json b/tasks/construct-rlve-min-harmonious-coloring-l5-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4fb2fcd52994e24271acbb0f77e3990b37f20a21
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l5-s0/tests/instance.json
@@ -0,0 +1 @@
+{"n": 15, "k": 9, "edges": [[0, 4], [0, 11], [1, 4], [1, 9], [1, 13], [2, 3], [2, 5], [2, 14], [3, 5], [4, 6], [4, 8], [4, 9], [5, 8], [5, 10], [5, 11], [5, 12], [5, 14], [6, 9], [6, 13], [6, 14], [7, 11], [9, 11], [10, 11], [10, 12], [11, 14], [12, 13]], "family": "rlve_min_harmonious_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l5-s0/tests/run.py b/tasks/construct-rlve-min-harmonious-coloring-l5-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l5-s0/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l5-s0/tests/test.sh b/tasks/construct-rlve-min-harmonious-coloring-l5-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l5-s0/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l5-s7/environment/Dockerfile b/tasks/construct-rlve-min-harmonious-coloring-l5-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l5-s7/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l5-s7/environment/instance.json b/tasks/construct-rlve-min-harmonious-coloring-l5-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3e59385644ae86e67e9b145cd2fe2423e22502e7
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l5-s7/environment/instance.json
@@ -0,0 +1 @@
+{"n": 15, "k": 8, "edges": [[0, 5], [0, 6], [0, 8], [0, 9], [0, 12], [1, 4], [1, 7], [1, 10], [1, 13], [1, 14], [2, 12], [2, 14], [3, 8], [4, 9], [5, 9], [8, 9], [9, 12], [9, 14], [11, 12], [11, 14], [12, 13]], "family": "rlve_min_harmonious_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l5-s7/solution/solve.sh b/tasks/construct-rlve-min-harmonious-coloring-l5-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..90d1dd2d2ad8430de0aac7f3b3ac2d3bb0d2aa2b
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l5-s7/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"colors": [0, 1, 2, 1, 2, 3, 1, 4, 5, 4, 6, 5, 6, 3, 7]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l5-s7/tests/check.py b/tasks/construct-rlve-min-harmonious-coloring-l5-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..91282a31c765b6103e6575180adb47abddb6a506
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l5-s7/tests/check.py
@@ -0,0 +1,15 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    c = int_list(ans.get("colors"), n, 0, k - 1, "colors")
+    seen = set()
+    for u, v in inst["edges"]:
+        if c[u] == c[v]:
+            raise Invalid(f"edge ({u},{v}) is monochromatic")
+        pair = (min(c[u], c[v]), max(c[u], c[v]))
+        if pair in seen:
+            raise Invalid(f"colour pair {pair} appears on two edges")
+        seen.add(pair)
+    return None, {}
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l5-s7/tests/instance.json b/tasks/construct-rlve-min-harmonious-coloring-l5-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3e59385644ae86e67e9b145cd2fe2423e22502e7
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l5-s7/tests/instance.json
@@ -0,0 +1 @@
+{"n": 15, "k": 8, "edges": [[0, 5], [0, 6], [0, 8], [0, 9], [0, 12], [1, 4], [1, 7], [1, 10], [1, 13], [1, 14], [2, 12], [2, 14], [3, 8], [4, 9], [5, 9], [8, 9], [9, 12], [9, 14], [11, 12], [11, 14], [12, 13]], "family": "rlve_min_harmonious_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l5-s7/tests/run.py b/tasks/construct-rlve-min-harmonious-coloring-l5-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l5-s7/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l5-s7/tests/test.sh b/tasks/construct-rlve-min-harmonious-coloring-l5-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l5-s7/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l5-s9/environment/Dockerfile b/tasks/construct-rlve-min-harmonious-coloring-l5-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l5-s9/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l5-s9/environment/instance.json b/tasks/construct-rlve-min-harmonious-coloring-l5-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..34e447eeefb6798be451c2b14c8f25ad3651445f
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l5-s9/environment/instance.json
@@ -0,0 +1 @@
+{"n": 15, "k": 9, "edges": [[0, 10], [0, 11], [0, 12], [1, 6], [1, 10], [1, 13], [2, 10], [2, 12], [3, 6], [3, 7], [3, 9], [3, 12], [4, 6], [4, 13], [5, 10], [5, 11], [5, 12], [5, 14], [6, 7], [8, 10], [8, 11], [9, 14], [10, 12], [10, 13], [10, 14], [11, 13]], "family": "rlve_min_harmonious_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l5-s9/solution/solve.sh b/tasks/construct-rlve-min-harmonious-coloring-l5-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..6ccc7b036a625fa431e9056b24e8b9c9908fb7d2
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l5-s9/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"colors": [0, 1, 2, 3, 0, 4, 4, 5, 3, 0, 6, 2, 7, 5, 8]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l5-s9/tests/check.py b/tasks/construct-rlve-min-harmonious-coloring-l5-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..91282a31c765b6103e6575180adb47abddb6a506
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l5-s9/tests/check.py
@@ -0,0 +1,15 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    c = int_list(ans.get("colors"), n, 0, k - 1, "colors")
+    seen = set()
+    for u, v in inst["edges"]:
+        if c[u] == c[v]:
+            raise Invalid(f"edge ({u},{v}) is monochromatic")
+        pair = (min(c[u], c[v]), max(c[u], c[v]))
+        if pair in seen:
+            raise Invalid(f"colour pair {pair} appears on two edges")
+        seen.add(pair)
+    return None, {}
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l5-s9/tests/instance.json b/tasks/construct-rlve-min-harmonious-coloring-l5-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..34e447eeefb6798be451c2b14c8f25ad3651445f
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l5-s9/tests/instance.json
@@ -0,0 +1 @@
+{"n": 15, "k": 9, "edges": [[0, 10], [0, 11], [0, 12], [1, 6], [1, 10], [1, 13], [2, 10], [2, 12], [3, 6], [3, 7], [3, 9], [3, 12], [4, 6], [4, 13], [5, 10], [5, 11], [5, 12], [5, 14], [6, 7], [8, 10], [8, 11], [9, 14], [10, 12], [10, 13], [10, 14], [11, 13]], "family": "rlve_min_harmonious_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l5-s9/tests/run.py b/tasks/construct-rlve-min-harmonious-coloring-l5-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l5-s9/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l5-s9/tests/test.sh b/tasks/construct-rlve-min-harmonious-coloring-l5-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l5-s9/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l3-s1/environment/Dockerfile b/tasks/construct-rlve-pairwise-sums-recovery-l3-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l3-s1/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l3-s1/environment/instance.json b/tasks/construct-rlve-pairwise-sums-recovery-l3-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..fdc68b328c6f3d6092dab75abf299deaba161c79
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l3-s1/environment/instance.json
@@ -0,0 +1 @@
+{"n": 8, "sums": [6, 29, 16, 18, 19, 21, 10, 22, 3, 16, 13, 24, 10, 15, 17, 28, 17, 11, 27, 23, 23, 5, 12, 14, 22, 9, 15, 17], "family": "rlve_pairwise_sums_recovery", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l3-s1/solution/solve.sh b/tasks/construct-rlve-pairwise-sums-recovery-l3-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..b9caa26d9914983e00257ec1ede1acaf54188469
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l3-s1/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"numbers": [1, 2, 4, 8, 9, 13, 14, 15]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l3-s1/tests/check.py b/tasks/construct-rlve-pairwise-sums-recovery-l3-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..01b5b475e927688c0209fb93478a95f6dd1d3407
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l3-s1/tests/check.py
@@ -0,0 +1,14 @@
+from collections import Counter
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n = inst["n"]
+    x = int_list(ans.get("numbers"), n, 1, None, "numbers")
+    if len(set(x)) != n:
+        raise Invalid("numbers must be distinct")
+    got = Counter(x[i] + x[j] for i in range(n) for j in range(i + 1, n))
+    if got != Counter(inst["sums"]):
+        raise Invalid("pairwise sums do not match")
+    return None, {}
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l3-s1/tests/instance.json b/tasks/construct-rlve-pairwise-sums-recovery-l3-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..fdc68b328c6f3d6092dab75abf299deaba161c79
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l3-s1/tests/instance.json
@@ -0,0 +1 @@
+{"n": 8, "sums": [6, 29, 16, 18, 19, 21, 10, 22, 3, 16, 13, 24, 10, 15, 17, 28, 17, 11, 27, 23, 23, 5, 12, 14, 22, 9, 15, 17], "family": "rlve_pairwise_sums_recovery", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l3-s1/tests/run.py b/tasks/construct-rlve-pairwise-sums-recovery-l3-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l3-s1/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l3-s1/tests/test.sh b/tasks/construct-rlve-pairwise-sums-recovery-l3-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l3-s1/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l3-s8/environment/Dockerfile b/tasks/construct-rlve-pairwise-sums-recovery-l3-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l3-s8/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l3-s8/environment/instance.json b/tasks/construct-rlve-pairwise-sums-recovery-l3-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a48a91b2b500546823bddd268c64672cf432357e
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l3-s8/environment/instance.json
@@ -0,0 +1 @@
+{"n": 8, "sums": [21, 22, 11, 12, 5, 13, 23, 23, 9, 24, 6, 16, 12, 14, 11, 9, 10, 13, 15, 17, 7, 8, 20, 20, 19, 31, 21, 22], "family": "rlve_pairwise_sums_recovery", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l3-s8/solution/solve.sh b/tasks/construct-rlve-pairwise-sums-recovery-l3-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..3c189332dc546b8ee7c8ef65efb35ba8c4a8e68a
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l3-s8/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"numbers": [1, 4, 5, 6, 7, 8, 15, 16]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l3-s8/tests/check.py b/tasks/construct-rlve-pairwise-sums-recovery-l3-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..01b5b475e927688c0209fb93478a95f6dd1d3407
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l3-s8/tests/check.py
@@ -0,0 +1,14 @@
+from collections import Counter
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n = inst["n"]
+    x = int_list(ans.get("numbers"), n, 1, None, "numbers")
+    if len(set(x)) != n:
+        raise Invalid("numbers must be distinct")
+    got = Counter(x[i] + x[j] for i in range(n) for j in range(i + 1, n))
+    if got != Counter(inst["sums"]):
+        raise Invalid("pairwise sums do not match")
+    return None, {}
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l3-s8/tests/instance.json b/tasks/construct-rlve-pairwise-sums-recovery-l3-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a48a91b2b500546823bddd268c64672cf432357e
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l3-s8/tests/instance.json
@@ -0,0 +1 @@
+{"n": 8, "sums": [21, 22, 11, 12, 5, 13, 23, 23, 9, 24, 6, 16, 12, 14, 11, 9, 10, 13, 15, 17, 7, 8, 20, 20, 19, 31, 21, 22], "family": "rlve_pairwise_sums_recovery", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l3-s8/tests/run.py b/tasks/construct-rlve-pairwise-sums-recovery-l3-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l3-s8/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l3-s8/tests/test.sh b/tasks/construct-rlve-pairwise-sums-recovery-l3-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l3-s8/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-pan-solar-panels-l2-s3/environment/Dockerfile b/tasks/construct-rlve-pan-solar-panels-l2-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l2-s3/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-pan-solar-panels-l2-s3/environment/instance.json b/tasks/construct-rlve-pan-solar-panels-l2-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e1a544c69f50b9e8647627908220f6c2af1f09ff
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l2-s3/environment/instance.json
@@ -0,0 +1 @@
+{"A": 791, "B": 822, "C": 119, "D": 729, "target": 411, "family": "rlve_pan_solar_panels", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pan-solar-panels-l2-s3/solution/solve.sh b/tasks/construct-rlve-pan-solar-panels-l2-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..71a3eef4b71e17751d25b27eac086a02a3a347ef
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l2-s3/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"X": 822, "Y": 411}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-pan-solar-panels-l2-s3/tests/check.py b/tasks/construct-rlve-pan-solar-panels-l2-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..8bb2e7cc0b0280ab877755cc5d97b510680f7aa3
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l2-s3/tests/check.py
@@ -0,0 +1,12 @@
+import math
+
+from run import Invalid, as_int
+
+
+def check(inst, ans):
+    X = as_int(ans.get("X"), "X", lo=inst["A"], hi=inst["B"])
+    Y = as_int(ans.get("Y"), "Y", lo=inst["C"], hi=inst["D"])
+    g = math.gcd(X, Y)
+    if g < inst["target"]:
+        raise Invalid(f"gcd(X, Y) = {g} < {inst['target']}")
+    return None, {"gcd": g}
diff --git a/tasks/construct-rlve-pan-solar-panels-l2-s3/tests/instance.json b/tasks/construct-rlve-pan-solar-panels-l2-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e1a544c69f50b9e8647627908220f6c2af1f09ff
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l2-s3/tests/instance.json
@@ -0,0 +1 @@
+{"A": 791, "B": 822, "C": 119, "D": 729, "target": 411, "family": "rlve_pan_solar_panels", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pan-solar-panels-l2-s3/tests/run.py b/tasks/construct-rlve-pan-solar-panels-l2-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l2-s3/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-pan-solar-panels-l2-s3/tests/test.sh b/tasks/construct-rlve-pan-solar-panels-l2-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l2-s3/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-pan-solar-panels-l6-s2/environment/Dockerfile b/tasks/construct-rlve-pan-solar-panels-l6-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l6-s2/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-pan-solar-panels-l6-s2/environment/instance.json b/tasks/construct-rlve-pan-solar-panels-l6-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..6edd830a3c569fc7b43c7f61d10a79b59a45a73f
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l6-s2/environment/instance.json
@@ -0,0 +1 @@
+{"A": 75902618468, "B": 93763097709, "C": 27284805205, "D": 37650187878, "target": 31254365903, "family": "rlve_pan_solar_panels", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pan-solar-panels-l6-s2/solution/solve.sh b/tasks/construct-rlve-pan-solar-panels-l6-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..6bf22c3f94eaf1c60d1ede823e4f510a2fb1ff5f
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l6-s2/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"X": 93763097709, "Y": 31254365903}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-pan-solar-panels-l6-s2/tests/check.py b/tasks/construct-rlve-pan-solar-panels-l6-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..8bb2e7cc0b0280ab877755cc5d97b510680f7aa3
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l6-s2/tests/check.py
@@ -0,0 +1,12 @@
+import math
+
+from run import Invalid, as_int
+
+
+def check(inst, ans):
+    X = as_int(ans.get("X"), "X", lo=inst["A"], hi=inst["B"])
+    Y = as_int(ans.get("Y"), "Y", lo=inst["C"], hi=inst["D"])
+    g = math.gcd(X, Y)
+    if g < inst["target"]:
+        raise Invalid(f"gcd(X, Y) = {g} < {inst['target']}")
+    return None, {"gcd": g}
diff --git a/tasks/construct-rlve-pan-solar-panels-l6-s2/tests/instance.json b/tasks/construct-rlve-pan-solar-panels-l6-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..6edd830a3c569fc7b43c7f61d10a79b59a45a73f
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l6-s2/tests/instance.json
@@ -0,0 +1 @@
+{"A": 75902618468, "B": 93763097709, "C": 27284805205, "D": 37650187878, "target": 31254365903, "family": "rlve_pan_solar_panels", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pan-solar-panels-l6-s2/tests/run.py b/tasks/construct-rlve-pan-solar-panels-l6-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l6-s2/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-pan-solar-panels-l6-s2/tests/test.sh b/tasks/construct-rlve-pan-solar-panels-l6-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l6-s2/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-pan-solar-panels-l6-s5/environment/Dockerfile b/tasks/construct-rlve-pan-solar-panels-l6-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l6-s5/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-pan-solar-panels-l6-s5/environment/instance.json b/tasks/construct-rlve-pan-solar-panels-l6-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..7531eea23565ac9eaa768bc3fca564c41b4e13fe
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l6-s5/environment/instance.json
@@ -0,0 +1 @@
+{"A": 63502323648, "B": 89787137638, "C": 35308634788, "D": 38651163703, "target": 38651163703, "family": "rlve_pan_solar_panels", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pan-solar-panels-l6-s5/solution/solve.sh b/tasks/construct-rlve-pan-solar-panels-l6-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..728aa6e2954ebfeaab3dd452aa990951a92c237b
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l6-s5/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"X": 77302327406, "Y": 38651163703}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-pan-solar-panels-l6-s5/tests/check.py b/tasks/construct-rlve-pan-solar-panels-l6-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..8bb2e7cc0b0280ab877755cc5d97b510680f7aa3
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l6-s5/tests/check.py
@@ -0,0 +1,12 @@
+import math
+
+from run import Invalid, as_int
+
+
+def check(inst, ans):
+    X = as_int(ans.get("X"), "X", lo=inst["A"], hi=inst["B"])
+    Y = as_int(ans.get("Y"), "Y", lo=inst["C"], hi=inst["D"])
+    g = math.gcd(X, Y)
+    if g < inst["target"]:
+        raise Invalid(f"gcd(X, Y) = {g} < {inst['target']}")
+    return None, {"gcd": g}
diff --git a/tasks/construct-rlve-pan-solar-panels-l6-s5/tests/instance.json b/tasks/construct-rlve-pan-solar-panels-l6-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..7531eea23565ac9eaa768bc3fca564c41b4e13fe
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l6-s5/tests/instance.json
@@ -0,0 +1 @@
+{"A": 63502323648, "B": 89787137638, "C": 35308634788, "D": 38651163703, "target": 38651163703, "family": "rlve_pan_solar_panels", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pan-solar-panels-l6-s5/tests/run.py b/tasks/construct-rlve-pan-solar-panels-l6-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l6-s5/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-pan-solar-panels-l6-s5/tests/test.sh b/tasks/construct-rlve-pan-solar-panels-l6-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l6-s5/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l3-s0/environment/Dockerfile b/tasks/construct-rlve-polynomial-global-minimum-l3-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l3-s0/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l3-s0/environment/instance.json b/tasks/construct-rlve-polynomial-global-minimum-l3-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..7a5e3adf59d36a2c79ae2bac52a563db79f0d8d0
