diff --git a/tasks/construct-ae-p38-factorial-factors-l3-s3/environment/Dockerfile b/tasks/construct-ae-p38-factorial-factors-l3-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-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-ae-p38-factorial-factors-l3-s3/environment/instance.json b/tasks/construct-ae-p38-factorial-factors-l3-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b1b22e06d71f6107862ae8c7646d2c5d1f7f52c0
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l3-s3/environment/instance.json
@@ -0,0 +1 @@
+{"N": 275, "t": 83, "family": "ae_p38_factorial_factors", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-ae-p38-factorial-factors-l3-s3/solution/solve.sh b/tasks/construct-ae-p38-factorial-factors-l3-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..f4f517b4c7f6c3e0e9bd290d28d55477dd1dad40
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-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'
+{"factors": [83, 83, 83, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 86, 86, 86, 86, 86, 86, 87, 87, 87, 87, 87, 87, 87, 87, 87, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 89, 89, 89, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 92, 92, 92, 92, 92, 92, 92, 92, 92, 92, 92, 93, 93, 93, 93, 93, 93, 93, 93, 94, 94, 94, 94, 94, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 97, 97, 101, 101, 103, 103, 104, 104, 104, 104, 104, 104, 104, 106, 106, 106, 106, 106, 107, 107, 109, 109, 111, 111, 111, 111, 111, 111, 111, 113, 113, 118, 118, 118, 118, 122, 122, 122, 122, 123, 123, 123, 123, 123, 123, 127, 127, 128, 128, 131, 131, 134, 134, 134, 134, 137, 137, 139, 142, 142, 142, 146, 146, 146, 149, 151, 157, 158, 158, 158, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271]}
+RLMATH_EOF
diff --git a/tasks/construct-ae-p38-factorial-factors-l3-s3/tests/check.py b/tasks/construct-ae-p38-factorial-factors-l3-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..93ec67a19039afa8b4ce6e39811691b802132562
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l3-s3/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-l3-s3/tests/instance.json b/tasks/construct-ae-p38-factorial-factors-l3-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b1b22e06d71f6107862ae8c7646d2c5d1f7f52c0
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l3-s3/tests/instance.json
@@ -0,0 +1 @@
+{"N": 275, "t": 83, "family": "ae_p38_factorial_factors", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-ae-p38-factorial-factors-l3-s3/tests/run.py b/tasks/construct-ae-p38-factorial-factors-l3-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-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-ae-p38-factorial-factors-l3-s3/tests/test.sh b/tasks/construct-ae-p38-factorial-factors-l3-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-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-ae-p38-factorial-factors-l3-s4/environment/Dockerfile b/tasks/construct-ae-p38-factorial-factors-l3-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-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-ae-p38-factorial-factors-l3-s4/environment/instance.json b/tasks/construct-ae-p38-factorial-factors-l3-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a42ef11d56f0b8bd29bd9258494df31530a8a9c5
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l3-s4/environment/instance.json
@@ -0,0 +1 @@
+{"N": 284, "t": 85, "family": "ae_p38_factorial_factors", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-ae-p38-factorial-factors-l3-s4/solution/solve.sh b/tasks/construct-ae-p38-factorial-factors-l3-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..9924ff4e5a4b0d1a2b68ae8b68eaa35eb9a4885c
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-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'
+{"factors": [85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 86, 86, 86, 86, 86, 86, 87, 87, 87, 87, 87, 87, 87, 87, 87, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 88, 89, 89, 89, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 90, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 91, 92, 92, 92, 92, 92, 92, 92, 92, 92, 92, 92, 92, 93, 93, 93, 93, 93, 93, 93, 93, 93, 94, 94, 94, 94, 94, 94, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 97, 97, 98, 98, 98, 98, 98, 98, 98, 98, 98, 98, 98, 101, 101, 102, 102, 102, 102, 102, 103, 103, 106, 106, 106, 106, 106, 107, 107, 109, 109, 111, 111, 111, 111, 111, 111, 111, 113, 113, 118, 118, 118, 118, 122, 122, 122, 122, 123, 123, 123, 123, 123, 123, 126, 127, 127, 128, 131, 131, 134, 134, 134, 134, 137, 137, 139, 139, 142, 142, 142, 142, 146, 146, 146, 149, 151, 157, 158, 158, 158, 163, 166, 166, 166, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283]}
+RLMATH_EOF
diff --git a/tasks/construct-ae-p38-factorial-factors-l3-s4/tests/check.py b/tasks/construct-ae-p38-factorial-factors-l3-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..93ec67a19039afa8b4ce6e39811691b802132562
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l3-s4/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-l3-s4/tests/instance.json b/tasks/construct-ae-p38-factorial-factors-l3-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a42ef11d56f0b8bd29bd9258494df31530a8a9c5
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l3-s4/tests/instance.json
@@ -0,0 +1 @@
+{"N": 284, "t": 85, "family": "ae_p38_factorial_factors", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-ae-p38-factorial-factors-l3-s4/tests/run.py b/tasks/construct-ae-p38-factorial-factors-l3-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-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-ae-p38-factorial-factors-l3-s4/tests/test.sh b/tasks/construct-ae-p38-factorial-factors-l3-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-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-ae-p52-erdos-squarefree-l3-s8/environment/Dockerfile b/tasks/construct-ae-p52-erdos-squarefree-l3-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-ae-p52-erdos-squarefree-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-ae-p52-erdos-squarefree-l3-s8/environment/instance.json b/tasks/construct-ae-p52-erdos-squarefree-l3-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..50abcb7cc3e71d0046e921550f952b93fccc58ea
--- /dev/null
+++ b/tasks/construct-ae-p52-erdos-squarefree-l3-s8/environment/instance.json
@@ -0,0 +1 @@
+{"N": 1067, "t": 43, "family": "ae_p52_erdos_squarefree", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-ae-p52-erdos-squarefree-l3-s8/solution/solve.sh b/tasks/construct-ae-p52-erdos-squarefree-l3-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..9d1b3e3f2f522672d14833a4f3a1d4551bb66972
--- /dev/null
+++ b/tasks/construct-ae-p52-erdos-squarefree-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'
+{"A": [7, 32, 57, 82, 107, 132, 157, 182, 207, 232, 257, 282, 307, 332, 357, 382, 407, 432, 457, 482, 507, 532, 557, 582, 607, 632, 657, 682, 707, 732, 757, 782, 807, 832, 857, 882, 907, 932, 957, 982, 1007, 1032, 1057]}
+RLMATH_EOF
diff --git a/tasks/construct-ae-p52-erdos-squarefree-l3-s8/tests/check.py b/tasks/construct-ae-p52-erdos-squarefree-l3-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..87e15b242f211059193ba122baecdee9add99130
--- /dev/null
+++ b/tasks/construct-ae-p52-erdos-squarefree-l3-s8/tests/check.py
@@ -0,0 +1,38 @@
+"""Erdos squarefree problem (AlphaEvolve problem 52): A in [1, N], |A| >= t, ab+1 NOT squarefree for all a, b in A
+(a = b included). The repo's problem page says "square-free"; the notebook and the Erdos problem ask for
+"never squarefree", which is what we check. Exact: for every pair we find a prime p with p^2 | ab+1."""
+import numpy as np
+
+from run import Invalid, int_list
+
+
+def _primes(n):
+    s = np.ones(n + 1, dtype=bool)
+    s[:2] = False
+    for i in range(2, int(n ** .5) + 1):
+        if s[i]:
+            s[i * i::i] = False
+    return np.nonzero(s)[0]
+
+
+def check(inst, ans):
+    n, t = inst["N"], inst["t"]
+    a = ans.get("A")
+    if not isinstance(a, list) or not (t <= len(a) <= 2 * t + 20):
+        raise Invalid(f"A must be a list of {t}..{2 * t + 20} integers")
+    int_list(a, None, 1, n, "A")
+    if len(set(a)) != len(a):
+        raise Invalid("A must consist of distinct integers")
+    x = np.array(sorted(a), dtype=np.int64)
+    iu, ju = np.triu_indices(len(x))
+    va, vb = x[iu], x[ju]
+    left = np.ones(len(iu), dtype=bool)  # pairs without a square divisor found yet
+    for p in _primes(int((n * n + 1) ** .5) + 1):
+        q = int(p) * int(p)
+        idx = np.nonzero(left)[0]
+        hit = ((va[idx] % q) * (vb[idx] % q) + 1) % q == 0
+        left[idx[hit]] = False
+        if not left.any():
+            return None, {"size": len(a)}
+    k = int(np.nonzero(left)[0][0])
+    raise Invalid(f"{int(va[k])}*{int(vb[k])}+1 = {int(va[k]) * int(vb[k]) + 1} is squarefree")
diff --git a/tasks/construct-ae-p52-erdos-squarefree-l3-s8/tests/instance.json b/tasks/construct-ae-p52-erdos-squarefree-l3-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..50abcb7cc3e71d0046e921550f952b93fccc58ea
--- /dev/null
+++ b/tasks/construct-ae-p52-erdos-squarefree-l3-s8/tests/instance.json
@@ -0,0 +1 @@
+{"N": 1067, "t": 43, "family": "ae_p52_erdos_squarefree", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-ae-p52-erdos-squarefree-l3-s8/tests/run.py b/tasks/construct-ae-p52-erdos-squarefree-l3-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-ae-p52-erdos-squarefree-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-ae-p52-erdos-squarefree-l3-s8/tests/test.sh b/tasks/construct-ae-p52-erdos-squarefree-l3-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-ae-p52-erdos-squarefree-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-mc-bibd-l4-s1/environment/Dockerfile b/tasks/construct-mc-bibd-l4-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-bibd-l4-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-bibd-l4-s1/environment/instance.json b/tasks/construct-mc-bibd-l4-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..6036b18bf14c0d9945c7afce57b25eba4b52ffb3
--- /dev/null
+++ b/tasks/construct-mc-bibd-l4-s1/environment/instance.json
@@ -0,0 +1 @@
+{"v": 13, "k": 3, "lambda": 2, "b": 52, "family": "mc_bibd", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-bibd-l4-s1/solution/solve.sh b/tasks/construct-mc-bibd-l4-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..7ca267da3f960f3b0aa9e0e6bbff798247df5f31
--- /dev/null
+++ b/tasks/construct-mc-bibd-l4-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'
+{"blocks": [[2, 5, 7], [5, 10, 11], [0, 11, 12], [0, 10, 12], [0, 1, 5], [0, 3, 10], [0, 5, 11], [0, 3, 9], [0, 1, 2], [5, 10, 12], [1, 7, 12], [1, 5, 9], [5, 8, 9], [1, 7, 11], [1, 9, 11], [1, 3, 12], [2, 9, 11], [2, 5, 12], [7, 9, 10], [2, 6, 7], [3, 7, 11], [3, 7, 9], [0, 2, 6], [1, 3, 6], [3, 6, 12], [2, 3, 10], [2, 3, 8], [2, 11, 12], [8, 10, 11], [1, 2, 8], [3, 8, 11], [4, 8, 12], [0, 8, 9], [0, 4, 7], [3, 4, 5], [3, 4, 5], [1, 4, 10], [4, 9, 12], [2, 4, 10], [1, 4, 8], [4, 6, 11], [2, 4, 9], [0, 4, 7], [4, 6, 11], [5, 6, 7], [5, 6, 8], [6, 9, 10], [1, 6, 10], [7, 8, 12], [7, 8, 10], [0, 6, 8], [6, 9, 12]]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-bibd-l4-s1/tests/check.py b/tasks/construct-mc-bibd-l4-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..c4b5da62477026f27ff9869fe33602fcc41e2ea9
--- /dev/null
+++ b/tasks/construct-mc-bibd-l4-s1/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-l4-s1/tests/instance.json b/tasks/construct-mc-bibd-l4-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..6036b18bf14c0d9945c7afce57b25eba4b52ffb3
--- /dev/null
+++ b/tasks/construct-mc-bibd-l4-s1/tests/instance.json
@@ -0,0 +1 @@
+{"v": 13, "k": 3, "lambda": 2, "b": 52, "family": "mc_bibd", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-bibd-l4-s1/tests/run.py b/tasks/construct-mc-bibd-l4-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-bibd-l4-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-bibd-l4-s1/tests/test.sh b/tasks/construct-mc-bibd-l4-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-bibd-l4-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-costas-array-l1-s3/environment/Dockerfile b/tasks/construct-mc-costas-array-l1-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-costas-array-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-costas-array-l1-s3/environment/instance.json b/tasks/construct-mc-costas-array-l1-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e54bc14552ab244d0e9f401f3655b10ea70ec837
--- /dev/null
+++ b/tasks/construct-mc-costas-array-l1-s3/environment/instance.json
@@ -0,0 +1 @@
+{"n": 5, "fixed": [], "family": "mc_costas_array", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-costas-array-l1-s3/solution/solve.sh b/tasks/construct-mc-costas-array-l1-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..ee9b9e28e7e460cb3427cad46d3aa70c31e645b1
--- /dev/null
+++ b/tasks/construct-mc-costas-array-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'
+{"x": [0, 2, 3, 1, 4]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-costas-array-l1-s3/tests/check.py b/tasks/construct-mc-costas-array-l1-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..c86323a5b639b23787ecd4d5f358b986bbd91b6c
--- /dev/null
+++ b/tasks/construct-mc-costas-array-l1-s3/tests/check.py
@@ -0,0 +1,22 @@
+"""Costas array of order n (MathConstraint costas_array, CC-BY-4.0). Accepts any valid array.
+Adapted from pilot/grader.py check_costas; adds the optional fixed entries (upstream partial assignment)."""
+import itertools
+
+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")
+    for name, i, v in inst.get("fixed", []):
+        if x[i] != v:
+            raise Invalid(f"fixed value {name}[{i}] = {v} not respected")
+    vecs = set()
+    for i, j in itertools.combinations(range(n), 2):
+        d = (j - i, x[j] - x[i])
+        if d in vecs:
+            raise Invalid(f"repeated displacement vector {d}")
+        vecs.add(d)
+    return None, {}
diff --git a/tasks/construct-mc-costas-array-l1-s3/tests/instance.json b/tasks/construct-mc-costas-array-l1-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e54bc14552ab244d0e9f401f3655b10ea70ec837
--- /dev/null
+++ b/tasks/construct-mc-costas-array-l1-s3/tests/instance.json
@@ -0,0 +1 @@
+{"n": 5, "fixed": [], "family": "mc_costas_array", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-costas-array-l1-s3/tests/run.py b/tasks/construct-mc-costas-array-l1-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-costas-array-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-costas-array-l1-s3/tests/test.sh b/tasks/construct-mc-costas-array-l1-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-costas-array-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-costas-array-l1-s4/environment/Dockerfile b/tasks/construct-mc-costas-array-l1-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-costas-array-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-costas-array-l1-s4/environment/instance.json b/tasks/construct-mc-costas-array-l1-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..69459ce0daed749c61e8eb9c54233d125b9e43b5
--- /dev/null
+++ b/tasks/construct-mc-costas-array-l1-s4/environment/instance.json
@@ -0,0 +1 @@
+{"n": 7, "fixed": [["x", 1, 1]], "family": "mc_costas_array", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-costas-array-l1-s4/solution/solve.sh b/tasks/construct-mc-costas-array-l1-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..e914f80957917faeaaf1012ae77e306259ed9abb
--- /dev/null
+++ b/tasks/construct-mc-costas-array-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'
+{"x": [0, 1, 6, 4, 3, 5, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-costas-array-l1-s4/tests/check.py b/tasks/construct-mc-costas-array-l1-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..c86323a5b639b23787ecd4d5f358b986bbd91b6c
--- /dev/null
+++ b/tasks/construct-mc-costas-array-l1-s4/tests/check.py
@@ -0,0 +1,22 @@
+"""Costas array of order n (MathConstraint costas_array, CC-BY-4.0). Accepts any valid array.
+Adapted from pilot/grader.py check_costas; adds the optional fixed entries (upstream partial assignment)."""
+import itertools
+
+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")
+    for name, i, v in inst.get("fixed", []):
+        if x[i] != v:
+            raise Invalid(f"fixed value {name}[{i}] = {v} not respected")
+    vecs = set()
+    for i, j in itertools.combinations(range(n), 2):
+        d = (j - i, x[j] - x[i])
+        if d in vecs:
+            raise Invalid(f"repeated displacement vector {d}")
+        vecs.add(d)
+    return None, {}
diff --git a/tasks/construct-mc-costas-array-l1-s4/tests/instance.json b/tasks/construct-mc-costas-array-l1-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..69459ce0daed749c61e8eb9c54233d125b9e43b5
--- /dev/null
+++ b/tasks/construct-mc-costas-array-l1-s4/tests/instance.json
@@ -0,0 +1 @@
+{"n": 7, "fixed": [["x", 1, 1]], "family": "mc_costas_array", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-costas-array-l1-s4/tests/run.py b/tasks/construct-mc-costas-array-l1-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-costas-array-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-costas-array-l1-s4/tests/test.sh b/tasks/construct-mc-costas-array-l1-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-costas-array-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-mc-golomb-l2-s0/environment/Dockerfile b/tasks/construct-mc-golomb-l2-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-golomb-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-mc-golomb-l2-s0/environment/instance.json b/tasks/construct-mc-golomb-l2-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..fb1e19f55d2e7ee39d0185d0e0fdc41f66b331bd
--- /dev/null
+++ b/tasks/construct-mc-golomb-l2-s0/environment/instance.json
@@ -0,0 +1 @@
+{"n": 9, "max_length": 44, "fixed": [], "family": "mc_golomb", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-golomb-l2-s0/solution/solve.sh b/tasks/construct-mc-golomb-l2-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..7d899199777d5c2f5410231d0bcb663c96da1d74
--- /dev/null
+++ b/tasks/construct-mc-golomb-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'
+{"marks": [0, 1, 5, 12, 25, 27, 35, 41, 44]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-golomb-l2-s0/tests/check.py b/tasks/construct-mc-golomb-l2-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..84aedd6e33fee7271543bfa49ab05128e23c31ee
--- /dev/null
+++ b/tasks/construct-mc-golomb-l2-s0/tests/check.py
@@ -0,0 +1,27 @@
+"""Golomb ruler with n marks and length <= L (MathConstraint golomb, CC-BY-4.0, reframed as a decision
+problem with the target length stated). Accepts any valid ruler.
+Upstream note: the upstream prompt states no length bound and its verifier re-solves the optimisation
+model with the marks fixed, which returns OPTIMUM for any Golomb ruler, so any ruler (e.g. 0,1,3,7,15,..)
+was accepted. Here the length bound is part of the instance and enforced."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, L = inst["n"], inst["max_length"]
+    a = int_list(ans.get("marks"), n, 0, L, "marks")
+    if a[0] != 0:
+        raise Invalid("first mark must be 0")
+    if any(a[i] >= a[i + 1] for i in range(n - 1)):
+        raise Invalid("marks must be strictly increasing")
+    for name, i, v in inst.get("fixed", []):
+        if a[i] != v:
+            raise Invalid(f"fixed value {name}[{i}] = {v} not respected")
+    seen = set()
+    for i, j in itertools.combinations(range(n), 2):
+        d = a[j] - a[i]
+        if d in seen:
+            raise Invalid(f"difference {d} occurs twice")
+        seen.add(d)
+    return None, {"length": a[-1]}
diff --git a/tasks/construct-mc-golomb-l2-s0/tests/instance.json b/tasks/construct-mc-golomb-l2-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..fb1e19f55d2e7ee39d0185d0e0fdc41f66b331bd
--- /dev/null
+++ b/tasks/construct-mc-golomb-l2-s0/tests/instance.json
@@ -0,0 +1 @@
+{"n": 9, "max_length": 44, "fixed": [], "family": "mc_golomb", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-golomb-l2-s0/tests/run.py b/tasks/construct-mc-golomb-l2-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-golomb-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-mc-golomb-l2-s0/tests/test.sh b/tasks/construct-mc-golomb-l2-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-golomb-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-mc-golomb-l6-s1/environment/Dockerfile b/tasks/construct-mc-golomb-l6-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-golomb-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-mc-golomb-l6-s1/environment/instance.json b/tasks/construct-mc-golomb-l6-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..005b756cbc744217eb96ba2faddd57f3cbb58d5e
--- /dev/null
+++ b/tasks/construct-mc-golomb-l6-s1/environment/instance.json
@@ -0,0 +1 @@
+{"n": 19, "max_length": 246, "fixed": [], "family": "mc_golomb", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-golomb-l6-s1/solution/solve.sh b/tasks/construct-mc-golomb-l6-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..6cc2256e454d735bd1981ac9a3044e94ef19cda8
--- /dev/null
+++ b/tasks/construct-mc-golomb-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'
+{"marks": [0, 1, 6, 25, 32, 72, 100, 108, 120, 130, 153, 169, 187, 190, 204, 231, 233, 242, 246]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-golomb-l6-s1/tests/check.py b/tasks/construct-mc-golomb-l6-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..84aedd6e33fee7271543bfa49ab05128e23c31ee
--- /dev/null
+++ b/tasks/construct-mc-golomb-l6-s1/tests/check.py
@@ -0,0 +1,27 @@
+"""Golomb ruler with n marks and length <= L (MathConstraint golomb, CC-BY-4.0, reframed as a decision
+problem with the target length stated). Accepts any valid ruler.
+Upstream note: the upstream prompt states no length bound and its verifier re-solves the optimisation
+model with the marks fixed, which returns OPTIMUM for any Golomb ruler, so any ruler (e.g. 0,1,3,7,15,..)
+was accepted. Here the length bound is part of the instance and enforced."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, L = inst["n"], inst["max_length"]
+    a = int_list(ans.get("marks"), n, 0, L, "marks")
+    if a[0] != 0:
+        raise Invalid("first mark must be 0")
+    if any(a[i] >= a[i + 1] for i in range(n - 1)):
+        raise Invalid("marks must be strictly increasing")
+    for name, i, v in inst.get("fixed", []):
+        if a[i] != v:
+            raise Invalid(f"fixed value {name}[{i}] = {v} not respected")
+    seen = set()
+    for i, j in itertools.combinations(range(n), 2):
+        d = a[j] - a[i]
+        if d in seen:
+            raise Invalid(f"difference {d} occurs twice")
+        seen.add(d)
+    return None, {"length": a[-1]}
diff --git a/tasks/construct-mc-golomb-l6-s1/tests/instance.json b/tasks/construct-mc-golomb-l6-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..005b756cbc744217eb96ba2faddd57f3cbb58d5e
--- /dev/null
+++ b/tasks/construct-mc-golomb-l6-s1/tests/instance.json
@@ -0,0 +1 @@
+{"n": 19, "max_length": 246, "fixed": [], "family": "mc_golomb", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-golomb-l6-s1/tests/run.py b/tasks/construct-mc-golomb-l6-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-golomb-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-mc-golomb-l6-s1/tests/test.sh b/tasks/construct-mc-golomb-l6-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-golomb-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-mc-hadamard-l4-s3/environment/Dockerfile b/tasks/construct-mc-hadamard-l4-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-hadamard-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-hadamard-l4-s3/environment/instance.json b/tasks/construct-mc-hadamard-l4-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..22a097d6b1a9137a499a78dc071e27b6cd6fd917
--- /dev/null
+++ b/tasks/construct-mc-hadamard-l4-s3/environment/instance.json
@@ -0,0 +1 @@
+{"n": 27, "family": "mc_hadamard", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-hadamard-l4-s3/solution/solve.sh b/tasks/construct-mc-hadamard-l4-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..6bfb73e643d405cbafdb79696f2903addb0b3433
--- /dev/null
+++ b/tasks/construct-mc-hadamard-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, 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]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-hadamard-l4-s3/tests/check.py b/tasks/construct-mc-hadamard-l4-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..fc4b93160fd7473dd14935db9d712e52abdd00c2
--- /dev/null
+++ b/tasks/construct-mc-hadamard-l4-s3/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-l4-s3/tests/instance.json b/tasks/construct-mc-hadamard-l4-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..22a097d6b1a9137a499a78dc071e27b6cd6fd917
--- /dev/null
+++ b/tasks/construct-mc-hadamard-l4-s3/tests/instance.json
@@ -0,0 +1 @@
+{"n": 27, "family": "mc_hadamard", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-hadamard-l4-s3/tests/run.py b/tasks/construct-mc-hadamard-l4-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-hadamard-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-hadamard-l4-s3/tests/test.sh b/tasks/construct-mc-hadamard-l4-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-hadamard-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-hadamard-l4-s4/environment/Dockerfile b/tasks/construct-mc-hadamard-l4-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-hadamard-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-hadamard-l4-s4/environment/instance.json b/tasks/construct-mc-hadamard-l4-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e85070d7fc8b32f65c74575c7654595675ab469b
--- /dev/null
+++ b/tasks/construct-mc-hadamard-l4-s4/environment/instance.json
@@ -0,0 +1 @@
+{"n": 25, "family": "mc_hadamard", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-hadamard-l4-s4/solution/solve.sh b/tasks/construct-mc-hadamard-l4-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..509a9d183286c19875d173960dae97ce049d7c6e
--- /dev/null
+++ b/tasks/construct-mc-hadamard-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": [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]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-hadamard-l4-s4/tests/check.py b/tasks/construct-mc-hadamard-l4-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..fc4b93160fd7473dd14935db9d712e52abdd00c2
--- /dev/null
+++ b/tasks/construct-mc-hadamard-l4-s4/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-l4-s4/tests/instance.json b/tasks/construct-mc-hadamard-l4-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e85070d7fc8b32f65c74575c7654595675ab469b
--- /dev/null
+++ b/tasks/construct-mc-hadamard-l4-s4/tests/instance.json
@@ -0,0 +1 @@
+{"n": 25, "family": "mc_hadamard", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-hadamard-l4-s4/tests/run.py b/tasks/construct-mc-hadamard-l4-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-hadamard-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-hadamard-l4-s4/tests/test.sh b/tasks/construct-mc-hadamard-l4-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-hadamard-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-low-autocorrelation-l2-s1/environment/Dockerfile b/tasks/construct-mc-low-autocorrelation-l2-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-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-low-autocorrelation-l2-s1/environment/instance.json b/tasks/construct-mc-low-autocorrelation-l2-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..cd575ed522aad2521cb17a30eb2d3d784bcc29b7
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-l2-s1/environment/instance.json
@@ -0,0 +1 @@
+{"n": 17, "bound": 40, "fixed": [["seq", 1, -1], ["seq", 6, 1], ["seq", 9, -1], ["c", 7, 3], ["c", 8, -2], ["c", 12, 0]], "family": "mc_low_autocorrelation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-low-autocorrelation-l2-s1/solution/solve.sh b/tasks/construct-mc-low-autocorrelation-l2-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..9bd454e4b5e7e755e61237bd50eebf5daa97f2c5
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-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'
+{"seq": [-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-l2-s1/tests/check.py b/tasks/construct-mc-low-autocorrelation-l2-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..8456f7b1c0ff0c9fb4bc9ae5a49074c8afa7d1aa
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-l2-s1/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-l2-s1/tests/instance.json b/tasks/construct-mc-low-autocorrelation-l2-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..cd575ed522aad2521cb17a30eb2d3d784bcc29b7
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-l2-s1/tests/instance.json
@@ -0,0 +1 @@
+{"n": 17, "bound": 40, "fixed": [["seq", 1, -1], ["seq", 6, 1], ["seq", 9, -1], ["c", 7, 3], ["c", 8, -2], ["c", 12, 0]], "family": "mc_low_autocorrelation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-low-autocorrelation-l2-s1/tests/run.py b/tasks/construct-mc-low-autocorrelation-l2-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-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-low-autocorrelation-l2-s1/tests/test.sh b/tasks/construct-mc-low-autocorrelation-l2-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-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-low-autocorrelation-l6-s0/environment/Dockerfile b/tasks/construct-mc-low-autocorrelation-l6-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-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-low-autocorrelation-l6-s0/environment/instance.json b/tasks/construct-mc-low-autocorrelation-l6-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..ab11e00d0c46cda7b1b8170981cf0dbd0f89b185
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-l6-s0/environment/instance.json
@@ -0,0 +1 @@
+{"n": 44, "bound": 134, "fixed": [], "family": "mc_low_autocorrelation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-low-autocorrelation-l6-s0/solution/solve.sh b/tasks/construct-mc-low-autocorrelation-l6-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..43036a9f8b6d7725acefcd3cff6fa9545194a4af
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-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'
+{"seq": [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-low-autocorrelation-l6-s0/tests/check.py b/tasks/construct-mc-low-autocorrelation-l6-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..8456f7b1c0ff0c9fb4bc9ae5a49074c8afa7d1aa
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-l6-s0/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-l6-s0/tests/instance.json b/tasks/construct-mc-low-autocorrelation-l6-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..ab11e00d0c46cda7b1b8170981cf0dbd0f89b185
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-l6-s0/tests/instance.json
@@ -0,0 +1 @@
+{"n": 44, "bound": 134, "fixed": [], "family": "mc_low_autocorrelation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-low-autocorrelation-l6-s0/tests/run.py b/tasks/construct-mc-low-autocorrelation-l6-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-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-mc-low-autocorrelation-l6-s0/tests/test.sh b/tasks/construct-mc-low-autocorrelation-l6-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-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-magic-sequence-l5-s0/environment/Dockerfile b/tasks/construct-mc-magic-sequence-l5-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-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-magic-sequence-l5-s0/environment/instance.json b/tasks/construct-mc-magic-sequence-l5-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8c3599a16972234c93b7c9a530535dd8a31c13ae
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-l5-s0/environment/instance.json
@@ -0,0 +1 @@
+{"n": 128, "fixed": [], "family": "mc_magic_sequence", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-magic-sequence-l5-s0/solution/solve.sh b/tasks/construct-mc-magic-sequence-l5-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..3dba021f00fd66ddc02422a6a153587a943a6fe8
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-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'
+{"x": [124, 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, 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, 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, 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, 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-l5-s0/tests/check.py b/tasks/construct-mc-magic-sequence-l5-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..27f636ddd6cfe523924ae01c6b9f37ee392f0332
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-l5-s0/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-l5-s0/tests/instance.json b/tasks/construct-mc-magic-sequence-l5-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8c3599a16972234c93b7c9a530535dd8a31c13ae
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-l5-s0/tests/instance.json
@@ -0,0 +1 @@
+{"n": 128, "fixed": [], "family": "mc_magic_sequence", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-magic-sequence-l5-s0/tests/run.py b/tasks/construct-mc-magic-sequence-l5-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-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-magic-sequence-l5-s0/tests/test.sh b/tasks/construct-mc-magic-sequence-l5-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-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-max-clique-l2-s1/environment/Dockerfile b/tasks/construct-mc-max-clique-l2-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-max-clique-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-max-clique-l2-s1/environment/instance.json b/tasks/construct-mc-max-clique-l2-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e35aa9564eb88c30e1f2558907eb3405cefc9c59
--- /dev/null
+++ b/tasks/construct-mc-max-clique-l2-s1/environment/instance.json
@@ -0,0 +1 @@
+{"n": 40, "k": 8, "edges": [[0, 1], [0, 2], [0, 4], [0, 5], [0, 7], [0, 8], [0, 10], [0, 11], [0, 17], [0, 18], [0, 20], [0, 21], [0, 23], [0, 25], [0, 26], [0, 30], [0, 31], [0, 33], [0, 34], [0, 35], [0, 36], [0, 38], [0, 39], [1, 2], [1, 4], [1, 6], [1, 7], [1, 10], [1, 12], [1, 13], [1, 14], [1, 15], [1, 16], [1, 18], [1, 20], [1, 21], [1, 27], [1, 31], [1, 35], [1, 38], [1, 39], [2, 3], [2, 10], [2, 11], [2, 15], [2, 18], [2, 20], [2, 21], [2, 23], [2, 26], [2, 28], [2, 29], [2, 33], [2, 36], [2, 37], [2, 38], [2, 39], [3, 7], [3, 8], [3, 10], [3, 11], [3, 12], [3, 14], [3, 19], [3, 21], [3, 23], [3, 25], [3, 27], [3, 30], [3, 31], [3, 37], [3, 38], [4, 6], [4, 13], [4, 14], [4, 15], [4, 18], [4, 19], [4, 20], [4, 21], [4, 23], [4, 27], [4, 28], [4, 30], [4, 31], [4, 34], [4, 35], [4, 37], [5, 6], [5, 9], [5, 11], [5, 12], [5, 13], [5, 14], [5, 16], [5, 20], [5, 21], [5, 22], [5, 23], [5, 24], [5, 26], [5, 31], [5, 34], [5, 35], [5, 37], [5, 39], [6, 7], [6, 9], [6, 11], [6, 12], [6, 13], [6, 15], [6, 17], [6, 20], [6, 21], [6, 22], [6, 23], [6, 25], [6, 27], [6, 28], [6, 29], [6, 32], [6, 35], [6, 37], [6, 39], [7, 9], [7, 10], [7, 12], [7, 13], [7, 16], [7, 17], [7, 18], [7, 21], [7, 25], [7, 26], [7, 30], [7, 31], [7, 37], [7, 38], [7, 39], [8, 9], [8, 10], [8, 11], [8, 12], [8, 15], [8, 17], [8, 19], [8, 20], [8, 21], [8, 22], [8, 23], [8, 29], [8, 30], [8, 31], [8, 33], [8, 36], [8, 37], [8, 38], [8, 39], [9, 10], [9, 11], [9, 12], [9, 16], [9, 21], [9, 23], [9, 24], [9, 25], [9, 29], [9, 31], [9, 33], [9, 34], [9, 36], [9, 39], [10, 12], [10, 13], [10, 15], [10, 16], [10, 19], [10, 20], [10, 21], [10, 22], [10, 24], [10, 25], [10, 27], [10, 29], [10, 31], [10, 32], [10, 33], [10, 34], [10, 35], [10, 36], [10, 37], [11, 13], [11, 14], [11, 15], [11, 16], [11, 17], [11, 19], [11, 20], [11, 22], [11, 23], [11, 25], [11, 29], [11, 30], [11, 34], [11, 37], [11, 39], [12, 13], [12, 15], [12, 16], [12, 17], [12, 18], [12, 21], [12, 22], [12, 23], [12, 24], [12, 25], [12, 26], [12, 28], [12, 29], [12, 31], [12, 32], [12, 33], [12, 36], [12, 37], [13, 15], [13, 18], [13, 19], [13, 20], [13, 22], [13, 26], [13, 27], [13, 29], [13, 30], [13, 35], [13, 38], [13, 39], [14, 16], [14, 17], [14, 18], [14, 22], [14, 25], [14, 26], [14, 27], [14, 28], [14, 29], [14, 31], [15, 17], [15, 21], [15, 22], [15, 24], [15, 26], [15, 27], [15, 29], [15, 30], [15, 34], [15, 35], [15, 36], [16, 17], [16, 19], [16, 20], [16, 21], [16, 24], [16, 25], [16, 26], [16, 27], [16, 30], [16, 31], [16, 33], [16, 37], [16, 38], [16, 39], [17, 20], [17, 21], [17, 23], [17, 24], [17, 32], [17, 33], [17, 36], [17, 37], [17, 38], [18, 19], [18, 20], [18, 21], [18, 22], [18, 26], [18, 30], [18, 33], [18, 35], [18, 38], [18, 39], [19, 22], [19, 24], [19, 25], [19, 27], [19, 31], [19, 33], [19, 34], [19, 37], [19, 39], [20, 23], [20, 26], [20, 28], [20, 30], [20, 32], [20, 35], [20, 36], [20, 37], [20, 39], [21, 25], [21, 26], [21, 27], [21, 28], [21, 29], [21, 31], [21, 33], [21, 34], [21, 35], [21, 36], [21, 38], [21, 39], [22, 23], [22, 28], [22, 29], [22, 31], [22, 33], [22, 35], [22, 37], [22, 38], [23, 25], [23, 28], [23, 29], [23, 30], [23, 31], [23, 34], [23, 35], [23, 36], [23, 39], [24, 25], [24, 26], [24, 32], [24, 33], [24, 34], [24, 39], [25, 27], [25, 29], [25, 30], [25, 34], [25, 39], [26, 27], [26, 30], [26, 31], [26, 36], [26, 38], [26, 39], [27, 32], [27, 33], [27, 38], [27, 39], [28, 29], [28, 33], [28, 35], [28, 36], [28, 39], [29, 30], [29, 32], [29, 33], [29, 35], [29, 36], [29, 38], [29, 39], [30, 31], [30, 35], [30, 37], [30, 38], [31, 32], [31, 33], [31, 36], [31, 37], [32, 34], [32, 35], [32, 37], [32, 38], [32, 39], [33, 35], [33, 36], [33, 37], [35, 36], [35, 38], [35, 39], [36, 39], [37, 38]], "family": "mc_max_clique", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-max-clique-l2-s1/solution/solve.sh b/tasks/construct-mc-max-clique-l2-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..d67f4e131a1744e6d2127911dcae2746a4ee9932
--- /dev/null
+++ b/tasks/construct-mc-max-clique-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'
+{"clique": [8, 9, 10, 12, 21, 29, 33, 36]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-max-clique-l2-s1/tests/check.py b/tasks/construct-mc-max-clique-l2-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..e3f70dbd5b155a3514c2102aa4ad460201495210
--- /dev/null
+++ b/tasks/construct-mc-max-clique-l2-s1/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-l2-s1/tests/instance.json b/tasks/construct-mc-max-clique-l2-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e35aa9564eb88c30e1f2558907eb3405cefc9c59
--- /dev/null
+++ b/tasks/construct-mc-max-clique-l2-s1/tests/instance.json
@@ -0,0 +1 @@
+{"n": 40, "k": 8, "edges": [[0, 1], [0, 2], [0, 4], [0, 5], [0, 7], [0, 8], [0, 10], [0, 11], [0, 17], [0, 18], [0, 20], [0, 21], [0, 23], [0, 25], [0, 26], [0, 30], [0, 31], [0, 33], [0, 34], [0, 35], [0, 36], [0, 38], [0, 39], [1, 2], [1, 4], [1, 6], [1, 7], [1, 10], [1, 12], [1, 13], [1, 14], [1, 15], [1, 16], [1, 18], [1, 20], [1, 21], [1, 27], [1, 31], [1, 35], [1, 38], [1, 39], [2, 3], [2, 10], [2, 11], [2, 15], [2, 18], [2, 20], [2, 21], [2, 23], [2, 26], [2, 28], [2, 29], [2, 33], [2, 36], [2, 37], [2, 38], [2, 39], [3, 7], [3, 8], [3, 10], [3, 11], [3, 12], [3, 14], [3, 19], [3, 21], [3, 23], [3, 25], [3, 27], [3, 30], [3, 31], [3, 37], [3, 38], [4, 6], [4, 13], [4, 14], [4, 15], [4, 18], [4, 19], [4, 20], [4, 21], [4, 23], [4, 27], [4, 28], [4, 30], [4, 31], [4, 34], [4, 35], [4, 37], [5, 6], [5, 9], [5, 11], [5, 12], [5, 13], [5, 14], [5, 16], [5, 20], [5, 21], [5, 22], [5, 23], [5, 24], [5, 26], [5, 31], [5, 34], [5, 35], [5, 37], [5, 39], [6, 7], [6, 9], [6, 11], [6, 12], [6, 13], [6, 15], [6, 17], [6, 20], [6, 21], [6, 22], [6, 23], [6, 25], [6, 27], [6, 28], [6, 29], [6, 32], [6, 35], [6, 37], [6, 39], [7, 9], [7, 10], [7, 12], [7, 13], [7, 16], [7, 17], [7, 18], [7, 21], [7, 25], [7, 26], [7, 30], [7, 31], [7, 37], [7, 38], [7, 39], [8, 9], [8, 10], [8, 11], [8, 12], [8, 15], [8, 17], [8, 19], [8, 20], [8, 21], [8, 22], [8, 23], [8, 29], [8, 30], [8, 31], [8, 33], [8, 36], [8, 37], [8, 38], [8, 39], [9, 10], [9, 11], [9, 12], [9, 16], [9, 21], [9, 23], [9, 24], [9, 25], [9, 29], [9, 31], [9, 33], [9, 34], [9, 36], [9, 39], [10, 12], [10, 13], [10, 15], [10, 16], [10, 19], [10, 20], [10, 21], [10, 22], [10, 24], [10, 25], [10, 27], [10, 29], [10, 31], [10, 32], [10, 33], [10, 34], [10, 35], [10, 36], [10, 37], [11, 13], [11, 14], [11, 15], [11, 16], [11, 17], [11, 19], [11, 20], [11, 22], [11, 23], [11, 25], [11, 29], [11, 30], [11, 34], [11, 37], [11, 39], [12, 13], [12, 15], [12, 16], [12, 17], [12, 18], [12, 21], [12, 22], [12, 23], [12, 24], [12, 25], [12, 26], [12, 28], [12, 29], [12, 31], [12, 32], [12, 33], [12, 36], [12, 37], [13, 15], [13, 18], [13, 19], [13, 20], [13, 22], [13, 26], [13, 27], [13, 29], [13, 30], [13, 35], [13, 38], [13, 39], [14, 16], [14, 17], [14, 18], [14, 22], [14, 25], [14, 26], [14, 27], [14, 28], [14, 29], [14, 31], [15, 17], [15, 21], [15, 22], [15, 24], [15, 26], [15, 27], [15, 29], [15, 30], [15, 34], [15, 35], [15, 36], [16, 17], [16, 19], [16, 20], [16, 21], [16, 24], [16, 25], [16, 26], [16, 27], [16, 30], [16, 31], [16, 33], [16, 37], [16, 38], [16, 39], [17, 20], [17, 21], [17, 23], [17, 24], [17, 32], [17, 33], [17, 36], [17, 37], [17, 38], [18, 19], [18, 20], [18, 21], [18, 22], [18, 26], [18, 30], [18, 33], [18, 35], [18, 38], [18, 39], [19, 22], [19, 24], [19, 25], [19, 27], [19, 31], [19, 33], [19, 34], [19, 37], [19, 39], [20, 23], [20, 26], [20, 28], [20, 30], [20, 32], [20, 35], [20, 36], [20, 37], [20, 39], [21, 25], [21, 26], [21, 27], [21, 28], [21, 29], [21, 31], [21, 33], [21, 34], [21, 35], [21, 36], [21, 38], [21, 39], [22, 23], [22, 28], [22, 29], [22, 31], [22, 33], [22, 35], [22, 37], [22, 38], [23, 25], [23, 28], [23, 29], [23, 30], [23, 31], [23, 34], [23, 35], [23, 36], [23, 39], [24, 25], [24, 26], [24, 32], [24, 33], [24, 34], [24, 39], [25, 27], [25, 29], [25, 30], [25, 34], [25, 39], [26, 27], [26, 30], [26, 31], [26, 36], [26, 38], [26, 39], [27, 32], [27, 33], [27, 38], [27, 39], [28, 29], [28, 33], [28, 35], [28, 36], [28, 39], [29, 30], [29, 32], [29, 33], [29, 35], [29, 36], [29, 38], [29, 39], [30, 31], [30, 35], [30, 37], [30, 38], [31, 32], [31, 33], [31, 36], [31, 37], [32, 34], [32, 35], [32, 37], [32, 38], [32, 39], [33, 35], [33, 36], [33, 37], [35, 36], [35, 38], [35, 39], [36, 39], [37, 38]], "family": "mc_max_clique", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-max-clique-l2-s1/tests/run.py b/tasks/construct-mc-max-clique-l2-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-max-clique-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-max-clique-l2-s1/tests/test.sh b/tasks/construct-mc-max-clique-l2-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-max-clique-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-max-clique-l2-s6/environment/Dockerfile b/tasks/construct-mc-max-clique-l2-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-max-clique-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-mc-max-clique-l2-s6/environment/instance.json b/tasks/construct-mc-max-clique-l2-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4a26298ecea9cd4f23ed0d47834f106413592710
--- /dev/null
+++ b/tasks/construct-mc-max-clique-l2-s6/environment/instance.json
@@ -0,0 +1 @@
+{"n": 40, "k": 8, "edges": [[0, 1], [0, 7], [0, 8], [0, 9], [0, 10], [0, 11], [0, 12], [0, 13], [0, 14], [0, 16], [0, 18], [0, 19], [0, 23], [0, 27], [0, 28], [0, 29], [0, 30], [0, 34], [0, 37], [1, 2], [1, 4], [1, 5], [1, 10], [1, 12], [1, 13], [1, 14], [1, 15], [1, 17], [1, 18], [1, 19], [1, 24], [1, 27], [1, 30], [1, 31], [1, 33], [1, 34], [1, 36], [2, 7], [2, 8], [2, 11], [2, 12], [2, 16], [2, 17], [2, 23], [2, 24], [2, 25], [2, 26], [2, 29], [2, 30], [2, 31], [2, 32], [2, 34], [2, 38], [2, 39], [3, 4], [3, 5], [3, 7], [3, 9], [3, 10], [3, 11], [3, 12], [3, 13], [3, 15], [3, 18], [3, 19], [3, 20], [3, 25], [3, 28], [3, 31], [3, 35], [3, 36], [3, 37], [3, 38], [4, 7], [4, 9], [4, 10], [4, 11], [4, 12], [4, 17], [4, 18], [4, 19], [4, 20], [4, 21], [4, 22], [4, 23], [4, 25], [4, 27], [4, 31], [4, 32], [4, 33], [4, 36], [4, 37], [4, 38], [5, 7], [5, 10], [5, 11], [5, 12], [5, 13], [5, 16], [5, 17], [5, 20], [5, 21], [5, 23], [5, 26], [5, 27], [5, 28], [5, 29], [5, 31], [5, 32], [5, 33], [5, 34], [5, 35], [6, 8], [6, 9], [6, 10], [6, 11], [6, 12], [6, 13], [6, 14], [6, 18], [6, 20], [6, 22], [6, 23], [6, 24], [6, 25], [6, 26], [6, 30], [6, 31], [6, 33], [6, 37], [6, 38], [6, 39], [7, 8], [7, 10], [7, 11], [7, 14], [7, 15], [7, 16], [7, 17], [7, 21], [7, 23], [7, 24], [7, 25], [7, 27], [7, 29], [7, 31], [7, 34], [7, 35], [7, 36], [7, 38], [8, 9], [8, 10], [8, 11], [8, 12], [8, 13], [8, 14], [8, 15], [8, 17], [8, 20], [8, 22], [8, 23], [8, 25], [8, 27], [8, 30], [8, 31], [8, 33], [8, 37], [8, 38], [8, 39], [9, 10], [9, 12], [9, 19], [9, 21], [9, 24], [9, 31], [9, 33], [9, 34], [9, 36], [9, 37], [9, 39], [10, 12], [10, 13], [10, 14], [10, 17], [10, 20], [10, 21], [10, 25], [10, 29], [10, 30], [10, 31], [10, 33], [10, 35], [10, 36], [10, 37], [11, 12], [11, 14], [11, 18], [11, 20], [11, 21], [11, 23], [11, 24], [11, 28], [11, 31], [11, 33], [11, 35], [12, 13], [12, 14], [12, 20], [12, 23], [12, 26], [12, 27], [12, 30], [12, 34], [12, 36], [12, 38], [12, 39], [13, 14], [13, 15], [13, 17], [13, 18], [13, 19], [13, 23], [13, 24], [13, 29], [13, 30], [13, 34], [13, 37], [13, 38], [14, 16], [14, 18], [14, 19], [14, 20], [14, 23], [14, 25], [14, 26], [14, 28], [14, 30], [14, 32], [14, 37], [15, 16], [15, 17], [15, 20], [15, 22], [15, 23], [15, 24], [15, 27], [15, 30], [15, 32], [15, 33], [15, 34], [15, 36], [15, 37], [15, 39], [16, 18], [16, 19], [16, 20], [16, 21], [16, 23], [16, 24], [16, 30], [16, 33], [17, 19], [17, 21], [17, 22], [17, 23], [17, 24], [17, 26], [17, 27], [17, 28], [17, 29], [17, 31], [17, 33], [17, 34], [17, 37], [17, 38], [18, 19], [18, 23], [18, 25], [18, 27], [18, 28], [18, 34], [18, 36], [18, 37], [18, 38], [19, 22], [19, 23], [19, 25], [19, 26], [19, 29], [19, 31], [19, 32], [19, 33], [19, 37], [19, 38], [19, 39], [20, 22], [20, 23], [20, 25], [20, 27], [20, 29], [20, 31], [20, 35], [20, 36], [20, 38], [20, 39], [21, 22], [21, 25], [21, 27], [21, 28], [21, 31], [21, 33], [21, 34], [21, 36], [22, 23], [22, 24], [22, 26], [22, 27], [22, 28], [22, 30], [22, 31], [22, 33], [22, 34], [22, 38], [22, 39], [23, 25], [23, 27], [23, 29], [23, 30], [23, 31], [23, 32], [23, 33], [23, 34], [23, 35], [23, 36], [23, 37], [23, 38], [24, 27], [24, 28], [24, 30], [24, 34], [24, 35], [24, 37], [24, 38], [24, 39], [25, 28], [25, 29], [25, 30], [25, 31], [25, 34], [25, 35], [25, 38], [26, 27], [26, 29], [26, 31], [26, 37], [27, 29], [27, 30], [27, 31], [27, 32], [27, 37], [27, 38], [28, 30], [28, 31], [28, 32], [28, 33], [28, 36], [28, 37], [28, 38], [28, 39], [29, 32], [29, 34], [29, 36], [29, 37], [29, 38], [29, 39], [30, 31], [30, 33], [30, 34], [30, 38], [31, 33], [31, 36], [31, 38], [31, 39], [32, 33], [32, 35], [32, 36], [32, 39], [33, 34], [33, 35], [33, 37], [33, 38], [34, 36], [34, 37], [34, 39], [35, 36], [35, 38], [36, 39], [38, 39]], "family": "mc_max_clique", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-max-clique-l2-s6/solution/solve.sh b/tasks/construct-mc-max-clique-l2-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..5939c33cb0ad5286185c44e9ab78bdaea3403c49
--- /dev/null
+++ b/tasks/construct-mc-max-clique-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'
+{"clique": [6, 8, 22, 23, 30, 31, 33, 38]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-max-clique-l2-s6/tests/check.py b/tasks/construct-mc-max-clique-l2-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..e3f70dbd5b155a3514c2102aa4ad460201495210
--- /dev/null
+++ b/tasks/construct-mc-max-clique-l2-s6/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-l2-s6/tests/instance.json b/tasks/construct-mc-max-clique-l2-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4a26298ecea9cd4f23ed0d47834f106413592710
--- /dev/null
+++ b/tasks/construct-mc-max-clique-l2-s6/tests/instance.json
@@ -0,0 +1 @@