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l3-s0/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [5, -2, 43, -32, 32, -12, 2], "fmin": "4.97633548420288359252975785584755474318034134", "family": "rlve_polynomial_global_minimum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l3-s0/solution/solve.sh b/tasks/construct-rlve-polynomial-global-minimum-l3-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..6f82159abdff0c83664764cd923822b004b20e54
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l3-s0/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x0": 0.023871924416206353}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l3-s0/tests/check.py b/tasks/construct-rlve-polynomial-global-minimum-l3-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3be068f4391ad0838c29a4f7d521b8072cf39991
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l3-s0/tests/check.py
@@ -0,0 +1,19 @@
+from fractions import Fraction
+
+from run import Invalid, num
+
+
+def check(inst, ans):
+    x0 = Fraction(num(ans.get("x0"), "x0"))
+    if abs(x0) > 10 ** 6:
+        raise Invalid("x0 out of range")
+    v = Fraction(0)
+    for a in reversed(inst["coeffs"]):
+        v = v * x0 + a
+    fmin = Fraction(inst["fmin"])
+    tol = Fraction(1, 10 ** 9) * max(Fraction(1), abs(fmin))
+    if v > fmin + tol:
+        raise Invalid(f"f(x0) = {float(v)} is above the global minimum {float(fmin)}")
+    if v < fmin - tol:  # cannot happen for a correct f_min; guards against a bad instance
+        raise Invalid("internal: f(x0) below recorded minimum")
+    return None, {"f_x0": float(v)}
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l3-s0/tests/instance.json b/tasks/construct-rlve-polynomial-global-minimum-l3-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..7a5e3adf59d36a2c79ae2bac52a563db79f0d8d0
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l3-s0/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [5, -2, 43, -32, 32, -12, 2], "fmin": "4.97633548420288359252975785584755474318034134", "family": "rlve_polynomial_global_minimum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l3-s0/tests/run.py b/tasks/construct-rlve-polynomial-global-minimum-l3-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l3-s0/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l3-s0/tests/test.sh b/tasks/construct-rlve-polynomial-global-minimum-l3-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l3-s0/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l3-s7/environment/Dockerfile b/tasks/construct-rlve-polynomial-global-minimum-l3-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l3-s7/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l3-s7/environment/instance.json b/tasks/construct-rlve-polynomial-global-minimum-l3-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8f0afb8341af649833e506142407f28827632134
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l3-s7/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [146, 356, 506, 312, 121, 24, 2], "fmin": "61.0494148059365849200539747958503145605441076", "family": "rlve_polynomial_global_minimum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l3-s7/solution/solve.sh b/tasks/construct-rlve-polynomial-global-minimum-l3-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..34ab7f21d5626a80b1502ba51af6e5fd8cd29371
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l3-s7/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x0": -0.5869653520946588}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l3-s7/tests/check.py b/tasks/construct-rlve-polynomial-global-minimum-l3-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3be068f4391ad0838c29a4f7d521b8072cf39991
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l3-s7/tests/check.py
@@ -0,0 +1,19 @@
+from fractions import Fraction
+
+from run import Invalid, num
+
+
+def check(inst, ans):
+    x0 = Fraction(num(ans.get("x0"), "x0"))
+    if abs(x0) > 10 ** 6:
+        raise Invalid("x0 out of range")
+    v = Fraction(0)
+    for a in reversed(inst["coeffs"]):
+        v = v * x0 + a
+    fmin = Fraction(inst["fmin"])
+    tol = Fraction(1, 10 ** 9) * max(Fraction(1), abs(fmin))
+    if v > fmin + tol:
+        raise Invalid(f"f(x0) = {float(v)} is above the global minimum {float(fmin)}")
+    if v < fmin - tol:  # cannot happen for a correct f_min; guards against a bad instance
+        raise Invalid("internal: f(x0) below recorded minimum")
+    return None, {"f_x0": float(v)}
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l3-s7/tests/instance.json b/tasks/construct-rlve-polynomial-global-minimum-l3-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8f0afb8341af649833e506142407f28827632134
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l3-s7/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [146, 356, 506, 312, 121, 24, 2], "fmin": "61.0494148059365849200539747958503145605441076", "family": "rlve_polynomial_global_minimum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l3-s7/tests/run.py b/tasks/construct-rlve-polynomial-global-minimum-l3-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l3-s7/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l3-s7/tests/test.sh b/tasks/construct-rlve-polynomial-global-minimum-l3-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l3-s7/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l3-s9/environment/Dockerfile b/tasks/construct-rlve-polynomial-global-minimum-l3-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l3-s9/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l3-s9/environment/instance.json b/tasks/construct-rlve-polynomial-global-minimum-l3-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..01a7092c9d43b29bc64d6b38aa7c2d8f83869adf
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l3-s9/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [65, 190, 241, 160, 62, 12, 1], "fmin": "6.69375361817812410334478160913312142495056204", "family": "rlve_polynomial_global_minimum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l3-s9/solution/solve.sh b/tasks/construct-rlve-polynomial-global-minimum-l3-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..e2934a039c9342dbc1ded9705ceaf8b325805ba1
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l3-s9/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x0": -0.900475393694498}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l3-s9/tests/check.py b/tasks/construct-rlve-polynomial-global-minimum-l3-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3be068f4391ad0838c29a4f7d521b8072cf39991
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l3-s9/tests/check.py
@@ -0,0 +1,19 @@
+from fractions import Fraction
+
+from run import Invalid, num
+
+
+def check(inst, ans):
+    x0 = Fraction(num(ans.get("x0"), "x0"))
+    if abs(x0) > 10 ** 6:
+        raise Invalid("x0 out of range")
+    v = Fraction(0)
+    for a in reversed(inst["coeffs"]):
+        v = v * x0 + a
+    fmin = Fraction(inst["fmin"])
+    tol = Fraction(1, 10 ** 9) * max(Fraction(1), abs(fmin))
+    if v > fmin + tol:
+        raise Invalid(f"f(x0) = {float(v)} is above the global minimum {float(fmin)}")
+    if v < fmin - tol:  # cannot happen for a correct f_min; guards against a bad instance
+        raise Invalid("internal: f(x0) below recorded minimum")
+    return None, {"f_x0": float(v)}
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l3-s9/tests/instance.json b/tasks/construct-rlve-polynomial-global-minimum-l3-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..01a7092c9d43b29bc64d6b38aa7c2d8f83869adf
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l3-s9/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [65, 190, 241, 160, 62, 12, 1], "fmin": "6.69375361817812410334478160913312142495056204", "family": "rlve_polynomial_global_minimum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l3-s9/tests/run.py b/tasks/construct-rlve-polynomial-global-minimum-l3-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l3-s9/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l3-s9/tests/test.sh b/tasks/construct-rlve-polynomial-global-minimum-l3-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l3-s9/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l4-s2/environment/Dockerfile b/tasks/construct-rlve-polynomial-integer-roots-l4-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l4-s2/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l4-s2/environment/instance.json b/tasks/construct-rlve-polynomial-integer-roots-l4-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..fb98e131e086034f3a114d76b9edd3b816c718f9
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l4-s2/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [0, -101039400, -98597250, 19947951, 16543665, -1930980, -911816, 79714, 21638, -1292, -238, 7, 1], "family": "rlve_polynomial_integer_roots", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l4-s2/solution/solve.sh b/tasks/construct-rlve-polynomial-integer-roots-l4-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..af9ab1046e05fc92f04b37a0973c7c8b03f6b3b0
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l4-s2/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"roots": [-11, -9, -9, -5, -3, -1, 0, 4, 5, 6, 7, 9]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l4-s2/tests/check.py b/tasks/construct-rlve-polynomial-integer-roots-l4-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2d30553cb884c5e03c8afe73ef652efc483ab791
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l4-s2/tests/check.py
@@ -0,0 +1,20 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    c = inst["coeffs"]  # c[i] = coefficient of x^i, monic of degree len(c) - 1
+    n = len(c) - 1
+    r = int_list(ans.get("roots"), n, name="roots")
+    bound = 1 + max(abs(v) for v in c)  # Cauchy bound on roots
+    if any(abs(a) > bound for a in r):  # any integer root divides the constant term / is bounded by max|c_i|
+        raise Invalid("root out of range")
+    prod = [1]
+    for a in r:  # multiply by (x - a)
+        nxt = [0] * (len(prod) + 1)
+        for i, v in enumerate(prod):
+            nxt[i + 1] += v
+            nxt[i] -= a * v
+        prod = nxt
+    if prod != c:
+        raise Invalid("the product does not expand to P(x)")
+    return None, {}
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l4-s2/tests/instance.json b/tasks/construct-rlve-polynomial-integer-roots-l4-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..fb98e131e086034f3a114d76b9edd3b816c718f9
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l4-s2/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [0, -101039400, -98597250, 19947951, 16543665, -1930980, -911816, 79714, 21638, -1292, -238, 7, 1], "family": "rlve_polynomial_integer_roots", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l4-s2/tests/run.py b/tasks/construct-rlve-polynomial-integer-roots-l4-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l4-s2/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l4-s2/tests/test.sh b/tasks/construct-rlve-polynomial-integer-roots-l4-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l4-s2/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l4-s5/environment/Dockerfile b/tasks/construct-rlve-polynomial-integer-roots-l4-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l4-s5/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l4-s5/environment/instance.json b/tasks/construct-rlve-polynomial-integer-roots-l4-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..6ace925170d7136a48eaf6cdc3fe834d5e38cba7
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l4-s5/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [-1428840000, -642978000, 929505600, 31855680, -105826072, 9363241, 3011639, -420075, -24549, 5739, -59, -25, 1], "family": "rlve_polynomial_integer_roots", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l4-s5/solution/solve.sh b/tasks/construct-rlve-polynomial-integer-roots-l4-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..7b59d5b30814ced8ea0a9a5d597fdd2b9c252bbb
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l4-s5/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"roots": [-10, -9, -7, -3, -1, 2, 5, 7, 9, 10, 10, 12]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l4-s5/tests/check.py b/tasks/construct-rlve-polynomial-integer-roots-l4-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2d30553cb884c5e03c8afe73ef652efc483ab791
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l4-s5/tests/check.py
@@ -0,0 +1,20 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    c = inst["coeffs"]  # c[i] = coefficient of x^i, monic of degree len(c) - 1
+    n = len(c) - 1
+    r = int_list(ans.get("roots"), n, name="roots")
+    bound = 1 + max(abs(v) for v in c)  # Cauchy bound on roots
+    if any(abs(a) > bound for a in r):  # any integer root divides the constant term / is bounded by max|c_i|
+        raise Invalid("root out of range")
+    prod = [1]
+    for a in r:  # multiply by (x - a)
+        nxt = [0] * (len(prod) + 1)
+        for i, v in enumerate(prod):
+            nxt[i + 1] += v
+            nxt[i] -= a * v
+        prod = nxt
+    if prod != c:
+        raise Invalid("the product does not expand to P(x)")
+    return None, {}
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l4-s5/tests/instance.json b/tasks/construct-rlve-polynomial-integer-roots-l4-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..6ace925170d7136a48eaf6cdc3fe834d5e38cba7
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l4-s5/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [-1428840000, -642978000, 929505600, 31855680, -105826072, 9363241, 3011639, -420075, -24549, 5739, -59, -25, 1], "family": "rlve_polynomial_integer_roots", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l4-s5/tests/run.py b/tasks/construct-rlve-polynomial-integer-roots-l4-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l4-s5/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l4-s5/tests/test.sh b/tasks/construct-rlve-polynomial-integer-roots-l4-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l4-s5/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-polynomial-interpolation-l3-s2/environment/Dockerfile b/tasks/construct-rlve-polynomial-interpolation-l3-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l3-s2/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-polynomial-interpolation-l3-s2/environment/instance.json b/tasks/construct-rlve-polynomial-interpolation-l3-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..807067d560683dd3aca39c750443c9ce6b76b281
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l3-s2/environment/instance.json
@@ -0,0 +1 @@
+{"X": [0, 2, -2, -3, 3, 6, -4], "Y": [-4, 86, 450, 3821, 1265, 105962, 18632], "family": "rlve_polynomial_interpolation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-interpolation-l3-s2/solution/solve.sh b/tasks/construct-rlve-polynomial-interpolation-l3-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..7113fe73cd32e1a95aba84b5437ee6e41251acab
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l3-s2/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"coeffs": [-4, -3, 4, -2, 4, -5, 3]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-interpolation-l3-s2/tests/check.py b/tasks/construct-rlve-polynomial-interpolation-l3-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..f05264e3fd4584db7cb3768800e4a2497564b4f5
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l3-s2/tests/check.py
@@ -0,0 +1,13 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    X, Y = inst["X"], inst["Y"]
+    c = int_list(ans.get("coeffs"), len(X), name="coeffs")
+    for x, y in zip(X, Y):
+        v = 0
+        for a in reversed(c):
+            v = v * x + a
+        if v != y:
+            raise Invalid(f"f({x}) = {v}, need {y}")
+    return None, {}
diff --git a/tasks/construct-rlve-polynomial-interpolation-l3-s2/tests/instance.json b/tasks/construct-rlve-polynomial-interpolation-l3-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..807067d560683dd3aca39c750443c9ce6b76b281
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l3-s2/tests/instance.json
@@ -0,0 +1 @@
+{"X": [0, 2, -2, -3, 3, 6, -4], "Y": [-4, 86, 450, 3821, 1265, 105962, 18632], "family": "rlve_polynomial_interpolation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-interpolation-l3-s2/tests/run.py b/tasks/construct-rlve-polynomial-interpolation-l3-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l3-s2/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-polynomial-interpolation-l3-s2/tests/test.sh b/tasks/construct-rlve-polynomial-interpolation-l3-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l3-s2/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-prefix-product-permutation-l3-s0/environment/Dockerfile b/tasks/construct-rlve-prefix-product-permutation-l3-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l3-s0/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-prefix-product-permutation-l3-s0/environment/instance.json b/tasks/construct-rlve-prefix-product-permutation-l3-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..6750c901be7fbbc3235de12282df0896a7a4a46e
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l3-s0/environment/instance.json
@@ -0,0 +1 @@
+{"n": 281, "family": "rlve_prefix_product_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-product-permutation-l3-s0/solution/solve.sh b/tasks/construct-rlve-prefix-product-permutation-l3-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..064dc90735d13e8f3cceb932f722710da62b15f8
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l3-s0/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"perm": [1, 2, 142, 95, 212, 226, 48, 242, 247, 126, 254, 231, 165, 174, 262, 76, 124, 249, 204, 75, 268, 175, 116, 111, 83, 46, 228, 230, 272, 127, 179, 137, 203, 265, 125, 274, 243, 39, 38, 246, 275, 49, 88, 184, 199, 26, 56, 7, 42, 196, 164, 271, 255, 229, 256, 47, 277, 213, 64, 182, 90, 130, 69, 59, 102, 148, 133, 152, 63, 225, 278, 96, 122, 78, 20, 16, 160, 74, 264, 250, 138, 171, 25, 238, 185, 163, 233, 43, 100, 61, 154, 106, 169, 140, 4, 72, 162, 198, 239, 89, 223, 65, 136, 252, 128, 92, 115, 261, 269, 166, 24, 201, 139, 190, 107, 23, 173, 270, 232, 197, 186, 73, 206, 17, 35, 10, 30, 105, 192, 62, 215, 134, 67, 132, 217, 103, 32, 81, 113, 94, 280, 3, 189, 170, 202, 251, 180, 66, 151, 216, 149, 68, 221, 91, 178, 253, 273, 248, 266, 77, 210, 97, 86, 51, 13, 110, 260, 176, 93, 144, 82, 259, 117, 14, 22, 168, 191, 155, 31, 147, 218, 60, 194, 44, 85, 121, 211, 279, 143, 114, 177, 129, 222, 183, 240, 50, 120, 98, 45, 258, 112, 145, 33, 19, 209, 123, 267, 263, 205, 161, 187, 5, 58, 220, 131, 150, 135, 181, 224, 214, 153, 193, 101, 219, 70, 6, 236, 27, 54, 28, 12, 119, 87, 241, 276, 227, 257, 84, 99, 195, 234, 8, 37, 245, 244, 40, 9, 158, 18, 80, 146, 104, 156, 11, 53, 55, 237, 200, 172, 167, 108, 15, 208, 79, 34, 159, 207, 21, 109, 118, 52, 29, 157, 36, 41, 235, 57, 71, 188, 141, 281]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-prefix-product-permutation-l3-s0/tests/check.py b/tasks/construct-rlve-prefix-product-permutation-l3-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3d93850acd79cc0f8fff05cdbb0c63aabc484815