+{"n": 40, "k": 8, "edges": [[0, 1], [0, 7], [0, 8], [0, 9], [0, 10], [0, 11], [0, 12], [0, 13], [0, 14], [0, 16], [0, 18], [0, 19], [0, 23], [0, 27], [0, 28], [0, 29], [0, 30], [0, 34], [0, 37], [1, 2], [1, 4], [1, 5], [1, 10], [1, 12], [1, 13], [1, 14], [1, 15], [1, 17], [1, 18], [1, 19], [1, 24], [1, 27], [1, 30], [1, 31], [1, 33], [1, 34], [1, 36], [2, 7], [2, 8], [2, 11], [2, 12], [2, 16], [2, 17], [2, 23], [2, 24], [2, 25], [2, 26], [2, 29], [2, 30], [2, 31], [2, 32], [2, 34], [2, 38], [2, 39], [3, 4], [3, 5], [3, 7], [3, 9], [3, 10], [3, 11], [3, 12], [3, 13], [3, 15], [3, 18], [3, 19], [3, 20], [3, 25], [3, 28], [3, 31], [3, 35], [3, 36], [3, 37], [3, 38], [4, 7], [4, 9], [4, 10], [4, 11], [4, 12], [4, 17], [4, 18], [4, 19], [4, 20], [4, 21], [4, 22], [4, 23], [4, 25], [4, 27], [4, 31], [4, 32], [4, 33], [4, 36], [4, 37], [4, 38], [5, 7], [5, 10], [5, 11], [5, 12], [5, 13], [5, 16], [5, 17], [5, 20], [5, 21], [5, 23], [5, 26], [5, 27], [5, 28], [5, 29], [5, 31], [5, 32], [5, 33], [5, 34], [5, 35], [6, 8], [6, 9], [6, 10], [6, 11], [6, 12], [6, 13], [6, 14], [6, 18], [6, 20], [6, 22], [6, 23], [6, 24], [6, 25], [6, 26], [6, 30], [6, 31], [6, 33], [6, 37], [6, 38], [6, 39], [7, 8], [7, 10], [7, 11], [7, 14], [7, 15], [7, 16], [7, 17], [7, 21], [7, 23], [7, 24], [7, 25], [7, 27], [7, 29], [7, 31], [7, 34], [7, 35], [7, 36], [7, 38], [8, 9], [8, 10], [8, 11], [8, 12], [8, 13], [8, 14], [8, 15], [8, 17], [8, 20], [8, 22], [8, 23], [8, 25], [8, 27], [8, 30], [8, 31], [8, 33], [8, 37], [8, 38], [8, 39], [9, 10], [9, 12], [9, 19], [9, 21], [9, 24], [9, 31], [9, 33], [9, 34], [9, 36], [9, 37], [9, 39], [10, 12], [10, 13], [10, 14], [10, 17], [10, 20], [10, 21], [10, 25], [10, 29], [10, 30], [10, 31], [10, 33], [10, 35], [10, 36], [10, 37], [11, 12], [11, 14], [11, 18], [11, 20], [11, 21], [11, 23], [11, 24], [11, 28], [11, 31], [11, 33], [11, 35], [12, 13], [12, 14], [12, 20], [12, 23], [12, 26], [12, 27], [12, 30], [12, 34], [12, 36], [12, 38], [12, 39], [13, 14], [13, 15], [13, 17], [13, 18], [13, 19], [13, 23], [13, 24], [13, 29], [13, 30], [13, 34], [13, 37], [13, 38], [14, 16], [14, 18], [14, 19], [14, 20], [14, 23], [14, 25], [14, 26], [14, 28], [14, 30], [14, 32], [14, 37], [15, 16], [15, 17], [15, 20], [15, 22], [15, 23], [15, 24], [15, 27], [15, 30], [15, 32], [15, 33], [15, 34], [15, 36], [15, 37], [15, 39], [16, 18], [16, 19], [16, 20], [16, 21], [16, 23], [16, 24], [16, 30], [16, 33], [17, 19], [17, 21], [17, 22], [17, 23], [17, 24], [17, 26], [17, 27], [17, 28], [17, 29], [17, 31], [17, 33], [17, 34], [17, 37], [17, 38], [18, 19], [18, 23], [18, 25], [18, 27], [18, 28], [18, 34], [18, 36], [18, 37], [18, 38], [19, 22], [19, 23], [19, 25], [19, 26], [19, 29], [19, 31], [19, 32], [19, 33], [19, 37], [19, 38], [19, 39], [20, 22], [20, 23], [20, 25], [20, 27], [20, 29], [20, 31], [20, 35], [20, 36], [20, 38], [20, 39], [21, 22], [21, 25], [21, 27], [21, 28], [21, 31], [21, 33], [21, 34], [21, 36], [22, 23], [22, 24], [22, 26], [22, 27], [22, 28], [22, 30], [22, 31], [22, 33], [22, 34], [22, 38], [22, 39], [23, 25], [23, 27], [23, 29], [23, 30], [23, 31], [23, 32], [23, 33], [23, 34], [23, 35], [23, 36], [23, 37], [23, 38], [24, 27], [24, 28], [24, 30], [24, 34], [24, 35], [24, 37], [24, 38], [24, 39], [25, 28], [25, 29], [25, 30], [25, 31], [25, 34], [25, 35], [25, 38], [26, 27], [26, 29], [26, 31], [26, 37], [27, 29], [27, 30], [27, 31], [27, 32], [27, 37], [27, 38], [28, 30], [28, 31], [28, 32], [28, 33], [28, 36], [28, 37], [28, 38], [28, 39], [29, 32], [29, 34], [29, 36], [29, 37], [29, 38], [29, 39], [30, 31], [30, 33], [30, 34], [30, 38], [31, 33], [31, 36], [31, 38], [31, 39], [32, 33], [32, 35], [32, 36], [32, 39], [33, 34], [33, 35], [33, 37], [33, 38], [34, 36], [34, 37], [34, 39], [35, 36], [35, 38], [36, 39], [38, 39]], "family": "mc_max_clique", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-max-clique-l2-s6/tests/run.py b/tasks/construct-mc-max-clique-l2-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-max-clique-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-mc-max-clique-l2-s6/tests/test.sh b/tasks/construct-mc-max-clique-l2-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-max-clique-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-mc-max-clique-l6-s9/environment/Dockerfile b/tasks/construct-mc-max-clique-l6-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-max-clique-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-mc-max-clique-l6-s9/environment/instance.json b/tasks/construct-mc-max-clique-l6-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..404bc8c9488d8d2ba7f859c709dff328687e5659
--- /dev/null
+++ b/tasks/construct-mc-max-clique-l6-s9/environment/instance.json
@@ -0,0 +1 @@
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118], [117, 119]], "family": "mc_max_clique", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-max-clique-l6-s9/solution/solve.sh b/tasks/construct-mc-max-clique-l6-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..14881f57f8f805cf530215b1663657bcf31f4967
--- /dev/null
+++ b/tasks/construct-mc-max-clique-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'
+{"clique": [2, 17, 23, 40, 49, 53, 61, 65, 83, 88, 101, 111, 115]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-max-clique-l6-s9/tests/check.py b/tasks/construct-mc-max-clique-l6-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..e3f70dbd5b155a3514c2102aa4ad460201495210
--- /dev/null
+++ b/tasks/construct-mc-max-clique-l6-s9/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-l6-s9/tests/instance.json b/tasks/construct-mc-max-clique-l6-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..404bc8c9488d8d2ba7f859c709dff328687e5659
--- /dev/null
+++ b/tasks/construct-mc-max-clique-l6-s9/tests/instance.json
@@ -0,0 +1 @@
+{"n": 120, "k": 13, "edges": [[0, 2], [0, 4], [0, 6], [0, 7], [0, 8], [0, 9], [0, 10], [0, 13], [0, 14], [0, 16], [0, 17], [0, 20], [0, 21], [0, 24], [0, 27], [0, 29], [0, 30], [0, 35], [0, 38], [0, 39], [0, 40], [0, 42], [0, 43], [0, 44], [0, 45], [0, 46], [0, 47], [0, 48], [0, 49], [0, 50], [0, 51], [0, 54], [0, 56], [0, 57], [0, 58], [0, 59], [0, 60], [0, 61], [0, 62], [0, 63], [0, 66], [0, 67], [0, 68], [0, 69], [0, 71], [0, 72], [0, 74], [0, 78], [0, 81], [0, 82], [0, 84], [0, 94], [0, 95], [0, 96], [0, 97], [0, 98], [0, 99], [0, 102], [0, 104], [0, 105], [0, 106], [0, 110], [0, 111], [0, 112], [0, 114], [0, 115], [0, 118], [1, 2], [1, 4], [1, 7], [1, 9], [1, 10], [1, 13], [1, 16], [1, 23], [1, 24], [1, 27], [1, 28], [1, 29], [1, 32], [1, 33], [1, 35], [1, 38], [1, 39], [1, 40], [1, 41], [1, 44], [1, 49], [1, 51], [1, 52], [1, 53], [1, 54], [1, 56], [1, 59], [1, 60], [1, 62], [1, 64], [1, 65], [1, 68], [1, 71], [1, 72], [1, 74], [1, 75], [1, 77], [1, 79], [1, 83], [1, 85], [1, 86], [1, 87], [1, 93], [1, 95], [1, 97], [1, 99], [1, 100], [1, 102], [1, 107], [1, 109], [1, 111], [1, 113], [1, 118], [1, 119], [2, 3], [2, 5], [2, 10], [2, 12], [2, 13], [2, 14], [2, 15], [2, 16], [2, 17], [2, 19], [2, 20], [2, 22], [2, 23], [2, 25], [2, 29], [2, 30], [2, 31], [2, 32], [2, 33], [2, 34], [2, 36], [2, 37], [2, 38], [2, 39], [2, 40], [2, 43], [2, 44], [2, 45], [2, 46], [2, 47], [2, 48], [2, 49], [2, 52], [2, 53], [2, 55], [2, 56], [2, 57], [2, 60], [2, 61], [2, 62], [2, 63], [2, 65], [2, 67], [2, 72], [2, 74], [2, 78], [2, 79], [2, 82], [2, 83], [2, 85], [2, 86], [2, 87], [2, 88], [2, 89], [2, 91], [2, 92], [2, 95], [2, 99], [2, 101], [2, 103], [2, 104], [2, 105], [2, 109], [2, 111], [2, 112], [2, 113], [2, 114], [2, 115], [2, 117], [3, 5], [3, 6], [3, 8], [3, 9], [3, 10], [3, 11], [3, 12], [3, 15], [3, 16], [3, 17], [3, 21], [3, 22], [3, 27], [3, 28], [3, 29], [3, 31], [3, 32], [3, 33], [3, 34], [3, 39], [3, 40], [3, 46], [3, 47], [3, 48], [3, 49], [3, 51], [3, 56], 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119], [102, 103], [102, 104], [102, 109], [102, 110], [102, 111], [102, 113], [102, 116], [103, 104], [103, 107], [103, 108], [103, 111], [103, 115], [103, 116], [103, 117], [103, 119], [104, 106], [104, 113], [104, 114], [104, 115], [104, 117], [104, 118], [104, 119], [105, 107], [105, 108], [105, 110], [105, 114], [105, 116], [105, 117], [105, 118], [105, 119], [106, 107], [106, 108], [106, 111], [106, 112], [106, 114], [106, 115], [106, 117], [106, 118], [106, 119], [107, 108], [107, 110], [107, 111], [107, 112], [107, 113], [107, 116], [107, 117], [107, 118], [107, 119], [108, 110], [108, 111], [108, 112], [108, 113], [109, 111], [109, 116], [109, 117], [109, 118], [109, 119], [110, 112], [110, 113], [110, 114], [110, 115], [110, 116], [110, 117], [111, 112], [111, 113], [111, 115], [111, 119], [112, 114], [112, 115], [112, 116], [112, 117], [113, 115], [113, 117], [113, 118], [114, 115], [114, 116], [114, 118], [115, 116], [115, 117], [115, 118], [116, 118], [116, 119], [117, 118], [117, 119]], "family": "mc_max_clique", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-max-clique-l6-s9/tests/run.py b/tasks/construct-mc-max-clique-l6-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-max-clique-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-mc-max-clique-l6-s9/tests/test.sh b/tasks/construct-mc-max-clique-l6-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-max-clique-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-mc-max-independent-set-l5-s2/environment/Dockerfile b/tasks/construct-mc-max-independent-set-l5-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-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-max-independent-set-l5-s2/environment/instance.json b/tasks/construct-mc-max-independent-set-l5-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..72a32a445165bacde52b0f0fa23f6cb80461e275
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-l5-s2/environment/instance.json
@@ -0,0 +1 @@
+{"n": 90, "k": 16, "edges": [[0, 3], [0, 5], [0, 6], [0, 8], [0, 9], [0, 10], [0, 12], [0, 13], [0, 15], [0, 26], [0, 27], [0, 41], [0, 47], [0, 49], [0, 52], [0, 53], [0, 54], [0, 57], [0, 58], [0, 61], [0, 64], [0, 66], [0, 73], [0, 79], [0, 80], [0, 82], [1, 7], [1, 9], [1, 15], [1, 16], [1, 17], [1, 27], [1, 31], [1, 32], [1, 33], [1, 37], [1, 42], [1, 43], [1, 45], [1, 49], [1, 50], [1, 52], [1, 54], [1, 61], [1, 62], [1, 63], [1, 68], [1, 69], [1, 75], [1, 81], [1, 84], [1, 89], [2, 3], [2, 7], [2, 11], [2, 16], [2, 17], [2, 23], [2, 27], [2, 28], [2, 30], [2, 32], [2, 35], [2, 42], [2, 44], [2, 45], [2, 49], [2, 51], [2, 52], [2, 58], [2, 59], [2, 60], [2, 62], [2, 63], [2, 70], [2, 74], [2, 77], [2, 78], [2, 84], [2, 87], [3, 4], [3, 7], [3, 8], [3, 13], [3, 15], [3, 21], [3, 32], [3, 35], [3, 36], [3, 40], [3, 41], [3, 43], [3, 46], [3, 48], [3, 49], [3, 50], [3, 51], [3, 53], [3, 55], [3, 58], [3, 59], [3, 60], [3, 61], [3, 68], [3, 69], [3, 89], [4, 9], [4, 10], [4, 11], 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72], [69, 73], [69, 76], [69, 79], [69, 81], [69, 83], [69, 87], [70, 73], [70, 75], [70, 76], [70, 81], [70, 87], [70, 89], [71, 75], [71, 82], [71, 87], [71, 88], [72, 74], [72, 77], [72, 79], [72, 81], [72, 84], [72, 86], [72, 88], [73, 74], [73, 77], [73, 79], [73, 88], [74, 83], [74, 85], [74, 89], [75, 76], [75, 77], [75, 82], [75, 86], [75, 87], [76, 79], [76, 82], [76, 89], [77, 82], [77, 88], [77, 89], [78, 81], [78, 82], [78, 84], [78, 88], [79, 82], [79, 85], [81, 82], [81, 84], [83, 86], [83, 88], [84, 86], [85, 86], [86, 88], [86, 89], [87, 89]], "family": "mc_max_independent_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-max-independent-set-l5-s2/solution/solve.sh b/tasks/construct-mc-max-independent-set-l5-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..79cf20984ca0bd1cf39e61b623f4f6f5cebeeadf
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-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'
+{"independent_set": [1, 3, 11, 12, 22, 25, 26, 34, 39, 44, 65, 76, 80, 83, 85, 87]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-max-independent-set-l5-s2/tests/check.py b/tasks/construct-mc-max-independent-set-l5-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..adce413e8578d996549583b990e4ff559f38ed41
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-l5-s2/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-l5-s2/tests/instance.json b/tasks/construct-mc-max-independent-set-l5-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..72a32a445165bacde52b0f0fa23f6cb80461e275
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-l5-s2/tests/instance.json
@@ -0,0 +1 @@
+{"n": 90, "k": 16, "edges": [[0, 3], [0, 5], [0, 6], [0, 8], [0, 9], [0, 10], [0, 12], [0, 13], [0, 15], [0, 26], [0, 27], [0, 41], [0, 47], [0, 49], [0, 52], [0, 53], [0, 54], [0, 57], [0, 58], [0, 61], [0, 64], [0, 66], [0, 73], [0, 79], [0, 80], [0, 82], [1, 7], [1, 9], [1, 15], [1, 16], [1, 17], [1, 27], [1, 31], [1, 32], [1, 33], [1, 37], [1, 42], [1, 43], [1, 45], [1, 49], [1, 50], [1, 52], [1, 54], [1, 61], [1, 62], [1, 63], [1, 68], [1, 69], [1, 75], [1, 81], [1, 84], [1, 89], [2, 3], [2, 7], [2, 11], [2, 16], [2, 17], [2, 23], [2, 27], [2, 28], [2, 30], [2, 32], [2, 35], [2, 42], [2, 44], [2, 45], [2, 49], [2, 51], [2, 52], [2, 58], [2, 59], [2, 60], [2, 62], [2, 63], [2, 70], [2, 74], [2, 77], [2, 78], [2, 84], [2, 87], [3, 4], [3, 7], [3, 8], [3, 13], [3, 15], [3, 21], [3, 32], [3, 35], [3, 36], [3, 40], [3, 41], [3, 43], [3, 46], [3, 48], [3, 49], [3, 50], [3, 51], [3, 53], [3, 55], [3, 58], [3, 59], [3, 60], [3, 61], [3, 68], [3, 69], [3, 89], [4, 9], [4, 10], [4, 11], 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72], [69, 73], [69, 76], [69, 79], [69, 81], [69, 83], [69, 87], [70, 73], [70, 75], [70, 76], [70, 81], [70, 87], [70, 89], [71, 75], [71, 82], [71, 87], [71, 88], [72, 74], [72, 77], [72, 79], [72, 81], [72, 84], [72, 86], [72, 88], [73, 74], [73, 77], [73, 79], [73, 88], [74, 83], [74, 85], [74, 89], [75, 76], [75, 77], [75, 82], [75, 86], [75, 87], [76, 79], [76, 82], [76, 89], [77, 82], [77, 88], [77, 89], [78, 81], [78, 82], [78, 84], [78, 88], [79, 82], [79, 85], [81, 82], [81, 84], [83, 86], [83, 88], [84, 86], [85, 86], [86, 88], [86, 89], [87, 89]], "family": "mc_max_independent_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-max-independent-set-l5-s2/tests/run.py b/tasks/construct-mc-max-independent-set-l5-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-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-max-independent-set-l5-s2/tests/test.sh b/tasks/construct-mc-max-independent-set-l5-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-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-max-independent-set-l5-s5/environment/Dockerfile b/tasks/construct-mc-max-independent-set-l5-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-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-mc-max-independent-set-l5-s5/environment/instance.json b/tasks/construct-mc-max-independent-set-l5-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..91092b4f613ef033707321c450203a327cc40123
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-l5-s5/environment/instance.json
@@ -0,0 +1 @@
+{"n": 90, "k": 16, "edges": [[0, 1], [0, 2], [0, 6], [0, 7], [0, 8], [0, 11], [0, 18], [0, 20], [0, 21], [0, 33], [0, 34], [0, 38], [0, 46], [0, 47], [0, 49], [0, 60], [0, 63], [0, 66], [0, 72], [0, 77], [0, 81], [0, 82], [0, 84], [1, 5], [1, 6], [1, 10], [1, 11], [1, 13], [1, 14], [1, 20], [1, 23], [1, 24], [1, 25], [1, 40], [1, 42], [1, 51], [1, 52], [1, 53], [1, 58], [1, 59], [1, 62], [1, 63], [1, 64], [1, 65], [1, 66], [1, 68], [1, 75], [1, 77], [1, 78], [1, 85], [1, 86], [1, 87], [2, 5], [2, 7], [2, 10], [2, 12], [2, 15], [2, 16], [2, 19], [2, 23], [2, 28], [2, 30], [2, 31], [2, 33], [2, 34], [2, 36], [2, 39], [2, 48], [2, 49], [2, 52], [2, 54], [2, 66], [2, 68], [2, 69], [2, 70], [2, 77], [2, 78], [2, 80], [2, 81], [2, 82], [2, 84], [3, 4], [3, 5], [3, 6], [3, 7], [3, 8], [3, 11], [3, 16], [3, 17], [3, 21], [3, 23], [3, 28], [3, 29], [3, 33], [3, 40], [3, 42], [3, 44], [3, 52], [3, 54], [3, 55], [3, 58], [3, 59], [3, 60], [3, 62], [3, 65], [3, 66], [3, 68], [3, 78], [3, 83], [3, 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\ No newline at end of file
diff --git a/tasks/construct-mc-max-independent-set-l5-s5/solution/solve.sh b/tasks/construct-mc-max-independent-set-l5-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..6365e2b22379a1627479bd5e7c2f5e26813fc4be
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-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'
+{"independent_set": [21, 24, 33, 34, 35, 45, 46, 53, 54, 62, 66, 69, 75, 77, 80, 89]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-max-independent-set-l5-s5/tests/check.py b/tasks/construct-mc-max-independent-set-l5-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..adce413e8578d996549583b990e4ff559f38ed41
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-l5-s5/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-l5-s5/tests/instance.json b/tasks/construct-mc-max-independent-set-l5-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..91092b4f613ef033707321c450203a327cc40123
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-l5-s5/tests/instance.json
@@ -0,0 +1 @@
+{"n": 90, "k": 16, "edges": [[0, 1], [0, 2], [0, 6], [0, 7], [0, 8], [0, 11], [0, 18], [0, 20], [0, 21], [0, 33], [0, 34], [0, 38], [0, 46], [0, 47], [0, 49], [0, 60], [0, 63], [0, 66], [0, 72], [0, 77], [0, 81], [0, 82], [0, 84], [1, 5], [1, 6], [1, 10], [1, 11], [1, 13], [1, 14], [1, 20], [1, 23], [1, 24], [1, 25], [1, 40], [1, 42], [1, 51], [1, 52], [1, 53], [1, 58], [1, 59], [1, 62], [1, 63], [1, 64], [1, 65], [1, 66], [1, 68], [1, 75], [1, 77], [1, 78], [1, 85], [1, 86], [1, 87], [2, 5], [2, 7], [2, 10], [2, 12], [2, 15], [2, 16], [2, 19], [2, 23], [2, 28], [2, 30], [2, 31], [2, 33], [2, 34], [2, 36], [2, 39], [2, 48], [2, 49], [2, 52], [2, 54], [2, 66], [2, 68], [2, 69], [2, 70], [2, 77], [2, 78], [2, 80], [2, 81], [2, 82], [2, 84], [3, 4], [3, 5], [3, 6], [3, 7], [3, 8], [3, 11], [3, 16], [3, 17], [3, 21], [3, 23], [3, 28], [3, 29], [3, 33], [3, 40], [3, 42], [3, 44], [3, 52], [3, 54], [3, 55], [3, 58], [3, 59], [3, 60], [3, 62], [3, 65], [3, 66], [3, 68], [3, 78], [3, 83], [3, 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63], [29, 64], [29, 67], [29, 70], [29, 71], [29, 73], [29, 75], [29, 77], [29, 84], [29, 85], [29, 86], [29, 88], [30, 31], [30, 33], [30, 40], [30, 48], [30, 53], [30, 60], [30, 61], [30, 64], [30, 65], [30, 66], [30, 71], [30, 72], [30, 75], [30, 76], [30, 86], [31, 33], [31, 35], [31, 41], [31, 42], [31, 43], [31, 44], [31, 49], [31, 50], [31, 57], [31, 59], [31, 64], [31, 70], [31, 72], [31, 77], [31, 79], [31, 89], [32, 34], [32, 38], [32, 44], [32, 48], [32, 57], [32, 63], [32, 69], [32, 74], [32, 75], [32, 81], [32, 83], [32, 88], [32, 89], [33, 38], [33, 40], [33, 41], [33, 42], [33, 44], [33, 48], [33, 56], [33, 79], [33, 82], [33, 84], [33, 88], [34, 39], [34, 42], [34, 47], [34, 48], [34, 67], [34, 68], [34, 78], [34, 79], [34, 84], [34, 85], [34, 86], [34, 87], [35, 38], [35, 41], [35, 43], [35, 48], [35, 50], [35, 52], [35, 55], [35, 58], [35, 64], [35, 65], [35, 67], [35, 71], [35, 72], [35, 79], [35, 84], [35, 85], [36, 40], [36, 43], [36, 46], [36, 48], [36, 50], [36, 51], [36, 52], [36, 53], [36, 54], [36, 56], [36, 57], [36, 60], [36, 64], [36, 69], [36, 79], [36, 81], [36, 85], [37, 38], [37, 41], [37, 46], [37, 47], [37, 51], [37, 57], [37, 58], [37, 63], [37, 64], [37, 66], [37, 68], [37, 70], [37, 73], [37, 74], [37, 78], [37, 85], [37, 86], [38, 40], [38, 41], [38, 42], [38, 45], [38, 55], [38, 58], [38, 63], [38, 74], [38, 78], [38, 81], [38, 82], [38, 85], [38, 88], [39, 43], [39, 55], [39, 57], [39, 60], [39, 64], [39, 66], [39, 69], [39, 70], [39, 72], [39, 75], [39, 77], [39, 78], [39, 84], [40, 42], [40, 57], [40, 58], [40, 59], [40, 60], [40, 61], [40, 63], [40, 64], [40, 69], [40, 70], [40, 71], [40, 73], [40, 75], [40, 86], [41, 43], [41, 51], [41, 53], [41, 54], [41, 56], [41, 57], [41, 58], [41, 60], [41, 62], [41, 68], [41, 73], [42, 48], [42, 50], [42, 51], [42, 58], [42, 62], [42, 70], [42, 74], [42, 77], [42, 85], [43, 45], [43, 48], [43, 49], [43, 60], [43, 63], [43, 65], [43, 66], [43, 72], [43, 78], [43, 81], [43, 83], [43, 85], [43, 88], [44, 45], [44, 47], [44, 53], [44, 61], [44, 62], [44, 68], [44, 72], [44, 74], [44, 78], [44, 79], [44, 84], [44, 87], [45, 47], [45, 55], [45, 56], [45, 60], [45, 63], [45, 70], [45, 73], [45, 78], [45, 79], [45, 83], [45, 84], [46, 49], [46, 57], [46, 60], [46, 63], [46, 70], [46, 74], [46, 76], [46, 78], [46, 81], [46, 88], [47, 48], [47, 54], [47, 58], [47, 59], [47, 60], [47, 63], [47, 65], [47, 67], [47, 69], [47, 70], [47, 78], [47, 79], [47, 81], [47, 83], [47, 85], [47, 87], [48, 51], [48, 52], [48, 53], [48, 55], [48, 68], [48, 69], [48, 75], [48, 76], [48, 77], [48, 84], [49, 55], [49, 58], [49, 62], [49, 65], [49, 81], [49, 82], [49, 88], [50, 53], [50, 56], [50, 58], [50, 60], [50, 67], [50, 68], [50, 69], [50, 73], [50, 75], [50, 80], [50, 81], [50, 88], [50, 89], [51, 57], [51, 59], [51, 65], [51, 67], [51, 77], [51, 83], [51, 87], [52, 59], [52, 61], [52, 64], [52, 65], [52, 66], [52, 68], [52, 69], [52, 70], [52, 71], [52, 75], [52, 79], [52, 85], [52, 87], [52, 88], [53, 55], [53, 60], [53, 61], [53, 68], [53, 79], [53, 84], [53, 85], [54, 59], [54, 65], [54, 68], [54, 70], [54, 82], [54, 84], [54, 86], [54, 88], [55, 58], [55, 61], [55, 69], [55, 70], [55, 75], [55, 76], [55, 77], [55, 78], [55, 83], [56, 63], [56, 65], [56, 75], [56, 78], [56, 81], [56, 82], [57, 60], [57, 69], [57, 74], [57, 76], [57, 81], [57, 82], [57, 84], [57, 86], [57, 88], [58, 65], [58, 66], [58, 68], [58, 73], [58, 74], [58, 79], [58, 82], [59, 63], [59, 65], [59, 68], [59, 80], [59, 82], [59, 85], [59, 89], [60, 64], [60, 67], [60, 70], [60, 83], [60, 84], [60, 86], [60, 89], [61, 62], [61, 64], [61, 67], [61, 70], [61, 74], [61, 78], [61, 80], [61, 81], [61, 82], [61, 83], [61, 89], [62, 64], [62, 67], [62, 70], [62, 78], [62, 79], [62, 87], [63, 64], [63, 66], [63, 70], [63, 75], [63, 84], [63, 86], [63, 88], [64, 65], [64, 67], [64, 75], [64, 79], [64, 81], [64, 82], [64, 84], [64, 85], [64, 89], [65, 69], [65, 70], [65, 71], [65, 74], [65, 75], [65, 77], [65, 80], [65, 87], [66, 67], [66, 71], [66, 73], [66, 76], [66, 78], [66, 81], [66, 84], [66, 85], [66, 87], [67, 72], [67, 81], [67, 88], [68, 69], [68, 75], [68, 79], [68, 80], [68, 83], [68, 84], [68, 86], [68, 88], [69, 70], [69, 72], [70, 74], [70, 75], [70, 81], [71, 73], [71, 74], [71, 75], [71, 78], [71, 80], [71, 81], [71, 83], [72, 73], [72, 75], [72, 82], [72, 84], [73, 76], [73, 82], [73, 83], [73, 85], [73, 86], [73, 87], [73, 88], [74, 79], [74, 80], [74, 81], [74, 83], [74, 85], [74, 88], [75, 83], [75, 84], [76, 83], [76, 86], [76, 87], [76, 89], [77, 79], [77, 81], [77, 82], [78, 80], [78, 82], [78, 84], [78, 86], [78, 87], [78, 88], [78, 89], [79, 81], [79, 82], [79, 84], [79, 85], [79, 88], [80, 81], [80, 82], [80, 86], [80, 87], [81, 86], [81, 87], [81, 89], [82, 83], [82, 84], [82, 87], [82, 89], [85, 88], [85, 89], [86, 87], [87, 89]], "family": "mc_max_independent_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-max-independent-set-l5-s5/tests/run.py b/tasks/construct-mc-max-independent-set-l5-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-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-mc-max-independent-set-l5-s5/tests/test.sh b/tasks/construct-mc-max-independent-set-l5-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-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-mc-non-transitive-dice-l5-s2/environment/Dockerfile b/tasks/construct-mc-non-transitive-dice-l5-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-non-transitive-dice-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-non-transitive-dice-l5-s2/environment/instance.json b/tasks/construct-mc-non-transitive-dice-l5-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e4addd221adaa9fce583e2556aacac110357f518
--- /dev/null
+++ b/tasks/construct-mc-non-transitive-dice-l5-s2/environment/instance.json
@@ -0,0 +1 @@
+{"n": 7, "m": 7, "family": "mc_non_transitive_dice", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-non-transitive-dice-l5-s2/solution/solve.sh b/tasks/construct-mc-non-transitive-dice-l5-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..918f17182b43aee14c0a3a76171d945e1724d262
--- /dev/null
+++ b/tasks/construct-mc-non-transitive-dice-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'
+{"dice": [[0, 0, 1, 9, 9, 9, 10], [0, 5, 6, 6, 7, 8, 11], [1, 2, 2, 3, 10, 11, 13], [0, 1, 7, 7, 9, 9, 12], [0, 0, 5, 6, 10, 10, 11], [0, 2, 6, 6, 9, 9, 9], [3, 4, 5, 5, 7, 7, 12]]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-non-transitive-dice-l5-s2/tests/check.py b/tasks/construct-mc-non-transitive-dice-l5-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..e3d5c04fa6e2819bf32f34fafe13a383437dc62a
--- /dev/null
+++ b/tasks/construct-mc-non-transitive-dice-l5-s2/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-l5-s2/tests/instance.json b/tasks/construct-mc-non-transitive-dice-l5-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e4addd221adaa9fce583e2556aacac110357f518
--- /dev/null
+++ b/tasks/construct-mc-non-transitive-dice-l5-s2/tests/instance.json
@@ -0,0 +1 @@
+{"n": 7, "m": 7, "family": "mc_non_transitive_dice", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-non-transitive-dice-l5-s2/tests/run.py b/tasks/construct-mc-non-transitive-dice-l5-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-non-transitive-dice-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-non-transitive-dice-l5-s2/tests/test.sh b/tasks/construct-mc-non-transitive-dice-l5-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-non-transitive-dice-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-number-partitioning-l1-s0/environment/Dockerfile b/tasks/construct-mc-number-partitioning-l1-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-number-partitioning-l1-s0/environment/instance.json b/tasks/construct-mc-number-partitioning-l1-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0092e8b334cf67745856d5766c2c09afa874679c
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l1-s0/environment/instance.json
@@ -0,0 +1 @@
+{"n": 9, "k": 5, "family": "mc_number_partitioning", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-number-partitioning-l1-s0/solution/solve.sh b/tasks/construct-mc-number-partitioning-l1-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..e880db5676fdb382101bdbbc7e9113013590ca3f
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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'
+{"x": [1, 2, 3, 4, 4, 3, 2, 1, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-number-partitioning-l1-s0/tests/check.py b/tasks/construct-mc-number-partitioning-l1-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..67793721db8ba7d3c3979ea0e1e07091491ffa7e
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l1-s0/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-l1-s0/tests/instance.json b/tasks/construct-mc-number-partitioning-l1-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0092e8b334cf67745856d5766c2c09afa874679c
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l1-s0/tests/instance.json
@@ -0,0 +1 @@
+{"n": 9, "k": 5, "family": "mc_number_partitioning", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-number-partitioning-l1-s0/tests/run.py b/tasks/construct-mc-number-partitioning-l1-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-number-partitioning-l1-s0/tests/test.sh b/tasks/construct-mc-number-partitioning-l1-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-number-partitioning-l1-s7/environment/Dockerfile b/tasks/construct-mc-number-partitioning-l1-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-mc-number-partitioning-l1-s7/environment/instance.json b/tasks/construct-mc-number-partitioning-l1-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..522031bd98670057427e22b1d98e47c6e6591c5c
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l1-s7/environment/instance.json
@@ -0,0 +1 @@
+{"n": 10, "k": 5, "family": "mc_number_partitioning", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-number-partitioning-l1-s7/solution/solve.sh b/tasks/construct-mc-number-partitioning-l1-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..5c921da5220c7e3fa9a3f7173dc4fa2930cf5d31
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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'
+{"x": [0, 1, 2, 3, 4, 4, 3, 2, 1, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-number-partitioning-l1-s7/tests/check.py b/tasks/construct-mc-number-partitioning-l1-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..67793721db8ba7d3c3979ea0e1e07091491ffa7e
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l1-s7/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-l1-s7/tests/instance.json b/tasks/construct-mc-number-partitioning-l1-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..522031bd98670057427e22b1d98e47c6e6591c5c
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l1-s7/tests/instance.json
@@ -0,0 +1 @@
+{"n": 10, "k": 5, "family": "mc_number_partitioning", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-number-partitioning-l1-s7/tests/run.py b/tasks/construct-mc-number-partitioning-l1-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-mc-number-partitioning-l1-s7/tests/test.sh b/tasks/construct-mc-number-partitioning-l1-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-mc-number-partitioning-l5-s1/environment/Dockerfile b/tasks/construct-mc-number-partitioning-l5-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-number-partitioning-l5-s1/environment/instance.json b/tasks/construct-mc-number-partitioning-l5-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e7ea2acdf11ed326933338822fb2709268f4d423