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l3-s0/tests/check.py
@@ -0,0 +1,15 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n = inst["n"]
+    p = int_list(ans.get("perm"), n, 1, n, "perm")
+    if len(set(p)) != n:
+        raise Invalid("not a permutation")
+    seen, acc = set(), 1
+    for v in p:
+        acc = acc * v % n
+        if acc in seen:
+            raise Invalid(f"prefix product residue {acc} repeats")
+        seen.add(acc)
+    return None, {}
diff --git a/tasks/construct-rlve-prefix-product-permutation-l3-s0/tests/instance.json b/tasks/construct-rlve-prefix-product-permutation-l3-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..6750c901be7fbbc3235de12282df0896a7a4a46e
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l3-s0/tests/instance.json
@@ -0,0 +1 @@
+{"n": 281, "family": "rlve_prefix_product_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-product-permutation-l3-s0/tests/run.py b/tasks/construct-rlve-prefix-product-permutation-l3-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l3-s0/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-prefix-product-permutation-l3-s0/tests/test.sh b/tasks/construct-rlve-prefix-product-permutation-l3-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l3-s0/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-prefix-product-permutation-l3-s7/environment/Dockerfile b/tasks/construct-rlve-prefix-product-permutation-l3-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l3-s7/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-prefix-product-permutation-l3-s7/environment/instance.json b/tasks/construct-rlve-prefix-product-permutation-l3-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3e89b2dcc36b942220edff12794e9d908841c070
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l3-s7/environment/instance.json
@@ -0,0 +1 @@
+{"n": 191, "family": "rlve_prefix_product_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-product-permutation-l3-s7/solution/solve.sh b/tasks/construct-rlve-prefix-product-permutation-l3-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..633716740de5ee17cbed9d23a4f70356cafa14db
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l3-s7/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"perm": [1, 2, 97, 65, 49, 154, 33, 83, 25, 86, 173, 140, 17, 148, 42, 52, 13, 46, 139, 182, 87, 92, 166, 109, 9, 108, 170, 93, 117, 113, 122, 38, 7, 111, 119, 132, 70, 32, 187, 50, 44, 15, 142, 41, 179, 18, 55, 127, 5, 40, 150, 16, 181, 174, 47, 67, 59, 125, 57, 69, 157, 120, 115, 95, 4, 145, 56, 135, 60, 37, 162, 114, 131, 158, 112, 164, 94, 130, 121, 163, 118, 159, 8, 169, 167, 10, 21, 102, 90, 89, 105, 22, 28, 77, 64, 190, 3, 129, 116, 165, 171, 88, 104, 103, 91, 172, 183, 26, 24, 185, 34, 75, 30, 72, 63, 99, 29, 81, 35, 62, 79, 31, 156, 133, 58, 137, 48, 189, 98, 78, 73, 36, 124, 136, 68, 134, 126, 146, 19, 12, 177, 43, 153, 188, 66, 138, 175, 14, 152, 51, 178, 149, 143, 6, 161, 123, 61, 74, 82, 186, 155, 71, 80, 76, 100, 23, 85, 184, 84, 27, 101, 106, 11, 54, 147, 180, 141, 151, 45, 176, 53, 20, 107, 168, 110, 160, 39, 144, 128, 96, 191]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-prefix-product-permutation-l3-s7/tests/check.py b/tasks/construct-rlve-prefix-product-permutation-l3-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3d93850acd79cc0f8fff05cdbb0c63aabc484815
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l3-s7/tests/check.py
@@ -0,0 +1,15 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n = inst["n"]
+    p = int_list(ans.get("perm"), n, 1, n, "perm")
+    if len(set(p)) != n:
+        raise Invalid("not a permutation")
+    seen, acc = set(), 1
+    for v in p:
+        acc = acc * v % n
+        if acc in seen:
+            raise Invalid(f"prefix product residue {acc} repeats")
+        seen.add(acc)
+    return None, {}
diff --git a/tasks/construct-rlve-prefix-product-permutation-l3-s7/tests/instance.json b/tasks/construct-rlve-prefix-product-permutation-l3-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3e89b2dcc36b942220edff12794e9d908841c070
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l3-s7/tests/instance.json
@@ -0,0 +1 @@
+{"n": 191, "family": "rlve_prefix_product_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-product-permutation-l3-s7/tests/run.py b/tasks/construct-rlve-prefix-product-permutation-l3-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l3-s7/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-prefix-product-permutation-l3-s7/tests/test.sh b/tasks/construct-rlve-prefix-product-permutation-l3-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l3-s7/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-prefix-product-permutation-l3-s9/environment/Dockerfile b/tasks/construct-rlve-prefix-product-permutation-l3-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l3-s9/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-prefix-product-permutation-l3-s9/environment/instance.json b/tasks/construct-rlve-prefix-product-permutation-l3-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..829acf37169ec95b74d88cb54f5d00fed411070a
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l3-s9/environment/instance.json
@@ -0,0 +1 @@
+{"n": 197, "family": "rlve_prefix_product_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-product-permutation-l3-s9/solution/solve.sh b/tasks/construct-rlve-prefix-product-permutation-l3-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..295f6ff13861ce61d68db064efd9bb1589f45ff6
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l3-s9/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"perm": [1, 2, 100, 67, 149, 80, 34, 170, 75, 23, 139, 19, 116, 92, 184, 93, 38, 59, 12, 84, 70, 123, 10, 61, 157, 135, 145, 74, 191, 35, 47, 90, 118, 7, 30, 153, 105, 17, 141, 97, 134, 174, 62, 56, 104, 163, 31, 110, 79, 194, 68, 86, 73, 172, 136, 44, 96, 160, 18, 188, 24, 43, 144, 173, 158, 98, 4, 51, 114, 21, 77, 112, 53, 28, 9, 177, 71, 88, 49, 6, 166, 91, 186, 20, 130, 52, 127, 78, 151, 32, 82, 14, 16, 162, 154, 57, 40, 66, 196, 3, 133, 159, 142, 45, 37, 183, 185, 117, 167, 48, 121, 72, 147, 69, 179, 13, 108, 33, 193, 150, 111, 128, 22, 190, 171, 146, 87, 122, 178, 85, 148, 195, 101, 41, 26, 55, 156, 175, 11, 181, 39, 103, 155, 63, 27, 126, 113, 131, 5, 120, 89, 168, 36, 95, 143, 137, 25, 65, 102, 58, 182, 94, 46, 169, 192, 81, 109, 152, 164, 8, 125, 54, 64, 42, 138, 189, 76, 129, 115, 187, 140, 161, 106, 15, 107, 83, 180, 60, 176, 124, 29, 165, 119, 50, 132, 99, 197]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-prefix-product-permutation-l3-s9/tests/check.py b/tasks/construct-rlve-prefix-product-permutation-l3-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3d93850acd79cc0f8fff05cdbb0c63aabc484815
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l3-s9/tests/check.py
@@ -0,0 +1,15 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n = inst["n"]
+    p = int_list(ans.get("perm"), n, 1, n, "perm")
+    if len(set(p)) != n:
+        raise Invalid("not a permutation")
+    seen, acc = set(), 1
+    for v in p:
+        acc = acc * v % n
+        if acc in seen:
+            raise Invalid(f"prefix product residue {acc} repeats")
+        seen.add(acc)
+    return None, {}
diff --git a/tasks/construct-rlve-prefix-product-permutation-l3-s9/tests/instance.json b/tasks/construct-rlve-prefix-product-permutation-l3-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..829acf37169ec95b74d88cb54f5d00fed411070a
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l3-s9/tests/instance.json
@@ -0,0 +1 @@
+{"n": 197, "family": "rlve_prefix_product_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-product-permutation-l3-s9/tests/run.py b/tasks/construct-rlve-prefix-product-permutation-l3-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l3-s9/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-prefix-product-permutation-l3-s9/tests/test.sh b/tasks/construct-rlve-prefix-product-permutation-l3-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l3-s9/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l2-s6/environment/Dockerfile b/tasks/construct-rlve-prefix-sum-permutation-l2-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l2-s6/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l2-s6/environment/instance.json b/tasks/construct-rlve-prefix-sum-permutation-l2-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..cb3b20548cddcc5188b9b0ed77bcfed8203951c2
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l2-s6/environment/instance.json
@@ -0,0 +1 @@
+{"n": 84, "family": "rlve_prefix_sum_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l2-s6/solution/solve.sh b/tasks/construct-rlve-prefix-sum-permutation-l2-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..c7782bc954ec52db30fbbfdf0dc4e28ebf22be06
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l2-s6/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"perm": [84, 1, 82, 3, 80, 5, 78, 7, 76, 9, 74, 11, 72, 13, 70, 15, 68, 17, 66, 19, 64, 21, 62, 23, 60, 25, 58, 27, 56, 29, 54, 31, 52, 33, 50, 35, 48, 37, 46, 39, 44, 41, 42, 43, 40, 45, 38, 47, 36, 49, 34, 51, 32, 53, 30, 55, 28, 57, 26, 59, 24, 61, 22, 63, 20, 65, 18, 67, 16, 69, 14, 71, 12, 73, 10, 75, 8, 77, 6, 79, 4, 81, 2, 83]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l2-s6/tests/check.py b/tasks/construct-rlve-prefix-sum-permutation-l2-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..60c3e0996cdfa8673fd7f34ad70cccb789b17654
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l2-s6/tests/check.py
@@ -0,0 +1,15 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n = inst["n"]
+    p = int_list(ans.get("perm"), n, 1, n, "perm")
+    if len(set(p)) != n:
+        raise Invalid("not a permutation")
+    seen, acc = set(), 0
+    for v in p:
+        acc = (acc + v) % n
+        if acc in seen:
+            raise Invalid(f"prefix sum residue {acc} repeats")
+        seen.add(acc)
+    return None, {}
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l2-s6/tests/instance.json b/tasks/construct-rlve-prefix-sum-permutation-l2-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..cb3b20548cddcc5188b9b0ed77bcfed8203951c2
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l2-s6/tests/instance.json
@@ -0,0 +1 @@
+{"n": 84, "family": "rlve_prefix_sum_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l2-s6/tests/run.py b/tasks/construct-rlve-prefix-sum-permutation-l2-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l2-s6/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l2-s6/tests/test.sh b/tasks/construct-rlve-prefix-sum-permutation-l2-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l2-s6/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l6-s8/environment/Dockerfile b/tasks/construct-rlve-prefix-sum-permutation-l6-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l6-s8/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l6-s8/environment/instance.json b/tasks/construct-rlve-prefix-sum-permutation-l6-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b4b5099b48a769f6ac0836be5abd3b69b5663e45
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l6-s8/environment/instance.json
@@ -0,0 +1 @@
+{"n": 10712, "family": "rlve_prefix_sum_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l6-s8/solution/solve.sh b/tasks/construct-rlve-prefix-sum-permutation-l6-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..946b03b5c3b1222a4867a0c8a1f8da887cf20b9e
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l6-s8/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"perm": [10712, 1, 10710, 3, 10708, 5, 10706, 7, 10704, 9, 10702, 11, 10700, 13, 10698, 15, 10696, 17, 10694, 19, 10692, 21, 10690, 23, 10688, 25, 10686, 27, 10684, 29, 10682, 31, 10680, 33, 10678, 35, 10676, 37, 10674, 39, 10672, 41, 10670, 43, 10668, 45, 10666, 47, 10664, 49, 10662, 51, 10660, 53, 10658, 55, 10656, 57, 10654, 59, 10652, 61, 10650, 63, 10648, 65, 10646, 67, 10644, 69, 10642, 71, 10640, 73, 10638, 75, 10636, 77, 10634, 79, 10632, 81, 10630, 83, 10628, 85, 10626, 87, 10624, 89, 10622, 91, 10620, 93, 10618, 95, 10616, 97, 10614, 99, 10612, 101, 10610, 103, 10608, 105, 10606, 107, 10604, 109, 10602, 111, 10600, 113, 10598, 115, 10596, 117, 10594, 119, 10592, 121, 10590, 123, 10588, 125, 10586, 127, 10584, 129, 10582, 131, 10580, 133, 10578, 135, 10576, 137, 10574, 139, 10572, 141, 10570, 143, 10568, 145, 10566, 147, 10564, 149, 10562, 151, 10560, 153, 10558, 155, 10556, 157, 10554, 159, 10552, 161, 10550, 163, 10548, 165, 10546, 167, 10544, 169, 10542, 171, 10540, 173, 10538, 175, 10536, 177, 10534, 179, 10532, 181, 10530, 183, 10528, 185, 10526, 187, 10524, 189, 10522, 191, 10520, 193, 10518, 195, 10516, 197, 10514, 199, 10512, 201, 10510, 203, 10508, 205, 10506, 207, 10504, 209, 10502, 211, 10500, 213, 10498, 215, 10496, 217, 10494, 219, 10492, 221, 10490, 223, 10488, 225, 10486, 227, 10484, 229, 10482, 231, 10480, 233, 10478, 235, 10476, 237, 10474, 239, 10472, 241, 10470, 243, 10468, 245, 10466, 247, 10464, 249, 10462, 251, 10460, 253, 10458, 255, 10456, 257, 10454, 259, 10452, 261, 10450, 263, 10448, 265, 10446, 267, 10444, 269, 10442, 271, 10440, 273, 10438, 275, 10436, 277, 10434, 279, 10432, 281, 10430, 283, 10428, 285, 10426, 287, 10424, 289, 10422, 291, 10420, 293, 10418, 295, 10416, 297, 10414, 299, 10412, 301, 10410, 303, 10408, 305, 10406, 307, 10404, 309, 10402, 311, 10400, 313, 10398, 315, 10396, 317, 10394, 319, 10392, 321, 10390, 323, 10388, 325, 10386, 327, 10384, 329, 10382, 331, 10380, 333, 10378, 335, 10376, 337, 10374, 339, 10372, 341, 10370, 343, 10368, 345, 10366, 347, 10364, 349, 10362, 351, 10360, 353, 10358, 355, 10356, 357, 10354, 359, 10352, 361, 10350, 363, 10348, 365, 10346, 367, 10344, 369, 10342, 371, 10340, 373, 10338, 375, 10336, 377, 10334, 379, 10332, 381, 10330, 383, 10328, 385, 10326, 387, 10324, 389, 10322, 391, 10320, 393, 10318, 395, 10316, 397, 10314, 399, 10312, 401, 10310, 403, 10308, 405, 10306, 407, 10304, 409, 10302, 411, 10300, 413, 10298, 415, 10296, 417, 10294, 419, 10292, 421, 10290, 423, 10288, 425, 10286, 427, 10284, 429, 10282, 431, 10280, 433, 10278, 435, 10276, 437, 10274, 439, 10272, 441, 10270, 443, 10268, 445, 10266, 447, 10264, 449, 10262, 451, 10260, 453, 10258, 455, 10256, 457, 10254, 459, 10252, 461, 10250, 463, 10248, 465, 10246, 467, 10244, 469, 10242, 471, 10240, 473, 10238, 475, 10236, 477, 10234, 479, 10232, 481, 10230, 483, 10228, 485, 10226, 487, 10224, 489, 10222, 491, 10220, 493, 10218, 495, 10216, 497, 10214, 499, 10212, 501, 10210, 503, 10208, 505, 10206, 507, 10204, 509, 10202, 511, 10200, 513, 10198, 515, 10196, 517, 10194, 519, 10192, 521, 10190, 523, 10188, 525, 10186, 527, 10184, 529, 10182, 531, 10180, 533, 10178, 535, 10176, 537, 10174, 539, 10172, 541, 10170, 543, 10168, 545, 10166, 547, 10164, 549, 10162, 551, 10160, 553, 10158, 555, 10156, 557, 10154, 559, 10152, 561, 10150, 563, 10148, 565, 10146, 567, 10144, 569, 10142, 571, 10140, 573, 10138, 575, 10136, 577, 10134, 579, 10132, 581, 10130, 583, 10128, 585, 10126, 587, 10124, 589, 10122, 591, 10120, 593, 10118, 595, 10116, 597, 10114, 599, 10112, 601, 10110, 603, 10108, 605, 10106, 607, 10104, 609, 10102, 611, 10100, 613, 10098, 615, 10096, 617, 10094, 619, 10092, 621, 10090, 623, 10088, 625, 10086, 627, 10084, 629, 10082, 631, 10080, 633, 10078, 635, 10076, 637, 10074, 639, 10072, 641, 10070, 643, 10068, 645, 10066, 647, 10064, 649, 10062, 651, 10060, 653, 10058, 655, 10056, 657, 10054, 659, 10052, 661, 10050, 663, 10048, 665, 10046, 667, 10044, 669, 10042, 671, 10040, 673, 10038, 675, 10036, 677, 10034, 679, 10032, 681, 10030, 683, 10028, 685, 10026, 687, 10024, 689, 10022, 691, 10020, 693, 10018, 695, 10016, 697, 10014, 699, 10012, 701, 10010, 703, 10008, 705, 10006, 707, 10004, 709, 10002, 711, 10000, 713, 9998, 715, 9996, 717, 9994, 719, 9992, 721, 9990, 723, 9988, 725, 9986, 727, 9984, 729, 9982, 731, 9980, 733, 9978, 735, 9976, 737, 9974, 739, 9972, 741, 9970, 743, 9968, 745, 9966, 747, 9964, 749, 9962, 751, 9960, 753, 9958, 755, 9956, 757, 9954, 759, 9952, 761, 9950, 763, 9948, 765, 9946, 767, 9944, 769, 9942, 771, 9940, 773, 9938, 775, 9936, 777, 9934, 779, 9932, 781, 9930, 783, 9928, 785, 9926, 787, 9924, 789, 9922, 791, 9920, 793, 9918, 795, 9916, 797, 9914, 799, 9912, 801, 9910, 803, 9908, 805, 9906, 807, 9904, 809, 9902, 811, 9900, 813, 9898, 815, 9896, 817, 9894, 819, 9892, 821, 9890, 823, 9888, 825, 9886, 827, 9884, 829, 9882, 831, 9880, 833, 9878, 835, 9876, 837, 9874, 839, 9872, 841, 9870, 843, 9868, 845, 9866, 847, 9864, 849, 9862, 851, 9860, 853, 9858, 855, 9856, 857, 9854, 859, 9852, 861, 9850, 863, 9848, 865, 9846, 867, 9844, 869, 9842, 871, 9840, 873, 9838, 875, 9836, 877, 9834, 879, 9832, 881, 9830, 883, 9828, 885, 9826, 887, 9824, 889, 9822, 891, 9820, 893, 9818, 895, 9816, 897, 9814, 899, 9812, 901, 9810, 903, 9808, 905, 9806, 907, 9804, 909, 9802, 911, 9800, 913, 9798, 915, 9796, 917, 9794, 919, 9792, 921, 9790, 923, 9788, 925, 9786, 927, 9784, 929, 9782, 931, 9780, 933, 9778, 935, 9776, 937, 9774, 939, 9772, 941, 9770, 943, 9768, 945, 9766, 947, 9764, 949, 9762, 951, 9760, 953, 9758, 955, 9756, 957, 9754, 959, 9752, 961, 9750, 963, 9748, 965, 9746, 967, 9744, 969, 9742, 971, 9740, 973, 9738, 975, 9736, 977, 9734, 979, 9732, 981, 9730, 983, 9728, 985, 9726, 987, 9724, 989, 9722, 991, 9720, 993, 9718, 995, 9716, 997, 9714, 999, 9712, 1001, 9710, 1003, 9708, 1005, 9706, 1007, 9704, 1009, 9702, 1011, 9700, 1013, 9698, 1015, 9696, 1017, 9694, 1019, 9692, 1021, 9690, 1023, 9688, 1025, 9686, 1027, 9684, 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8831, 1880, 8833, 1878, 8835, 1876, 8837, 1874, 8839, 1872, 8841, 1870, 8843, 1868, 8845, 1866, 8847, 1864, 8849, 1862, 8851, 1860, 8853, 1858, 8855, 1856, 8857, 1854, 8859, 1852, 8861, 1850, 8863, 1848, 8865, 1846, 8867, 1844, 8869, 1842, 8871, 1840, 8873, 1838, 8875, 1836, 8877, 1834, 8879, 1832, 8881, 1830, 8883, 1828, 8885, 1826, 8887, 1824, 8889, 1822, 8891, 1820, 8893, 1818, 8895, 1816, 8897, 1814, 8899, 1812, 8901, 1810, 8903, 1808, 8905, 1806, 8907, 1804, 8909, 1802, 8911, 1800, 8913, 1798, 8915, 1796, 8917, 1794, 8919, 1792, 8921, 1790, 8923, 1788, 8925, 1786, 8927, 1784, 8929, 1782, 8931, 1780, 8933, 1778, 8935, 1776, 8937, 1774, 8939, 1772, 8941, 1770, 8943, 1768, 8945, 1766, 8947, 1764, 8949, 1762, 8951, 1760, 8953, 1758, 8955, 1756, 8957, 1754, 8959, 1752, 8961, 1750, 8963, 1748, 8965, 1746, 8967, 1744, 8969, 1742, 8971, 1740, 8973, 1738, 8975, 1736, 8977, 1734, 8979, 1732, 8981, 1730, 8983, 1728, 8985, 1726, 8987, 1724, 8989, 1722, 8991, 1720, 8993, 1718, 8995, 1716, 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694, 10019, 692, 10021, 690, 10023, 688, 10025, 686, 10027, 684, 10029, 682, 10031, 680, 10033, 678, 10035, 676, 10037, 674, 10039, 672, 10041, 670, 10043, 668, 10045, 666, 10047, 664, 10049, 662, 10051, 660, 10053, 658, 10055, 656, 10057, 654, 10059, 652, 10061, 650, 10063, 648, 10065, 646, 10067, 644, 10069, 642, 10071, 640, 10073, 638, 10075, 636, 10077, 634, 10079, 632, 10081, 630, 10083, 628, 10085, 626, 10087, 624, 10089, 622, 10091, 620, 10093, 618, 10095, 616, 10097, 614, 10099, 612, 10101, 610, 10103, 608, 10105, 606, 10107, 604, 10109, 602, 10111, 600, 10113, 598, 10115, 596, 10117, 594, 10119, 592, 10121, 590, 10123, 588, 10125, 586, 10127, 584, 10129, 582, 10131, 580, 10133, 578, 10135, 576, 10137, 574, 10139, 572, 10141, 570, 10143, 568, 10145, 566, 10147, 564, 10149, 562, 10151, 560, 10153, 558, 10155, 556, 10157, 554, 10159, 552, 10161, 550, 10163, 548, 10165, 546, 10167, 544, 10169, 542, 10171, 540, 10173, 538, 10175, 536, 10177, 534, 10179, 532, 10181, 530, 10183, 528, 10185, 526, 10187, 524, 10189, 522, 10191, 520, 10193, 518, 10195, 516, 10197, 514, 10199, 512, 10201, 510, 10203, 508, 10205, 506, 10207, 504, 10209, 502, 10211, 500, 10213, 498, 10215, 496, 10217, 494, 10219, 492, 10221, 490, 10223, 488, 10225, 486, 10227, 484, 10229, 482, 10231, 480, 10233, 478, 10235, 476, 10237, 474, 10239, 472, 10241, 470, 10243, 468, 10245, 466, 10247, 464, 10249, 462, 10251, 460, 10253, 458, 10255, 456, 10257, 454, 10259, 452, 10261, 450, 10263, 448, 10265, 446, 10267, 444, 10269, 442, 10271, 440, 10273, 438, 10275, 436, 10277, 434, 10279, 432, 10281, 430, 10283, 428, 10285, 426, 10287, 424, 10289, 422, 10291, 420, 10293, 418, 10295, 416, 10297, 414, 10299, 412, 10301, 410, 10303, 408, 10305, 406, 10307, 404, 10309, 402, 10311, 400, 10313, 398, 10315, 396, 10317, 394, 10319, 392, 10321, 390, 10323, 388, 10325, 386, 10327, 384, 10329, 382, 10331, 380, 10333, 378, 10335, 376, 10337, 374, 10339, 372, 10341, 370, 10343, 368, 10345, 366, 10347, 364, 10349, 362, 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22, 10691, 20, 10693, 18, 10695, 16, 10697, 14, 10699, 12, 10701, 10, 10703, 8, 10705, 6, 10707, 4, 10709, 2, 10711]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l6-s8/tests/check.py b/tasks/construct-rlve-prefix-sum-permutation-l6-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..60c3e0996cdfa8673fd7f34ad70cccb789b17654