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l5-s1/environment/instance.json
@@ -0,0 +1 @@
+{"n": 69, "k": 3, "family": "mc_number_partitioning", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-number-partitioning-l5-s1/solution/solve.sh b/tasks/construct-mc-number-partitioning-l5-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..7ba4fb47b21c7e30739d825798eeef79821bf074
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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": [0, 2, 1, 1, 0, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-number-partitioning-l5-s1/tests/check.py b/tasks/construct-mc-number-partitioning-l5-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..67793721db8ba7d3c3979ea0e1e07091491ffa7e
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l5-s1/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-l5-s1/tests/instance.json b/tasks/construct-mc-number-partitioning-l5-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e7ea2acdf11ed326933338822fb2709268f4d423
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l5-s1/tests/instance.json
@@ -0,0 +1 @@
+{"n": 69, "k": 3, "family": "mc_number_partitioning", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-number-partitioning-l5-s1/tests/run.py b/tasks/construct-mc-number-partitioning-l5-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-number-partitioning-l5-s1/tests/test.sh b/tasks/construct-mc-number-partitioning-l5-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-number-partitioning-l5-s6/environment/Dockerfile b/tasks/construct-mc-number-partitioning-l5-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-mc-number-partitioning-l5-s6/environment/instance.json b/tasks/construct-mc-number-partitioning-l5-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a29748171c68837a46ba0c363ea6a9bfae080964
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l5-s6/environment/instance.json
@@ -0,0 +1 @@
+{"n": 50, "k": 5, "family": "mc_number_partitioning", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-number-partitioning-l5-s6/solution/solve.sh b/tasks/construct-mc-number-partitioning-l5-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..244777e16659feb9ce39dc812d426878205099d0
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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'
+{"x": [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, 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-l5-s6/tests/check.py b/tasks/construct-mc-number-partitioning-l5-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..67793721db8ba7d3c3979ea0e1e07091491ffa7e
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l5-s6/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-l5-s6/tests/instance.json b/tasks/construct-mc-number-partitioning-l5-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a29748171c68837a46ba0c363ea6a9bfae080964
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l5-s6/tests/instance.json
@@ -0,0 +1 @@
+{"n": 50, "k": 5, "family": "mc_number_partitioning", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-number-partitioning-l5-s6/tests/run.py b/tasks/construct-mc-number-partitioning-l5-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-mc-number-partitioning-l5-s6/tests/test.sh b/tasks/construct-mc-number-partitioning-l5-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-mc-number-partitioning-l5-s8/environment/Dockerfile b/tasks/construct-mc-number-partitioning-l5-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-mc-number-partitioning-l5-s8/environment/instance.json b/tasks/construct-mc-number-partitioning-l5-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2fbe46e7ecbe116911cf17ce8caf46a2dfd7029d
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l5-s8/environment/instance.json
@@ -0,0 +1 @@
+{"n": 68, "k": 3, "family": "mc_number_partitioning", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-number-partitioning-l5-s8/solution/solve.sh b/tasks/construct-mc-number-partitioning-l5-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..fb74fc15e034e9b13dc7f81f93cc9db27885c738
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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'
+{"x": [2, 1, 1, 0, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-number-partitioning-l5-s8/tests/check.py b/tasks/construct-mc-number-partitioning-l5-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..67793721db8ba7d3c3979ea0e1e07091491ffa7e
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l5-s8/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-l5-s8/tests/instance.json b/tasks/construct-mc-number-partitioning-l5-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2fbe46e7ecbe116911cf17ce8caf46a2dfd7029d
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l5-s8/tests/instance.json
@@ -0,0 +1 @@
+{"n": 68, "k": 3, "family": "mc_number_partitioning", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-number-partitioning-l5-s8/tests/run.py b/tasks/construct-mc-number-partitioning-l5-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-mc-number-partitioning-l5-s8/tests/test.sh b/tasks/construct-mc-number-partitioning-l5-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-mc-ortholatin-l3-s0/environment/Dockerfile b/tasks/construct-mc-ortholatin-l3-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-ortholatin-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-mc-ortholatin-l3-s0/environment/instance.json b/tasks/construct-mc-ortholatin-l3-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..64320cbb64481fc888bb0cc32d0c4831c193a780
--- /dev/null
+++ b/tasks/construct-mc-ortholatin-l3-s0/environment/instance.json
@@ -0,0 +1 @@
+{"n": 10, "family": "mc_ortholatin", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-ortholatin-l3-s0/solution/solve.sh b/tasks/construct-mc-ortholatin-l3-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..7ee633a9f35e5537d5b2e500bd64b4ba19510277
--- /dev/null
+++ b/tasks/construct-mc-ortholatin-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'
+{"X": [[0, 6, 2, 3, 4, 5, 1, 7, 8, 9], [1, 7, 8, 9, 0, 6, 2, 3, 4, 5], [7, 8, 4, 0, 6, 2, 3, 9, 5, 1], [8, 9, 5, 1, 2, 3, 4, 0, 6, 7], [9, 0, 1, 7, 8, 4, 5, 6, 2, 3], [5, 1, 7, 8, 9, 0, 6, 2, 3, 4], [6, 2, 3, 4, 5, 1, 7, 8, 9, 0], [2, 3, 9, 5, 1, 7, 8, 4, 0, 6], [3, 4, 0, 6, 7, 8, 9, 5, 1, 2], [4, 5, 6, 2, 3, 9, 0, 1, 7, 8]], "Y": [[5, 0, 6, 7, 9, 2, 8, 4, 1, 3], [3, 1, 4, 8, 2, 6, 5, 0, 7, 9], [6, 3, 1, 4, 8, 7, 2, 9, 0, 5], [2, 5, 3, 6, 1, 4, 0, 8, 9, 7], [7, 9, 0, 5, 6, 3, 4, 2, 8, 1], [8, 2, 9, 0, 4, 1, 7, 3, 5, 6], [1, 4, 8, 2, 5, 9, 3, 7, 6, 0], [0, 6, 2, 1, 7, 8, 9, 5, 3, 4], [9, 8, 7, 3, 0, 5, 1, 6, 4, 2], [4, 7, 5, 9, 3, 0, 6, 1, 2, 8]]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-ortholatin-l3-s0/tests/check.py b/tasks/construct-mc-ortholatin-l3-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..a5418a59fae1c86318eee903e90d75b36d8a3306
--- /dev/null
+++ b/tasks/construct-mc-ortholatin-l3-s0/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-l3-s0/tests/instance.json b/tasks/construct-mc-ortholatin-l3-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..64320cbb64481fc888bb0cc32d0c4831c193a780
--- /dev/null
+++ b/tasks/construct-mc-ortholatin-l3-s0/tests/instance.json
@@ -0,0 +1 @@
+{"n": 10, "family": "mc_ortholatin", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-ortholatin-l3-s0/tests/run.py b/tasks/construct-mc-ortholatin-l3-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-ortholatin-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-mc-ortholatin-l3-s0/tests/test.sh b/tasks/construct-mc-ortholatin-l3-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-ortholatin-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-mc-pysms-chromatic-girth-l3-s0/environment/Dockerfile b/tasks/construct-mc-pysms-chromatic-girth-l3-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-pysms-chromatic-girth-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-mc-pysms-chromatic-girth-l3-s0/environment/instance.json b/tasks/construct-mc-pysms-chromatic-girth-l3-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0cae8648e6011de56ecdfc99a3b1e34d6f66ef08
--- /dev/null
+++ b/tasks/construct-mc-pysms-chromatic-girth-l3-s0/environment/instance.json
@@ -0,0 +1 @@
+{"vertices": 23, "required_edges": [], "max_chromatic_number": 3, "min_edges": 44, "min_girth": 5, "family": "mc_pysms_chromatic_girth", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-chromatic-girth-l3-s0/solution/solve.sh b/tasks/construct-mc-pysms-chromatic-girth-l3-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..c76aa1ee224363a33a489f0520963eb8382cca3a
--- /dev/null
+++ b/tasks/construct-mc-pysms-chromatic-girth-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'
+{"edges": [[0, 20], [0, 21], [0, 22], [1, 17], [1, 18], [1, 19], [2, 15], [2, 16], [2, 19], [2, 22], [3, 13], [3, 14], [3, 19], [3, 21], [4, 11], [4, 12], [4, 18], [4, 20], [5, 10], [5, 12], [5, 16], [5, 21], [6, 9], [6, 12], [6, 14], [6, 15], [7, 8], [7, 11], [7, 13], [7, 15], [8, 10], [8, 14], [8, 22], [9, 10], [9, 11], [9, 19], [10, 20], [11, 21], [12, 17], [13, 16], [13, 17], [14, 18], [15, 20], [16, 18], [17, 22]], "coloring": [0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 2, 2, 1, 2, 1, 2, 1, 0, 2, 2, 1, 1, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-pysms-chromatic-girth-l3-s0/tests/check.py b/tasks/construct-mc-pysms-chromatic-girth-l3-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..4ab0b12508cf59b733f007a9d1f50831aec9efdf
--- /dev/null
+++ b/tasks/construct-mc-pysms-chromatic-girth-l3-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-chromatic-girth-l3-s0/tests/instance.json b/tasks/construct-mc-pysms-chromatic-girth-l3-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0cae8648e6011de56ecdfc99a3b1e34d6f66ef08
--- /dev/null
+++ b/tasks/construct-mc-pysms-chromatic-girth-l3-s0/tests/instance.json
@@ -0,0 +1 @@
+{"vertices": 23, "required_edges": [], "max_chromatic_number": 3, "min_edges": 44, "min_girth": 5, "family": "mc_pysms_chromatic_girth", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-chromatic-girth-l3-s0/tests/run.py b/tasks/construct-mc-pysms-chromatic-girth-l3-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-pysms-chromatic-girth-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-mc-pysms-chromatic-girth-l3-s0/tests/test.sh b/tasks/construct-mc-pysms-chromatic-girth-l3-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-pysms-chromatic-girth-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-mc-pysms-degree-bounds-l3-s0/environment/Dockerfile b/tasks/construct-mc-pysms-degree-bounds-l3-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-pysms-degree-bounds-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-mc-pysms-degree-bounds-l3-s0/environment/instance.json b/tasks/construct-mc-pysms-degree-bounds-l3-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0efcab3d04e84123d20ec898ee8830a2638d0ad6
--- /dev/null
+++ b/tasks/construct-mc-pysms-degree-bounds-l3-s0/environment/instance.json
@@ -0,0 +1 @@
+{"vertices": 11, "required_edges": [[5, 6], [3, 8], [0, 9], [3, 7], [3, 9], [4, 9]], "max_degree": 6, "min_degree": 4, "family": "mc_pysms_degree_bounds", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-degree-bounds-l3-s0/solution/solve.sh b/tasks/construct-mc-pysms-degree-bounds-l3-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..12ba5a6b930dcf86d462ed6efd022d8b7214a34e
--- /dev/null
+++ b/tasks/construct-mc-pysms-degree-bounds-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'
+{"edges": [[0, 6], [0, 7], [0, 8], [0, 9], [0, 10], [1, 6], [1, 7], [1, 8], [1, 9], [1, 10], [2, 5], [2, 6], [2, 7], [2, 8], [2, 9], [2, 10], [3, 4], [3, 6], [3, 7], [3, 8], [3, 9], [3, 10], [4, 6], [4, 7], [4, 8], [4, 9], [4, 10], [5, 6], [5, 7], [5, 8], [5, 9], [5, 10]]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-pysms-degree-bounds-l3-s0/tests/check.py b/tasks/construct-mc-pysms-degree-bounds-l3-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..4ab0b12508cf59b733f007a9d1f50831aec9efdf
--- /dev/null
+++ b/tasks/construct-mc-pysms-degree-bounds-l3-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-degree-bounds-l3-s0/tests/instance.json b/tasks/construct-mc-pysms-degree-bounds-l3-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0efcab3d04e84123d20ec898ee8830a2638d0ad6
--- /dev/null
+++ b/tasks/construct-mc-pysms-degree-bounds-l3-s0/tests/instance.json
@@ -0,0 +1 @@
+{"vertices": 11, "required_edges": [[5, 6], [3, 8], [0, 9], [3, 7], [3, 9], [4, 9]], "max_degree": 6, "min_degree": 4, "family": "mc_pysms_degree_bounds", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-degree-bounds-l3-s0/tests/run.py b/tasks/construct-mc-pysms-degree-bounds-l3-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-pysms-degree-bounds-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-mc-pysms-degree-bounds-l3-s0/tests/test.sh b/tasks/construct-mc-pysms-degree-bounds-l3-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-pysms-degree-bounds-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-mc-pysms-independent-connectivity-l1-s2/environment/Dockerfile b/tasks/construct-mc-pysms-independent-connectivity-l1-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-pysms-independent-connectivity-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-mc-pysms-independent-connectivity-l1-s2/environment/instance.json b/tasks/construct-mc-pysms-independent-connectivity-l1-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8eb7b99321c93b9d222bb5cc9dc16dcd85d4ec6e
--- /dev/null
+++ b/tasks/construct-mc-pysms-independent-connectivity-l1-s2/environment/instance.json
@@ -0,0 +1 @@
+{"vertices": 9, "required_edges": [[3, 6], [1, 4], [0, 2], [3, 8], [1, 3], [7, 8], [5, 6]], "max_independent_set": 2, "min_connectivity": 2, "family": "mc_pysms_independent_connectivity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-independent-connectivity-l1-s2/solution/solve.sh b/tasks/construct-mc-pysms-independent-connectivity-l1-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..266b10696d549d81ec1db10a95e1067a0f575ccd
--- /dev/null
+++ b/tasks/construct-mc-pysms-independent-connectivity-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'
+{"edges": [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [0, 6], [0, 7], [0, 8], [1, 2], [1, 3], [1, 4], [1, 5], [1, 6], [1, 7], [1, 8], [2, 3], [2, 4], [2, 5], [2, 6], [2, 7], [2, 8], [3, 4], [3, 5], [3, 6], [3, 7], [3, 8], [4, 5], [4, 6], [4, 7], [4, 8], [5, 6], [5, 7], [5, 8], [6, 7], [6, 8], [7, 8]]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-pysms-independent-connectivity-l1-s2/tests/check.py b/tasks/construct-mc-pysms-independent-connectivity-l1-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..4ab0b12508cf59b733f007a9d1f50831aec9efdf
--- /dev/null
+++ b/tasks/construct-mc-pysms-independent-connectivity-l1-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-independent-connectivity-l1-s2/tests/instance.json b/tasks/construct-mc-pysms-independent-connectivity-l1-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8eb7b99321c93b9d222bb5cc9dc16dcd85d4ec6e
--- /dev/null
+++ b/tasks/construct-mc-pysms-independent-connectivity-l1-s2/tests/instance.json
@@ -0,0 +1 @@
+{"vertices": 9, "required_edges": [[3, 6], [1, 4], [0, 2], [3, 8], [1, 3], [7, 8], [5, 6]], "max_independent_set": 2, "min_connectivity": 2, "family": "mc_pysms_independent_connectivity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-independent-connectivity-l1-s2/tests/run.py b/tasks/construct-mc-pysms-independent-connectivity-l1-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-pysms-independent-connectivity-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-mc-pysms-independent-connectivity-l1-s2/tests/test.sh b/tasks/construct-mc-pysms-independent-connectivity-l1-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-pysms-independent-connectivity-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-mc-pysms-min-degree-l4-s0/environment/Dockerfile b/tasks/construct-mc-pysms-min-degree-l4-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-degree-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-pysms-min-degree-l4-s0/environment/instance.json b/tasks/construct-mc-pysms-min-degree-l4-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d8b3db2e3e50f45eb0fed3dff91cd197786db696
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-degree-l4-s0/environment/instance.json
@@ -0,0 +1 @@
+{"vertices": 13, "required_edges": [], "min_degree": 6, "family": "mc_pysms_min_degree", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-min-degree-l4-s0/solution/solve.sh b/tasks/construct-mc-pysms-min-degree-l4-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..4cafa9813841741ebbf193085ce3e45b85fa4e68
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-degree-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'
+{"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], [1, 2], [1, 3], [1, 4], [1, 5], [1, 6], [1, 7], [1, 8], [1, 9], [1, 10], [1, 11], [1, 12], [2, 3], [2, 4], [2, 5], [2, 6], [2, 7], [2, 8], [2, 9], [2, 10], [2, 11], [2, 12], [3, 4], [3, 5], [3, 6], [3, 7], [3, 8], [3, 9], [3, 10], [3, 11], [3, 12], [4, 5], [4, 6], [4, 7], [4, 8], [4, 9], [4, 10], [4, 11], [4, 12], [5, 6], [5, 7], [5, 8], [5, 9], [5, 10], [5, 11], [5, 12], [6, 7], [6, 8], [6, 9], [6, 10], [6, 11], [6, 12], [7, 8], [7, 9], [7, 10], [7, 11], [7, 12], [8, 9], [8, 10], [8, 11], [8, 12], [9, 10], [9, 11], [9, 12], [10, 11], [10, 12], [11, 12]]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-pysms-min-degree-l4-s0/tests/check.py b/tasks/construct-mc-pysms-min-degree-l4-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..4ab0b12508cf59b733f007a9d1f50831aec9efdf
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-degree-l4-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-min-degree-l4-s0/tests/instance.json b/tasks/construct-mc-pysms-min-degree-l4-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d8b3db2e3e50f45eb0fed3dff91cd197786db696
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-degree-l4-s0/tests/instance.json
@@ -0,0 +1 @@
+{"vertices": 13, "required_edges": [], "min_degree": 6, "family": "mc_pysms_min_degree", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-min-degree-l4-s0/tests/run.py b/tasks/construct-mc-pysms-min-degree-l4-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-degree-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-pysms-min-degree-l4-s0/tests/test.sh b/tasks/construct-mc-pysms-min-degree-l4-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-degree-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-ramsey-l1-s3/environment/Dockerfile b/tasks/construct-mc-ramsey-l1-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-ramsey-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-ramsey-l1-s3/environment/instance.json b/tasks/construct-mc-ramsey-l1-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e42b98d5ab39c2b8fc3ee9c1e5d0c1964ca8a1d6
--- /dev/null
+++ b/tasks/construct-mc-ramsey-l1-s3/environment/instance.json
@@ -0,0 +1 @@
+{"n": 5, "r": 4, "s": 3, "fixed": [], "family": "mc_ramsey", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-ramsey-l1-s3/solution/solve.sh b/tasks/construct-mc-ramsey-l1-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..88922d11a5ef2128237794cdf1c7d4f42a478ede
--- /dev/null
+++ b/tasks/construct-mc-ramsey-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'
+{"colors": [1, 0, 0, 1, 1, 0, 0, 1, 0, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-ramsey-l1-s3/tests/check.py b/tasks/construct-mc-ramsey-l1-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..53526c0bc87af6298219d3e1eac5f95f01b57541
--- /dev/null
+++ b/tasks/construct-mc-ramsey-l1-s3/tests/check.py
@@ -0,0 +1,31 @@
+"""Two-colouring of the edges of K_n with no colour-0 K_r and no colour-1 K_s (MathConstraint ramsey,
+CC-BY-4.0). Adapted from pilot/grader.py check_ramsey; adds the optional fixed edge colours."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, r, s = inst["n"], inst["r"], inst["s"]
+    edges = list(itertools.combinations(range(n), 2))
+    col = int_list(ans.get("colors"), len(edges), 0, 1, "colors")
+    for name, i, v in inst.get("fixed", []):
+        if col[i] != v:
+            raise Invalid(f"fixed value {name}[{i}] = {v} not respected")
+    c = {e: col[i] for i, e in enumerate(edges)}
+    for k, colour in ((r, 0), (s, 1)):
+        adj = {v: {u for u in range(n) if u != v and c[(min(u, v), max(u, v))] == colour} for v in range(n)}
+
+        def grow(size, cand):
+            if size == k:
+                return True
+            if size + len(cand) < k:
+                return False
+            for v in sorted(cand):
+                if grow(size + 1, {u for u in cand if u > v} & adj[v]):
+                    return True
+            return False
+
+        if grow(0, set(range(n))):
+            raise Invalid(f"monochromatic K_{k} in colour {colour}")
+    return None, {}
diff --git a/tasks/construct-mc-ramsey-l1-s3/tests/instance.json b/tasks/construct-mc-ramsey-l1-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e42b98d5ab39c2b8fc3ee9c1e5d0c1964ca8a1d6
--- /dev/null
+++ b/tasks/construct-mc-ramsey-l1-s3/tests/instance.json
@@ -0,0 +1 @@
+{"n": 5, "r": 4, "s": 3, "fixed": [], "family": "mc_ramsey", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-ramsey-l1-s3/tests/run.py b/tasks/construct-mc-ramsey-l1-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-ramsey-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-ramsey-l1-s3/tests/test.sh b/tasks/construct-mc-ramsey-l1-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-ramsey-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-ramsey-l1-s4/environment/Dockerfile b/tasks/construct-mc-ramsey-l1-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-ramsey-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-ramsey-l1-s4/environment/instance.json b/tasks/construct-mc-ramsey-l1-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4593ea5429ca8f3552b4e145c095db6650f41bf2
--- /dev/null
+++ b/tasks/construct-mc-ramsey-l1-s4/environment/instance.json
@@ -0,0 +1 @@
+{"n": 5, "r": 3, "s": 3, "fixed": [["colors", 3, 1], ["colors", 9, 0]], "family": "mc_ramsey", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-ramsey-l1-s4/solution/solve.sh b/tasks/construct-mc-ramsey-l1-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..785c1e78489f87a5c57f1ac662d92e371d3abc46
--- /dev/null
+++ b/tasks/construct-mc-ramsey-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'
+{"colors": [0, 0, 1, 1, 1, 0, 1, 1, 0, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-ramsey-l1-s4/tests/check.py b/tasks/construct-mc-ramsey-l1-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..53526c0bc87af6298219d3e1eac5f95f01b57541
--- /dev/null
+++ b/tasks/construct-mc-ramsey-l1-s4/tests/check.py
@@ -0,0 +1,31 @@
+"""Two-colouring of the edges of K_n with no colour-0 K_r and no colour-1 K_s (MathConstraint ramsey,
+CC-BY-4.0). Adapted from pilot/grader.py check_ramsey; adds the optional fixed edge colours."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, r, s = inst["n"], inst["r"], inst["s"]
+    edges = list(itertools.combinations(range(n), 2))
+    col = int_list(ans.get("colors"), len(edges), 0, 1, "colors")
+    for name, i, v in inst.get("fixed", []):
+        if col[i] != v:
+            raise Invalid(f"fixed value {name}[{i}] = {v} not respected")
+    c = {e: col[i] for i, e in enumerate(edges)}
+    for k, colour in ((r, 0), (s, 1)):
+        adj = {v: {u for u in range(n) if u != v and c[(min(u, v), max(u, v))] == colour} for v in range(n)}
+
+        def grow(size, cand):
+            if size == k:
+                return True
+            if size + len(cand) < k:
+                return False
+            for v in sorted(cand):
+                if grow(size + 1, {u for u in cand if u > v} & adj[v]):
+                    return True
+            return False
+
+        if grow(0, set(range(n))):
+            raise Invalid(f"monochromatic K_{k} in colour {colour}")
+    return None, {}
diff --git a/tasks/construct-mc-ramsey-l1-s4/tests/instance.json b/tasks/construct-mc-ramsey-l1-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4593ea5429ca8f3552b4e145c095db6650f41bf2
--- /dev/null
+++ b/tasks/construct-mc-ramsey-l1-s4/tests/instance.json
@@ -0,0 +1 @@
+{"n": 5, "r": 3, "s": 3, "fixed": [["colors", 3, 1], ["colors", 9, 0]], "family": "mc_ramsey", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-ramsey-l1-s4/tests/run.py b/tasks/construct-mc-ramsey-l1-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-ramsey-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-ramsey-l1-s4/tests/test.sh b/tasks/construct-mc-ramsey-l1-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-ramsey-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-mc-social-golfers-l4-s1/environment/Dockerfile b/tasks/construct-mc-social-golfers-l4-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-social-golfers-l4-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-social-golfers-l4-s1/environment/instance.json b/tasks/construct-mc-social-golfers-l4-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3400326b43f867af87bae35dc44b4a352cfe511a
--- /dev/null
+++ b/tasks/construct-mc-social-golfers-l4-s1/environment/instance.json
@@ -0,0 +1 @@
+{"n_groups": 7, "group_size": 3, "n_weeks": 5, "family": "mc_social_golfers", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-social-golfers-l4-s1/solution/solve.sh b/tasks/construct-mc-social-golfers-l4-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..bb4fb6aef13da17d8fc0df60df435c898e872044
--- /dev/null
+++ b/tasks/construct-mc-social-golfers-l4-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'
+{"schedule": [[0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6], [3, 5, 6, 5, 4, 6, 5, 4, 6, 1, 3, 0, 1, 0, 2, 2, 4, 1, 3, 2, 0], [1, 2, 3, 1, 2, 4, 4, 5, 6, 5, 6, 0, 3, 5, 2, 1, 3, 0, 0, 6, 4], [1, 3, 4, 4, 5, 6, 5, 6, 3, 4, 6, 5, 1, 2, 0, 2, 3, 0, 2, 1, 0], [3, 4, 5, 6, 3, 4, 5, 6, 3, 4, 5, 6, 1, 2, 0, 1, 0, 2, 0, 2, 1]]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-social-golfers-l4-s1/tests/check.py b/tasks/construct-mc-social-golfers-l4-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..73977815b7b6094a8491075b6dc2ae1293e44930
--- /dev/null
+++ b/tasks/construct-mc-social-golfers-l4-s1/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-l4-s1/tests/instance.json b/tasks/construct-mc-social-golfers-l4-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3400326b43f867af87bae35dc44b4a352cfe511a
--- /dev/null
+++ b/tasks/construct-mc-social-golfers-l4-s1/tests/instance.json
@@ -0,0 +1 @@
+{"n_groups": 7, "group_size": 3, "n_weeks": 5, "family": "mc_social_golfers", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-social-golfers-l4-s1/tests/run.py b/tasks/construct-mc-social-golfers-l4-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-social-golfers-l4-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-social-golfers-l4-s1/tests/test.sh b/tasks/construct-mc-social-golfers-l4-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-social-golfers-l4-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-van-der-waerden-l1-s1/environment/Dockerfile b/tasks/construct-mc-van-der-waerden-l1-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-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-van-der-waerden-l1-s1/environment/instance.json b/tasks/construct-mc-van-der-waerden-l1-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..48a79635724859883ac121ffd0fed42b5e87ea77
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l1-s1/environment/instance.json
@@ -0,0 +1 @@
+{"k": 2, "L": 3, "n": 7, "family": "mc_van_der_waerden", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-van-der-waerden-l1-s1/solution/solve.sh b/tasks/construct-mc-van-der-waerden-l1-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..e3e7e5ad611c2a47fbcd96942d0844a170f710f8
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-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'
+{"colors": [0, 1, 1, 0, 0, 1, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-van-der-waerden-l1-s1/tests/check.py b/tasks/construct-mc-van-der-waerden-l1-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..9853b089b6ac1773d80eac8e9e02ceb4019ad41f
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l1-s1/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-l1-s1/tests/instance.json b/tasks/construct-mc-van-der-waerden-l1-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..48a79635724859883ac121ffd0fed42b5e87ea77
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l1-s1/tests/instance.json
@@ -0,0 +1 @@
+{"k": 2, "L": 3, "n": 7, "family": "mc_van_der_waerden", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-van-der-waerden-l1-s1/tests/run.py b/tasks/construct-mc-van-der-waerden-l1-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-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-van-der-waerden-l1-s1/tests/test.sh b/tasks/construct-mc-van-der-waerden-l1-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-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-van-der-waerden-l5-s0/environment/Dockerfile b/tasks/construct-mc-van-der-waerden-l5-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-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-van-der-waerden-l5-s0/environment/instance.json b/tasks/construct-mc-van-der-waerden-l5-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a512a05f1411f777878f39b8adaffc5cead4ce8d
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l5-s0/environment/instance.json
@@ -0,0 +1 @@
+{"k": 2, "L": 5, "n": 120, "family": "mc_van_der_waerden", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-van-der-waerden-l5-s0/solution/solve.sh b/tasks/construct-mc-van-der-waerden-l5-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..4e8330d8497c217388b4661f80ecdbb120ea91cd
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-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, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-van-der-waerden-l5-s0/tests/check.py b/tasks/construct-mc-van-der-waerden-l5-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..9853b089b6ac1773d80eac8e9e02ceb4019ad41f
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l5-s0/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-l5-s0/tests/instance.json b/tasks/construct-mc-van-der-waerden-l5-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a512a05f1411f777878f39b8adaffc5cead4ce8d
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l5-s0/tests/instance.json
@@ -0,0 +1 @@
+{"k": 2, "L": 5, "n": 120, "family": "mc_van_der_waerden", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-van-der-waerden-l5-s0/tests/run.py b/tasks/construct-mc-van-der-waerden-l5-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-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-van-der-waerden-l5-s0/tests/test.sh b/tasks/construct-mc-van-der-waerden-l5-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-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-van-der-waerden-l5-s7/environment/Dockerfile b/tasks/construct-mc-van-der-waerden-l5-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-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-mc-van-der-waerden-l5-s7/environment/instance.json b/tasks/construct-mc-van-der-waerden-l5-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..523c9d8e3a06f7672890a73e63a0cc03cc88ad97
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l5-s7/environment/instance.json
@@ -0,0 +1 @@
+{"k": 4, "L": 3, "n": 55, "family": "mc_van_der_waerden", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-van-der-waerden-l5-s7/solution/solve.sh b/tasks/construct-mc-van-der-waerden-l5-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..43354e16af493b012f013f2208a19bbad98b54b8
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-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, 1, 3, 1, 3, 3, 0, 0, 2, 1, 0, 0, 3, 1, 1, 2, 2, 0, 2, 0, 0, 1, 0, 2, 3, 3, 0, 2, 3, 2, 3, 1, 2, 2, 3, 1, 0, 1, 1, 3, 1, 3, 3, 0, 0, 2, 1, 0, 0, 3, 1, 1, 2, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-van-der-waerden-l5-s7/tests/check.py b/tasks/construct-mc-van-der-waerden-l5-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..9853b089b6ac1773d80eac8e9e02ceb4019ad41f
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l5-s7/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-l5-s7/tests/instance.json b/tasks/construct-mc-van-der-waerden-l5-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..523c9d8e3a06f7672890a73e63a0cc03cc88ad97
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l5-s7/tests/instance.json
@@ -0,0 +1 @@
+{"k": 4, "L": 3, "n": 55, "family": "mc_van_der_waerden", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-van-der-waerden-l5-s7/tests/run.py b/tasks/construct-mc-van-der-waerden-l5-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-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-mc-van-der-waerden-l5-s7/tests/test.sh b/tasks/construct-mc-van-der-waerden-l5-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-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-mc-vertex-cover-l4-s9/environment/Dockerfile b/tasks/construct-mc-vertex-cover-l4-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-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-mc-vertex-cover-l4-s9/environment/instance.json b/tasks/construct-mc-vertex-cover-l4-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..174e0cbd83449a0ea3ea0bfd1d6045afcab705a0
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-l4-s9/environment/instance.json
@@ -0,0 +1 @@
+{"n": 60, "k": 41, "edges": [[0, 6], [0, 7], [0, 10], [0, 12], [0, 14], [0, 16], [0, 24], [0, 25], [0, 26], [0, 31], [0, 34], [0, 46], [0, 56], [1, 6], [1, 8], [1, 11], [1, 12], [1, 13], [1, 17], [1, 28], [1, 32], [1, 33], [1, 35], [1, 36], [1, 43], [1, 56], [1, 59], [2, 5], [2, 6], [2, 15], [2, 24], [2, 27], [2, 31], [2, 34], [2, 35], [2, 39], [2, 40], [2, 46], [2, 47], [3, 4], [3, 9], [3, 12], [3, 25], [3, 26], [3, 28], [3, 34], [3, 39], [3, 40], [3, 41], [3, 44], [3, 51], [3, 58], [4, 5], [4, 16], [4, 17], [4, 18], [4, 27], [4, 37], [4, 38], [4, 39], [4, 40], [4, 46], [4, 47], [4, 49], [4, 51], [4, 52], [5, 8], [5, 11], [5, 13], [5, 14], [5, 18], [5, 19], [5, 24], [5, 28], [5, 41], [5, 43], [5, 46], [5, 49], [5, 53], [5, 58], [6, 13], [6, 15], [6, 17], [6, 21], [6, 27], [6, 38], [6, 44], [6, 45], [6, 49], [6, 52], [6, 57], [7, 8], [7, 12], [7, 17], [7, 18], [7, 21], [7, 28], [7, 36], [7, 45], [7, 50], [7, 58], [8, 9], [8, 20], [8, 22], [8, 26], [9, 12], [9, 16], [9, 19], [9, 30], [9, 34], [9, 36], [9, 39], [9, 42], [9, 43], [9, 44], [10, 12], [10, 20], [10, 22], [10, 25], [10, 29], [10, 46], [10, 49], [10, 51], [10, 53], [11, 18], [11, 22], [11, 28], [11, 36], [11, 37], [11, 40], [11, 47], [11, 50], [11, 54], [11, 56], [12, 13], [12, 14], [12, 18], [12, 19], [12, 23], [12, 29], [12, 40], [12, 54], [13, 18], [13, 20], [13, 22], [13, 31], [13, 38], [13, 59], [14, 23], [14, 27], [14, 29], [14, 31], [14, 32], [14, 35], [14, 36], [14, 44], [14, 56], [14, 58], [15, 16], [15, 20], [15, 22], [15, 25], [15, 33], [15, 34], [15, 39], [15, 41], [15, 46], [15, 52], [15, 53], [15, 56], [16, 18], [16, 28], [16, 31], [16, 37], [16, 39], [16, 43], [16, 46], [16, 47], [16, 59], [17, 25], [17, 26], [17, 40], [17, 42], [17, 51], [18, 31], [18, 35], [18, 36], [18, 43], [18, 46], [18, 58], [18, 59], [19, 20], [19, 25], [19, 40], [19, 51], [20, 23], [20, 26], [20, 28], [20, 35], [20, 36], [20, 41], [20, 43], [20, 46], [20, 51], [21, 26], [21, 27], [21, 37], [21, 40], [21, 44], [21, 51], [22, 24], [22, 26], [22, 29], [22, 34], [22, 40], [22, 41], [22, 46], [22, 49], [22, 55], [23, 26], [23, 28], [23, 31], [23, 32], [23, 33], [23, 37], [23, 38], [23, 41], [23, 46], [23, 50], [23, 57], [23, 58], [23, 59], [24, 27], [24, 28], [24, 42], [24, 44], [24, 59], [25, 27], [25, 38], [25, 46], [25, 48], [25, 53], [26, 32], [26, 33], [26, 34], [26, 53], [26, 54], [26, 55], [26, 56], [26, 57], [27, 30], [27, 34], [27, 36], [27, 38], [27, 42], [27, 48], [27, 49], [27, 50], [27, 54], [28, 29], [28, 30], [28, 35], [28, 40], [28, 41], [28, 49], [28, 54], [29, 37], [29, 49], [30, 53], [31, 32], [31, 36], [31, 41], [31, 45], [31, 49], [31, 52], [32, 36], [32, 39], [32, 49], [32, 51], [33, 37], [33, 40], [33, 42], [33, 46], [33, 51], [33, 56], [33, 58], [34, 39], [34, 47], [34, 49], [34, 56], [34, 57], [35, 37], [35, 51], [35, 59], [36, 42], [36, 46], [36, 50], [36, 54], [36, 56], [36, 57], [36, 58], [36, 59], [37, 39], [37, 42], [37, 44], [37, 47], [37, 48], [37, 56], [38, 43], [38, 45], [38, 46], [38, 50], [39, 47], [39, 51], [39, 52], [39, 59], [40, 41], [40, 45], [40, 54], [40, 55], [41, 43], [41, 45], [41, 49], [41, 57], [41, 59], [42, 45], [42, 49], [42, 51], [42, 52], [42, 53], [43, 56], [43, 59], [44, 53], [44, 59], [45, 51], [45, 52], [46, 48], [46, 51], [46, 55], [48, 53], [48, 59], [49, 56], [49, 58], [51, 54], [51, 56], [51, 57], [51, 59], [52, 53], [53, 54]], "family": "mc_vertex_cover", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-vertex-cover-l4-s9/solution/solve.sh b/tasks/construct-mc-vertex-cover-l4-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..58b7f00808699148d14d7a7c141a47681b814d96
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-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'
+{"cover": [1, 2, 4, 5, 6, 7, 9, 10, 11, 12, 14, 15, 16, 18, 20, 22, 23, 24, 25, 26, 27, 28, 31, 34, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 49, 51, 53, 56, 57, 58, 59]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-vertex-cover-l4-s9/tests/check.py b/tasks/construct-mc-vertex-cover-l4-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..e3accb3da1c773fd2126b8b5b0a97b099e29c83b
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-l4-s9/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-l4-s9/tests/instance.json b/tasks/construct-mc-vertex-cover-l4-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..174e0cbd83449a0ea3ea0bfd1d6045afcab705a0
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-l4-s9/tests/instance.json
@@ -0,0 +1 @@
+{"n": 60, "k": 41, "edges": [[0, 6], [0, 7], [0, 10], [0, 12], [0, 14], [0, 16], [0, 24], [0, 25], [0, 26], [0, 31], [0, 34], [0, 46], [0, 56], [1, 6], [1, 8], [1, 11], [1, 12], [1, 13], [1, 17], [1, 28], [1, 32], [1, 33], [1, 35], [1, 36], [1, 43], [1, 56], [1, 59], [2, 5], [2, 6], [2, 15], [2, 24], [2, 27], [2, 31], [2, 34], [2, 35], [2, 39], [2, 40], [2, 46], [2, 47], [3, 4], [3, 9], [3, 12], [3, 25], [3, 26], [3, 28], [3, 34], [3, 39], [3, 40], [3, 41], [3, 44], [3, 51], [3, 58], [4, 5], [4, 16], [4, 17], [4, 18], [4, 27], [4, 37], [4, 38], [4, 39], [4, 40], [4, 46], [4, 47], [4, 49], [4, 51], [4, 52], [5, 8], [5, 11], [5, 13], [5, 14], [5, 18], [5, 19], [5, 24], [5, 28], [5, 41], [5, 43], [5, 46], [5, 49], [5, 53], [5, 58], [6, 13], [6, 15], [6, 17], [6, 21], [6, 27], [6, 38], [6, 44], [6, 45], [6, 49], [6, 52], [6, 57], [7, 8], [7, 12], [7, 17], [7, 18], [7, 21], [7, 28], [7, 36], [7, 45], [7, 50], [7, 58], [8, 9], [8, 20], [8, 22], [8, 26], [9, 12], [9, 16], [9, 19], [9, 30], [9, 34], [9, 36], [9, 39], [9, 42], [9, 43], [9, 44], [10, 12], [10, 20], [10, 22], [10, 25], [10, 29], [10, 46], [10, 49], [10, 51], [10, 53], [11, 18], [11, 22], [11, 28], [11, 36], [11, 37], [11, 40], [11, 47], [11, 50], [11, 54], [11, 56], [12, 13], [12, 14], [12, 18], [12, 19], [12, 23], [12, 29], [12, 40], [12, 54], [13, 18], [13, 20], [13, 22], [13, 31], [13, 38], [13, 59], [14, 23], [14, 27], [14, 29], [14, 31], [14, 32], [14, 35], [14, 36], [14, 44], [14, 56], [14, 58], [15, 16], [15, 20], [15, 22], [15, 25], [15, 33], [15, 34], [15, 39], [15, 41], [15, 46], [15, 52], [15, 53], [15, 56], [16, 18], [16, 28], [16, 31], [16, 37], [16, 39], [16, 43], [16, 46], [16, 47], [16, 59], [17, 25], [17, 26], [17, 40], [17, 42], [17, 51], [18, 31], [18, 35], [18, 36], [18, 43], [18, 46], [18, 58], [18, 59], [19, 20], [19, 25], [19, 40], [19, 51], [20, 23], [20, 26], [20, 28], [20, 35], [20, 36], [20, 41], [20, 43], [20, 46], [20, 51], [21, 26], [21, 27], [21, 37], [21, 40], [21, 44], [21, 51], [22, 24], [22, 26], [22, 29], [22, 34], [22, 40], [22, 41], [22, 46], [22, 49], [22, 55], [23, 26], [23, 28], [23, 31], [23, 32], [23, 33], [23, 37], [23, 38], [23, 41], [23, 46], [23, 50], [23, 57], [23, 58], [23, 59], [24, 27], [24, 28], [24, 42], [24, 44], [24, 59], [25, 27], [25, 38], [25, 46], [25, 48], [25, 53], [26, 32], [26, 33], [26, 34], [26, 53], [26, 54], [26, 55], [26, 56], [26, 57], [27, 30], [27, 34], [27, 36], [27, 38], [27, 42], [27, 48], [27, 49], [27, 50], [27, 54], [28, 29], [28, 30], [28, 35], [28, 40], [28, 41], [28, 49], [28, 54], [29, 37], [29, 49], [30, 53], [31, 32], [31, 36], [31, 41], [31, 45], [31, 49], [31, 52], [32, 36], [32, 39], [32, 49], [32, 51], [33, 37], [33, 40], [33, 42], [33, 46], [33, 51], [33, 56], [33, 58], [34, 39], [34, 47], [34, 49], [34, 56], [34, 57], [35, 37], [35, 51], [35, 59], [36, 42], [36, 46], [36, 50], [36, 54], [36, 56], [36, 57], [36, 58], [36, 59], [37, 39], [37, 42], [37, 44], [37, 47], [37, 48], [37, 56], [38, 43], [38, 45], [38, 46], [38, 50], [39, 47], [39, 51], [39, 52], [39, 59], [40, 41], [40, 45], [40, 54], [40, 55], [41, 43], [41, 45], [41, 49], [41, 57], [41, 59], [42, 45], [42, 49], [42, 51], [42, 52], [42, 53], [43, 56], [43, 59], [44, 53], [44, 59], [45, 51], [45, 52], [46, 48], [46, 51], [46, 55], [48, 53], [48, 59], [49, 56], [49, 58], [51, 54], [51, 56], [51, 57], [51, 59], [52, 53], [53, 54]], "family": "mc_vertex_cover", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-vertex-cover-l4-s9/tests/run.py b/tasks/construct-mc-vertex-cover-l4-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-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-mc-vertex-cover-l4-s9/tests/test.sh b/tasks/construct-mc-vertex-cover-l4-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-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-nppc-3col-l4-s0/environment/Dockerfile b/tasks/construct-nppc-3col-l4-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-3col-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-nppc-3col-l4-s0/environment/instance.json b/tasks/construct-nppc-3col-l4-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2d9fe28642676def8ce48ffb054a9eb771dd6dd8
--- /dev/null
+++ b/tasks/construct-nppc-3col-l4-s0/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 30, "edges": [[0, 14], [0, 18], [0, 22], [0, 28], [1, 3], [1, 8], [1, 10], [1, 12], [1, 13], [1, 14], [1, 19], [1, 23], [2, 17], [2, 24], [3, 22], [3, 27], [4, 17], [4, 22], [4, 25], [4, 26], [5, 11], [5, 12], [5, 14], [5, 16], [5, 18], [6, 12], [6, 18], [6, 22], [6, 25], [6, 29], [7, 12], [7, 13], [8, 10], [8, 13], [8, 20], [8, 22], [8, 24], [9, 12], [9, 14], [9, 19], [9, 28], [10, 17], [10, 26], [10, 28], [10, 29], [11, 19], [11, 25], [11, 29], [12, 19], [12, 20], [12, 23], [12, 25], [12, 27], [13, 17], [13, 21], [14, 17], [14, 19], [14, 22], [14, 25], [14, 27], [16, 24], [16, 28], [17, 26], [17, 27], [18, 19], [18, 20], [18, 21], [19, 21], [19, 28], [20, 21], [20, 22], [20, 25], [20, 26], [20, 29], [21, 27], [21, 28], [22, 28], [23, 29], [26, 29], [28, 29]], "family": "nppc_three_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-3col-l4-s0/solution/solve.sh b/tasks/construct-nppc-3col-l4-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..4a1a6b0ddd7fd92defb42a18b23000d3d042293d
--- /dev/null
+++ b/tasks/construct-nppc-3col-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'
+{"coloring": [0, 1, 2, 2, 0, 1, 0, 1, 2, 1, 0, 2, 2, 0, 2, 2, 0, 1, 2, 0, 0, 1, 1, 0, 1, 1, 2, 0, 2, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-3col-l4-s0/tests/check.py b/tasks/construct-nppc-3col-l4-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..72362ed9180250b6c53ce0c71867ca476ac2af20
--- /dev/null
+++ b/tasks/construct-nppc-3col-l4-s0/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-l4-s0/tests/instance.json b/tasks/construct-nppc-3col-l4-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2d9fe28642676def8ce48ffb054a9eb771dd6dd8
--- /dev/null
+++ b/tasks/construct-nppc-3col-l4-s0/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 30, "edges": [[0, 14], [0, 18], [0, 22], [0, 28], [1, 3], [1, 8], [1, 10], [1, 12], [1, 13], [1, 14], [1, 19], [1, 23], [2, 17], [2, 24], [3, 22], [3, 27], [4, 17], [4, 22], [4, 25], [4, 26], [5, 11], [5, 12], [5, 14], [5, 16], [5, 18], [6, 12], [6, 18], [6, 22], [6, 25], [6, 29], [7, 12], [7, 13], [8, 10], [8, 13], [8, 20], [8, 22], [8, 24], [9, 12], [9, 14], [9, 19], [9, 28], [10, 17], [10, 26], [10, 28], [10, 29], [11, 19], [11, 25], [11, 29], [12, 19], [12, 20], [12, 23], [12, 25], [12, 27], [13, 17], [13, 21], [14, 17], [14, 19], [14, 22], [14, 25], [14, 27], [16, 24], [16, 28], [17, 26], [17, 27], [18, 19], [18, 20], [18, 21], [19, 21], [19, 28], [20, 21], [20, 22], [20, 25], [20, 26], [20, 29], [21, 27], [21, 28], [22, 28], [23, 29], [26, 29], [28, 29]], "family": "nppc_three_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-3col-l4-s0/tests/run.py b/tasks/construct-nppc-3col-l4-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-3col-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-nppc-3col-l4-s0/tests/test.sh b/tasks/construct-nppc-3col-l4-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-3col-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-nppc-3col-l4-s7/environment/Dockerfile b/tasks/construct-nppc-3col-l4-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-3col-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-nppc-3col-l4-s7/environment/instance.json b/tasks/construct-nppc-3col-l4-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..ef5fa0fc6327e8596176657761de45f4dc4145f5
--- /dev/null
+++ b/tasks/construct-nppc-3col-l4-s7/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 30, "edges": [[0, 8], [0, 11], [0, 13], [0, 14], [0, 18], [0, 28], [1, 3], [1, 5], [1, 9], [1, 16], [1, 18], [1, 22], [1, 28], [2, 4], [2, 20], [2, 26], [2, 29], [3, 13], [3, 14], [3, 15], [3, 19], [3, 24], [3, 28], [4, 26], [5, 11], [5, 14], [5, 15], [5, 24], [6, 9], [6, 11], [6, 14], [6, 15], [6, 18], [6, 26], [6, 29], [7, 8], [7, 21], [7, 22], [8, 20], [9, 21], [9, 22], [9, 23], [9, 27], [10, 11], [10, 19], [10, 28], [11, 20], [11, 21], [11, 26], [11, 27], [12, 26], [12, 27], [13, 16], [13, 21], [13, 27], [13, 28], [14, 20], [14, 21], [14, 23], [14, 24], [14, 27], [16, 17], [16, 24], [17, 18], [17, 29], [18, 23], [18, 24], [19, 25], [21, 23], [21, 24], [22, 23], [22, 24], [22, 26], [22, 29], [23, 25], [23, 27], [24, 28], [25, 29], [26, 27], [27, 28]], "family": "nppc_three_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-3col-l4-s7/solution/solve.sh b/tasks/construct-nppc-3col-l4-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..df4db46298023d8b24f56bf670d5a39bbf65784f
--- /dev/null
+++ b/tasks/construct-nppc-3col-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'
+{"coloring": [1, 2, 1, 1, 0, 1, 1, 2, 0, 0, 2, 0, 0, 2, 0, 2, 0, 1, 0, 0, 2, 1, 1, 2, 2, 1, 2, 1, 0, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-3col-l4-s7/tests/check.py b/tasks/construct-nppc-3col-l4-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..72362ed9180250b6c53ce0c71867ca476ac2af20
--- /dev/null
+++ b/tasks/construct-nppc-3col-l4-s7/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-l4-s7/tests/instance.json b/tasks/construct-nppc-3col-l4-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..ef5fa0fc6327e8596176657761de45f4dc4145f5
--- /dev/null
+++ b/tasks/construct-nppc-3col-l4-s7/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 30, "edges": [[0, 8], [0, 11], [0, 13], [0, 14], [0, 18], [0, 28], [1, 3], [1, 5], [1, 9], [1, 16], [1, 18], [1, 22], [1, 28], [2, 4], [2, 20], [2, 26], [2, 29], [3, 13], [3, 14], [3, 15], [3, 19], [3, 24], [3, 28], [4, 26], [5, 11], [5, 14], [5, 15], [5, 24], [6, 9], [6, 11], [6, 14], [6, 15], [6, 18], [6, 26], [6, 29], [7, 8], [7, 21], [7, 22], [8, 20], [9, 21], [9, 22], [9, 23], [9, 27], [10, 11], [10, 19], [10, 28], [11, 20], [11, 21], [11, 26], [11, 27], [12, 26], [12, 27], [13, 16], [13, 21], [13, 27], [13, 28], [14, 20], [14, 21], [14, 23], [14, 24], [14, 27], [16, 17], [16, 24], [17, 18], [17, 29], [18, 23], [18, 24], [19, 25], [21, 23], [21, 24], [22, 23], [22, 24], [22, 26], [22, 29], [23, 25], [23, 27], [24, 28], [25, 29], [26, 27], [27, 28]], "family": "nppc_three_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-3col-l4-s7/tests/run.py b/tasks/construct-nppc-3col-l4-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-3col-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-nppc-3col-l4-s7/tests/test.sh b/tasks/construct-nppc-3col-l4-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-3col-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-nppc-bandwidth-l2-s2/environment/Dockerfile b/tasks/construct-nppc-bandwidth-l2-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-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-nppc-bandwidth-l2-s2/environment/instance.json b/tasks/construct-nppc-bandwidth-l2-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c9e5effbd9c1bed1a67b627dde60d36ad606894c
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l2-s2/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 8, "bandwidth": 3, "edges": [[0, 4], [1, 2], [2, 3], [3, 4], [3, 5], [3, 7], [4, 5], [5, 6]], "family": "nppc_bandwidth", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-bandwidth-l2-s2/solution/solve.sh b/tasks/construct-nppc-bandwidth-l2-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..7b8e36f8333f25bd211414a830cc7c481cafd9fc
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-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'
+{"layout": [7, 3, 1, 2, 4, 5, 6, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-bandwidth-l2-s2/tests/check.py b/tasks/construct-nppc-bandwidth-l2-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..0f847cded257e6c404063dfafde33fa44163afa2
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l2-s2/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-l2-s2/tests/instance.json b/tasks/construct-nppc-bandwidth-l2-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c9e5effbd9c1bed1a67b627dde60d36ad606894c
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l2-s2/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 8, "bandwidth": 3, "edges": [[0, 4], [1, 2], [2, 3], [3, 4], [3, 5], [3, 7], [4, 5], [5, 6]], "family": "nppc_bandwidth", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-bandwidth-l2-s2/tests/run.py b/tasks/construct-nppc-bandwidth-l2-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-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-nppc-bandwidth-l2-s2/tests/test.sh b/tasks/construct-nppc-bandwidth-l2-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-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-nppc-bandwidth-l2-s5/environment/Dockerfile b/tasks/construct-nppc-bandwidth-l2-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l2-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-bandwidth-l2-s5/environment/instance.json b/tasks/construct-nppc-bandwidth-l2-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..6e9d4c18bb96a99d39f82fd985a27e27c81d090a
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l2-s5/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 8, "bandwidth": 3, "edges": [[0, 1], [1, 3], [2, 4], [3, 6], [4, 5], [4, 7], [5, 6], [6, 7]], "family": "nppc_bandwidth", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-bandwidth-l2-s5/solution/solve.sh b/tasks/construct-nppc-bandwidth-l2-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..3d5709a32d6dfaaa06d21cd5966badba74e8c0cd
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l2-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'
+{"layout": [4, 6, 0, 7, 1, 3, 5, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-bandwidth-l2-s5/tests/check.py b/tasks/construct-nppc-bandwidth-l2-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..0f847cded257e6c404063dfafde33fa44163afa2
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l2-s5/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-l2-s5/tests/instance.json b/tasks/construct-nppc-bandwidth-l2-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..6e9d4c18bb96a99d39f82fd985a27e27c81d090a
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l2-s5/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 8, "bandwidth": 3, "edges": [[0, 1], [1, 3], [2, 4], [3, 6], [4, 5], [4, 7], [5, 6], [6, 7]], "family": "nppc_bandwidth", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-bandwidth-l2-s5/tests/run.py b/tasks/construct-nppc-bandwidth-l2-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l2-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-bandwidth-l2-s5/tests/test.sh b/tasks/construct-nppc-bandwidth-l2-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l2-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-dominating-set-l3-s2/environment/Dockerfile b/tasks/construct-nppc-dominating-set-l3-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-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-nppc-dominating-set-l3-s2/environment/instance.json b/tasks/construct-nppc-dominating-set-l3-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d5d5835a2c9212ac54a8eef1c54410a31fb1711e
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l3-s2/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 50, "k": 20, "edges": [[0, 2], [0, 6], [0, 9], [0, 10], [0, 11], [0, 14], [0, 22], [0, 26], [0, 30], [0, 36], [0, 39], [0, 43], [0, 47], [1, 2], [1, 5], [1, 6], [1, 11], [1, 12], [1, 13], [1, 14], [1, 18], [1, 21], [1, 23], [1, 25], [1, 30], [1, 33], [1, 34], [1, 35], [1, 37], [1, 38], [1, 42], [2, 3], [2, 4], [2, 15], [2, 16], [2, 18], [2, 19], [2, 21], [2, 25], [2, 26], [2, 32], [2, 34], [2, 42], [2, 44], [2, 46], [3, 6], [3, 9], [3, 16], [3, 17], [3, 20], [3, 22], [3, 27], [3, 29], [3, 35], [3, 36], [3, 39], [3, 48], [4, 5], [4, 6], [4, 8], [4, 9], [4, 11], [4, 15], [4, 24], [4, 27], [4, 28], [4, 29], [4, 32], [4, 34], [4, 35], [4, 37], [4, 40], [4, 41], [4, 43], [5, 7], [5, 9], [5, 14], [5, 20], [5, 21], [5, 22], [5, 26], [5, 27], [5, 28], [5, 30], [5, 32], [5, 34], [5, 39], [5, 41], [5, 45], [5, 47], [6, 8], [6, 9], [6, 13], [6, 16], [6, 21], [6, 23], [6, 26], [6, 29], [6, 31], [6, 32], [6, 36], [6, 38], [6, 45], [7, 11], [7, 23], [7, 29], [7, 30], [7, 32], [7, 40], [7, 41], [7, 42], [7, 43], [7, 44], [7, 46], [8, 13], [8, 15], [8, 16], [8, 17], [8, 18], [8, 21], [8, 24], [8, 28], [8, 38], [8, 43], [8, 46], [8, 47], [8, 48], [9, 11], [9, 18], [9, 19], [9, 23], [9, 26], [9, 33], [9, 37], [9, 41], [9, 42], [9, 46], [9, 47], [9, 49], [10, 11], [10, 13], [10, 17], [10, 24], [10, 28], [10, 32], [10, 37], [10, 41], [10, 42], [10, 45], [10, 46], [10, 48], [11, 14], [11, 18], [11, 20], [11, 22], [11, 25], [11, 27], [11, 31], [11, 32], [11, 33], [11, 39], [11, 40], [11, 41], [11, 43], [11, 47], [11, 48], [12, 13], [12, 19], [12, 25], [12, 27], [12, 30], [12, 31], [12, 33], [12, 38], [12, 41], [12, 47], [13, 14], [13, 15], [13, 17], [13, 19], [13, 20], [13, 21], [13, 22], [13, 25], [13, 29], [13, 34], [13, 35], [13, 37], [13, 40], [13, 41], [13, 43], [13, 45], [13, 46], [13, 48], [14, 17], [14, 21], [14, 23], [14, 25], [14, 30], [14, 31], [14, 32], [14, 37], [14, 39], [14, 40], [14, 43], [14, 44], [14, 45], [14, 46], [15, 16], [15, 17], [15, 18], [15, 26], [15, 31], [15, 37], [15, 38], [15, 40], [15, 41], [15, 48], [16, 19], [16, 24], [16, 27], [16, 32], [16, 39], [16, 40], [16, 44], [16, 45], [16, 47], [17, 20], [17, 23], [17, 28], [17, 30], [17, 31], [17, 37], [17, 40], [17, 42], [17, 44], [17, 48], [18, 26], [18, 28], [18, 34], [18, 35], [18, 40], [18, 47], [18, 48], [19, 26], [19, 34], [19, 38], [19, 41], [19, 45], [19, 47], [19, 49], [20, 23], [20, 29], [20, 30], [20, 32], [20, 37], [20, 38], [20, 39], [20, 40], [20, 41], [20, 47], [21, 22], [21, 24], [21, 27], [21, 29], [21, 30], [21, 32], [21, 36], [21, 41], [21, 44], [21, 46], [21, 47], [22, 25], [22, 27], [22, 29], [22, 30], [22, 32], [22, 45], [22, 47], [23, 25], [23, 26], [23, 28], [23, 30], [23, 31], [23, 33], [23, 38], [23, 42], [23, 44], [23, 45], [24, 27], [24, 35], [24, 36], [24, 37], [24, 39], [24, 41], [24, 42], [24, 43], [24, 46], [25, 26], [25, 30], [25, 32], [25, 33], [25, 40], [25, 44], [25, 47], [25, 48], [26, 28], [26, 33], [26, 35], [26, 37], [26, 40], [26, 46], [27, 28], [27, 32], [27, 36], [27, 38], [27, 48], [28, 32], [28, 33], [28, 40], [28, 41], [28, 42], [28, 43], [28, 46], [29, 33], [29, 36], [29, 37], [29, 38], [29, 46], [29, 49], [30, 34], [30, 35], [30, 37], [30, 41], [30, 43], [30, 48], [30, 49], [31, 36], [31, 37], [31, 41], [31, 42], [31, 47], [31, 48], [31, 49], [32, 35], [32, 37], [32, 42], [32, 46], [33, 34], [33, 43], [33, 49], [34, 38], [34, 39], [34, 42], [34, 44], [34, 45], [34, 46], [35, 36], [35, 37], [35, 41], [35, 43], [35, 45], [36, 37], [36, 41], [36, 42], [36, 45], [36, 46], [36, 47], [36, 49], [37, 38], [37, 39], [37, 40], [37, 48], [38, 46], [38, 47], [39, 44], [39, 46], [39, 48], [40, 41], [40, 45], [40, 46], [40, 48], [41, 42], [41, 46], [41, 48], [41, 49], [42, 44], [42, 46], [42, 47], [43, 46], [43, 48], [43, 49], [44, 48], [45, 46], [45, 48], [45, 49], [46, 47], [46, 48], [46, 49]], "family": "nppc_dominating_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-dominating-set-l3-s2/solution/solve.sh b/tasks/construct-nppc-dominating-set-l3-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..574aaa835f7ea79e5954c7e0c9692e03c16f1b49