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l6-s8/tests/check.py
@@ -0,0 +1,15 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n = inst["n"]
+    p = int_list(ans.get("perm"), n, 1, n, "perm")
+    if len(set(p)) != n:
+        raise Invalid("not a permutation")
+    seen, acc = set(), 0
+    for v in p:
+        acc = (acc + v) % n
+        if acc in seen:
+            raise Invalid(f"prefix sum residue {acc} repeats")
+        seen.add(acc)
+    return None, {}
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l6-s8/tests/instance.json b/tasks/construct-rlve-prefix-sum-permutation-l6-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b4b5099b48a769f6ac0836be5abd3b69b5663e45
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l6-s8/tests/instance.json
@@ -0,0 +1 @@
+{"n": 10712, "family": "rlve_prefix_sum_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l6-s8/tests/run.py b/tasks/construct-rlve-prefix-sum-permutation-l6-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l6-s8/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l6-s8/tests/test.sh b/tasks/construct-rlve-prefix-sum-permutation-l6-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l6-s8/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l6-s9/environment/Dockerfile b/tasks/construct-rlve-prefix-sum-permutation-l6-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l6-s9/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l6-s9/environment/instance.json b/tasks/construct-rlve-prefix-sum-permutation-l6-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..bc9ec44bf226d9d07eb20478da9905e764d5e276
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l6-s9/environment/instance.json
@@ -0,0 +1 @@
+{"n": 22966, "family": "rlve_prefix_sum_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l6-s9/solution/solve.sh b/tasks/construct-rlve-prefix-sum-permutation-l6-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..54c3336a07eef43b4e96406f08afab97e6e06d52
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l6-s9/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"perm": [22966, 1, 22964, 3, 22962, 5, 22960, 7, 22958, 9, 22956, 11, 22954, 13, 22952, 15, 22950, 17, 22948, 19, 22946, 21, 22944, 23, 22942, 25, 22940, 27, 22938, 29, 22936, 31, 22934, 33, 22932, 35, 22930, 37, 22928, 39, 22926, 41, 22924, 43, 22922, 45, 22920, 47, 22918, 49, 22916, 51, 22914, 53, 22912, 55, 22910, 57, 22908, 59, 22906, 61, 22904, 63, 22902, 65, 22900, 67, 22898, 69, 22896, 71, 22894, 73, 22892, 75, 22890, 77, 22888, 79, 22886, 81, 22884, 83, 22882, 85, 22880, 87, 22878, 89, 22876, 91, 22874, 93, 22872, 95, 22870, 97, 22868, 99, 22866, 101, 22864, 103, 22862, 105, 22860, 107, 22858, 109, 22856, 111, 22854, 113, 22852, 115, 22850, 117, 22848, 119, 22846, 121, 22844, 123, 22842, 125, 22840, 127, 22838, 129, 22836, 131, 22834, 133, 22832, 135, 22830, 137, 22828, 139, 22826, 141, 22824, 143, 22822, 145, 22820, 147, 22818, 149, 22816, 151, 22814, 153, 22812, 155, 22810, 157, 22808, 159, 22806, 161, 22804, 163, 22802, 165, 22800, 167, 22798, 169, 22796, 171, 22794, 173, 22792, 175, 22790, 177, 22788, 179, 22786, 181, 22784, 183, 22782, 185, 22780, 187, 22778, 189, 22776, 191, 22774, 193, 22772, 195, 22770, 197, 22768, 199, 22766, 201, 22764, 203, 22762, 205, 22760, 207, 22758, 209, 22756, 211, 22754, 213, 22752, 215, 22750, 217, 22748, 219, 22746, 221, 22744, 223, 22742, 225, 22740, 227, 22738, 229, 22736, 231, 22734, 233, 22732, 235, 22730, 237, 22728, 239, 22726, 241, 22724, 243, 22722, 245, 22720, 247, 22718, 249, 22716, 251, 22714, 253, 22712, 255, 22710, 257, 22708, 259, 22706, 261, 22704, 263, 22702, 265, 22700, 267, 22698, 269, 22696, 271, 22694, 273, 22692, 275, 22690, 277, 22688, 279, 22686, 281, 22684, 283, 22682, 285, 22680, 287, 22678, 289, 22676, 291, 22674, 293, 22672, 295, 22670, 297, 22668, 299, 22666, 301, 22664, 303, 22662, 305, 22660, 307, 22658, 309, 22656, 311, 22654, 313, 22652, 315, 22650, 317, 22648, 319, 22646, 321, 22644, 323, 22642, 325, 22640, 327, 22638, 329, 22636, 331, 22634, 333, 22632, 335, 22630, 337, 22628, 339, 22626, 341, 22624, 343, 22622, 345, 22620, 347, 22618, 349, 22616, 351, 22614, 353, 22612, 355, 22610, 357, 22608, 359, 22606, 361, 22604, 363, 22602, 365, 22600, 367, 22598, 369, 22596, 371, 22594, 373, 22592, 375, 22590, 377, 22588, 379, 22586, 381, 22584, 383, 22582, 385, 22580, 387, 22578, 389, 22576, 391, 22574, 393, 22572, 395, 22570, 397, 22568, 399, 22566, 401, 22564, 403, 22562, 405, 22560, 407, 22558, 409, 22556, 411, 22554, 413, 22552, 415, 22550, 417, 22548, 419, 22546, 421, 22544, 423, 22542, 425, 22540, 427, 22538, 429, 22536, 431, 22534, 433, 22532, 435, 22530, 437, 22528, 439, 22526, 441, 22524, 443, 22522, 445, 22520, 447, 22518, 449, 22516, 451, 22514, 453, 22512, 455, 22510, 457, 22508, 459, 22506, 461, 22504, 463, 22502, 465, 22500, 467, 22498, 469, 22496, 471, 22494, 473, 22492, 475, 22490, 477, 22488, 479, 22486, 481, 22484, 483, 22482, 485, 22480, 487, 22478, 489, 22476, 491, 22474, 493, 22472, 495, 22470, 497, 22468, 499, 22466, 501, 22464, 503, 22462, 505, 22460, 507, 22458, 509, 22456, 511, 22454, 513, 22452, 515, 22450, 517, 22448, 519, 22446, 521, 22444, 523, 22442, 525, 22440, 527, 22438, 529, 22436, 531, 22434, 533, 22432, 535, 22430, 537, 22428, 539, 22426, 541, 22424, 543, 22422, 545, 22420, 547, 22418, 549, 22416, 551, 22414, 553, 22412, 555, 22410, 557, 22408, 559, 22406, 561, 22404, 563, 22402, 565, 22400, 567, 22398, 569, 22396, 571, 22394, 573, 22392, 575, 22390, 577, 22388, 579, 22386, 581, 22384, 583, 22382, 585, 22380, 587, 22378, 589, 22376, 591, 22374, 593, 22372, 595, 22370, 597, 22368, 599, 22366, 601, 22364, 603, 22362, 605, 22360, 607, 22358, 609, 22356, 611, 22354, 613, 22352, 615, 22350, 617, 22348, 619, 22346, 621, 22344, 623, 22342, 625, 22340, 627, 22338, 629, 22336, 631, 22334, 633, 22332, 635, 22330, 637, 22328, 639, 22326, 641, 22324, 643, 22322, 645, 22320, 647, 22318, 649, 22316, 651, 22314, 653, 22312, 655, 22310, 657, 22308, 659, 22306, 661, 22304, 663, 22302, 665, 22300, 667, 22298, 669, 22296, 671, 22294, 673, 22292, 675, 22290, 677, 22288, 679, 22286, 681, 22284, 683, 22282, 685, 22280, 687, 22278, 689, 22276, 691, 22274, 693, 22272, 695, 22270, 697, 22268, 699, 22266, 701, 22264, 703, 22262, 705, 22260, 707, 22258, 709, 22256, 711, 22254, 713, 22252, 715, 22250, 717, 22248, 719, 22246, 721, 22244, 723, 22242, 725, 22240, 727, 22238, 729, 22236, 731, 22234, 733, 22232, 735, 22230, 737, 22228, 739, 22226, 741, 22224, 743, 22222, 745, 22220, 747, 22218, 749, 22216, 751, 22214, 753, 22212, 755, 22210, 757, 22208, 759, 22206, 761, 22204, 763, 22202, 765, 22200, 767, 22198, 769, 22196, 771, 22194, 773, 22192, 775, 22190, 777, 22188, 779, 22186, 781, 22184, 783, 22182, 785, 22180, 787, 22178, 789, 22176, 791, 22174, 793, 22172, 795, 22170, 797, 22168, 799, 22166, 801, 22164, 803, 22162, 805, 22160, 807, 22158, 809, 22156, 811, 22154, 813, 22152, 815, 22150, 817, 22148, 819, 22146, 821, 22144, 823, 22142, 825, 22140, 827, 22138, 829, 22136, 831, 22134, 833, 22132, 835, 22130, 837, 22128, 839, 22126, 841, 22124, 843, 22122, 845, 22120, 847, 22118, 849, 22116, 851, 22114, 853, 22112, 855, 22110, 857, 22108, 859, 22106, 861, 22104, 863, 22102, 865, 22100, 867, 22098, 869, 22096, 871, 22094, 873, 22092, 875, 22090, 877, 22088, 879, 22086, 881, 22084, 883, 22082, 885, 22080, 887, 22078, 889, 22076, 891, 22074, 893, 22072, 895, 22070, 897, 22068, 899, 22066, 901, 22064, 903, 22062, 905, 22060, 907, 22058, 909, 22056, 911, 22054, 913, 22052, 915, 22050, 917, 22048, 919, 22046, 921, 22044, 923, 22042, 925, 22040, 927, 22038, 929, 22036, 931, 22034, 933, 22032, 935, 22030, 937, 22028, 939, 22026, 941, 22024, 943, 22022, 945, 22020, 947, 22018, 949, 22016, 951, 22014, 953, 22012, 955, 22010, 957, 22008, 959, 22006, 961, 22004, 963, 22002, 965, 22000, 967, 21998, 969, 21996, 971, 21994, 973, 21992, 975, 21990, 977, 21988, 979, 21986, 981, 21984, 983, 21982, 985, 21980, 987, 21978, 989, 21976, 991, 21974, 993, 21972, 995, 21970, 997, 21968, 999, 21966, 1001, 21964, 1003, 21962, 1005, 21960, 1007, 21958, 1009, 21956, 1011, 21954, 1013, 21952, 1015, 21950, 1017, 21948, 1019, 21946, 1021, 21944, 1023, 21942, 1025, 21940, 1027, 21938, 1029, 21936, 1031, 21934, 1033, 21932, 1035, 21930, 1037, 21928, 1039, 21926, 1041, 21924, 1043, 21922, 1045, 21920, 1047, 21918, 1049, 21916, 1051, 21914, 1053, 21912, 1055, 21910, 1057, 21908, 1059, 21906, 1061, 21904, 1063, 21902, 1065, 21900, 1067, 21898, 1069, 21896, 1071, 21894, 1073, 21892, 1075, 21890, 1077, 21888, 1079, 21886, 1081, 21884, 1083, 21882, 1085, 21880, 1087, 21878, 1089, 21876, 1091, 21874, 1093, 21872, 1095, 21870, 1097, 21868, 1099, 21866, 1101, 21864, 1103, 21862, 1105, 21860, 1107, 21858, 1109, 21856, 1111, 21854, 1113, 21852, 1115, 21850, 1117, 21848, 1119, 21846, 1121, 21844, 1123, 21842, 1125, 21840, 1127, 21838, 1129, 21836, 1131, 21834, 1133, 21832, 1135, 21830, 1137, 21828, 1139, 21826, 1141, 21824, 1143, 21822, 1145, 21820, 1147, 21818, 1149, 21816, 1151, 21814, 1153, 21812, 1155, 21810, 1157, 21808, 1159, 21806, 1161, 21804, 1163, 21802, 1165, 21800, 1167, 21798, 1169, 21796, 1171, 21794, 1173, 21792, 1175, 21790, 1177, 21788, 1179, 21786, 1181, 21784, 1183, 21782, 1185, 21780, 1187, 21778, 1189, 21776, 1191, 21774, 1193, 21772, 1195, 21770, 1197, 21768, 1199, 21766, 1201, 21764, 1203, 21762, 1205, 21760, 1207, 21758, 1209, 21756, 1211, 21754, 1213, 21752, 1215, 21750, 1217, 21748, 1219, 21746, 1221, 21744, 1223, 21742, 1225, 21740, 1227, 21738, 1229, 21736, 1231, 21734, 1233, 21732, 1235, 21730, 1237, 21728, 1239, 21726, 1241, 21724, 1243, 21722, 1245, 21720, 1247, 21718, 1249, 21716, 1251, 21714, 1253, 21712, 1255, 21710, 1257, 21708, 1259, 21706, 1261, 21704, 1263, 21702, 1265, 21700, 1267, 21698, 1269, 21696, 1271, 21694, 1273, 21692, 1275, 21690, 1277, 21688, 1279, 21686, 1281, 21684, 1283, 21682, 1285, 21680, 1287, 21678, 1289, 21676, 1291, 21674, 1293, 21672, 1295, 21670, 1297, 21668, 1299, 21666, 1301, 21664, 1303, 21662, 1305, 21660, 1307, 21658, 1309, 21656, 1311, 21654, 1313, 21652, 1315, 21650, 1317, 21648, 1319, 21646, 1321, 21644, 1323, 21642, 1325, 21640, 1327, 21638, 1329, 21636, 1331, 21634, 1333, 21632, 1335, 21630, 1337, 21628, 1339, 21626, 1341, 21624, 1343, 21622, 1345, 21620, 1347, 21618, 1349, 21616, 1351, 21614, 1353, 21612, 1355, 21610, 1357, 21608, 1359, 21606, 1361, 21604, 1363, 21602, 1365, 21600, 1367, 21598, 1369, 21596, 1371, 21594, 1373, 21592, 1375, 21590, 1377, 21588, 1379, 21586, 1381, 21584, 1383, 21582, 1385, 21580, 1387, 21578, 1389, 21576, 1391, 21574, 1393, 21572, 1395, 21570, 1397, 21568, 1399, 21566, 1401, 21564, 1403, 21562, 1405, 21560, 1407, 21558, 1409, 21556, 1411, 21554, 1413, 21552, 1415, 21550, 1417, 21548, 1419, 21546, 1421, 21544, 1423, 21542, 1425, 21540, 1427, 21538, 1429, 21536, 1431, 21534, 1433, 21532, 1435, 21530, 1437, 21528, 1439, 21526, 1441, 21524, 1443, 21522, 1445, 21520, 1447, 21518, 1449, 21516, 1451, 21514, 1453, 21512, 1455, 21510, 1457, 21508, 1459, 21506, 1461, 21504, 1463, 21502, 1465, 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22861, 104, 22863, 102, 22865, 100, 22867, 98, 22869, 96, 22871, 94, 22873, 92, 22875, 90, 22877, 88, 22879, 86, 22881, 84, 22883, 82, 22885, 80, 22887, 78, 22889, 76, 22891, 74, 22893, 72, 22895, 70, 22897, 68, 22899, 66, 22901, 64, 22903, 62, 22905, 60, 22907, 58, 22909, 56, 22911, 54, 22913, 52, 22915, 50, 22917, 48, 22919, 46, 22921, 44, 22923, 42, 22925, 40, 22927, 38, 22929, 36, 22931, 34, 22933, 32, 22935, 30, 22937, 28, 22939, 26, 22941, 24, 22943, 22, 22945, 20, 22947, 18, 22949, 16, 22951, 14, 22953, 12, 22955, 10, 22957, 8, 22959, 6, 22961, 4, 22963, 2, 22965]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l6-s9/tests/check.py b/tasks/construct-rlve-prefix-sum-permutation-l6-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..60c3e0996cdfa8673fd7f34ad70cccb789b17654
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l6-s9/tests/check.py
@@ -0,0 +1,15 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n = inst["n"]
+    p = int_list(ans.get("perm"), n, 1, n, "perm")
+    if len(set(p)) != n:
+        raise Invalid("not a permutation")
+    seen, acc = set(), 0
+    for v in p:
+        acc = (acc + v) % n
+        if acc in seen:
+            raise Invalid(f"prefix sum residue {acc} repeats")
+        seen.add(acc)
+    return None, {}
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l6-s9/tests/instance.json b/tasks/construct-rlve-prefix-sum-permutation-l6-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..bc9ec44bf226d9d07eb20478da9905e764d5e276
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l6-s9/tests/instance.json
@@ -0,0 +1 @@
+{"n": 22966, "family": "rlve_prefix_sum_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l6-s9/tests/run.py b/tasks/construct-rlve-prefix-sum-permutation-l6-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l6-s9/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l6-s9/tests/test.sh b/tasks/construct-rlve-prefix-sum-permutation-l6-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l6-s9/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-prime-graph-coloring-l3-s0/environment/Dockerfile b/tasks/construct-rlve-prime-graph-coloring-l3-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-l3-s0/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-prime-graph-coloring-l3-s0/environment/instance.json b/tasks/construct-rlve-prime-graph-coloring-l3-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..cc2cb59529f3f7a8f05391723d97e769549f7ef8
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-l3-s0/environment/instance.json
@@ -0,0 +1 @@
+{"n": 43, "k": 4, "family": "rlve_prime_graph_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prime-graph-coloring-l3-s0/solution/solve.sh b/tasks/construct-rlve-prime-graph-coloring-l3-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..9992307075de7a0ca2941dfbefdd11b5eb979c46
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-l3-s0/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"colors": [1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-prime-graph-coloring-l3-s0/tests/check.py b/tasks/construct-rlve-prime-graph-coloring-l3-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..b3736f1375f1862241999142c975ced361b8fcb5
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-l3-s0/tests/check.py
@@ -0,0 +1,20 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    c = int_list(ans.get("colors"), n, 0, k - 1, "colors")
+    sieve = [True] * (n + 1)
+    sieve[0] = False
+    if n >= 1:
+        sieve[1] = False
+    for i in range(2, int(n ** 0.5) + 1):
+        if sieve[i]:
+            for j in range(i * i, n + 1, i):
+                sieve[j] = False
+    for p in range(2, n):
+        if sieve[p]:
+            for u in range(n - p):
+                if c[u] == c[u + p]:
+                    raise Invalid(f"vertices {u + 1} and {u + p + 1} (difference {p}) share a colour")
+    return None, {}
diff --git a/tasks/construct-rlve-prime-graph-coloring-l3-s0/tests/instance.json b/tasks/construct-rlve-prime-graph-coloring-l3-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..cc2cb59529f3f7a8f05391723d97e769549f7ef8
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-l3-s0/tests/instance.json
@@ -0,0 +1 @@
+{"n": 43, "k": 4, "family": "rlve_prime_graph_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prime-graph-coloring-l3-s0/tests/run.py b/tasks/construct-rlve-prime-graph-coloring-l3-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-l3-s0/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-prime-graph-coloring-l3-s0/tests/test.sh b/tasks/construct-rlve-prime-graph-coloring-l3-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-l3-s0/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l1-s3/environment/Dockerfile b/tasks/construct-rlve-recursive-sequence-sum-l1-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l1-s3/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l1-s3/environment/instance.json b/tasks/construct-rlve-recursive-sequence-sum-l1-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..bd1fe5851784deccd75196935903f3ee82b32a7f
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l1-s3/environment/instance.json
@@ -0,0 +1 @@
+{"F0": 116, "A": 12, "B": 8992, "S": 10384, "family": "rlve_recursive_sequence_sum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l1-s3/solution/solve.sh b/tasks/construct-rlve-recursive-sequence-sum-l1-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..a9905329f0545a88db99d9b47a6808bfe6bf637d
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l1-s3/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"indices": [1]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l1-s3/tests/check.py b/tasks/construct-rlve-recursive-sequence-sum-l1-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..feda076c3f95e5e12198151dee30e1b8223924bc
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l1-s3/tests/check.py
@@ -0,0 +1,29 @@
+from run import Invalid, int_list
+
+
+def F(inst, n):
+    F0, A, B = inst["F0"], inst["A"], inst["B"]
+    if A == 1:
+        return F0 + n * B
+    An = A ** n
+    return An * F0 + B * (An - 1) // (A - 1)
+
+
+def check(inst, ans):
+    S = inst["S"]
+    idx = int_list(ans.get("indices"), None, 1, None, "indices")
+    if not idx:
+        raise Invalid("need at least one index")
+    if len(set(idx)) != len(idx):
+        raise Invalid("indices must be distinct")
+    total = 0
+    for n in idx:
+        # F is strictly increasing with F(n) >= n and, when A >= 2, F(n) >= 2^(n-1); bail out before big powers