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-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'
+{"dominating_set": [0, 1, 5, 6, 8, 12, 13, 19, 23, 28, 30, 32, 36, 37, 39, 40, 41, 43, 44, 46]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-dominating-set-l3-s2/tests/check.py b/tasks/construct-nppc-dominating-set-l3-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..0677a7bcbccb02297d022ffa4b5dd15da6371bff
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l3-s2/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-l3-s2/tests/instance.json b/tasks/construct-nppc-dominating-set-l3-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d5d5835a2c9212ac54a8eef1c54410a31fb1711e
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l3-s2/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 50, "k": 20, "edges": [[0, 2], [0, 6], [0, 9], [0, 10], [0, 11], [0, 14], [0, 22], [0, 26], [0, 30], [0, 36], [0, 39], [0, 43], [0, 47], [1, 2], [1, 5], [1, 6], [1, 11], [1, 12], [1, 13], [1, 14], [1, 18], [1, 21], [1, 23], [1, 25], [1, 30], [1, 33], [1, 34], [1, 35], [1, 37], [1, 38], [1, 42], [2, 3], [2, 4], [2, 15], [2, 16], [2, 18], [2, 19], [2, 21], [2, 25], [2, 26], [2, 32], [2, 34], [2, 42], [2, 44], [2, 46], [3, 6], [3, 9], [3, 16], [3, 17], [3, 20], [3, 22], [3, 27], [3, 29], [3, 35], [3, 36], [3, 39], [3, 48], [4, 5], [4, 6], [4, 8], [4, 9], [4, 11], [4, 15], [4, 24], [4, 27], [4, 28], [4, 29], [4, 32], [4, 34], [4, 35], [4, 37], [4, 40], [4, 41], [4, 43], [5, 7], [5, 9], [5, 14], [5, 20], [5, 21], [5, 22], [5, 26], [5, 27], [5, 28], [5, 30], [5, 32], [5, 34], [5, 39], [5, 41], [5, 45], [5, 47], [6, 8], [6, 9], [6, 13], [6, 16], [6, 21], [6, 23], [6, 26], [6, 29], [6, 31], [6, 32], [6, 36], [6, 38], [6, 45], [7, 11], [7, 23], [7, 29], [7, 30], [7, 32], [7, 40], [7, 41], [7, 42], [7, 43], [7, 44], [7, 46], [8, 13], [8, 15], [8, 16], [8, 17], [8, 18], [8, 21], [8, 24], [8, 28], [8, 38], [8, 43], [8, 46], [8, 47], [8, 48], [9, 11], [9, 18], [9, 19], [9, 23], [9, 26], [9, 33], [9, 37], [9, 41], [9, 42], [9, 46], [9, 47], [9, 49], [10, 11], [10, 13], [10, 17], [10, 24], [10, 28], [10, 32], [10, 37], [10, 41], [10, 42], [10, 45], [10, 46], [10, 48], [11, 14], [11, 18], [11, 20], [11, 22], [11, 25], [11, 27], [11, 31], [11, 32], [11, 33], [11, 39], [11, 40], [11, 41], [11, 43], [11, 47], [11, 48], [12, 13], [12, 19], [12, 25], [12, 27], [12, 30], [12, 31], [12, 33], [12, 38], [12, 41], [12, 47], [13, 14], [13, 15], [13, 17], [13, 19], [13, 20], [13, 21], [13, 22], [13, 25], [13, 29], [13, 34], [13, 35], [13, 37], [13, 40], [13, 41], [13, 43], [13, 45], [13, 46], [13, 48], [14, 17], [14, 21], [14, 23], [14, 25], [14, 30], [14, 31], [14, 32], [14, 37], [14, 39], [14, 40], [14, 43], [14, 44], [14, 45], [14, 46], [15, 16], [15, 17], [15, 18], [15, 26], [15, 31], [15, 37], [15, 38], [15, 40], [15, 41], [15, 48], [16, 19], [16, 24], [16, 27], [16, 32], [16, 39], [16, 40], [16, 44], [16, 45], [16, 47], [17, 20], [17, 23], [17, 28], [17, 30], [17, 31], [17, 37], [17, 40], [17, 42], [17, 44], [17, 48], [18, 26], [18, 28], [18, 34], [18, 35], [18, 40], [18, 47], [18, 48], [19, 26], [19, 34], [19, 38], [19, 41], [19, 45], [19, 47], [19, 49], [20, 23], [20, 29], [20, 30], [20, 32], [20, 37], [20, 38], [20, 39], [20, 40], [20, 41], [20, 47], [21, 22], [21, 24], [21, 27], [21, 29], [21, 30], [21, 32], [21, 36], [21, 41], [21, 44], [21, 46], [21, 47], [22, 25], [22, 27], [22, 29], [22, 30], [22, 32], [22, 45], [22, 47], [23, 25], [23, 26], [23, 28], [23, 30], [23, 31], [23, 33], [23, 38], [23, 42], [23, 44], [23, 45], [24, 27], [24, 35], [24, 36], [24, 37], [24, 39], [24, 41], [24, 42], [24, 43], [24, 46], [25, 26], [25, 30], [25, 32], [25, 33], [25, 40], [25, 44], [25, 47], [25, 48], [26, 28], [26, 33], [26, 35], [26, 37], [26, 40], [26, 46], [27, 28], [27, 32], [27, 36], [27, 38], [27, 48], [28, 32], [28, 33], [28, 40], [28, 41], [28, 42], [28, 43], [28, 46], [29, 33], [29, 36], [29, 37], [29, 38], [29, 46], [29, 49], [30, 34], [30, 35], [30, 37], [30, 41], [30, 43], [30, 48], [30, 49], [31, 36], [31, 37], [31, 41], [31, 42], [31, 47], [31, 48], [31, 49], [32, 35], [32, 37], [32, 42], [32, 46], [33, 34], [33, 43], [33, 49], [34, 38], [34, 39], [34, 42], [34, 44], [34, 45], [34, 46], [35, 36], [35, 37], [35, 41], [35, 43], [35, 45], [36, 37], [36, 41], [36, 42], [36, 45], [36, 46], [36, 47], [36, 49], [37, 38], [37, 39], [37, 40], [37, 48], [38, 46], [38, 47], [39, 44], [39, 46], [39, 48], [40, 41], [40, 45], [40, 46], [40, 48], [41, 42], [41, 46], [41, 48], [41, 49], [42, 44], [42, 46], [42, 47], [43, 46], [43, 48], [43, 49], [44, 48], [45, 46], [45, 48], [45, 49], [46, 47], [46, 48], [46, 49]], "family": "nppc_dominating_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-dominating-set-l3-s2/tests/run.py b/tasks/construct-nppc-dominating-set-l3-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-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-nppc-dominating-set-l3-s2/tests/test.sh b/tasks/construct-nppc-dominating-set-l3-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-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-nppc-dominating-set-l3-s5/environment/Dockerfile b/tasks/construct-nppc-dominating-set-l3-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l3-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-dominating-set-l3-s5/environment/instance.json b/tasks/construct-nppc-dominating-set-l3-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d97861139737aad4901dcdc6f83bb2d0aa37a856
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l3-s5/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 50, "k": 20, "edges": [[0, 2], [0, 4], [0, 5], [0, 8], [0, 10], [0, 19], [0, 20], [0, 22], [0, 23], [0, 25], [0, 27], [0, 28], [0, 31], [0, 35], [0, 36], [0, 38], [0, 39], [0, 40], [0, 42], [0, 49], [1, 6], [1, 9], [1, 10], [1, 13], [1, 17], [1, 19], [1, 23], [1, 27], [1, 36], [1, 37], [1, 39], [1, 41], [1, 47], [2, 5], [2, 17], [2, 20], [2, 24], [2, 28], [2, 31], [2, 37], [2, 42], [2, 43], [2, 45], [3, 4], [3, 8], [3, 13], [3, 15], [3, 17], [3, 18], [3, 19], [3, 24], [3, 27], [3, 35], [3, 40], [3, 41], [3, 44], [3, 47], [4, 6], [4, 16], [4, 22], [4, 25], [4, 29], [4, 30], [4, 31], [4, 32], [4, 33], [4, 37], [4, 42], [4, 43], [4, 45], [5, 6], [5, 11], [5, 23], [5, 25], [5, 26], [5, 31], [5, 34], [5, 35], [5, 36], [5, 45], [5, 46], [6, 12], [6, 13], [6, 17], [6, 19], [6, 20], [6, 22], [6, 23], [6, 24], [6, 25], [6, 32], [6, 33], [6, 34], [6, 35], [6, 36], [6, 37], [6, 38], [6, 40], [6, 49], [7, 11], [7, 14], [7, 16], [7, 17], [7, 20], [7, 30], [7, 33], [7, 35], [7, 36], [7, 41], [7, 44], [7, 48], [8, 9], [8, 10], [8, 13], [8, 16], [8, 23], [8, 29], [8, 31], [8, 33], [8, 34], [8, 41], [8, 43], [8, 47], [8, 48], [9, 11], [9, 15], [9, 19], [9, 22], [9, 26], [9, 27], [9, 31], [9, 32], [9, 34], [9, 36], [9, 45], [9, 48], [9, 49], [10, 12], [10, 15], [10, 17], [10, 27], [10, 32], [10, 36], [10, 38], [10, 39], [10, 46], [11, 12], [11, 17], [11, 20], [11, 23], [11, 24], [11, 25], [11, 33], [11, 34], [11, 36], [11, 40], [11, 42], [11, 45], [11, 49], [12, 14], [12, 20], [12, 22], [12, 26], [12, 31], [12, 32], [12, 34], [12, 36], [12, 37], [12, 38], [12, 47], [13, 14], [13, 16], [13, 22], [13, 23], [13, 28], [13, 29], [13, 32], [13, 34], [13, 36], [13, 44], [13, 48], [14, 15], [14, 18], [14, 19], [14, 20], [14, 22], [14, 23], [14, 26], [14, 30], [14, 32], [14, 34], [14, 42], [14, 48], [15, 22], [15, 26], [15, 27], [15, 28], [15, 33], [15, 34], [15, 35], [15, 36], [15, 39], [15, 40], [16, 17], [16, 24], [16, 30], [16, 34], [16, 37], [16, 38], [16, 39], [16, 40], [16, 41], [16, 43], [16, 44], [16, 45], [16, 46], [16, 47], [16, 48], [17, 18], [17, 21], [17, 29], [17, 30], [17, 35], [17, 43], [17, 46], [18, 24], [18, 34], [18, 35], [18, 37], [18, 38], [18, 43], [18, 47], [18, 49], [19, 22], [19, 27], [19, 32], [19, 33], [19, 37], [19, 44], [19, 47], [20, 21], [20, 23], [20, 29], [20, 32], [20, 33], [20, 43], [20, 47], [21, 22], [21, 30], [21, 31], [21, 33], [21, 34], [21, 35], [21, 39], [21, 41], [22, 23], [22, 29], [22, 34], [22, 39], [22, 41], [22, 47], [22, 48], [23, 24], [23, 25], [23, 29], [23, 30], [23, 32], [23, 36], [23, 37], [23, 40], [23, 42], [23, 48], [24, 26], [24, 31], [24, 32], [24, 34], [24, 35], [24, 40], [24, 41], [24, 46], [24, 48], [24, 49], [25, 27], [25, 31], [25, 35], [25, 37], [25, 39], [25, 46], [26, 31], [26, 33], [26, 34], [26, 35], [26, 36], [26, 41], [26, 43], [26, 47], [26, 49], [27, 28], [27, 33], [27, 34], [27, 35], [27, 38], [27, 42], [27, 45], [27, 46], [28, 32], [28, 34], [28, 36], [28, 39], [28, 43], [28, 44], [28, 47], [28, 48], [29, 30], [29, 32], [29, 33], [29, 34], [29, 35], [29, 38], [29, 41], [29, 49], [30, 31], [30, 32], [30, 38], [30, 43], [30, 45], [30, 46], [31, 32], [31, 34], [31, 40], [31, 44], [31, 45], [31, 46], [31, 49], [32, 33], [32, 35], [32, 38], [32, 47], [33, 34], [33, 37], [33, 38], [33, 40], [33, 45], [33, 48], [33, 49], [34, 37], [34, 41], [34, 43], [34, 44], [34, 45], [34, 49], [35, 38], [35, 39], [35, 40], [35, 41], [35, 44], [35, 48], [36, 39], [36, 44], [37, 39], [37, 40], [37, 47], [37, 49], [38, 41], [38, 46], [38, 48], [39, 45], [39, 46], [39, 49], [40, 44], [40, 45], [40, 46], [40, 48], [41, 42], [41, 47], [41, 48], [42, 46], [42, 47], [42, 49], [43, 47], [44, 48], [45, 46]], "family": "nppc_dominating_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-dominating-set-l3-s5/solution/solve.sh b/tasks/construct-nppc-dominating-set-l3-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..5508ab1e85baea38e05776e642ede20aabe597be
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l3-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'
+{"dominating_set": [1, 2, 4, 5, 9, 10, 11, 12, 13, 16, 17, 19, 20, 23, 26, 28, 33, 34, 42, 48]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-dominating-set-l3-s5/tests/check.py b/tasks/construct-nppc-dominating-set-l3-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..0677a7bcbccb02297d022ffa4b5dd15da6371bff
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l3-s5/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-l3-s5/tests/instance.json b/tasks/construct-nppc-dominating-set-l3-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d97861139737aad4901dcdc6f83bb2d0aa37a856
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l3-s5/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 50, "k": 20, "edges": [[0, 2], [0, 4], [0, 5], [0, 8], [0, 10], [0, 19], [0, 20], [0, 22], [0, 23], [0, 25], [0, 27], [0, 28], [0, 31], [0, 35], [0, 36], [0, 38], [0, 39], [0, 40], [0, 42], [0, 49], [1, 6], [1, 9], [1, 10], [1, 13], [1, 17], [1, 19], [1, 23], [1, 27], [1, 36], [1, 37], [1, 39], [1, 41], [1, 47], [2, 5], [2, 17], [2, 20], [2, 24], [2, 28], [2, 31], [2, 37], [2, 42], [2, 43], [2, 45], [3, 4], [3, 8], [3, 13], [3, 15], [3, 17], [3, 18], [3, 19], [3, 24], [3, 27], [3, 35], [3, 40], [3, 41], [3, 44], [3, 47], [4, 6], [4, 16], [4, 22], [4, 25], [4, 29], [4, 30], [4, 31], [4, 32], [4, 33], [4, 37], [4, 42], [4, 43], [4, 45], [5, 6], [5, 11], [5, 23], [5, 25], [5, 26], [5, 31], [5, 34], [5, 35], [5, 36], [5, 45], [5, 46], [6, 12], [6, 13], [6, 17], [6, 19], [6, 20], [6, 22], [6, 23], [6, 24], [6, 25], [6, 32], [6, 33], [6, 34], [6, 35], [6, 36], [6, 37], [6, 38], [6, 40], [6, 49], [7, 11], [7, 14], [7, 16], [7, 17], [7, 20], [7, 30], [7, 33], [7, 35], [7, 36], [7, 41], [7, 44], [7, 48], [8, 9], [8, 10], [8, 13], [8, 16], [8, 23], [8, 29], [8, 31], [8, 33], [8, 34], [8, 41], [8, 43], [8, 47], [8, 48], [9, 11], [9, 15], [9, 19], [9, 22], [9, 26], [9, 27], [9, 31], [9, 32], [9, 34], [9, 36], [9, 45], [9, 48], [9, 49], [10, 12], [10, 15], [10, 17], [10, 27], [10, 32], [10, 36], [10, 38], [10, 39], [10, 46], [11, 12], [11, 17], [11, 20], [11, 23], [11, 24], [11, 25], [11, 33], [11, 34], [11, 36], [11, 40], [11, 42], [11, 45], [11, 49], [12, 14], [12, 20], [12, 22], [12, 26], [12, 31], [12, 32], [12, 34], [12, 36], [12, 37], [12, 38], [12, 47], [13, 14], [13, 16], [13, 22], [13, 23], [13, 28], [13, 29], [13, 32], [13, 34], [13, 36], [13, 44], [13, 48], [14, 15], [14, 18], [14, 19], [14, 20], [14, 22], [14, 23], [14, 26], [14, 30], [14, 32], [14, 34], [14, 42], [14, 48], [15, 22], [15, 26], [15, 27], [15, 28], [15, 33], [15, 34], [15, 35], [15, 36], [15, 39], [15, 40], [16, 17], [16, 24], [16, 30], [16, 34], [16, 37], [16, 38], [16, 39], [16, 40], [16, 41], [16, 43], [16, 44], [16, 45], [16, 46], [16, 47], [16, 48], [17, 18], [17, 21], [17, 29], [17, 30], [17, 35], [17, 43], [17, 46], [18, 24], [18, 34], [18, 35], [18, 37], [18, 38], [18, 43], [18, 47], [18, 49], [19, 22], [19, 27], [19, 32], [19, 33], [19, 37], [19, 44], [19, 47], [20, 21], [20, 23], [20, 29], [20, 32], [20, 33], [20, 43], [20, 47], [21, 22], [21, 30], [21, 31], [21, 33], [21, 34], [21, 35], [21, 39], [21, 41], [22, 23], [22, 29], [22, 34], [22, 39], [22, 41], [22, 47], [22, 48], [23, 24], [23, 25], [23, 29], [23, 30], [23, 32], [23, 36], [23, 37], [23, 40], [23, 42], [23, 48], [24, 26], [24, 31], [24, 32], [24, 34], [24, 35], [24, 40], [24, 41], [24, 46], [24, 48], [24, 49], [25, 27], [25, 31], [25, 35], [25, 37], [25, 39], [25, 46], [26, 31], [26, 33], [26, 34], [26, 35], [26, 36], [26, 41], [26, 43], [26, 47], [26, 49], [27, 28], [27, 33], [27, 34], [27, 35], [27, 38], [27, 42], [27, 45], [27, 46], [28, 32], [28, 34], [28, 36], [28, 39], [28, 43], [28, 44], [28, 47], [28, 48], [29, 30], [29, 32], [29, 33], [29, 34], [29, 35], [29, 38], [29, 41], [29, 49], [30, 31], [30, 32], [30, 38], [30, 43], [30, 45], [30, 46], [31, 32], [31, 34], [31, 40], [31, 44], [31, 45], [31, 46], [31, 49], [32, 33], [32, 35], [32, 38], [32, 47], [33, 34], [33, 37], [33, 38], [33, 40], [33, 45], [33, 48], [33, 49], [34, 37], [34, 41], [34, 43], [34, 44], [34, 45], [34, 49], [35, 38], [35, 39], [35, 40], [35, 41], [35, 44], [35, 48], [36, 39], [36, 44], [37, 39], [37, 40], [37, 47], [37, 49], [38, 41], [38, 46], [38, 48], [39, 45], [39, 46], [39, 49], [40, 44], [40, 45], [40, 46], [40, 48], [41, 42], [41, 47], [41, 48], [42, 46], [42, 47], [42, 49], [43, 47], [44, 48], [45, 46]], "family": "nppc_dominating_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-dominating-set-l3-s5/tests/run.py b/tasks/construct-nppc-dominating-set-l3-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l3-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-dominating-set-l3-s5/tests/test.sh b/tasks/construct-nppc-dominating-set-l3-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l3-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-independent-set-l6-s3/environment/Dockerfile b/tasks/construct-nppc-independent-set-l6-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-independent-set-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-nppc-independent-set-l6-s3/environment/instance.json b/tasks/construct-nppc-independent-set-l6-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f7d8577047e0fa9a8eadb1881f3be4b6f0d62c7e
--- /dev/null
+++ b/tasks/construct-nppc-independent-set-l6-s3/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 100, "k": 30, "edges": [[0, 3], [0, 4], [0, 11], [0, 26], [0, 30], [0, 33], [0, 38], [0, 51], [0, 53], [0, 58], [0, 62], [0, 63], [0, 64], [0, 65], [0, 67], [0, 86], [0, 89], [0, 94], [1, 4], [1, 5], [1, 14], [1, 17], [1, 19], [1, 22], [1, 28], [1, 29], [1, 40], [1, 41], [1, 42], [1, 43], [1, 46], [1, 48], [1, 55], [1, 59], [1, 60], [1, 63], [1, 66], [1, 68], [1, 69], [1, 71], [1, 75], [1, 76], [1, 81], [1, 82], [1, 85], [1, 87], [1, 89], [1, 95], [2, 3], [2, 4], [2, 5], [2, 10], [2, 17], [2, 21], [2, 22], [2, 24], [2, 25], [2, 32], [2, 35], [2, 37], [2, 46], [2, 49], [2, 51], [2, 55], [2, 58], [2, 60], [2, 69], [2, 71], [2, 73], [2, 75], [2, 79], [2, 81], [2, 83], [2, 86], [2, 88], [2, 89], [2, 92], [2, 93], [2, 99], [3, 5], [3, 10], [3, 16], [3, 17], [3, 31], [3, 33], [3, 40], [3, 43], [3, 78], [3, 84], [3, 87], [4, 5], [4, 9], [4, 10], [4, 14], [4, 16], [4, 17], [4, 29], [4, 30], [4, 33], [4, 35], [4, 40], [4, 42], [4, 46], [4, 48], [4, 52], [4, 54], [4, 55], [4, 59], 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92], [80, 93], [80, 95], [80, 98], [80, 99], [81, 89], [81, 90], [81, 96], [81, 97], [82, 84], [82, 92], [82, 95], [82, 96], [83, 90], [83, 94], [83, 95], [83, 97], [84, 87], [84, 93], [84, 94], [84, 98], [85, 88], [85, 89], [85, 92], [85, 94], [85, 96], [86, 90], [86, 92], [86, 97], [87, 89], [87, 95], [87, 96], [88, 89], [88, 95], [89, 90], [89, 92], [89, 98], [90, 91], [90, 93], [90, 94], [91, 94], [92, 96], [93, 97], [94, 99], [95, 96], [95, 98], [96, 97]], "family": "nppc_independent_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-independent-set-l6-s3/solution/solve.sh b/tasks/construct-nppc-independent-set-l6-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..0da27d9a659a58919ee86167a7e115ccf871cbe4
--- /dev/null
+++ b/tasks/construct-nppc-independent-set-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'
+{"independent_set": [3, 6, 9, 14, 15, 19, 23, 25, 27, 29, 32, 35, 38, 41, 47, 48, 55, 56, 58, 61, 66, 67, 72, 81, 82, 83, 88, 93, 98, 99]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-independent-set-l6-s3/tests/check.py b/tasks/construct-nppc-independent-set-l6-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..92279a931488c7f39c08a3b11162b3f54a9c662a
--- /dev/null
+++ b/tasks/construct-nppc-independent-set-l6-s3/tests/check.py
@@ -0,0 +1,19 @@
+"""Independent set of size >= k. Accepts any valid independent set."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["num_nodes"], inst["k"]
+    s = int_list(ans.get("independent_set"), None, 0, n - 1, "independent_set")
+    # upstream NPPC verifier checks len(list) >= size but only tests pairs u < v, so [v]*k passed
+    if len(set(s)) != len(s):
+        raise Invalid("independent_set contains duplicates")
+    if len(s) < k:
+        raise Invalid(f"independent_set has {len(s)} < {k} vertices")
+    es = {(min(u, v), max(u, v)) for u, v in inst["edges"]}
+    for u, v in itertools.combinations(sorted(s), 2):
+        if (u, v) in es:
+            raise Invalid(f"edge between {u} and {v}")
+    return None, {"size": len(s)}
diff --git a/tasks/construct-nppc-independent-set-l6-s3/tests/instance.json b/tasks/construct-nppc-independent-set-l6-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f7d8577047e0fa9a8eadb1881f3be4b6f0d62c7e
--- /dev/null
+++ b/tasks/construct-nppc-independent-set-l6-s3/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 100, "k": 30, "edges": [[0, 3], [0, 4], [0, 11], [0, 26], [0, 30], [0, 33], [0, 38], [0, 51], [0, 53], [0, 58], [0, 62], [0, 63], [0, 64], [0, 65], [0, 67], [0, 86], [0, 89], [0, 94], [1, 4], [1, 5], [1, 14], [1, 17], [1, 19], [1, 22], [1, 28], [1, 29], [1, 40], [1, 41], [1, 42], [1, 43], [1, 46], [1, 48], [1, 55], [1, 59], [1, 60], [1, 63], [1, 66], [1, 68], [1, 69], [1, 71], [1, 75], [1, 76], [1, 81], [1, 82], [1, 85], [1, 87], [1, 89], [1, 95], [2, 3], [2, 4], [2, 5], [2, 10], [2, 17], [2, 21], [2, 22], [2, 24], [2, 25], [2, 32], [2, 35], [2, 37], [2, 46], [2, 49], [2, 51], [2, 55], [2, 58], [2, 60], [2, 69], [2, 71], [2, 73], [2, 75], [2, 79], [2, 81], [2, 83], [2, 86], [2, 88], [2, 89], [2, 92], [2, 93], [2, 99], [3, 5], [3, 10], [3, 16], [3, 17], [3, 31], [3, 33], [3, 40], [3, 43], [3, 78], [3, 84], [3, 87], [4, 5], [4, 9], [4, 10], [4, 14], [4, 16], [4, 17], [4, 29], [4, 30], [4, 33], [4, 35], [4, 40], [4, 42], [4, 46], [4, 48], [4, 52], [4, 54], [4, 55], [4, 59], 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92], [80, 93], [80, 95], [80, 98], [80, 99], [81, 89], [81, 90], [81, 96], [81, 97], [82, 84], [82, 92], [82, 95], [82, 96], [83, 90], [83, 94], [83, 95], [83, 97], [84, 87], [84, 93], [84, 94], [84, 98], [85, 88], [85, 89], [85, 92], [85, 94], [85, 96], [86, 90], [86, 92], [86, 97], [87, 89], [87, 95], [87, 96], [88, 89], [88, 95], [89, 90], [89, 92], [89, 98], [90, 91], [90, 93], [90, 94], [91, 94], [92, 96], [93, 97], [94, 99], [95, 96], [95, 98], [96, 97]], "family": "nppc_independent_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-independent-set-l6-s3/tests/run.py b/tasks/construct-nppc-independent-set-l6-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-independent-set-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-nppc-independent-set-l6-s3/tests/test.sh b/tasks/construct-nppc-independent-set-l6-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-independent-set-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-nppc-independent-set-l6-s4/environment/Dockerfile b/tasks/construct-nppc-independent-set-l6-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-independent-set-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-nppc-independent-set-l6-s4/environment/instance.json b/tasks/construct-nppc-independent-set-l6-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2cc41c3288dcbeae35b3ce3f48cf3193ff76b977
--- /dev/null
+++ b/tasks/construct-nppc-independent-set-l6-s4/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 100, "k": 30, "edges": [[0, 4], [0, 9], [0, 12], [0, 14], [0, 17], [0, 27], [0, 31], [0, 33], [0, 46], [0, 47], [0, 56], [0, 57], [0, 58], [0, 68], [0, 72], [0, 73], [0, 74], [0, 75], [0, 84], [0, 85], [0, 92], [1, 3], [1, 4], [1, 5], [1, 11], [1, 16], [1, 36], [1, 37], [1, 46], [1, 56], [1, 57], [1, 58], [1, 64], [1, 68], [1, 79], [1, 83], [1, 92], [1, 96], [1, 97], [2, 4], [2, 5], [2, 17], [2, 25], [2, 26], [2, 31], [2, 32], [2, 34], [2, 36], [2, 39], [2, 42], [2, 43], [2, 44], [2, 46], [2, 49], [2, 50], [2, 52], [2, 56], [2, 57], [2, 62], [2, 66], [2, 69], [2, 73], [2, 77], [2, 78], [2, 89], [2, 90], [2, 95], [2, 96], [2, 98], [2, 99], [3, 5], [3, 6], [3, 10], [3, 15], [3, 26], [3, 31], [3, 36], [3, 39], [3, 40], [3, 41], [3, 47], [3, 50], [3, 51], [3, 57], [3, 58], [3, 63], [3, 65], [3, 69], [3, 72], [3, 77], [3, 80], [3, 82], [3, 83], [3, 86], [3, 88], [3, 89], [3, 90], [3, 95], [3, 97], [4, 5], [4, 6], [4, 13], [4, 14], [4, 19], [4, 20], [4, 23], [4, 24], [4, 25], 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\ No newline at end of file
diff --git a/tasks/construct-nppc-independent-set-l6-s4/solution/solve.sh b/tasks/construct-nppc-independent-set-l6-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..260c44d5682151ffa7d54b3056551734a16e7d85
--- /dev/null
+++ b/tasks/construct-nppc-independent-set-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'
+{"independent_set": [0, 1, 7, 8, 10, 20, 23, 25, 26, 30, 32, 35, 42, 51, 52, 55, 60, 62, 65, 70, 76, 77, 78, 80, 86, 89, 90, 91, 94, 95]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-independent-set-l6-s4/tests/check.py b/tasks/construct-nppc-independent-set-l6-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..92279a931488c7f39c08a3b11162b3f54a9c662a
--- /dev/null
+++ b/tasks/construct-nppc-independent-set-l6-s4/tests/check.py
@@ -0,0 +1,19 @@
+"""Independent set of size >= k. Accepts any valid independent set."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["num_nodes"], inst["k"]
+    s = int_list(ans.get("independent_set"), None, 0, n - 1, "independent_set")
+    # upstream NPPC verifier checks len(list) >= size but only tests pairs u < v, so [v]*k passed
+    if len(set(s)) != len(s):
+        raise Invalid("independent_set contains duplicates")
+    if len(s) < k:
+        raise Invalid(f"independent_set has {len(s)} < {k} vertices")
+    es = {(min(u, v), max(u, v)) for u, v in inst["edges"]}
+    for u, v in itertools.combinations(sorted(s), 2):
+        if (u, v) in es:
+            raise Invalid(f"edge between {u} and {v}")
+    return None, {"size": len(s)}
diff --git a/tasks/construct-nppc-independent-set-l6-s4/tests/instance.json b/tasks/construct-nppc-independent-set-l6-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2cc41c3288dcbeae35b3ce3f48cf3193ff76b977
--- /dev/null
+++ b/tasks/construct-nppc-independent-set-l6-s4/tests/instance.json
@@ -0,0 +1 @@
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\ No newline at end of file
diff --git a/tasks/construct-nppc-independent-set-l6-s4/tests/run.py b/tasks/construct-nppc-independent-set-l6-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-independent-set-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-nppc-independent-set-l6-s4/tests/test.sh b/tasks/construct-nppc-independent-set-l6-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-independent-set-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-nppc-vertex-cover-l1-s1/environment/Dockerfile b/tasks/construct-nppc-vertex-cover-l1-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-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-nppc-vertex-cover-l1-s1/environment/instance.json b/tasks/construct-nppc-vertex-cover-l1-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..76288ba803fdc6f39c89b5272964ab2f3d225f1f
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-l1-s1/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 8, "k": 3, "edges": [[0, 3], [0, 4], [0, 5], [1, 4], [1, 7], [2, 4], [3, 4], [4, 5], [4, 6], [4, 7]], "family": "nppc_vertex_cover", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-vertex-cover-l1-s1/solution/solve.sh b/tasks/construct-nppc-vertex-cover-l1-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..1f51f29830b0cd9cd24b5d6e9ebfb1cf1af349a3
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-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'
+{"cover": [0, 1, 4]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-vertex-cover-l1-s1/tests/check.py b/tasks/construct-nppc-vertex-cover-l1-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..d8ed740cc660c4837d941a8d70ea394c3bc9f69e
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-l1-s1/tests/check.py
@@ -0,0 +1,18 @@
+"""Vertex cover of size <= k. Accepts any valid cover."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["num_nodes"], inst["k"]
+    # upstream NPPC verifier converts to a set without type checks (floats/bools slip through and
+    # duplicates are silently merged); here entries must be distinct ints in range
+    c = int_list(ans.get("cover"), None, 0, n - 1, "cover")
+    if len(set(c)) != len(c):
+        raise Invalid("cover contains duplicates")
+    if len(c) > k:
+        raise Invalid(f"cover has {len(c)} > {k} vertices")
+    s = set(c)
+    for u, v in inst["edges"]:
+        if u not in s and v not in s:
+            raise Invalid(f"edge ({u},{v}) not covered")