+        if n > S or (inst["A"] >= 2 and n - 1 > S.bit_length()):
+            raise Invalid(f"F({n}) exceeds S")
+        total += F(inst, n)
+        if total > S:
+            raise Invalid("sum exceeds S")
+    if total != S:
+        raise Invalid(f"sum is {total}, need {S}")
+    return None, {}
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l1-s3/tests/instance.json b/tasks/construct-rlve-recursive-sequence-sum-l1-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..bd1fe5851784deccd75196935903f3ee82b32a7f
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l1-s3/tests/instance.json
@@ -0,0 +1 @@
+{"F0": 116, "A": 12, "B": 8992, "S": 10384, "family": "rlve_recursive_sequence_sum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l1-s3/tests/run.py b/tasks/construct-rlve-recursive-sequence-sum-l1-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l1-s3/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l1-s3/tests/test.sh b/tasks/construct-rlve-recursive-sequence-sum-l1-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l1-s3/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l1-s4/environment/Dockerfile b/tasks/construct-rlve-recursive-sequence-sum-l1-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l1-s4/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l1-s4/environment/instance.json b/tasks/construct-rlve-recursive-sequence-sum-l1-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b4fbafade0ade291e5d22a8346964b80558e9d82
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l1-s4/environment/instance.json
@@ -0,0 +1 @@
+{"F0": 74, "A": 4, "B": 12677, "S": 348495, "family": "rlve_recursive_sequence_sum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l1-s4/solution/solve.sh b/tasks/construct-rlve-recursive-sequence-sum-l1-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..c621b4e4f01eab810b4e898c046839c8aa6ca435
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l1-s4/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"indices": [1, 2, 3]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l1-s4/tests/check.py b/tasks/construct-rlve-recursive-sequence-sum-l1-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..feda076c3f95e5e12198151dee30e1b8223924bc
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l1-s4/tests/check.py
@@ -0,0 +1,29 @@
+from run import Invalid, int_list
+
+
+def F(inst, n):
+    F0, A, B = inst["F0"], inst["A"], inst["B"]
+    if A == 1:
+        return F0 + n * B
+    An = A ** n
+    return An * F0 + B * (An - 1) // (A - 1)
+
+
+def check(inst, ans):
+    S = inst["S"]
+    idx = int_list(ans.get("indices"), None, 1, None, "indices")
+    if not idx:
+        raise Invalid("need at least one index")
+    if len(set(idx)) != len(idx):
+        raise Invalid("indices must be distinct")
+    total = 0
+    for n in idx:
+        # F is strictly increasing with F(n) >= n and, when A >= 2, F(n) >= 2^(n-1); bail out before big powers
+        if n > S or (inst["A"] >= 2 and n - 1 > S.bit_length()):
+            raise Invalid(f"F({n}) exceeds S")
+        total += F(inst, n)
+        if total > S:
+            raise Invalid("sum exceeds S")
+    if total != S:
+        raise Invalid(f"sum is {total}, need {S}")
+    return None, {}
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l1-s4/tests/instance.json b/tasks/construct-rlve-recursive-sequence-sum-l1-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b4fbafade0ade291e5d22a8346964b80558e9d82
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l1-s4/tests/instance.json
@@ -0,0 +1 @@
+{"F0": 74, "A": 4, "B": 12677, "S": 348495, "family": "rlve_recursive_sequence_sum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l1-s4/tests/run.py b/tasks/construct-rlve-recursive-sequence-sum-l1-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l1-s4/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l1-s4/tests/test.sh b/tasks/construct-rlve-recursive-sequence-sum-l1-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l1-s4/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l5-s2/environment/Dockerfile b/tasks/construct-rlve-recursive-sequence-sum-l5-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l5-s2/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l5-s2/environment/instance.json b/tasks/construct-rlve-recursive-sequence-sum-l5-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..fdb8a6e85d55b3454239495a4e976a936c9e96ad
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l5-s2/environment/instance.json
@@ -0,0 +1 @@
+{"F0": 37, "A": 16, "B": 3753, "S": 104219017961490983349818169994538658321432, "family": "rlve_recursive_sequence_sum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l5-s2/solution/solve.sh b/tasks/construct-rlve-recursive-sequence-sum-l5-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..24d951ba1fbfad931b91dd1924a0921c0e7513f0
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l5-s2/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"indices": [1, 3, 4, 5, 6, 7, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 30, 31, 32]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l5-s2/tests/check.py b/tasks/construct-rlve-recursive-sequence-sum-l5-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..feda076c3f95e5e12198151dee30e1b8223924bc
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l5-s2/tests/check.py
@@ -0,0 +1,29 @@
+from run import Invalid, int_list
+
+
+def F(inst, n):
+    F0, A, B = inst["F0"], inst["A"], inst["B"]
+    if A == 1:
+        return F0 + n * B
+    An = A ** n
+    return An * F0 + B * (An - 1) // (A - 1)
+
+
+def check(inst, ans):
+    S = inst["S"]
+    idx = int_list(ans.get("indices"), None, 1, None, "indices")
+    if not idx:
+        raise Invalid("need at least one index")
+    if len(set(idx)) != len(idx):
+        raise Invalid("indices must be distinct")
+    total = 0
+    for n in idx:
+        # F is strictly increasing with F(n) >= n and, when A >= 2, F(n) >= 2^(n-1); bail out before big powers
+        if n > S or (inst["A"] >= 2 and n - 1 > S.bit_length()):
+            raise Invalid(f"F({n}) exceeds S")
+        total += F(inst, n)
+        if total > S:
+            raise Invalid("sum exceeds S")
+    if total != S:
+        raise Invalid(f"sum is {total}, need {S}")
+    return None, {}
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l5-s2/tests/instance.json b/tasks/construct-rlve-recursive-sequence-sum-l5-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..fdb8a6e85d55b3454239495a4e976a936c9e96ad
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l5-s2/tests/instance.json
@@ -0,0 +1 @@
+{"F0": 37, "A": 16, "B": 3753, "S": 104219017961490983349818169994538658321432, "family": "rlve_recursive_sequence_sum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l5-s2/tests/run.py b/tasks/construct-rlve-recursive-sequence-sum-l5-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l5-s2/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l5-s2/tests/test.sh b/tasks/construct-rlve-recursive-sequence-sum-l5-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l5-s2/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l5-s5/environment/Dockerfile b/tasks/construct-rlve-recursive-sequence-sum-l5-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l5-s5/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l5-s5/environment/instance.json b/tasks/construct-rlve-recursive-sequence-sum-l5-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..9d3cfa7be94fe17d7a036eefc0758f93e88c7332
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l5-s5/environment/instance.json
@@ -0,0 +1 @@
+{"F0": 59, "A": 2, "B": 4657, "S": 20261247114260, "family": "rlve_recursive_sequence_sum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l5-s5/solution/solve.sh b/tasks/construct-rlve-recursive-sequence-sum-l5-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..18553bfc2739240d63227c0741cfe4e9ed5e57f4
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l5-s5/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"indices": [1, 18, 20, 32]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l5-s5/tests/check.py b/tasks/construct-rlve-recursive-sequence-sum-l5-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..feda076c3f95e5e12198151dee30e1b8223924bc
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l5-s5/tests/check.py
@@ -0,0 +1,29 @@
+from run import Invalid, int_list
+
+
+def F(inst, n):
+    F0, A, B = inst["F0"], inst["A"], inst["B"]
+    if A == 1:
+        return F0 + n * B
+    An = A ** n
+    return An * F0 + B * (An - 1) // (A - 1)
+
+
+def check(inst, ans):
+    S = inst["S"]
+    idx = int_list(ans.get("indices"), None, 1, None, "indices")
+    if not idx:
+        raise Invalid("need at least one index")
+    if len(set(idx)) != len(idx):
+        raise Invalid("indices must be distinct")
+    total = 0
+    for n in idx:
+        # F is strictly increasing with F(n) >= n and, when A >= 2, F(n) >= 2^(n-1); bail out before big powers
+        if n > S or (inst["A"] >= 2 and n - 1 > S.bit_length()):
+            raise Invalid(f"F({n}) exceeds S")
+        total += F(inst, n)
+        if total > S:
+            raise Invalid("sum exceeds S")
+    if total != S:
+        raise Invalid(f"sum is {total}, need {S}")
+    return None, {}
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l5-s5/tests/instance.json b/tasks/construct-rlve-recursive-sequence-sum-l5-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..9d3cfa7be94fe17d7a036eefc0758f93e88c7332
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l5-s5/tests/instance.json
@@ -0,0 +1 @@
+{"F0": 59, "A": 2, "B": 4657, "S": 20261247114260, "family": "rlve_recursive_sequence_sum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l5-s5/tests/run.py b/tasks/construct-rlve-recursive-sequence-sum-l5-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l5-s5/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l5-s5/tests/test.sh b/tasks/construct-rlve-recursive-sequence-sum-l5-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l5-s5/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-round-robin-scores-l1-s6/environment/Dockerfile b/tasks/construct-rlve-round-robin-scores-l1-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-l1-s6/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-round-robin-scores-l1-s6/environment/instance.json b/tasks/construct-rlve-round-robin-scores-l1-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..ffd6f6758cd079023623995d748920d83d750d52
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-l1-s6/environment/instance.json
@@ -0,0 +1 @@
+{"W": [4, 0, 7, 6], "family": "rlve_round_robin_scores", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-round-robin-scores-l1-s6/solution/solve.sh b/tasks/construct-rlve-round-robin-scores-l1-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..4300384feb1357c359a88bf54f80d7fe98271020
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-l1-s6/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"A": ["0210", "0000", "1202", "2200"]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-round-robin-scores-l1-s6/tests/check.py b/tasks/construct-rlve-round-robin-scores-l1-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..214446acaa90ce170daad3989b42317353060f9d
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-l1-s6/tests/check.py
@@ -0,0 +1,24 @@
+from run import Invalid
+
+
+def check(inst, ans):
+    W = inst["W"]
+    n = len(W)
+    A = ans.get("A")
+    if not isinstance(A, list) or len(A) != n:
+        raise Invalid(f"A must be a list of {n} strings")
+    for r in A:
+        if not isinstance(r, str) or len(r) != n or set(r) - {"0", "1", "2"}:
+            raise Invalid(f"each row must be a string of {n} digits 0/1/2")
+    for i in range(n):
+        if A[i][i] != "0":
+            raise Invalid("diagonal must be 0")
+        w = 0
+        for j in range(n):
+            if i != j:
+                if int(A[i][j]) + int(A[j][i]) != 2:
+                    raise Invalid(f"A[{i}][{j}] + A[{j}][{i}] != 2")
+                w += 3 * (A[i][j] == "2") + (A[i][j] == "1")
+        if w != W[i]:
+            raise Invalid(f"W[{i}] = {w}, need {W[i]}")
+    return None, {}
diff --git a/tasks/construct-rlve-round-robin-scores-l1-s6/tests/instance.json b/tasks/construct-rlve-round-robin-scores-l1-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..ffd6f6758cd079023623995d748920d83d750d52
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-l1-s6/tests/instance.json
@@ -0,0 +1 @@
+{"W": [4, 0, 7, 6], "family": "rlve_round_robin_scores", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-round-robin-scores-l1-s6/tests/run.py b/tasks/construct-rlve-round-robin-scores-l1-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-l1-s6/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-round-robin-scores-l1-s6/tests/test.sh b/tasks/construct-rlve-round-robin-scores-l1-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-l1-s6/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-round-robin-scores-l1-s7/environment/Dockerfile b/tasks/construct-rlve-round-robin-scores-l1-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-l1-s7/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-round-robin-scores-l1-s7/environment/instance.json b/tasks/construct-rlve-round-robin-scores-l1-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4e78c971f22cd9aea504d9f958a0f64f334f10e3
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-l1-s7/environment/instance.json
@@ -0,0 +1 @@
+{"W": [2, 3, 7, 2], "family": "rlve_round_robin_scores", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-round-robin-scores-l1-s7/solution/solve.sh b/tasks/construct-rlve-round-robin-scores-l1-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..a1f9a2f805cceb3d27ad995c59e5d4fc3a7656d6
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-l1-s7/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"A": ["0101", "1011", "2102", "1100"]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-round-robin-scores-l1-s7/tests/check.py b/tasks/construct-rlve-round-robin-scores-l1-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..214446acaa90ce170daad3989b42317353060f9d
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-l1-s7/tests/check.py
@@ -0,0 +1,24 @@
+from run import Invalid
+
+
+def check(inst, ans):
+    W = inst["W"]
+    n = len(W)
+    A = ans.get("A")
+    if not isinstance(A, list) or len(A) != n:
+        raise Invalid(f"A must be a list of {n} strings")
+    for r in A:
+        if not isinstance(r, str) or len(r) != n or set(r) - {"0", "1", "2"}:
+            raise Invalid(f"each row must be a string of {n} digits 0/1/2")
+    for i in range(n):
+        if A[i][i] != "0":
+            raise Invalid("diagonal must be 0")
+        w = 0
+        for j in range(n):
+            if i != j:
+                if int(A[i][j]) + int(A[j][i]) != 2:
+                    raise Invalid(f"A[{i}][{j}] + A[{j}][{i}] != 2")
+                w += 3 * (A[i][j] == "2") + (A[i][j] == "1")
+        if w != W[i]:
+            raise Invalid(f"W[{i}] = {w}, need {W[i]}")
+    return None, {}
diff --git a/tasks/construct-rlve-round-robin-scores-l1-s7/tests/instance.json b/tasks/construct-rlve-round-robin-scores-l1-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4e78c971f22cd9aea504d9f958a0f64f334f10e3
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-l1-s7/tests/instance.json
@@ -0,0 +1 @@
+{"W": [2, 3, 7, 2], "family": "rlve_round_robin_scores", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-round-robin-scores-l1-s7/tests/run.py b/tasks/construct-rlve-round-robin-scores-l1-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-l1-s7/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-round-robin-scores-l1-s7/tests/test.sh b/tasks/construct-rlve-round-robin-scores-l1-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-l1-s7/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-round-robin-scores-l5-s8/environment/Dockerfile b/tasks/construct-rlve-round-robin-scores-l5-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-l5-s8/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-round-robin-scores-l5-s8/environment/instance.json b/tasks/construct-rlve-round-robin-scores-l5-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a0dd739728c3bf17110f74e4d6dd45271095a6b4
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-l5-s8/environment/instance.json
@@ -0,0 +1 @@
+{"W": [21, 25, 17, 20, 17, 17, 29, 20, 16, 18, 20, 17, 18, 15, 19, 26, 15, 21], "family": "rlve_round_robin_scores", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-round-robin-scores-l5-s8/solution/solve.sh b/tasks/construct-rlve-round-robin-scores-l5-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..2c52075c2a235733bf16f0b1fe08101352c8a70e
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-l5-s8/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"A": ["001111102112111112", "201112021120112111", "110201110111111111", "110020121011112111", "112001211101111010", "101210021110110111", "121102012121221012", "201010101121112111", "012111010111111110", "111211111011111011", "101121001102112111", "021112111100010111", "111111011112011111", "111111011111101011", "101012101102110121", "111121211211121020", "111111111111110001", "011121012111111210"]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-round-robin-scores-l5-s8/tests/check.py b/tasks/construct-rlve-round-robin-scores-l5-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..214446acaa90ce170daad3989b42317353060f9d
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-l5-s8/tests/check.py
@@ -0,0 +1,24 @@
+from run import Invalid
+
+
+def check(inst, ans):
+    W = inst["W"]
+    n = len(W)
+    A = ans.get("A")
+    if not isinstance(A, list) or len(A) != n:
+        raise Invalid(f"A must be a list of {n} strings")
+    for r in A:
+        if not isinstance(r, str) or len(r) != n or set(r) - {"0", "1", "2"}:
+            raise Invalid(f"each row must be a string of {n} digits 0/1/2")
+    for i in range(n):
+        if A[i][i] != "0":
+            raise Invalid("diagonal must be 0")
+        w = 0
+        for j in range(n):
+            if i != j:
+                if int(A[i][j]) + int(A[j][i]) != 2:
+                    raise Invalid(f"A[{i}][{j}] + A[{j}][{i}] != 2")
+                w += 3 * (A[i][j] == "2") + (A[i][j] == "1")
+        if w != W[i]:
+            raise Invalid(f"W[{i}] = {w}, need {W[i]}")
+    return None, {}
diff --git a/tasks/construct-rlve-round-robin-scores-l5-s8/tests/instance.json b/tasks/construct-rlve-round-robin-scores-l5-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a0dd739728c3bf17110f74e4d6dd45271095a6b4
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-l5-s8/tests/instance.json
@@ -0,0 +1 @@
+{"W": [21, 25, 17, 20, 17, 17, 29, 20, 16, 18, 20, 17, 18, 15, 19, 26, 15, 21], "family": "rlve_round_robin_scores", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-round-robin-scores-l5-s8/tests/run.py b/tasks/construct-rlve-round-robin-scores-l5-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-l5-s8/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-round-robin-scores-l5-s8/tests/test.sh b/tasks/construct-rlve-round-robin-scores-l5-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-l5-s8/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-shortest-path-count-l6-s2/environment/Dockerfile b/tasks/construct-rlve-shortest-path-count-l6-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-l6-s2/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-shortest-path-count-l6-s2/environment/instance.json b/tasks/construct-rlve-shortest-path-count-l6-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d635b9ce1485259a9454c46853db4b7097d64de3