+    return None, {"size": len(c)}
diff --git a/tasks/construct-nppc-vertex-cover-l1-s1/tests/instance.json b/tasks/construct-nppc-vertex-cover-l1-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..76288ba803fdc6f39c89b5272964ab2f3d225f1f
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-l1-s1/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 8, "k": 3, "edges": [[0, 3], [0, 4], [0, 5], [1, 4], [1, 7], [2, 4], [3, 4], [4, 5], [4, 6], [4, 7]], "family": "nppc_vertex_cover", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-vertex-cover-l1-s1/tests/run.py b/tasks/construct-nppc-vertex-cover-l1-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-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-nppc-vertex-cover-l1-s1/tests/test.sh b/tasks/construct-nppc-vertex-cover-l1-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-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-nppc-vertex-cover-l1-s6/environment/Dockerfile b/tasks/construct-nppc-vertex-cover-l1-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-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-nppc-vertex-cover-l1-s6/environment/instance.json b/tasks/construct-nppc-vertex-cover-l1-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1082879a030dfe867e25d4976ba5a0cd586aec4d
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-l1-s6/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 8, "k": 3, "edges": [[0, 2], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [2, 6], [3, 4], [4, 5], [4, 6], [4, 7]], "family": "nppc_vertex_cover", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-vertex-cover-l1-s6/solution/solve.sh b/tasks/construct-nppc-vertex-cover-l1-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..3cd92f46e6f5e6a61e31950f8b15491e3764ab6d
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-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'
+{"cover": [2, 3, 4]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-vertex-cover-l1-s6/tests/check.py b/tasks/construct-nppc-vertex-cover-l1-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..d8ed740cc660c4837d941a8d70ea394c3bc9f69e
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-l1-s6/tests/check.py
@@ -0,0 +1,18 @@
+"""Vertex cover of size <= k. Accepts any valid cover."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["num_nodes"], inst["k"]
+    # upstream NPPC verifier converts to a set without type checks (floats/bools slip through and
+    # duplicates are silently merged); here entries must be distinct ints in range
+    c = int_list(ans.get("cover"), None, 0, n - 1, "cover")
+    if len(set(c)) != len(c):
+        raise Invalid("cover contains duplicates")
+    if len(c) > k:
+        raise Invalid(f"cover has {len(c)} > {k} vertices")
+    s = set(c)
+    for u, v in inst["edges"]:
+        if u not in s and v not in s:
+            raise Invalid(f"edge ({u},{v}) not covered")
+    return None, {"size": len(c)}
diff --git a/tasks/construct-nppc-vertex-cover-l1-s6/tests/instance.json b/tasks/construct-nppc-vertex-cover-l1-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1082879a030dfe867e25d4976ba5a0cd586aec4d
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-l1-s6/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 8, "k": 3, "edges": [[0, 2], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [2, 6], [3, 4], [4, 5], [4, 6], [4, 7]], "family": "nppc_vertex_cover", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-vertex-cover-l1-s6/tests/run.py b/tasks/construct-nppc-vertex-cover-l1-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-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-nppc-vertex-cover-l1-s6/tests/test.sh b/tasks/construct-nppc-vertex-cover-l1-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-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-nppc-vertex-cover-l5-s9/environment/Dockerfile b/tasks/construct-nppc-vertex-cover-l5-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-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-nppc-vertex-cover-l5-s9/environment/instance.json b/tasks/construct-nppc-vertex-cover-l5-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b03eb3eb3c00dc3ab8d8e90bc5cc626154a1fb72
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-l5-s9/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 80, "k": 40, "edges": [[0, 5], [0, 8], [0, 9], [0, 19], [0, 21], [0, 37], [0, 40], [0, 41], [0, 46], [0, 49], [0, 50], [0, 53], [0, 58], [0, 61], [0, 64], [0, 67], [0, 71], [0, 76], [0, 77], [1, 4], [1, 5], [1, 6], [1, 7], [1, 9], [1, 11], [1, 12], [1, 17], [1, 18], [1, 19], [1, 20], [1, 21], [1, 22], [1, 26], [1, 29], [1, 30], [1, 33], [1, 35], [1, 36], [1, 37], [1, 40], [1, 41], [1, 42], [1, 43], [1, 45], [1, 46], [1, 47], [1, 48], [1, 50], [1, 51], [1, 56], [1, 57], [1, 58], [1, 60], [1, 61], [1, 62], [1, 64], [1, 66], [1, 67], [1, 69], [1, 71], [1, 76], [1, 77], [1, 78], [1, 79], [2, 8], [2, 12], [2, 13], [2, 16], [2, 18], [2, 22], [2, 24], [2, 29], [2, 40], [2, 41], [2, 46], [2, 50], [2, 53], [2, 58], [2, 61], [2, 64], [2, 67], [2, 72], [2, 76], [2, 77], [2, 79], [3, 7], [3, 16], [3, 19], [3, 21], [3, 29], [3, 35], [3, 36], [3, 40], [3, 45], [3, 56], [3, 67], [3, 69], [3, 76], [4, 5], [4, 8], [4, 9], [4, 12], [4, 13], [4, 16], [4, 19], [4, 22], [4, 30], [4, 31], [4, 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\ No newline at end of file
diff --git a/tasks/construct-nppc-vertex-cover-l5-s9/solution/solve.sh b/tasks/construct-nppc-vertex-cover-l5-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..d008fa98d48a3939a3a9fa248677b99e3c7a3206
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-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'
+{"cover": [1, 5, 7, 8, 9, 12, 13, 16, 18, 19, 21, 22, 24, 29, 30, 31, 35, 36, 37, 40, 41, 43, 45, 46, 49, 50, 53, 54, 56, 58, 59, 61, 64, 67, 69, 71, 72, 76, 77, 79]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-vertex-cover-l5-s9/tests/check.py b/tasks/construct-nppc-vertex-cover-l5-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..d8ed740cc660c4837d941a8d70ea394c3bc9f69e
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-l5-s9/tests/check.py
@@ -0,0 +1,18 @@
+"""Vertex cover of size <= k. Accepts any valid cover."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["num_nodes"], inst["k"]
+    # upstream NPPC verifier converts to a set without type checks (floats/bools slip through and
+    # duplicates are silently merged); here entries must be distinct ints in range
+    c = int_list(ans.get("cover"), None, 0, n - 1, "cover")
+    if len(set(c)) != len(c):
+        raise Invalid("cover contains duplicates")
+    if len(c) > k:
+        raise Invalid(f"cover has {len(c)} > {k} vertices")
+    s = set(c)
+    for u, v in inst["edges"]:
+        if u not in s and v not in s:
+            raise Invalid(f"edge ({u},{v}) not covered")
+    return None, {"size": len(c)}
diff --git a/tasks/construct-nppc-vertex-cover-l5-s9/tests/instance.json b/tasks/construct-nppc-vertex-cover-l5-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b03eb3eb3c00dc3ab8d8e90bc5cc626154a1fb72
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-l5-s9/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 80, "k": 40, "edges": [[0, 5], [0, 8], [0, 9], [0, 19], [0, 21], [0, 37], [0, 40], [0, 41], [0, 46], [0, 49], [0, 50], [0, 53], [0, 58], [0, 61], [0, 64], [0, 67], [0, 71], [0, 76], [0, 77], [1, 4], [1, 5], [1, 6], [1, 7], [1, 9], [1, 11], [1, 12], [1, 17], [1, 18], [1, 19], [1, 20], [1, 21], [1, 22], [1, 26], [1, 29], [1, 30], [1, 33], [1, 35], [1, 36], [1, 37], [1, 40], [1, 41], [1, 42], [1, 43], [1, 45], [1, 46], [1, 47], [1, 48], [1, 50], [1, 51], [1, 56], [1, 57], [1, 58], [1, 60], [1, 61], [1, 62], [1, 64], [1, 66], [1, 67], [1, 69], [1, 71], [1, 76], [1, 77], [1, 78], [1, 79], [2, 8], [2, 12], [2, 13], [2, 16], [2, 18], [2, 22], [2, 24], [2, 29], [2, 40], [2, 41], [2, 46], [2, 50], [2, 53], [2, 58], [2, 61], [2, 64], [2, 67], [2, 72], [2, 76], [2, 77], [2, 79], [3, 7], [3, 16], [3, 19], [3, 21], [3, 29], [3, 35], [3, 36], [3, 40], [3, 45], [3, 56], [3, 67], [3, 69], [3, 76], [4, 5], [4, 8], [4, 9], [4, 12], [4, 13], [4, 16], [4, 19], [4, 22], [4, 30], [4, 31], [4, 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\ No newline at end of file
diff --git a/tasks/construct-nppc-vertex-cover-l5-s9/tests/run.py b/tasks/construct-nppc-vertex-cover-l5-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-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-nppc-vertex-cover-l5-s9/tests/test.sh b/tasks/construct-nppc-vertex-cover-l5-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-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-rg-graph-color-l2-s3/environment/Dockerfile b/tasks/construct-rg-graph-color-l2-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rg-graph-color-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-rg-graph-color-l2-s3/environment/instance.json b/tasks/construct-rg-graph-color-l2-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b01054ece7aff425316604283ba56f322c344195
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l2-s3/environment/instance.json
@@ -0,0 +1 @@
+{"num_vertices": 10, "num_colors": 4, "edges": [[0, 1], [0, 7], [0, 8], [1, 4], [2, 6], [3, 8], [5, 6], [5, 7], [8, 9]], "family": "rg_graph_color", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-graph-color-l2-s3/solution/solve.sh b/tasks/construct-rg-graph-color-l2-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..2590179b6ee07fd1c352a4b606a88be16e0a34a7
--- /dev/null
+++ b/tasks/construct-rg-graph-color-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'
+{"coloring": [1, 2, 1, 1, 1, 1, 2, 2, 2, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-rg-graph-color-l2-s3/tests/check.py b/tasks/construct-rg-graph-color-l2-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..e9328912405d1c894970c90415be62bf9556ed36
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l2-s3/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-l2-s3/tests/instance.json b/tasks/construct-rg-graph-color-l2-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b01054ece7aff425316604283ba56f322c344195
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l2-s3/tests/instance.json
@@ -0,0 +1 @@
+{"num_vertices": 10, "num_colors": 4, "edges": [[0, 1], [0, 7], [0, 8], [1, 4], [2, 6], [3, 8], [5, 6], [5, 7], [8, 9]], "family": "rg_graph_color", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-graph-color-l2-s3/tests/run.py b/tasks/construct-rg-graph-color-l2-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rg-graph-color-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-rg-graph-color-l2-s3/tests/test.sh b/tasks/construct-rg-graph-color-l2-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rg-graph-color-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-rg-graph-color-l6-s2/environment/Dockerfile b/tasks/construct-rg-graph-color-l6-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rg-graph-color-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-rg-graph-color-l6-s2/environment/instance.json b/tasks/construct-rg-graph-color-l6-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..47b95006820058a65cb9b0f67a9e1b6c8c080178
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l6-s2/environment/instance.json
@@ -0,0 +1 @@
+{"num_vertices": 40, "num_colors": 3, "edges": [[0, 1], [0, 19], [0, 30], [0, 33], [0, 34], [1, 5], [1, 15], [1, 18], [1, 21], [2, 9], [2, 11], [2, 20], [2, 22], [2, 28], [2, 29], [4, 17], [4, 20], [4, 29], [5, 9], [5, 17], [5, 21], [5, 32], [5, 36], [6, 18], [6, 34], [7, 10], [7, 15], [7, 24], [7, 35], [8, 9], [10, 21], [11, 16], [11, 22], [11, 27], [11, 28], [12, 15], [12, 18], [13, 14], [13, 18], [13, 23], [14, 23], [14, 26], [14, 27], [14, 31], [15, 20], [15, 32], [16, 19], [16, 30], [16, 37], [17, 22], [17, 35], [17, 39], [18, 39], [19, 21], [19, 23], [20, 31], [20, 35], [21, 31], [23, 24], [23, 31], [23, 36], [23, 37], [24, 27], [26, 28], [28, 29], [28, 33], [30, 38], [31, 32], [35, 36], [35, 38], [36, 39], [37, 39]], "family": "rg_graph_color", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-graph-color-l6-s2/solution/solve.sh b/tasks/construct-rg-graph-color-l6-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..bdd4535a8eeddae29c07f146caa3cf7f68c34e47
--- /dev/null
+++ b/tasks/construct-rg-graph-color-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'
+{"coloring": [1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 3, 1, 2, 3, 2, 2, 3, 3, 3, 2, 1, 1, 1, 3, 2, 2, 1, 2, 2, 2, 3, 2, 2, 1, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-rg-graph-color-l6-s2/tests/check.py b/tasks/construct-rg-graph-color-l6-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..e9328912405d1c894970c90415be62bf9556ed36
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l6-s2/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-l6-s2/tests/instance.json b/tasks/construct-rg-graph-color-l6-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..47b95006820058a65cb9b0f67a9e1b6c8c080178
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l6-s2/tests/instance.json
@@ -0,0 +1 @@
+{"num_vertices": 40, "num_colors": 3, "edges": [[0, 1], [0, 19], [0, 30], [0, 33], [0, 34], [1, 5], [1, 15], [1, 18], [1, 21], [2, 9], [2, 11], [2, 20], [2, 22], [2, 28], [2, 29], [4, 17], [4, 20], [4, 29], [5, 9], [5, 17], [5, 21], [5, 32], [5, 36], [6, 18], [6, 34], [7, 10], [7, 15], [7, 24], [7, 35], [8, 9], [10, 21], [11, 16], [11, 22], [11, 27], [11, 28], [12, 15], [12, 18], [13, 14], [13, 18], [13, 23], [14, 23], [14, 26], [14, 27], [14, 31], [15, 20], [15, 32], [16, 19], [16, 30], [16, 37], [17, 22], [17, 35], [17, 39], [18, 39], [19, 21], [19, 23], [20, 31], [20, 35], [21, 31], [23, 24], [23, 31], [23, 36], [23, 37], [24, 27], [26, 28], [28, 29], [28, 33], [30, 38], [31, 32], [35, 36], [35, 38], [36, 39], [37, 39]], "family": "rg_graph_color", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-graph-color-l6-s2/tests/run.py b/tasks/construct-rg-graph-color-l6-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rg-graph-color-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-rg-graph-color-l6-s2/tests/test.sh b/tasks/construct-rg-graph-color-l6-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rg-graph-color-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-rg-graph-color-l6-s5/environment/Dockerfile b/tasks/construct-rg-graph-color-l6-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rg-graph-color-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-rg-graph-color-l6-s5/environment/instance.json b/tasks/construct-rg-graph-color-l6-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..6284be497a53282fa3be34299819a1457ddc11f5
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l6-s5/environment/instance.json
@@ -0,0 +1 @@
+{"num_vertices": 40, "num_colors": 3, "edges": [[0, 6], [0, 12], [0, 18], [1, 10], [1, 19], [1, 20], [1, 27], [1, 39], [2, 14], [2, 23], [2, 30], [2, 35], [3, 5], [3, 8], [3, 30], [4, 5], [4, 18], [4, 24], [4, 30], [4, 38], [5, 24], [6, 7], [6, 11], [6, 16], [6, 26], [7, 19], [7, 20], [8, 24], [9, 23], [9, 35], [9, 39], [10, 20], [11, 38], [12, 22], [12, 36], [12, 39], [13, 27], [13, 31], [13, 37], [14, 17], [14, 21], [14, 22], [14, 35], [14, 38], [15, 27], [15, 35], [16, 17], [16, 25], [18, 20], [18, 23], [18, 26], [19, 24], [19, 32], [19, 36], [20, 25], [20, 27], [20, 36], [21, 37], [22, 27], [22, 37], [22, 39], [23, 29], [23, 30], [23, 36], [24, 27], [24, 36], [26, 33], [26, 34], [26, 39], [28, 37], [29, 33], [30, 31], [30, 32], [30, 35], [31, 34], [35, 36], [35, 37], [36, 39]], "family": "rg_graph_color", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-graph-color-l6-s5/solution/solve.sh b/tasks/construct-rg-graph-color-l6-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..92193b968bb735e999137ee61d8dee4ab5406e98
--- /dev/null
+++ b/tasks/construct-rg-graph-color-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'
+{"coloring": [1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 3, 2, 2, 3, 1, 1, 3, 3, 2, 1, 2, 1, 1, 2, 3, 1, 2, 2, 3, 1, 2, 3, 3]}
+RLMATH_EOF
diff --git a/tasks/construct-rg-graph-color-l6-s5/tests/check.py b/tasks/construct-rg-graph-color-l6-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..e9328912405d1c894970c90415be62bf9556ed36
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l6-s5/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-l6-s5/tests/instance.json b/tasks/construct-rg-graph-color-l6-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..6284be497a53282fa3be34299819a1457ddc11f5
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l6-s5/tests/instance.json
@@ -0,0 +1 @@
+{"num_vertices": 40, "num_colors": 3, "edges": [[0, 6], [0, 12], [0, 18], [1, 10], [1, 19], [1, 20], [1, 27], [1, 39], [2, 14], [2, 23], [2, 30], [2, 35], [3, 5], [3, 8], [3, 30], [4, 5], [4, 18], [4, 24], [4, 30], [4, 38], [5, 24], [6, 7], [6, 11], [6, 16], [6, 26], [7, 19], [7, 20], [8, 24], [9, 23], [9, 35], [9, 39], [10, 20], [11, 38], [12, 22], [12, 36], [12, 39], [13, 27], [13, 31], [13, 37], [14, 17], [14, 21], [14, 22], [14, 35], [14, 38], [15, 27], [15, 35], [16, 17], [16, 25], [18, 20], [18, 23], [18, 26], [19, 24], [19, 32], [19, 36], [20, 25], [20, 27], [20, 36], [21, 37], [22, 27], [22, 37], [22, 39], [23, 29], [23, 30], [23, 36], [24, 27], [24, 36], [26, 33], [26, 34], [26, 39], [28, 37], [29, 33], [30, 31], [30, 32], [30, 35], [31, 34], [35, 36], [35, 37], [36, 39]], "family": "rg_graph_color", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-graph-color-l6-s5/tests/run.py b/tasks/construct-rg-graph-color-l6-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rg-graph-color-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-rg-graph-color-l6-s5/tests/test.sh b/tasks/construct-rg-graph-color-l6-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rg-graph-color-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-rg-intermediate-integration-l4-s9/environment/Dockerfile b/tasks/construct-rg-intermediate-integration-l4-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-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-rg-intermediate-integration-l4-s9/environment/instance.json b/tasks/construct-rg-intermediate-integration-l4-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..548705986dedce3346462956263496019c3b5b36
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-l4-s9/environment/instance.json
@@ -0,0 +1 @@
+{"variable": "x", "integrand": "-2*log(x**3)", "problem_type": "log_inverse_trig", "family": "rg_intermediate_integration", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-intermediate-integration-l4-s9/tests/check.py b/tasks/construct-rg-intermediate-integration-l4-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..a9e364a8ede6c8215a16a04ee6b0d3529adc647d
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-l4-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-intermediate-integration-l4-s9/tests/instance.json b/tasks/construct-rg-intermediate-integration-l4-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..548705986dedce3346462956263496019c3b5b36
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-l4-s9/tests/instance.json
@@ -0,0 +1 @@
+{"variable": "x", "integrand": "-2*log(x**3)", "problem_type": "log_inverse_trig", "family": "rg_intermediate_integration", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-intermediate-integration-l4-s9/tests/run.py b/tasks/construct-rg-intermediate-integration-l4-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-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-rg-intermediate-integration-l4-s9/tests/test.sh b/tasks/construct-rg-intermediate-integration-l4-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-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-rg-simple-integration-l2-s2/environment/Dockerfile b/tasks/construct-rg-simple-integration-l2-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-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-rg-simple-integration-l2-s2/environment/instance.json b/tasks/construct-rg-simple-integration-l2-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1a2056251f75b043cd8336a5213e03b95633640f
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l2-s2/environment/instance.json
@@ -0,0 +1 @@
+{"variable": "x", "integrand": "48*x**5 - 20*x - 7/2", "family": "rg_simple_integration", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-simple-integration-l2-s2/solution/solve.sh b/tasks/construct-rg-simple-integration-l2-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..314457f2228e1c3916288a4b1c5f0ded0746747b
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-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'
+{"antiderivative": "8*x**6 - 10*x**2 - 7*x/2 + C"}
+RLMATH_EOF
diff --git a/tasks/construct-rg-simple-integration-l2-s2/tests/check.py b/tasks/construct-rg-simple-integration-l2-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..a9e364a8ede6c8215a16a04ee6b0d3529adc647d
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l2-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-simple-integration-l2-s2/tests/instance.json b/tasks/construct-rg-simple-integration-l2-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1a2056251f75b043cd8336a5213e03b95633640f
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l2-s2/tests/instance.json
@@ -0,0 +1 @@
+{"variable": "x", "integrand": "48*x**5 - 20*x - 7/2", "family": "rg_simple_integration", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-simple-integration-l2-s2/tests/run.py b/tasks/construct-rg-simple-integration-l2-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-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-rg-simple-integration-l2-s2/tests/test.sh b/tasks/construct-rg-simple-integration-l2-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-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-rg-simple-integration-l2-s5/environment/Dockerfile b/tasks/construct-rg-simple-integration-l2-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l2-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-rg-simple-integration-l2-s5/environment/instance.json b/tasks/construct-rg-simple-integration-l2-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3465d7973f04e735471f73ad3e03a33d85c2a454
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l2-s5/environment/instance.json
@@ -0,0 +1 @@
+{"variable": "x", "integrand": "5*x**4/2 - 9*x**2/7 + x", "family": "rg_simple_integration", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-simple-integration-l2-s5/solution/solve.sh b/tasks/construct-rg-simple-integration-l2-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..81f00d2f1d0638f9b393602ce8e5c51068a68195
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l2-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'
+{"antiderivative": "x**5/2 - 3*x**3/7 + x**2/2 + C"}
+RLMATH_EOF
diff --git a/tasks/construct-rg-simple-integration-l2-s5/tests/check.py b/tasks/construct-rg-simple-integration-l2-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..a9e364a8ede6c8215a16a04ee6b0d3529adc647d
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l2-s5/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-l2-s5/tests/instance.json b/tasks/construct-rg-simple-integration-l2-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3465d7973f04e735471f73ad3e03a33d85c2a454
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l2-s5/tests/instance.json
@@ -0,0 +1 @@
+{"variable": "x", "integrand": "5*x**4/2 - 9*x**2/7 + x", "family": "rg_simple_integration", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-simple-integration-l2-s5/tests/run.py b/tasks/construct-rg-simple-integration-l2-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l2-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-rg-simple-integration-l2-s5/tests/test.sh b/tasks/construct-rg-simple-integration-l2-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l2-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-rg-simple-integration-l6-s3/environment/Dockerfile b/tasks/construct-rg-simple-integration-l6-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-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-rg-simple-integration-l6-s3/environment/instance.json b/tasks/construct-rg-simple-integration-l6-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a784d31a85cbe7baff4b9c5adf63e0760c03d9b3
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l6-s3/environment/instance.json
@@ -0,0 +1 @@
+{"variable": "x", "integrand": "-25*x**9/4 - 3*x**2 - 14*x + 7/(2*x**2) + 2/x**3 + 9/(2*x**6) + 63/x**8", "family": "rg_simple_integration", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-simple-integration-l6-s3/solution/solve.sh b/tasks/construct-rg-simple-integration-l6-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..e66923ebc389ed877d04ce5fa7cd1362ed496df2
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-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'
+{"antiderivative": "-5*x**10/8 - x**3 - 7*x**2 - 7/(2*x) - 1/x**2 - 9/(10*x**5) - 9/x**7 + C"}
+RLMATH_EOF
diff --git a/tasks/construct-rg-simple-integration-l6-s3/tests/check.py b/tasks/construct-rg-simple-integration-l6-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..a9e364a8ede6c8215a16a04ee6b0d3529adc647d
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l6-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-simple-integration-l6-s3/tests/instance.json b/tasks/construct-rg-simple-integration-l6-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a784d31a85cbe7baff4b9c5adf63e0760c03d9b3
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l6-s3/tests/instance.json
@@ -0,0 +1 @@
+{"variable": "x", "integrand": "-25*x**9/4 - 3*x**2 - 14*x + 7/(2*x**2) + 2/x**3 + 9/(2*x**6) + 63/x**8", "family": "rg_simple_integration", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-simple-integration-l6-s3/tests/run.py b/tasks/construct-rg-simple-integration-l6-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-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-rg-simple-integration-l6-s3/tests/test.sh b/tasks/construct-rg-simple-integration-l6-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-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-rg-simple-integration-l6-s4/environment/Dockerfile b/tasks/construct-rg-simple-integration-l6-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-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-rg-simple-integration-l6-s4/environment/instance.json b/tasks/construct-rg-simple-integration-l6-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0e93cf65950b606601d78da7f594a2eb5cd1c658
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l6-s4/environment/instance.json
@@ -0,0 +1 @@
+{"variable": "X", "integrand": "-333*X**8/40 - 14*X**6 - 9/X**2 - 21/(2*X**4) - 12/X**5 + 36/X**10", "family": "rg_simple_integration", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-simple-integration-l6-s4/solution/solve.sh b/tasks/construct-rg-simple-integration-l6-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..8c2dcab6b73219eeff443727a83e4f486533b850
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-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'
+{"antiderivative": "-37*X**9/40 - 2*X**7 + 9/X + 7/(2*X**3) + 3/X**4 - 4/X**9 + C"}
+RLMATH_EOF
diff --git a/tasks/construct-rg-simple-integration-l6-s4/tests/check.py b/tasks/construct-rg-simple-integration-l6-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..a9e364a8ede6c8215a16a04ee6b0d3529adc647d
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l6-s4/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-l6-s4/tests/instance.json b/tasks/construct-rg-simple-integration-l6-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0e93cf65950b606601d78da7f594a2eb5cd1c658
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l6-s4/tests/instance.json
@@ -0,0 +1 @@