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-l6-s2/environment/instance.json
@@ -0,0 +1 @@
+{"K": 713669029456444787, "n_max": 363, "family": "rlve_shortest_path_count", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-shortest-path-count-l6-s2/solution/solve.sh b/tasks/construct-rlve-shortest-path-count-l6-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..86af56d35b81c5927a504d5983df2a1a599df9f7
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-l6-s2/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"n": 180, "adj": ["NNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNY", "YNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "YNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", 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"NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNY", "NYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYN"]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-shortest-path-count-l6-s2/tests/check.py b/tasks/construct-rlve-shortest-path-count-l6-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..c59d4c4f4bb4e824a2019380643a5b3ade0ca974
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-l6-s2/tests/check.py
@@ -0,0 +1,34 @@
+from run import Invalid, as_int
+
+
+def check(inst, ans):
+    n = as_int(ans.get("n"), "n", lo=2, hi=inst["n_max"])
+    adj = ans.get("adj")
+    if not isinstance(adj, list) or len(adj) != n:
+        raise Invalid(f"adj must be a list of {n} strings")
+    for r in adj:
+        if not isinstance(r, str) or len(r) != n or set(r) - {"Y", "N"}:
+            raise Invalid("each row must be a string of length n over {'Y','N'}")
+    for i in range(n):
+        if adj[i][i] != "N":
+            raise Invalid("self-loop")
+        for j in range(i + 1, n):
+            if adj[i][j] != adj[j][i]:
+                raise Invalid("matrix not symmetric")
+    nb = [[j for j in range(n) if adj[i][j] == "Y"] for i in range(n)]
+    dist = [-1] * n
+    cnt = [0] * n
+    dist[0], cnt[0] = 0, 1
+    q = [0]
+    for u in q:  # BFS with exact (Python int) path counts
+        for v in nb[u]:
+            if dist[v] < 0:
+                dist[v] = dist[u] + 1
+                q.append(v)
+            if dist[v] == dist[u] + 1:
+                cnt[v] += cnt[u]
+    if dist[1] < 0:
+        raise Invalid("vertex 2 is unreachable from vertex 1")
+    if cnt[1] != inst["K"]:
+        raise Invalid(f"{cnt[1]} shortest paths, need {inst['K']}")
+    return None, {}
diff --git a/tasks/construct-rlve-shortest-path-count-l6-s2/tests/instance.json b/tasks/construct-rlve-shortest-path-count-l6-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d635b9ce1485259a9454c46853db4b7097d64de3
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-l6-s2/tests/instance.json
@@ -0,0 +1 @@
+{"K": 713669029456444787, "n_max": 363, "family": "rlve_shortest_path_count", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-shortest-path-count-l6-s2/tests/run.py b/tasks/construct-rlve-shortest-path-count-l6-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-l6-s2/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-shortest-path-count-l6-s2/tests/test.sh b/tasks/construct-rlve-shortest-path-count-l6-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-l6-s2/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-shortest-path-count-l6-s4/environment/Dockerfile b/tasks/construct-rlve-shortest-path-count-l6-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-l6-s4/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-shortest-path-count-l6-s4/solution/solve.sh b/tasks/construct-rlve-shortest-path-count-l6-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..f53c931b56c0ab8b28bd83c9f8632d6d08881895
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-l6-s4/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"n": 174, "adj": ["NNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNY", "YNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "YNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", 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"NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNY", "NYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYN"]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-shortest-path-count-l6-s4/tests/check.py b/tasks/construct-rlve-shortest-path-count-l6-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..c59d4c4f4bb4e824a2019380643a5b3ade0ca974
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-l6-s4/tests/check.py
@@ -0,0 +1,34 @@
+from run import Invalid, as_int
+
+
+def check(inst, ans):
+    n = as_int(ans.get("n"), "n", lo=2, hi=inst["n_max"])
+    adj = ans.get("adj")
+    if not isinstance(adj, list) or len(adj) != n:
+        raise Invalid(f"adj must be a list of {n} strings")
+    for r in adj:
+        if not isinstance(r, str) or len(r) != n or set(r) - {"Y", "N"}:
+            raise Invalid("each row must be a string of length n over {'Y','N'}")
+    for i in range(n):
+        if adj[i][i] != "N":
+            raise Invalid("self-loop")
+        for j in range(i + 1, n):
+            if adj[i][j] != adj[j][i]:
+                raise Invalid("matrix not symmetric")
+    nb = [[j for j in range(n) if adj[i][j] == "Y"] for i in range(n)]
+    dist = [-1] * n
+    cnt = [0] * n
+    dist[0], cnt[0] = 0, 1
+    q = [0]
+    for u in q:  # BFS with exact (Python int) path counts
+        for v in nb[u]:
+            if dist[v] < 0:
+                dist[v] = dist[u] + 1
+                q.append(v)
+            if dist[v] == dist[u] + 1:
+                cnt[v] += cnt[u]
+    if dist[1] < 0:
+        raise Invalid("vertex 2 is unreachable from vertex 1")
+    if cnt[1] != inst["K"]:
+        raise Invalid(f"{cnt[1]} shortest paths, need {inst['K']}")
+    return None, {}
diff --git a/tasks/construct-rlve-shortest-path-count-l6-s4/tests/instance.json b/tasks/construct-rlve-shortest-path-count-l6-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b7385382052ffde35fbc823bb1265e281ea72ebd
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-l6-s4/tests/instance.json
@@ -0,0 +1 @@
+{"K": 187789265141101595, "n_max": 351, "family": "rlve_shortest_path_count", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-shortest-path-count-l6-s4/tests/run.py b/tasks/construct-rlve-shortest-path-count-l6-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-l6-s4/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-shortest-path-count-l6-s4/tests/test.sh b/tasks/construct-rlve-shortest-path-count-l6-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-l6-s4/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l1-s1/environment/Dockerfile b/tasks/construct-rlve-smallest-enclosing-circle-l1-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l1-s1/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l1-s1/environment/instance.json b/tasks/construct-rlve-smallest-enclosing-circle-l1-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..320f80f1e6c4bebc958a34a16088050715d4d939
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l1-s1/environment/instance.json
@@ -0,0 +1 @@
+{"points": [[0, 7], [1, 7], [4, 6], [6, 0]], "r_opt": 4.6097722286464435, "r2_num": 85, "r2_den": 4, "family": "rlve_smallest_enclosing_circle", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l1-s1/solution/solve.sh b/tasks/construct-rlve-smallest-enclosing-circle-l1-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..74c9c5a2dcc2b2c4ae06098787d4c663452e9290
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l1-s1/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": 3.0, "y": 3.5, "r": 4.609772229646444}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l1-s1/tests/check.py b/tasks/construct-rlve-smallest-enclosing-circle-l1-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..0eabf3219588f98db42bf5fd305dc24ce21ca495
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l1-s1/tests/check.py
@@ -0,0 +1,17 @@
+import math
+
+from run import Invalid, num
+
+TOL = 1e-6
+
+
+def check(inst, ans):
+    x, y, r = num(ans.get("x"), "x"), num(ans.get("y"), "y"), num(ans.get("r"), "r")
+    if r <= 0:
+        raise Invalid("r must be positive")
+    if r > inst["r_opt"] + TOL:
+        raise Invalid(f"r = {r} exceeds the minimum radius {inst['r_opt']} by more than {TOL}")
+    for px, py in inst["points"]:
+        if math.hypot(px - x, py - y) > r + TOL:
+            raise Invalid(f"point ({px}, {py}) is outside the circle")
+    return None, {"r": r}
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l1-s1/tests/instance.json b/tasks/construct-rlve-smallest-enclosing-circle-l1-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..320f80f1e6c4bebc958a34a16088050715d4d939
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l1-s1/tests/instance.json
@@ -0,0 +1 @@
+{"points": [[0, 7], [1, 7], [4, 6], [6, 0]], "r_opt": 4.6097722286464435, "r2_num": 85, "r2_den": 4, "family": "rlve_smallest_enclosing_circle", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l1-s1/tests/run.py b/tasks/construct-rlve-smallest-enclosing-circle-l1-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l1-s1/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l1-s1/tests/test.sh b/tasks/construct-rlve-smallest-enclosing-circle-l1-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l1-s1/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l1-s6/environment/Dockerfile b/tasks/construct-rlve-smallest-enclosing-circle-l1-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l1-s6/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l1-s6/environment/instance.json b/tasks/construct-rlve-smallest-enclosing-circle-l1-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e7f83bbdfc8b5ed99b0efb3fdca3d94cc6079463
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l1-s6/environment/instance.json
@@ -0,0 +1 @@
+{"points": [[1, 5], [2, 8], [5, 3], [7, 7]], "r_opt": 3.1622776601683795, "r2_num": 10, "r2_den": 1, "family": "rlve_smallest_enclosing_circle", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l1-s6/solution/solve.sh b/tasks/construct-rlve-smallest-enclosing-circle-l1-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..d8843f0b8c952c5afa49a2fb8beeff123f29afc8
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l1-s6/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": 4.0, "y": 6.0, "r": 3.1622776611683796}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l1-s6/tests/check.py b/tasks/construct-rlve-smallest-enclosing-circle-l1-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..0eabf3219588f98db42bf5fd305dc24ce21ca495
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l1-s6/tests/check.py
@@ -0,0 +1,17 @@
+import math
+
+from run import Invalid, num
+
+TOL = 1e-6
+
+
+def check(inst, ans):
+    x, y, r = num(ans.get("x"), "x"), num(ans.get("y"), "y"), num(ans.get("r"), "r")
+    if r <= 0:
+        raise Invalid("r must be positive")
+    if r > inst["r_opt"] + TOL:
+        raise Invalid(f"r = {r} exceeds the minimum radius {inst['r_opt']} by more than {TOL}")
+    for px, py in inst["points"]:
+        if math.hypot(px - x, py - y) > r + TOL:
+            raise Invalid(f"point ({px}, {py}) is outside the circle")
+    return None, {"r": r}
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l1-s6/tests/instance.json b/tasks/construct-rlve-smallest-enclosing-circle-l1-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e7f83bbdfc8b5ed99b0efb3fdca3d94cc6079463
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l1-s6/tests/instance.json
@@ -0,0 +1 @@
+{"points": [[1, 5], [2, 8], [5, 3], [7, 7]], "r_opt": 3.1622776601683795, "r2_num": 10, "r2_den": 1, "family": "rlve_smallest_enclosing_circle", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l1-s6/tests/run.py b/tasks/construct-rlve-smallest-enclosing-circle-l1-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l1-s6/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l1-s6/tests/test.sh b/tasks/construct-rlve-smallest-enclosing-circle-l1-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l1-s6/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l5-s9/environment/Dockerfile b/tasks/construct-rlve-smallest-enclosing-circle-l5-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l5-s9/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l5-s9/environment/instance.json b/tasks/construct-rlve-smallest-enclosing-circle-l5-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f98dc69fa304c7d589c2231a1a561456f4d79323
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l5-s9/environment/instance.json
@@ -0,0 +1 @@
+{"points": [[2, 73], [2, 99], [3, 95], [5, 42], [7, 100], [12, 27], [12, 111], [13, 7], [17, 35], [23, 35], [23, 115], [24, 10], [26, 89], [29, 55], [29, 70], [31, 117], [32, 27], [32, 36], [34, 20], [39, 80], [39, 96], [41, 110], [42, 71], [42, 100], [43, 31], [44, 33], [45, 83], [49, 15], [54, 107], [60, 107], [61, 46], [64, 29], [67, 75], [68, 90], [68, 95], [69, 0], [69, 9], [75, 30], [75, 62], [79, 24], [81, 58], [82, 15], [83, 29], [84, 44], [85, 60], [86, 90], [93, 75], [96, 49], [100, 31], [103, 97], [103, 101], [104, 14], [109, 87], [110, 46], [114, 16], [114, 74], [116, 9], [117, 87], [117, 99], [120, 103]], "r_opt": 72.86091408974517, "r2_num": 189632530, "r2_den": 35721, "family": "rlve_smallest_enclosing_circle", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l5-s9/solution/solve.sh b/tasks/construct-rlve-smallest-enclosing-circle-l5-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..1ced48b52046ac75a512c17ee05076784a034e81
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l5-s9/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": 62.65079365079365, "y": 58.62433862433863, "r": 72.86091409074517}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l5-s9/tests/check.py b/tasks/construct-rlve-smallest-enclosing-circle-l5-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..0eabf3219588f98db42bf5fd305dc24ce21ca495
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l5-s9/tests/check.py
@@ -0,0 +1,17 @@
+import math
+
+from run import Invalid, num
+
+TOL = 1e-6
+
+
+def check(inst, ans):
+    x, y, r = num(ans.get("x"), "x"), num(ans.get("y"), "y"), num(ans.get("r"), "r")
+    if r <= 0:
+        raise Invalid("r must be positive")
+    if r > inst["r_opt"] + TOL:
+        raise Invalid(f"r = {r} exceeds the minimum radius {inst['r_opt']} by more than {TOL}")
+    for px, py in inst["points"]:
+        if math.hypot(px - x, py - y) > r + TOL:
+            raise Invalid(f"point ({px}, {py}) is outside the circle")
+    return None, {"r": r}
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l5-s9/tests/instance.json b/tasks/construct-rlve-smallest-enclosing-circle-l5-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f98dc69fa304c7d589c2231a1a561456f4d79323
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l5-s9/tests/instance.json
@@ -0,0 +1 @@
+{"points": [[2, 73], [2, 99], [3, 95], [5, 42], [7, 100], [12, 27], [12, 111], [13, 7], [17, 35], [23, 35], [23, 115], [24, 10], [26, 89], [29, 55], [29, 70], [31, 117], [32, 27], [32, 36], [34, 20], [39, 80], [39, 96], [41, 110], [42, 71], [42, 100], [43, 31], [44, 33], [45, 83], [49, 15], [54, 107], [60, 107], [61, 46], [64, 29], [67, 75], [68, 90], [68, 95], [69, 0], [69, 9], [75, 30], [75, 62], [79, 24], [81, 58], [82, 15], [83, 29], [84, 44], [85, 60], [86, 90], [93, 75], [96, 49], [100, 31], [103, 97], [103, 101], [104, 14], [109, 87], [110, 46], [114, 16], [114, 74], [116, 9], [117, 87], [117, 99], [120, 103]], "r_opt": 72.86091408974517, "r2_num": 189632530, "r2_den": 35721, "family": "rlve_smallest_enclosing_circle", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l5-s9/tests/run.py b/tasks/construct-rlve-smallest-enclosing-circle-l5-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l5-s9/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l5-s9/tests/test.sh b/tasks/construct-rlve-smallest-enclosing-circle-l5-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l5-s9/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l1-s3/environment/Dockerfile b/tasks/construct-rlve-subgraph-isomorphism-l1-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l1-s3/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l1-s3/environment/instance.json b/tasks/construct-rlve-subgraph-isomorphism-l1-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..fce01a9654031380aea0811602794e5d2d765ce5
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l1-s3/environment/instance.json
@@ -0,0 +1 @@
+{"n1": 3, "n2": 5, "edges1": [[0, 1]], "edges2": [[2, 3]], "family": "rlve_subgraph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l1-s3/solution/solve.sh b/tasks/construct-rlve-subgraph-isomorphism-l1-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..8a151f37e8bfdb4e31034f525d2cd314797e5496
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l1-s3/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"p": [2, 3, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l1-s3/tests/check.py b/tasks/construct-rlve-subgraph-isomorphism-l1-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..b8467dd00c27c8902c06206c376ff7a11ac78ae0
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l1-s3/tests/check.py
@@ -0,0 +1,16 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n1, n2 = inst["n1"], inst["n2"]
+    p = int_list(ans.get("p"), n1, 0, n2 - 1, "p")
+    if len(set(p)) != n1:
+        raise Invalid("p is not injective")
+    e1 = {(min(u, v), max(u, v)) for u, v in inst["edges1"]}
+    e2 = {(min(u, v), max(u, v)) for u, v in inst["edges2"]}
+    for u in range(n1):
+        for v in range(u + 1, n1):
+            a, b = min(p[u], p[v]), max(p[u], p[v])
+            if ((u, v) in e1) != ((a, b) in e2):
+                raise Invalid(f"pair ({u}, {v}) is not preserved")
+    return None, {}
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l1-s3/tests/instance.json b/tasks/construct-rlve-subgraph-isomorphism-l1-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..fce01a9654031380aea0811602794e5d2d765ce5
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l1-s3/tests/instance.json
@@ -0,0 +1 @@
+{"n1": 3, "n2": 5, "edges1": [[0, 1]], "edges2": [[2, 3]], "family": "rlve_subgraph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l1-s3/tests/run.py b/tasks/construct-rlve-subgraph-isomorphism-l1-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l1-s3/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l1-s3/tests/test.sh b/tasks/construct-rlve-subgraph-isomorphism-l1-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l1-s3/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l1-s4/environment/Dockerfile b/tasks/construct-rlve-subgraph-isomorphism-l1-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l1-s4/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l1-s4/environment/instance.json b/tasks/construct-rlve-subgraph-isomorphism-l1-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..78d9ad8d934883d149cb0ae8c33543fd8efb2fd9
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l1-s4/environment/instance.json
@@ -0,0 +1 @@
+{"n1": 4, "n2": 5, "edges1": [[1, 2], [2, 3], [1, 3]], "edges2": [[0, 3], [0, 2], [2, 3]], "family": "rlve_subgraph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l1-s4/solution/solve.sh b/tasks/construct-rlve-subgraph-isomorphism-l1-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..0d1bd52f05ab11b11ec869932c9c87cc7ccc1682
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l1-s4/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"p": [1, 3, 2, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l1-s4/tests/check.py b/tasks/construct-rlve-subgraph-isomorphism-l1-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..b8467dd00c27c8902c06206c376ff7a11ac78ae0
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l1-s4/tests/check.py
@@ -0,0 +1,16 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n1, n2 = inst["n1"], inst["n2"]
+    p = int_list(ans.get("p"), n1, 0, n2 - 1, "p")
+    if len(set(p)) != n1:
+        raise Invalid("p is not injective")
+    e1 = {(min(u, v), max(u, v)) for u, v in inst["edges1"]}
+    e2 = {(min(u, v), max(u, v)) for u, v in inst["edges2"]}
+    for u in range(n1):
+        for v in range(u + 1, n1):
+            a, b = min(p[u], p[v]), max(p[u], p[v])
+            if ((u, v) in e1) != ((a, b) in e2):
+                raise Invalid(f"pair ({u}, {v}) is not preserved")
+    return None, {}
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l1-s4/tests/instance.json b/tasks/construct-rlve-subgraph-isomorphism-l1-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..78d9ad8d934883d149cb0ae8c33543fd8efb2fd9
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l1-s4/tests/instance.json
@@ -0,0 +1 @@
+{"n1": 4, "n2": 5, "edges1": [[1, 2], [2, 3], [1, 3]], "edges2": [[0, 3], [0, 2], [2, 3]], "family": "rlve_subgraph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l1-s4/tests/run.py b/tasks/construct-rlve-subgraph-isomorphism-l1-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l1-s4/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l1-s4/tests/test.sh b/tasks/construct-rlve-subgraph-isomorphism-l1-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l1-s4/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l5-s2/environment/Dockerfile b/tasks/construct-rlve-subgraph-isomorphism-l5-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l5-s2/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l5-s2/environment/instance.json b/tasks/construct-rlve-subgraph-isomorphism-l5-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4055ede82ef8f9a452a16cd05f4ba30851389e32
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l5-s2/environment/instance.json
@@ -0,0 +1 @@
+{"n1": 13, "n2": 26, "edges1": [[9, 12], [6, 9], [4, 11], [5, 11], [7, 10], [2, 3], [0, 3], [9, 10], [5, 8], [0, 7], [5, 9], [6, 8], [1, 4], [0, 2], [1, 2], [3, 12], [8, 10], [8, 11], [4, 6], [1, 7], [2, 5], [1, 6], [3, 4], [2, 8], [0, 1], [8, 9], [1, 10], [3, 6], [2, 10], [6, 7], [1, 9], [5, 10], [0, 4], [4, 12], [2, 12], [5, 6], [11, 12], [0, 10], [3, 5], [10, 11], [4, 9], [1, 8], [0, 12], [1, 12]], "edges2": [[6, 14], [22, 24], [15, 18], [12, 17], [4, 20], [15, 23], [7, 21], [21, 22], [4, 14], [23, 24], [3, 11], [19, 22], [22, 23], [23, 25], [1, 20], [7, 17], [1, 16], [1, 13], [4, 10], [2, 9], [10, 19], [0, 8], [0, 7], [18, 24], [12, 22], [7, 20], [2, 3], [0, 18], [17, 22], [11, 23], [11, 22], [8, 23], [4, 24], [9, 15], [3, 21], [8, 22], [15, 20], [13, 17], [5, 15], [3, 17], [2, 17], [6, 12], [5, 18], [3, 7], [11, 25], [24, 25], [9, 16], [0, 15], [5, 14], [21, 23], [10, 16], [17, 25], [6, 25], [5, 25], [20, 24], [19, 20], [8, 18], [1, 15], [7, 14], [10, 13], [4, 5], [0, 9], [13, 16], [16, 17], [20, 23], [6, 15], [3, 14], [0, 4], [16, 21], [4, 25], [13, 15], [0, 19], [0, 2], [2, 11], [10, 11], [2, 23], [20, 21], [3, 12], [9, 13], [1, 9], [14, 23], [0, 11], [2, 15], [10, 20], [10, 21], [3, 15], [20, 25], [6, 13], [1, 6], [7, 23], [13, 24], [17, 21], [9, 22], [0, 5], [6, 23], [11, 24], [7, 24], [12, 14], [11, 16], [9, 17], [7, 22], [6, 24], [4, 22], [8, 12], [13, 21], [8, 15], [5, 22], [8, 16], [5, 8], [1, 12], [1, 8], [7, 8], [2, 8], [5, 12], [17, 18], [2, 13], [10, 12], [12, 15], [12, 20], [3, 20], [0, 21], [6, 16], [8, 20], [15, 16], [7, 18], [5, 10], [12, 19], [18, 23], [16, 25], [3, 5], [3, 19], [5, 9], [10, 14], [2, 18], [16, 20], [16, 18], [9, 24], [4, 16], [14, 17], [8, 21], [11, 21], [2, 4], [14, 18], [4, 13], [2, 25], [3, 13], [3, 6], [5, 23], [7, 19], [13, 23], [0, 13], [2, 5], [3, 24], [0, 25], [5, 6], [4, 6], [19, 25], [14, 20], [0, 24], [7, 12], [10, 25], [1, 24], [9, 12], [0, 23], [2, 7], [5, 7], [9, 23], [12, 18], [19, 23], [2, 22], [11, 14], [19, 24], [17, 24], [12, 16], [8, 19], [0, 16], [13, 22], [0, 6]], "family": "rlve_subgraph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l5-s2/solution/solve.sh b/tasks/construct-rlve-subgraph-isomorphism-l5-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..56649eb2df6929dcaf764664b32bdaa8163849da
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l5-s2/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"p": [16, 20, 21, 11, 25, 22, 24, 1, 7, 19, 8, 5, 10]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l5-s2/tests/check.py b/tasks/construct-rlve-subgraph-isomorphism-l5-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..b8467dd00c27c8902c06206c376ff7a11ac78ae0
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l5-s2/tests/check.py
@@ -0,0 +1,16 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n1, n2 = inst["n1"], inst["n2"]
+    p = int_list(ans.get("p"), n1, 0, n2 - 1, "p")
+    if len(set(p)) != n1:
+        raise Invalid("p is not injective")
+    e1 = {(min(u, v), max(u, v)) for u, v in inst["edges1"]}
+    e2 = {(min(u, v), max(u, v)) for u, v in inst["edges2"]}
+    for u in range(n1):
+        for v in range(u + 1, n1):
+            a, b = min(p[u], p[v]), max(p[u], p[v])
+            if ((u, v) in e1) != ((a, b) in e2):
+                raise Invalid(f"pair ({u}, {v}) is not preserved")
+    return None, {}
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l5-s2/tests/instance.json b/tasks/construct-rlve-subgraph-isomorphism-l5-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4055ede82ef8f9a452a16cd05f4ba30851389e32
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l5-s2/tests/instance.json
@@ -0,0 +1 @@
+{"n1": 13, "n2": 26, "edges1": [[9, 12], [6, 9], [4, 11], [5, 11], [7, 10], [2, 3], [0, 3], [9, 10], [5, 8], [0, 7], [5, 9], [6, 8], [1, 4], [0, 2], [1, 2], [3, 12], [8, 10], [8, 11], [4, 6], [1, 7], [2, 5], [1, 6], [3, 4], [2, 8], [0, 1], [8, 9], [1, 10], [3, 6], [2, 10], [6, 7], [1, 9], [5, 10], [0, 4], [4, 12], [2, 12], [5, 6], [11, 12], [0, 10], [3, 5], [10, 11], [4, 9], [1, 8], [0, 12], [1, 12]], "edges2": [[6, 14], [22, 24], [15, 18], [12, 17], [4, 20], [15, 23], [7, 21], [21, 22], [4, 14], [23, 24], [3, 11], [19, 22], [22, 23], [23, 25], [1, 20], [7, 17], [1, 16], [1, 13], [4, 10], [2, 9], [10, 19], [0, 8], [0, 7], [18, 24], [12, 22], [7, 20], [2, 3], [0, 18], [17, 22], [11, 23], [11, 22], [8, 23], [4, 24], [9, 15], [3, 21], [8, 22], [15, 20], [13, 17], [5, 15], [3, 17], [2, 17], [6, 12], [5, 18], [3, 7], [11, 25], [24, 25], [9, 16], [0, 15], [5, 14], [21, 23], [10, 16], [17, 25], [6, 25], [5, 25], [20, 24], [19, 20], [8, 18], [1, 15], [7, 14], [10, 13], [4, 5], [0, 9], [13, 16], [16, 17], [20, 23], [6, 15], [3, 14], [0, 4], [16, 21], [4, 25], [13, 15], [0, 19], [0, 2], [2, 11], [10, 11], [2, 23], [20, 21], [3, 12], [9, 13], [1, 9], [14, 23], [0, 11], [2, 15], [10, 20], [10, 21], [3, 15], [20, 25], [6, 13], [1, 6], [7, 23], [13, 24], [17, 21], [9, 22], [0, 5], [6, 23], [11, 24], [7, 24], [12, 14], [11, 16], [9, 17], [7, 22], [6, 24], [4, 22], [8, 12], [13, 21], [8, 15], [5, 22], [8, 16], [5, 8], [1, 12], [1, 8], [7, 8], [2, 8], [5, 12], [17, 18], [2, 13], [10, 12], [12, 15], [12, 20], [3, 20], [0, 21], [6, 16], [8, 20], [15, 16], [7, 18], [5, 10], [12, 19], [18, 23], [16, 25], [3, 5], [3, 19], [5, 9], [10, 14], [2, 18], [16, 20], [16, 18], [9, 24], [4, 16], [14, 17], [8, 21], [11, 21], [2, 4], [14, 18], [4, 13], [2, 25], [3, 13], [3, 6], [5, 23], [7, 19], [13, 23], [0, 13], [2, 5], [3, 24], [0, 25], [5, 6], [4, 6], [19, 25], [14, 20], [0, 24], [7, 12], [10, 25], [1, 24], [9, 12], [0, 23], [2, 7], [5, 7], [9, 23], [12, 18], [19, 23], [2, 22], [11, 14], [19, 24], [17, 24], [12, 16], [8, 19], [0, 16], [13, 22], [0, 6]], "family": "rlve_subgraph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l5-s2/tests/run.py b/tasks/construct-rlve-subgraph-isomorphism-l5-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l5-s2/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l5-s2/tests/test.sh b/tasks/construct-rlve-subgraph-isomorphism-l5-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l5-s2/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l5-s5/environment/Dockerfile b/tasks/construct-rlve-subgraph-isomorphism-l5-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l5-s5/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l5-s5/environment/instance.json b/tasks/construct-rlve-subgraph-isomorphism-l5-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..7fbaa89c48d96a7166b7f1b046cf56ec4a5adbb1
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l5-s5/environment/instance.json
@@ -0,0 +1 @@
+{"n1": 24, "n2": 26, "edges1": [[6, 10], [7, 22], [4, 15], [2, 19], [9, 12], [8, 11], [7, 14], [14, 23], [2, 10], [14, 17], [0, 8], [13, 15], [3, 16], [0, 6], [1, 3], [5, 8], [18, 22], [6, 16], [1, 4], [19, 22], [8, 14], [7, 19], [15, 23], [8, 18], [10, 23], [8, 22], [3, 12], [1, 15], [14, 16], [3, 9], [19, 23], [12, 22], [12, 18], [3, 20], [9, 17], [8, 23], [11, 17], [3, 6], [4, 18], [5, 7], [15, 18], [10, 17], [1, 18], [17, 21], [4, 21], [4, 19], [11, 16], [6, 11], [4, 16], [12, 15], [2, 8], [0, 4], [6, 13], [3, 5], [2, 22], [20, 23], [16, 22], [5, 18], [6, 8], [0, 7], [10, 20], [7, 12], [4, 22], [17, 23], [14, 21], [11, 22], [0, 14], [8, 17], [6, 19], [2, 13], [15, 17], [8, 19], [4, 20], [14, 18], [5, 6], [1, 21], [0, 5], [4, 17], [3, 7], [11, 23], [11, 12], [0, 18], [5, 12], [4, 14], [15, 22], [4, 13], [5, 11], [12, 19], [0, 16], [10, 18], [9, 10], [4, 23], [7, 11], [0, 9], [3, 21], [1, 12], [7, 13], [18, 21], [1, 10], [2, 11], [5, 13], [6, 15], [14, 20], [12, 13], [15, 19], [9, 19], [0, 23], [4, 8], [9, 14], [6, 18], [0, 2], [1, 20], [10, 21], [18, 19], [3, 10], [3, 18], [1, 14], [20, 22], [1, 9], [11, 15], [9, 16], [10, 19], [5, 15], [12, 20], [10, 11], [0, 11], [7, 16], [1, 17], [22, 23], [1, 11], [0, 22], [1, 5], [11, 21], [13, 17], [5, 16], [14, 15], [3, 22], [8, 13], [19, 20], [9, 21], [3, 23], [7, 23], [7, 20], [2, 6], [13, 23], [2, 3]], "edges2": [[14, 21], [4, 25], [9, 21], [2, 16], [15, 21], [9, 13], [4, 9], [18, 21], [11, 12], [5, 12], [3, 19], [9, 25], [12, 23], [10, 18], [4, 24], [17, 23], [4, 18], [3, 20], [5, 15], [1, 21], [22, 25], [5, 23], [0, 21], [3, 7], [0, 17], [13, 19], [0, 8], [2, 14], [8, 23], [2, 25], [10, 24], [1, 19], [6, 25], [8, 9], [10, 21], [20, 24], [7, 8], [3, 8], [0, 3], [8, 20], [13, 16], [4, 5], [8, 24], [6, 14], [15, 23], [5, 25], [6, 7], [9, 22], [2, 5], [12, 24], [10, 17], [6, 13], [0, 10], [20, 22], [3, 6], [11, 15], [6, 12], [6, 15], [8, 13], [4, 11], [6, 10], [3, 18], [9, 15], [2, 13], [1, 4], [17, 25], [13, 18], [7, 21], [6, 16], [11, 13], [17, 22], [2, 7], [5, 9], [11, 14], [7, 12], [7, 18], [19, 23], [1, 15], [5, 10], [13, 24], [12, 25], [1, 3], [13, 15], [13, 20], [6, 18], [9, 23], [0, 1], [1, 25], [8, 11], [8, 19], [0, 6], [16, 25], [11, 18], [11, 20], [3, 16], [2, 4], [14, 15], [16, 22], [13, 22], [5, 7], [19, 20], [4, 7], [11, 21], [15, 25], [4, 23], [8, 17], [3, 12], [0, 13], [5, 16], [15, 19], [11, 22], [1, 5], [19, 25], [9, 12], [15, 22], [19, 22], [12, 13], [3, 17], [11, 19], [10, 20], [5, 13], [7, 22], [14, 23], [3, 15], [1, 6], [10, 25], [2, 22], [5, 24], [5, 8], [14, 18], [5, 20], [16, 24], [19, 24], [13, 17], [1, 17], [5, 18], [9, 17], [3, 10], [0, 4], [2, 15], [16, 23], [2, 21], [7, 17], [10, 14], [1, 22], [2, 24], [0, 18], [4, 21], [0, 14], [5, 19], [12, 20], [7, 9], [2, 23], [3, 22], [20, 21], [9, 14], [2, 12], [12, 16], [4, 15], [4, 14], [11, 23], [17, 19], [4, 20], [2, 8], [8, 12], [22, 24], [5, 6], [6, 11], [7, 25], [13, 14], [15, 24], [7, 24], [21, 24], [19, 21], [4, 22], [10, 11], [0, 9], [0, 7]], "family": "rlve_subgraph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l5-s5/solution/solve.sh b/tasks/construct-rlve-subgraph-isomorphism-l5-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..48f9d1493c22ea46cbb16a095e4c6f452604dde2
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l5-s5/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"p": [24, 0, 16, 3, 4, 8, 12, 19, 2, 10, 6, 13, 17, 23, 21, 9, 20, 14, 7, 25, 1, 18, 22, 15]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l5-s5/tests/check.py b/tasks/construct-rlve-subgraph-isomorphism-l5-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..b8467dd00c27c8902c06206c376ff7a11ac78ae0
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l5-s5/tests/check.py
@@ -0,0 +1,16 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n1, n2 = inst["n1"], inst["n2"]
+    p = int_list(ans.get("p"), n1, 0, n2 - 1, "p")
+    if len(set(p)) != n1:
+        raise Invalid("p is not injective")
+    e1 = {(min(u, v), max(u, v)) for u, v in inst["edges1"]}
+    e2 = {(min(u, v), max(u, v)) for u, v in inst["edges2"]}
+    for u in range(n1):
+        for v in range(u + 1, n1):
+            a, b = min(p[u], p[v]), max(p[u], p[v])
+            if ((u, v) in e1) != ((a, b) in e2):
+                raise Invalid(f"pair ({u}, {v}) is not preserved")
+    return None, {}
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l5-s5/tests/instance.json b/tasks/construct-rlve-subgraph-isomorphism-l5-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..7fbaa89c48d96a7166b7f1b046cf56ec4a5adbb1
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l5-s5/tests/instance.json
@@ -0,0 +1 @@
+{"n1": 24, "n2": 26, "edges1": [[6, 10], [7, 22], [4, 15], [2, 19], [9, 12], [8, 11], [7, 14], [14, 23], [2, 10], [14, 17], [0, 8], [13, 15], [3, 16], [0, 6], [1, 3], [5, 8], [18, 22], [6, 16], [1, 4], [19, 22], [8, 14], [7, 19], [15, 23], [8, 18], [10, 23], [8, 22], [3, 12], [1, 15], [14, 16], [3, 9], [19, 23], [12, 22], [12, 18], [3, 20], [9, 17], [8, 23], [11, 17], [3, 6], [4, 18], [5, 7], [15, 18], [10, 17], [1, 18], [17, 21], [4, 21], [4, 19], [11, 16], [6, 11], [4, 16], [12, 15], [2, 8], [0, 4], [6, 13], [3, 5], [2, 22], [20, 23], [16, 22], [5, 18], [6, 8], [0, 7], [10, 20], [7, 12], [4, 22], [17, 23], [14, 21], [11, 22], [0, 14], [8, 17], [6, 19], [2, 13], [15, 17], [8, 19], [4, 20], [14, 18], [5, 6], [1, 21], [0, 5], [4, 17], [3, 7], [11, 23], [11, 12], [0, 18], [5, 12], [4, 14], [15, 22], [4, 13], [5, 11], [12, 19], [0, 16], [10, 18], [9, 10], [4, 23], [7, 11], [0, 9], [3, 21], [1, 12], [7, 13], [18, 21], [1, 10], [2, 11], [5, 13], [6, 15], [14, 20], [12, 13], [15, 19], [9, 19], [0, 23], [4, 8], [9, 14], [6, 18], [0, 2], [1, 20], [10, 21], [18, 19], [3, 10], [3, 18], [1, 14], [20, 22], [1, 9], [11, 15], [9, 16], [10, 19], [5, 15], [12, 20], [10, 11], [0, 11], [7, 16], [1, 17], [22, 23], [1, 11], [0, 22], [1, 5], [11, 21], [13, 17], [5, 16], [14, 15], [3, 22], [8, 13], [19, 20], [9, 21], [3, 23], [7, 23], [7, 20], [2, 6], [13, 23], [2, 3]], "edges2": [[14, 21], [4, 25], [9, 21], [2, 16], [15, 21], [9, 13], [4, 9], [18, 21], [11, 12], [5, 12], [3, 19], [9, 25], [12, 23], [10, 18], [4, 24], [17, 23], [4, 18], [3, 20], [5, 15], [1, 21], [22, 25], [5, 23], [0, 21], [3, 7], [0, 17], [13, 19], [0, 8], [2, 14], [8, 23], [2, 25], [10, 24], [1, 19], [6, 25], [8, 9], [10, 21], [20, 24], [7, 8], [3, 8], [0, 3], [8, 20], [13, 16], [4, 5], [8, 24], [6, 14], [15, 23], [5, 25], [6, 7], [9, 22], [2, 5], [12, 24], [10, 17], [6, 13], [0, 10], [20, 22], [3, 6], [11, 15], [6, 12], [6, 15], [8, 13], [4, 11], [6, 10], [3, 18], [9, 15], [2, 13], [1, 4], [17, 25], [13, 18], [7, 21], [6, 16], [11, 13], [17, 22], [2, 7], [5, 9], [11, 14], [7, 12], [7, 18], [19, 23], [1, 15], [5, 10], [13, 24], [12, 25], [1, 3], [13, 15], [13, 20], [6, 18], [9, 23], [0, 1], [1, 25], [8, 11], [8, 19], [0, 6], [16, 25], [11, 18], [11, 20], [3, 16], [2, 4], [14, 15], [16, 22], [13, 22], [5, 7], [19, 20], [4, 7], [11, 21], [15, 25], [4, 23], [8, 17], [3, 12], [0, 13], [5, 16], [15, 19], [11, 22], [1, 5], [19, 25], [9, 12], [15, 22], [19, 22], [12, 13], [3, 17], [11, 19], [10, 20], [5, 13], [7, 22], [14, 23], [3, 15], [1, 6], [10, 25], [2, 22], [5, 24], [5, 8], [14, 18], [5, 20], [16, 24], [19, 24], [13, 17], [1, 17], [5, 18], [9, 17], [3, 10], [0, 4], [2, 15], [16, 23], [2, 21], [7, 17], [10, 14], [1, 22], [2, 24], [0, 18], [4, 21], [0, 14], [5, 19], [12, 20], [7, 9], [2, 23], [3, 22], [20, 21], [9, 14], [2, 12], [12, 16], [4, 15], [4, 14], [11, 23], [17, 19], [4, 20], [2, 8], [8, 12], [22, 24], [5, 6], [6, 11], [7, 25], [13, 14], [15, 24], [7, 24], [21, 24], [19, 21], [4, 22], [10, 11], [0, 9], [0, 7]], "family": "rlve_subgraph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l5-s5/tests/run.py b/tasks/construct-rlve-subgraph-isomorphism-l5-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l5-s5/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l5-s5/tests/test.sh b/tasks/construct-rlve-subgraph-isomorphism-l5-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l5-s5/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/optimize-ae-p31-furstenberg-sarkozy-k3/environment/Dockerfile b/tasks/optimize-ae-p31-furstenberg-sarkozy-k3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/optimize-ae-p31-furstenberg-sarkozy-k3/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/optimize-ae-p31-furstenberg-sarkozy-k3/environment/instance.json b/tasks/optimize-ae-p31-furstenberg-sarkozy-k3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..5db7f9f9fc3d42f2f4cdd23c4b09400329c9f5f6
--- /dev/null
+++ b/tasks/optimize-ae-p31-furstenberg-sarkozy-k3/environment/instance.json
@@ -0,0 +1 @@
+{"k": 3, "max_m": 1000000, "max_size": 3000, "direction": "maximize", "family": "ae_p31_furstenberg_sarkozy", "subset": "optimize"}
\ No newline at end of file
diff --git a/tasks/optimize-ae-p31-furstenberg-sarkozy-k3/solution/solve.sh b/tasks/optimize-ae-p31-furstenberg-sarkozy-k3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..a6b70d3980d6717b96cf259c20ad050910afdefb
--- /dev/null
+++ b/tasks/optimize-ae-p31-furstenberg-sarkozy-k3/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"m": 91, "residues": [0, 16, 18, 22, 38, 41, 44, 47, 63, 67, 69, 85, 87, 89]}
+RLMATH_EOF
diff --git a/tasks/optimize-ae-p31-furstenberg-sarkozy-k3/tests/check.py b/tasks/optimize-ae-p31-furstenberg-sarkozy-k3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..296ac248623e85e3ff5e54bab1732a30c7889e1a
--- /dev/null
+++ b/tasks/optimize-ae-p31-furstenberg-sarkozy-k3/tests/check.py
@@ -0,0 +1,47 @@
+"""Ruzsa-type modular construction for the Furstenberg-Sarkozy problem.
+
+A subset A of Z/mZ (m square-free) with no two distinct elements differing by a non-zero k-th power residue
+gives subsets of {1..N} with no k-th power differences of size N^{(k-1)/k + log|A|/(k log m)} (Ruzsa 1984).
+Score = that exponent.  Mirrors upstream `get_score` (furstenberg_sarkozy_problem.ipynb), which silently
+reduced/deduplicated the list; here entries must be distinct residues in [0, m).