+{"variable": "X", "integrand": "-333*X**8/40 - 14*X**6 - 9/X**2 - 21/(2*X**4) - 12/X**5 + 36/X**10", "family": "rg_simple_integration", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-simple-integration-l6-s4/tests/run.py b/tasks/construct-rg-simple-integration-l6-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-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-rg-simple-integration-l6-s4/tests/test.sh b/tasks/construct-rg-simple-integration-l6-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-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-antiderivative-l3-s2/environment/Dockerfile b/tasks/construct-rlve-antiderivative-l3-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-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-antiderivative-l3-s2/environment/instance.json b/tasks/construct-rlve-antiderivative-l3-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..12595f3043f6c7e8d48295d7d05910f748862cbf
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l3-s2/environment/instance.json
@@ -0,0 +1 @@
+{"f_prime": "4*x + 1/sqrt(x)", "max_ops": 44, "max_len": 760, "family": "rlve_antiderivative", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-antiderivative-l3-s2/solution/solve.sh b/tasks/construct-rlve-antiderivative-l3-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..ebcdbf28e6688fbbb736d232152df3d0595e7389
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-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'
+{"F": "2*sqrt(x) + 2*x**2"}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-antiderivative-l3-s2/tests/check.py b/tasks/construct-rlve-antiderivative-l3-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..c6b30b93bbbb7795b54d973e43f25b23c82cb7fc
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l3-s2/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-l3-s2/tests/instance.json b/tasks/construct-rlve-antiderivative-l3-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..12595f3043f6c7e8d48295d7d05910f748862cbf
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l3-s2/tests/instance.json
@@ -0,0 +1 @@
+{"f_prime": "4*x + 1/sqrt(x)", "max_ops": 44, "max_len": 760, "family": "rlve_antiderivative", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-antiderivative-l3-s2/tests/run.py b/tasks/construct-rlve-antiderivative-l3-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-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-antiderivative-l3-s2/tests/test.sh b/tasks/construct-rlve-antiderivative-l3-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-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-antiderivative-l3-s5/environment/Dockerfile b/tasks/construct-rlve-antiderivative-l3-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l3-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-antiderivative-l3-s5/environment/instance.json b/tasks/construct-rlve-antiderivative-l3-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c9dc9ec9f9ccc10f0f0c353575360a0f342976cc
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l3-s5/environment/instance.json
@@ -0,0 +1 @@
+{"f_prime": "(x**2/cos(x))**(1/4)*(x**2*sin(x)/(4*cos(x)**2) + x/(2*cos(x)))*cos(x)/x**2", "max_ops": 40, "max_len": 800, "family": "rlve_antiderivative", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-antiderivative-l3-s5/solution/solve.sh b/tasks/construct-rlve-antiderivative-l3-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..4946bc20ed258966de47bab0c8e528505d1ce17b
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l3-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'
+{"F": "(x**2/cos(x))**(1/4)"}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-antiderivative-l3-s5/tests/check.py b/tasks/construct-rlve-antiderivative-l3-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..c6b30b93bbbb7795b54d973e43f25b23c82cb7fc
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l3-s5/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-l3-s5/tests/instance.json b/tasks/construct-rlve-antiderivative-l3-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c9dc9ec9f9ccc10f0f0c353575360a0f342976cc
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l3-s5/tests/instance.json
@@ -0,0 +1 @@
+{"f_prime": "(x**2/cos(x))**(1/4)*(x**2*sin(x)/(4*cos(x)**2) + x/(2*cos(x)))*cos(x)/x**2", "max_ops": 40, "max_len": 800, "family": "rlve_antiderivative", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-antiderivative-l3-s5/tests/run.py b/tasks/construct-rlve-antiderivative-l3-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l3-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-antiderivative-l3-s5/tests/test.sh b/tasks/construct-rlve-antiderivative-l3-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l3-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-bezout-identity-l2-s1/environment/Dockerfile b/tasks/construct-rlve-bezout-identity-l2-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-rlve-bezout-identity-l2-s1/environment/instance.json b/tasks/construct-rlve-bezout-identity-l2-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f51ad5044a904b7040ad290980e1890b50148b1e
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l2-s1/environment/instance.json
@@ -0,0 +1 @@
+{"A": [72, 96, -108, 96], "target": 12, "family": "rlve_bezout_identity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l2-s1/solution/solve.sh b/tasks/construct-rlve-bezout-identity-l2-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..693da166cd6b0a38aab7007cd308116ad2b4d322
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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'
+{"X": [4, -4, -1, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-bezout-identity-l2-s1/tests/check.py b/tasks/construct-rlve-bezout-identity-l2-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..b4cb860ed36974a12b71adc9a0fa3f1ad85c814a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l2-s1/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-l2-s1/tests/instance.json b/tasks/construct-rlve-bezout-identity-l2-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f51ad5044a904b7040ad290980e1890b50148b1e
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l2-s1/tests/instance.json
@@ -0,0 +1 @@
+{"A": [72, 96, -108, 96], "target": 12, "family": "rlve_bezout_identity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l2-s1/tests/run.py b/tasks/construct-rlve-bezout-identity-l2-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-rlve-bezout-identity-l2-s1/tests/test.sh b/tasks/construct-rlve-bezout-identity-l2-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-rlve-bezout-identity-l2-s6/environment/Dockerfile b/tasks/construct-rlve-bezout-identity-l2-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-bezout-identity-l2-s6/environment/instance.json b/tasks/construct-rlve-bezout-identity-l2-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8b9c0062b8c2f73f5932e3da2bc4a761171a98ab
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l2-s6/environment/instance.json
@@ -0,0 +1 @@
+{"A": [-128, -66, -10, -4], "target": 2, "family": "rlve_bezout_identity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l2-s6/solution/solve.sh b/tasks/construct-rlve-bezout-identity-l2-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..7d9e125d709c27d97a091f40e512109337a58a02
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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'
+{"X": [16, -31, 0, -1]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-bezout-identity-l2-s6/tests/check.py b/tasks/construct-rlve-bezout-identity-l2-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..b4cb860ed36974a12b71adc9a0fa3f1ad85c814a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l2-s6/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-l2-s6/tests/instance.json b/tasks/construct-rlve-bezout-identity-l2-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8b9c0062b8c2f73f5932e3da2bc4a761171a98ab
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l2-s6/tests/instance.json
@@ -0,0 +1 @@
+{"A": [-128, -66, -10, -4], "target": 2, "family": "rlve_bezout_identity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l2-s6/tests/run.py b/tasks/construct-rlve-bezout-identity-l2-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-bezout-identity-l2-s6/tests/test.sh b/tasks/construct-rlve-bezout-identity-l2-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-bezout-identity-l2-s8/environment/Dockerfile b/tasks/construct-rlve-bezout-identity-l2-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-rlve-bezout-identity-l2-s8/environment/instance.json b/tasks/construct-rlve-bezout-identity-l2-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1fcac667c33046c98bf1a2e7e7034fb6b2993eb5
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l2-s8/environment/instance.json
@@ -0,0 +1 @@
+{"A": [124, -4, 42, -24], "target": 2, "family": "rlve_bezout_identity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l2-s8/solution/solve.sh b/tasks/construct-rlve-bezout-identity-l2-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..d493eaed1963102d2744fdf43c5741af751beb43
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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": [0, 10, 1, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-bezout-identity-l2-s8/tests/check.py b/tasks/construct-rlve-bezout-identity-l2-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..b4cb860ed36974a12b71adc9a0fa3f1ad85c814a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l2-s8/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-l2-s8/tests/instance.json b/tasks/construct-rlve-bezout-identity-l2-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1fcac667c33046c98bf1a2e7e7034fb6b2993eb5
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l2-s8/tests/instance.json
@@ -0,0 +1 @@
+{"A": [124, -4, 42, -24], "target": 2, "family": "rlve_bezout_identity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l2-s8/tests/run.py b/tasks/construct-rlve-bezout-identity-l2-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-rlve-bezout-identity-l2-s8/tests/test.sh b/tasks/construct-rlve-bezout-identity-l2-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-rlve-bezout-identity-l6-s0/environment/Dockerfile b/tasks/construct-rlve-bezout-identity-l6-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-bezout-identity-l6-s0/environment/instance.json b/tasks/construct-rlve-bezout-identity-l6-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..35729d04a476b707d00d6c91c36efce8a7274ed2
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l6-s0/environment/instance.json
@@ -0,0 +1 @@
+{"A": [18004, -28890, -13270, -18442, -15464, 31360, 4466, -4056, 29088, 2734, 15212, 17142], "target": 2, "family": "rlve_bezout_identity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l6-s0/solution/solve.sh b/tasks/construct-rlve-bezout-identity-l6-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..d2fc227d5022385089a6e522fa88fbc27604e288
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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": [4838, 3015, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-bezout-identity-l6-s0/tests/check.py b/tasks/construct-rlve-bezout-identity-l6-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..b4cb860ed36974a12b71adc9a0fa3f1ad85c814a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l6-s0/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-l6-s0/tests/instance.json b/tasks/construct-rlve-bezout-identity-l6-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..35729d04a476b707d00d6c91c36efce8a7274ed2
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l6-s0/tests/instance.json
@@ -0,0 +1 @@
+{"A": [18004, -28890, -13270, -18442, -15464, 31360, 4466, -4056, 29088, 2734, 15212, 17142], "target": 2, "family": "rlve_bezout_identity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l6-s0/tests/run.py b/tasks/construct-rlve-bezout-identity-l6-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-bezout-identity-l6-s0/tests/test.sh b/tasks/construct-rlve-bezout-identity-l6-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-bezout-identity-l6-s7/environment/Dockerfile b/tasks/construct-rlve-bezout-identity-l6-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-rlve-bezout-identity-l6-s7/environment/instance.json b/tasks/construct-rlve-bezout-identity-l6-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..35441aa226f273e65b4d014a2df13b70b815bb93
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l6-s7/environment/instance.json
@@ -0,0 +1 @@
+{"A": [-16307, 21091, 138, -13179, -23598, -18676, 2829, 29969, 5957, 15065, -25921, -4899], "target": 23, "family": "rlve_bezout_identity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l6-s7/solution/solve.sh b/tasks/construct-rlve-bezout-identity-l6-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..b6d6befe5669a50e80ffbc711e2440c09bb26b63
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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": [-410, -317, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-bezout-identity-l6-s7/tests/check.py b/tasks/construct-rlve-bezout-identity-l6-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..b4cb860ed36974a12b71adc9a0fa3f1ad85c814a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l6-s7/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-l6-s7/tests/instance.json b/tasks/construct-rlve-bezout-identity-l6-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..35441aa226f273e65b4d014a2df13b70b815bb93
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l6-s7/tests/instance.json
@@ -0,0 +1 @@
+{"A": [-16307, 21091, 138, -13179, -23598, -18676, 2829, 29969, 5957, 15065, -25921, -4899], "target": 23, "family": "rlve_bezout_identity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l6-s7/tests/run.py b/tasks/construct-rlve-bezout-identity-l6-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-rlve-bezout-identity-l6-s7/tests/test.sh b/tasks/construct-rlve-bezout-identity-l6-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-rlve-bezout-identity-l6-s9/environment/Dockerfile b/tasks/construct-rlve-bezout-identity-l6-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-bezout-identity-l6-s9/environment/instance.json b/tasks/construct-rlve-bezout-identity-l6-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c26f952cec765b7c0647ce0cbf4b1fda0c684411
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l6-s9/environment/instance.json
@@ -0,0 +1 @@
+{"A": [3396, -23268, 6122, 2866, -16970, -1924, -31354, 31672, -8478, 14886, 2974, -30864], "target": 2, "family": "rlve_bezout_identity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l6-s9/solution/solve.sh b/tasks/construct-rlve-bezout-identity-l6-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..5e6e5b973687848240681940cee6862061042918
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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'
+{"X": [447270, 65280, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-bezout-identity-l6-s9/tests/check.py b/tasks/construct-rlve-bezout-identity-l6-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..b4cb860ed36974a12b71adc9a0fa3f1ad85c814a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l6-s9/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-l6-s9/tests/instance.json b/tasks/construct-rlve-bezout-identity-l6-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c26f952cec765b7c0647ce0cbf4b1fda0c684411
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l6-s9/tests/instance.json
@@ -0,0 +1 @@
+{"A": [3396, -23268, 6122, 2866, -16970, -1924, -31354, 31672, -8478, 14886, 2974, -30864], "target": 2, "family": "rlve_bezout_identity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l6-s9/tests/run.py b/tasks/construct-rlve-bezout-identity-l6-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-bezout-identity-l6-s9/tests/test.sh b/tasks/construct-rlve-bezout-identity-l6-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-construct-hack-interval-l4-s3/environment/Dockerfile b/tasks/construct-rlve-construct-hack-interval-l4-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-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-construct-hack-interval-l4-s3/environment/instance.json b/tasks/construct-rlve-construct-hack-interval-l4-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..42b82fd6c185d695201c52749876a062ae430256
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l4-s3/environment/instance.json
@@ -0,0 +1 @@
+{"mod": 7686302393, "family": "rlve_construct_hack_interval", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-construct-hack-interval-l4-s3/solution/solve.sh b/tasks/construct-rlve-construct-hack-interval-l4-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..51fe27144cc3726264a2afa9d50634c122bbf8a9
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-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'
+{"L": 5617814358, "R": 6617814357}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-construct-hack-interval-l4-s3/tests/check.py b/tasks/construct-rlve-construct-hack-interval-l4-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..34b46e9758e3a50ed16c9c2a29cfc0abe663926b
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l4-s3/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-l4-s3/tests/instance.json b/tasks/construct-rlve-construct-hack-interval-l4-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..42b82fd6c185d695201c52749876a062ae430256
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l4-s3/tests/instance.json
@@ -0,0 +1 @@
+{"mod": 7686302393, "family": "rlve_construct_hack_interval", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-construct-hack-interval-l4-s3/tests/run.py b/tasks/construct-rlve-construct-hack-interval-l4-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-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-construct-hack-interval-l4-s3/tests/test.sh b/tasks/construct-rlve-construct-hack-interval-l4-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-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-construct-hack-interval-l4-s4/environment/Dockerfile b/tasks/construct-rlve-construct-hack-interval-l4-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-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-construct-hack-interval-l4-s4/environment/instance.json b/tasks/construct-rlve-construct-hack-interval-l4-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..895e8c6342c36ec3b7097714922b415c7ef351e4
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l4-s4/environment/instance.json
@@ -0,0 +1 @@
+{"mod": 8522692339, "family": "rlve_construct_hack_interval", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-construct-hack-interval-l4-s4/solution/solve.sh b/tasks/construct-rlve-construct-hack-interval-l4-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..f5d26dd8fe880a82af7831293545b378ba13e601
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-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'
+{"L": 2113461695, "R": 3113461694}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-construct-hack-interval-l4-s4/tests/check.py b/tasks/construct-rlve-construct-hack-interval-l4-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..34b46e9758e3a50ed16c9c2a29cfc0abe663926b
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l4-s4/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-l4-s4/tests/instance.json b/tasks/construct-rlve-construct-hack-interval-l4-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..895e8c6342c36ec3b7097714922b415c7ef351e4
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l4-s4/tests/instance.json
@@ -0,0 +1 @@
+{"mod": 8522692339, "family": "rlve_construct_hack_interval", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-construct-hack-interval-l4-s4/tests/run.py b/tasks/construct-rlve-construct-hack-interval-l4-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-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-construct-hack-interval-l4-s4/tests/test.sh b/tasks/construct-rlve-construct-hack-interval-l4-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-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-crt-l1-s3/environment/Dockerfile b/tasks/construct-rlve-crt-l1-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-crt-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-crt-l1-s3/environment/instance.json b/tasks/construct-rlve-crt-l1-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..52c6b45b4febdf093ed0333b5db26d13a2ffffdf
--- /dev/null
+++ b/tasks/construct-rlve-crt-l1-s3/environment/instance.json
@@ -0,0 +1 @@
+{"moduli": [15, 14], "residues": [4, 5], "family": "rlve_crt", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-crt-l1-s3/solution/solve.sh b/tasks/construct-rlve-crt-l1-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..2346383f8bef3a5e63c542bc4c1fddb7181d67fd
--- /dev/null
+++ b/tasks/construct-rlve-crt-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'
+{"x": 19}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-crt-l1-s3/tests/check.py b/tasks/construct-rlve-crt-l1-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..886d95606ef325930b24794c0959c2ba7b254ebd
--- /dev/null
+++ b/tasks/construct-rlve-crt-l1-s3/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-l1-s3/tests/instance.json b/tasks/construct-rlve-crt-l1-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..52c6b45b4febdf093ed0333b5db26d13a2ffffdf
--- /dev/null
+++ b/tasks/construct-rlve-crt-l1-s3/tests/instance.json
@@ -0,0 +1 @@
+{"moduli": [15, 14], "residues": [4, 5], "family": "rlve_crt", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-crt-l1-s3/tests/run.py b/tasks/construct-rlve-crt-l1-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-crt-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-crt-l1-s3/tests/test.sh b/tasks/construct-rlve-crt-l1-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-crt-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-crt-l1-s4/environment/Dockerfile b/tasks/construct-rlve-crt-l1-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-crt-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-crt-l1-s4/environment/instance.json b/tasks/construct-rlve-crt-l1-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d10d987276e11b7c18cb407cc4ec2c9b4c759c34
--- /dev/null
+++ b/tasks/construct-rlve-crt-l1-s4/environment/instance.json
@@ -0,0 +1 @@
+{"moduli": [4, 10], "residues": [1, 7], "family": "rlve_crt", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-crt-l1-s4/solution/solve.sh b/tasks/construct-rlve-crt-l1-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..2de3a92cc59ac9423eb380e4b67f3fee4a1c9d7a
--- /dev/null
+++ b/tasks/construct-rlve-crt-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'
+{"x": 17}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-crt-l1-s4/tests/check.py b/tasks/construct-rlve-crt-l1-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..886d95606ef325930b24794c0959c2ba7b254ebd
--- /dev/null
+++ b/tasks/construct-rlve-crt-l1-s4/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-l1-s4/tests/instance.json b/tasks/construct-rlve-crt-l1-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d10d987276e11b7c18cb407cc4ec2c9b4c759c34
--- /dev/null
+++ b/tasks/construct-rlve-crt-l1-s4/tests/instance.json
@@ -0,0 +1 @@
+{"moduli": [4, 10], "residues": [1, 7], "family": "rlve_crt", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-crt-l1-s4/tests/run.py b/tasks/construct-rlve-crt-l1-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-crt-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-crt-l1-s4/tests/test.sh b/tasks/construct-rlve-crt-l1-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-crt-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-crt-l5-s2/environment/Dockerfile b/tasks/construct-rlve-crt-l5-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-crt-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-crt-l5-s2/environment/instance.json b/tasks/construct-rlve-crt-l5-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4fc30f3cdf707ec2fd6000e35ccecb7c6e815b53
--- /dev/null
+++ b/tasks/construct-rlve-crt-l5-s2/environment/instance.json
@@ -0,0 +1 @@
+{"moduli": [3279395310, 6832203215, 4519823366, 2615421463, 933731197, 1438987691], "residues": [53617477, 3059600192, 852156675, 2045539018, 554491437, 1257877261], "family": "rlve_crt", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-crt-l5-s2/solution/solve.sh b/tasks/construct-rlve-crt-l5-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..007531212dd0d5fc4fb7d50455a7ace1b0b71667
--- /dev/null
+++ b/tasks/construct-rlve-crt-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'
+{"x": 9891803407}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-crt-l5-s2/tests/check.py b/tasks/construct-rlve-crt-l5-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..886d95606ef325930b24794c0959c2ba7b254ebd
--- /dev/null
+++ b/tasks/construct-rlve-crt-l5-s2/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-l5-s2/tests/instance.json b/tasks/construct-rlve-crt-l5-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4fc30f3cdf707ec2fd6000e35ccecb7c6e815b53
--- /dev/null
+++ b/tasks/construct-rlve-crt-l5-s2/tests/instance.json
@@ -0,0 +1 @@
+{"moduli": [3279395310, 6832203215, 4519823366, 2615421463, 933731197, 1438987691], "residues": [53617477, 3059600192, 852156675, 2045539018, 554491437, 1257877261], "family": "rlve_crt", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-crt-l5-s2/tests/run.py b/tasks/construct-rlve-crt-l5-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-crt-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-crt-l5-s2/tests/test.sh b/tasks/construct-rlve-crt-l5-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-crt-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-distinct-subset-sum-permutation-l4-s2/environment/Dockerfile b/tasks/construct-rlve-distinct-subset-sum-permutation-l4-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-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-distinct-subset-sum-permutation-l4-s2/environment/instance.json b/tasks/construct-rlve-distinct-subset-sum-permutation-l4-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..bb73cffe9969fe943556d4016b17db43b0bb7053
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l4-s2/environment/instance.json
@@ -0,0 +1 @@
+{"A": [27, 24, 2, 14, 4, 7, 5, 1, 6, 26, 16, 13, 0, 25], "family": "rlve_distinct_subset_sum_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l4-s2/solution/solve.sh b/tasks/construct-rlve-distinct-subset-sum-permutation-l4-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..b5c675358cfe97c7789184456678def023ff2202
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-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'
+{"B": [0, 25, 4, 16, 5, 13, 6, 2, 7, 27, 24, 14, 1, 26]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l4-s2/tests/check.py b/tasks/construct-rlve-distinct-subset-sum-permutation-l4-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..06bbbcf872c2fca05785f40fdfdcf85a5e5a0e7c
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l4-s2/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-l4-s2/tests/instance.json b/tasks/construct-rlve-distinct-subset-sum-permutation-l4-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..bb73cffe9969fe943556d4016b17db43b0bb7053
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l4-s2/tests/instance.json
@@ -0,0 +1 @@
+{"A": [27, 24, 2, 14, 4, 7, 5, 1, 6, 26, 16, 13, 0, 25], "family": "rlve_distinct_subset_sum_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l4-s2/tests/run.py b/tasks/construct-rlve-distinct-subset-sum-permutation-l4-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-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-distinct-subset-sum-permutation-l4-s2/tests/test.sh b/tasks/construct-rlve-distinct-subset-sum-permutation-l4-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-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-distinct-