+"""
+import math
+
+import numpy as np
+
+from run import Invalid, as_int, int_list
+
+
+def squarefree(m):
+    d = 2
+    while d * d <= m:
+        if m % (d * d) == 0:
+            return False
+        d += 1
+    return True
+
+
+def check(inst, ans):
+    k = inst["k"]
+    m = as_int(ans.get("m"), "m", 2, inst["max_m"])
+    if not squarefree(m):
+        raise Invalid("m must be square-free")
+    A = int_list(ans.get("residues"), None, 0, m - 1, "residues")
+    if not 1 <= len(A) <= inst["max_size"]:
+        raise Invalid(f"need 1 <= |A| <= {inst['max_size']}")
+    if len(set(A)) != len(A):
+        raise Invalid("duplicate residues")
+    y = np.arange(1, m, dtype=np.int64)
+    r = y % m
+    for _ in range(k - 1):
+        r = (r * y) % m
+    forb = np.zeros(m, dtype=bool)
+    forb[r] = True
+    forb[0] = False
+    a = np.array(A, dtype=np.int64)
+    diff = (a[:, None] - a[None, :]) % m
+    np.fill_diagonal(diff, 0)
+    if forb[diff].any():
+        i, j = np.argwhere(forb[diff])[0]
+        raise Invalid(f"{A[i]} - {A[j]} is a non-zero k-th power residue mod {m} (k = {k})")
+    return (k - 1) / k + math.log(len(A)) / (k * math.log(m)), {"m": m, "size": len(A)}
diff --git a/tasks/optimize-ae-p31-furstenberg-sarkozy-k3/tests/instance.json b/tasks/optimize-ae-p31-furstenberg-sarkozy-k3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2ff2f15419010180e0675396620641406df90dd8
--- /dev/null
+++ b/tasks/optimize-ae-p31-furstenberg-sarkozy-k3/tests/instance.json
@@ -0,0 +1 @@
+{"k": 3, "max_m": 1000000, "max_size": 3000, "direction": "maximize", "best_known": 0.8616817796994568, "baseline": 0.6666666666666666, "family": "ae_p31_furstenberg_sarkozy", "subset": "optimize"}
\ No newline at end of file
diff --git a/tasks/optimize-ae-p31-furstenberg-sarkozy-k3/tests/run.py b/tasks/optimize-ae-p31-furstenberg-sarkozy-k3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/optimize-ae-p31-furstenberg-sarkozy-k3/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/optimize-ae-p31-furstenberg-sarkozy-k3/tests/test.sh b/tasks/optimize-ae-p31-furstenberg-sarkozy-k3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/optimize-ae-p31-furstenberg-sarkozy-k3/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/optimize-ae-p41-sphere-max-volume-n100/environment/Dockerfile b/tasks/optimize-ae-p41-sphere-max-volume-n100/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/optimize-ae-p41-sphere-max-volume-n100/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/optimize-ae-p41-sphere-max-volume-n100/environment/instance.json b/tasks/optimize-ae-p41-sphere-max-volume-n100/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4f2b7ee6a90480da38ea60e9968113a1128afd56
--- /dev/null
+++ b/tasks/optimize-ae-p41-sphere-max-volume-n100/environment/instance.json
@@ -0,0 +1 @@
+{"n": 100, "direction": "maximize", "family": "ae_p41_sphere_max_volume", "subset": "optimize"}
\ No newline at end of file
diff --git a/tasks/optimize-ae-p41-sphere-max-volume-n100/solution/solve.sh b/tasks/optimize-ae-p41-sphere-max-volume-n100/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..27904ef4132e8d2d448a51a6f88692577cd9b35f
--- /dev/null
+++ b/tasks/optimize-ae-p41-sphere-max-volume-n100/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"points": [[0.073833351885545, -0.170104592095713, 0.982656126983034], [-0.177341695235486, 0.079857726287005, 0.98090400482579], [0.250587811960338, 0.131115839180673, 0.959173803443817], [-0.227862237897793, -0.309878157821803, 0.923067889077058], [-0.083553435910935, 0.411249076518349, 0.907685529470578], [0.391728946560412, -0.265694272009011, 0.880883071836593], [-0.518684105781253, -0.058201634384294, 0.852982630634938], [0.272771252654725, 0.447607779537711, 0.851612070970399], [0.101841516917694, -0.505343411316493, 0.856887590102058], [-0.448378965016579, 0.332084145931557, 0.829865304583563], [0.588673477109153, 0.016487494288508, 0.808202759139256], [-0.500669352411902, -0.425143053524028, 0.75404481537619], [0.0775253720939, 0.686117351905618, 0.723348322798722], [0.439798483422533, -0.595178262076013, 0.672562361667242], [-0.741327658798737, 0.16374266184003, 0.650862230424784], [0.579586405604204, 0.345309684567972, 0.738133334962164], [-0.217072467000234, -0.656388324204006, 0.722519142943001], [-0.320194644769304, 0.667161615208717, 0.672585138590762], [0.702743966796423, -0.282801620781659, 0.652820159319927], [-0.765439736450661, -0.264137834030181, 0.586799126189011], [0.425805386049638, 0.680400126018441, 0.596443997140727], [0.13635624878715, -0.769065973823772, 0.624455363755641], [-0.654545460142105, 0.524560419864287, 0.544432370932481], [0.840585951393698, 0.120074089027058, 0.52820211232432], [-0.543736909916453, -0.682975325863713, 0.487744684292779], [-0.068273662124464, 0.839239527708715, 0.539458730758311], [0.671042493465269, -0.614486891441663, 0.414858809970664], [-0.916197190588067, 0.042994924736717, 0.398414538396151], [0.724446852591152, 0.534533366142954, 0.43525950678952], [-0.126997799950947, -0.887695366717024, 0.442570327424634], [-0.499187665416743, 0.791079797799357, 0.353559652971759], [0.899303767268954, -0.251249463029269, 0.357947540155497], [-0.818802140801413, -0.484702750368082, 0.307613878107995], [0.210605090473684, 0.901973358239059, 0.376947684557852], [0.340346304255546, -0.87777491735898, 0.337158104804004], [-0.874601640367787, 0.381229881290518, 0.299559256704236], [0.912843048922076, 0.340951526394104, 0.224654456185893], [-0.406247576364625, -0.891301342101847, 0.20135745396015], [-0.208963211239092, 0.925853089840982, 0.314849856885077], [0.83323916212131, -0.537507315322373, 0.129608582595105], [-0.960222458695331, -0.188650420383565, 0.205873380275836], [0.563366346340131, 0.794299825002586, 0.227389858639808], [0.013337525017151, -0.991460522918563, 0.12972332835873], [-0.721616461431413, 0.67672654905118, 0.145982397570686], [0.990290584055841, 0.026367996031794, 0.136489149442761], [-0.71432081951494, -0.695320883089954, 0.07921260219505], [-0.011737949657895, 0.998943244551323, 0.04443663694638], [0.605429247913529, -0.79502494602716, 0.037292907708691], [-0.984254927758904, 0.173186459679005, 0.035336770737677], [0.805512009293496, 0.592425484158844, 0.013507353668068], [-0.267821997438708, -0.960560342101776, -0.074801115427914], [-0.414928518672711, 0.909642625001041, -0.019611710106074], [0.951914742085123, -0.301322850716091, -0.055343142640636], [-0.909387487826631, -0.408071469131885, -0.080573401721123], [0.331469800630283, 0.943003590586819, 0.029529636138749], [0.315368482814565, -0.944635271537407, -0.090592073685774], [-0.869129635139966, 0.477317913453876, -0.129542606186178], [0.942845098963281, 0.286600284109824, -0.170009989439118], [-0.560613492355427, -0.806172672415924, -0.189203949321411], [-0.134436792818281, 0.952159258737308, -0.274443973767971], [0.769175470744074, -0.586580644046366, -0.253559151354833], [-0.984450023989259, -0.081144892433427, -0.155800053592787], [0.582009585750837, 0.79132674479229, -0.187261381685003], [-0.003927381317941, -0.96211361628144, -0.27262055139302], [-0.650979170463935, 0.723497726780967, -0.229732799061998], [0.966333711632048, -0.03146337314202, -0.255360948294675], [-0.759810636564833, -0.546674462633584, -0.351901731264888], [0.22102525489299, 0.929268200357925, -0.295987243818094], [0.493453674077528, -0.771715283628738, -0.401196949830386], [-0.904402477974688, 0.204624393414098, -0.374412894346812], [0.760458653855903, 0.513907854616983, -0.396990368571629], [-0.308088983895785, -0.8671080761009, -0.391413799402446], [-0.394837781596147, 0.808577979363862, -0.436239358050242], [0.84432977473886, -0.286225342381516, -0.452970512139598], [-0.870405690688742, -0.191352622887731, -0.453627718873791], [0.436226877994615, 0.733496767369103, -0.521237568843755], [0.204818161181484, -0.851661657874034, -0.482412625619901], [-0.686244596107605, 0.489094541757721, -0.538381726599181], [0.816144609491396, 0.139267560895094, -0.560814160734613], [-0.49184126272536, -0.653950900336277, -0.574839449090002], [-0.000155876082976, 0.802653942811619, -0.596444988068144], [0.576970643538791, -0.519670637591562, -0.630116897821041], [-0.716911504537777, 0.082563788130419, -0.692257983378261], [0.563533939744869, 0.417373002849734, -0.712902009569213], [-0.113629077406762, -0.765296920585161, -0.633568509405697], [-0.351897268082952, 0.623355418371321, -0.698280985780703], [0.664077144014496, -0.176842977160439, -0.726448971522834], [-0.631999922888147, -0.331441807515543, -0.700515828300979], [0.23982484205379, 0.600677823730597, -0.762673060499811], [0.194853441070733, -0.604974067496623, -0.772035306290771], [-0.454942147560721, 0.299789507846574, -0.838542720055423], [0.512665979397536, 0.088982912717867, -0.853964656653078], [-0.260837209290139, -0.488339833875332, -0.832759363141805], [-0.096308391984418, 0.492856569622733, -0.86476418485798], [0.359811724898042, -0.295992076676284, -0.884830047619743], [-0.414934525157292, -0.105828869773243, -0.903675600066202], [0.221177826670129, 0.312629943926841, -0.923765601843768], [-0.035926984716091, -0.27691217846843, -0.960223357966821], [-0.139127049155097, 0.130251933242821, -0.981671074280944], [0.183914304662891, -0.013596066499935, -0.982848246432833]]}
+RLMATH_EOF
diff --git a/tasks/optimize-ae-p41-sphere-max-volume-n100/tests/check.py b/tasks/optimize-ae-p41-sphere-max-volume-n100/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..58b6521809868700c39ccbae7ef07cd7c99c8d83
--- /dev/null
+++ b/tasks/optimize-ae-p41-sphere-max-volume-n100/tests/check.py
@@ -0,0 +1,23 @@
+"""n points on the unit sphere maximizing the volume of their convex hull (AlphaEvolve problem 41).
+
+Upstream bug fixed: the notebook summed |det|/6 of origin-pyramids over (QJ-joggled) hull facets, which
+over-counts whenever the origin lies outside the hull (e.g. all points in one hemisphere). We use the exact
+hull volume (scipy ConvexHull.volume) of the normalized points, without joggling."""
+import numpy as np
+from scipy.spatial import ConvexHull
+
+from run import Invalid, float_matrix
+
+
+def check(inst, ans):
+    n = inst["n"]
+    p = np.array(float_matrix(ans.get("points"), n, 3, "points"))
+    nr = np.linalg.norm(p, axis=1)
+    if (nr < 1e-9).any() or (nr > 1e9).any():
+        raise Invalid("every point needs a non-zero, finite norm (it is projected to the unit sphere)")
+    p = p / nr[:, None]
+    try:
+        v = ConvexHull(p).volume
+    except Exception as e:  # degenerate (coplanar) input
+        raise Invalid(f"degenerate hull: {type(e).__name__}")
+    return float(v), {}
diff --git a/tasks/optimize-ae-p41-sphere-max-volume-n100/tests/instance.json b/tasks/optimize-ae-p41-sphere-max-volume-n100/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4681de63e6b7c9923b62ddb54dada89a20062135
--- /dev/null
+++ b/tasks/optimize-ae-p41-sphere-max-volume-n100/tests/instance.json
@@ -0,0 +1 @@
+{"n": 100, "direction": "maximize", "best_known": 3.961753339062, "baseline": 2.092960519369956, "family": "ae_p41_sphere_max_volume", "subset": "optimize"}
\ No newline at end of file
diff --git a/tasks/optimize-ae-p41-sphere-max-volume-n100/tests/run.py b/tasks/optimize-ae-p41-sphere-max-volume-n100/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/optimize-ae-p41-sphere-max-volume-n100/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/optimize-ae-p41-sphere-max-volume-n100/tests/test.sh b/tasks/optimize-ae-p41-sphere-max-volume-n100/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/optimize-ae-p41-sphere-max-volume-n100/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/optimize-ae-p41-sphere-max-volume-n16/environment/Dockerfile b/tasks/optimize-ae-p41-sphere-max-volume-n16/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/optimize-ae-p41-sphere-max-volume-n16/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/optimize-ae-p41-sphere-max-volume-n16/environment/instance.json b/tasks/optimize-ae-p41-sphere-max-volume-n16/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d5b92afb676add09a90f83e258eb84488bd87a2b
--- /dev/null
+++ b/tasks/optimize-ae-p41-sphere-max-volume-n16/environment/instance.json
@@ -0,0 +1 @@
+{"n": 16, "direction": "maximize", "family": "ae_p41_sphere_max_volume", "subset": "optimize"}
\ No newline at end of file
diff --git a/tasks/optimize-ae-p41-sphere-max-volume-n16/solution/solve.sh b/tasks/optimize-ae-p41-sphere-max-volume-n16/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..49d07c402d2830849ffaeb5eae52f0fd10f11d3a
--- /dev/null
+++ b/tasks/optimize-ae-p41-sphere-max-volume-n16/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"points": [[0.149416994548902, -0.460244767572493, 0.875128170991035], [-0.444778213064685, 0.272659865497073, 0.853128911085602], [0.641401381220424, 0.265747422692511, 0.719710063498364], [-0.624032371210396, -0.592063051062083, 0.509946019936021], [0.013098870583807, 0.905216729959373, 0.424748268261436], [0.763420814154356, -0.604174749830634, 0.228388993130124], [-0.998932243644909, -0.040725120306146, 0.021813692548444], [0.674653869700649, 0.730602206961526, -0.105178758696268], [0.045039015782498, -0.993432894450579, 0.105178758696268], [-0.737587355722845, 0.674899292854263, -0.021813692548444], [0.973107553316303, -0.030002624814606, -0.228388993130124], [-0.573348152244774, -0.700614590860351, -0.424748268261433], [-0.095612478844117, 0.854876196089641, -0.509946019936021], [0.319186797070964, -0.616552684751887, -0.719710063498364], [-0.515814041493742, 0.078146885208486, -0.853128911085601], [0.410854906636044, 0.255634367859592, -0.875128170991035]]}
+RLMATH_EOF
diff --git a/tasks/optimize-ae-p41-sphere-max-volume-n16/tests/check.py b/tasks/optimize-ae-p41-sphere-max-volume-n16/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..58b6521809868700c39ccbae7ef07cd7c99c8d83
--- /dev/null
+++ b/tasks/optimize-ae-p41-sphere-max-volume-n16/tests/check.py
@@ -0,0 +1,23 @@
+"""n points on the unit sphere maximizing the volume of their convex hull (AlphaEvolve problem 41).
+
+Upstream bug fixed: the notebook summed |det|/6 of origin-pyramids over (QJ-joggled) hull facets, which
+over-counts whenever the origin lies outside the hull (e.g. all points in one hemisphere). We use the exact
+hull volume (scipy ConvexHull.volume) of the normalized points, without joggling."""
+import numpy as np
+from scipy.spatial import ConvexHull
+
+from run import Invalid, float_matrix
+
+
+def check(inst, ans):
+    n = inst["n"]
+    p = np.array(float_matrix(ans.get("points"), n, 3, "points"))
+    nr = np.linalg.norm(p, axis=1)
+    if (nr < 1e-9).any() or (nr > 1e9).any():
+        raise Invalid("every point needs a non-zero, finite norm (it is projected to the unit sphere)")
+    p = p / nr[:, None]
+    try:
+        v = ConvexHull(p).volume
+    except Exception as e:  # degenerate (coplanar) input
+        raise Invalid(f"degenerate hull: {type(e).__name__}")
+    return float(v), {}
diff --git a/tasks/optimize-ae-p41-sphere-max-volume-n16/tests/instance.json b/tasks/optimize-ae-p41-sphere-max-volume-n16/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d163d2083c7d57d05c7a461980f588fee4c5c663
--- /dev/null
+++ b/tasks/optimize-ae-p41-sphere-max-volume-n16/tests/instance.json
@@ -0,0 +1 @@
+{"n": 16, "direction": "maximize", "best_known": 2.886455392275, "baseline": 2.024790782548605, "family": "ae_p41_sphere_max_volume", "subset": "optimize"}
\ No newline at end of file
diff --git a/tasks/optimize-ae-p41-sphere-max-volume-n16/tests/run.py b/tasks/optimize-ae-p41-sphere-max-volume-n16/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/optimize-ae-p41-sphere-max-volume-n16/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/optimize-ae-p41-sphere-max-volume-n16/tests/test.sh b/tasks/optimize-ae-p41-sphere-max-volume-n16/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/optimize-ae-p41-sphere-max-volume-n16/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/optimize-ae-p60-no5-on-sphere-n12/environment/Dockerfile b/tasks/optimize-ae-p60-no5-on-sphere-n12/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/optimize-ae-p60-no5-on-sphere-n12/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/optimize-ae-p60-no5-on-sphere-n12/environment/instance.json b/tasks/optimize-ae-p60-no5-on-sphere-n12/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e0ab3569b9e670c72cf173db525fc7598da2b276
--- /dev/null
+++ b/tasks/optimize-ae-p60-no5-on-sphere-n12/environment/instance.json
@@ -0,0 +1 @@
+{"n": 12, "max_points": 48, "direction": "maximize", "family": "ae_p60_no5_on_sphere", "subset": "optimize"}
\ No newline at end of file
diff --git a/tasks/optimize-ae-p60-no5-on-sphere-n12/solution/solve.sh b/tasks/optimize-ae-p60-no5-on-sphere-n12/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..8c902d28ca5b8740e4a7bccde99b92ba28213d7f
--- /dev/null
+++ b/tasks/optimize-ae-p60-no5-on-sphere-n12/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"points": [[5, 9, 10], [10, 5, 8], [11, 11, 2], [0, 2, 10], [4, 9, 0], [3, 8, 9], [6, 10, 6], [3, 1, 11], [1, 4, 6], [8, 8, 2], [8, 7, 6], [0, 10, 0], [4, 2, 1], [1, 10, 7], [10, 11, 7], [4, 3, 7], [9, 0, 9], [10, 6, 0], [4, 3, 5], [7, 2, 0], [2, 0, 11], [0, 5, 1], [9, 9, 8], [0, 9, 2], [11, 1, 1], [9, 6, 10], [7, 1, 2], [1, 2, 5], [7, 3, 10], [11, 7, 11], [1, 11, 9], [8, 0, 11], [11, 0, 3]]}
+RLMATH_EOF
diff --git a/tasks/optimize-ae-p60-no5-on-sphere-n12/tests/check.py b/tasks/optimize-ae-p60-no5-on-sphere-n12/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..535715270fb95983b4b3896cda93a8746da1a961
--- /dev/null
+++ b/tasks/optimize-ae-p60-no5-on-sphere-n12/tests/check.py
@@ -0,0 +1,53 @@
+"""No 5 points of an n x n x n grid subset on a common sphere or plane (AlphaEvolve problem 60).
+
+Exact int64 test: five points are co-spherical/co-planar iff det[x, y, z, x^2+y^2+z^2, 1] = 0, evaluated as the
+4x4 determinant of differences (entries <= 3(n-1)^2, so |det| < 2^40). Upstream used float det + np.isclose."""
+import itertools
+
+import numpy as np
+
+from run import Invalid, int_list
+
+
+def det4(m):
+    a, b, c, d = m[:, 0], m[:, 1], m[:, 2], m[:, 3]
+
+    def det3(r0, r1, r2, i, j, k):
+        return (r0[:, i] * (r1[:, j] * r2[:, k] - r1[:, k] * r2[:, j])
+                - r0[:, j] * (r1[:, i] * r2[:, k] - r1[:, k] * r2[:, i])
+                + r0[:, k] * (r1[:, i] * r2[:, j] - r1[:, j] * r2[:, i]))
+    return (a[:, 0] * det3(b, c, d, 1, 2, 3) - a[:, 1] * det3(b, c, d, 0, 2, 3)
+            + a[:, 2] * det3(b, c, d, 0, 1, 3) - a[:, 3] * det3(b, c, d, 0, 1, 2))
+
+
+def lifted(p):
+    a = np.array(p, dtype=np.int64)
+    return np.concatenate([a, (a * a).sum(1, keepdims=True)], axis=1)
+
+
+def bad_five(rows, combos):
+    """rows: (m, 4) lifted points; combos: (N, 5) index array. Returns mask of degenerate 5-sets."""
+    base = rows[combos[:, 0]]
+    m = np.stack([rows[combos[:, i]] - base for i in range(1, 5)], axis=1)
+    return det4(m) == 0
+
+
+def check(inst, ans):
+    n = inst["n"]
+    pts = ans.get("points")
+    if not isinstance(pts, list) or not (1 <= len(pts) <= inst["max_points"]):
+        raise Invalid(f"points must be a list of 1..{inst['max_points']} [x, y, z] triples")
+    p = [tuple(int_list(q, 3, 0, n - 1, "point")) for q in pts]
+    if len(set(p)) != len(p):
+        raise Invalid("points must be distinct")
+    rows = lifted(p)
+    it = itertools.combinations(range(len(p)), 5)
+    while True:
+        chunk = np.array(list(itertools.islice(it, 200000)), dtype=np.int64)
+        if len(chunk) == 0:
+            break
+        bad = bad_five(rows, chunk)
+        if bad.any():
+            q = [p[i] for i in chunk[np.argmax(bad)]]
+            raise Invalid(f"points {q} lie on a common sphere or plane")
+    return float(len(p)), {"size": len(p)}
diff --git a/tasks/optimize-ae-p60-no5-on-sphere-n12/tests/instance.json b/tasks/optimize-ae-p60-no5-on-sphere-n12/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a7cc03c424d493f8ec042ad21ffa920d3a9d6352
--- /dev/null
+++ b/tasks/optimize-ae-p60-no5-on-sphere-n12/tests/instance.json
@@ -0,0 +1 @@
+{"n": 12, "max_points": 48, "direction": "maximize", "best_known": 33.0, "baseline": 26.0, "family": "ae_p60_no5_on_sphere", "subset": "optimize"}
\ No newline at end of file
diff --git a/tasks/optimize-ae-p60-no5-on-sphere-n12/tests/run.py b/tasks/optimize-ae-p60-no5-on-sphere-n12/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/optimize-ae-p60-no5-on-sphere-n12/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/optimize-ae-p60-no5-on-sphere-n12/tests/test.sh b/tasks/optimize-ae-p60-no5-on-sphere-n12/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/optimize-ae-p60-no5-on-sphere-n12/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt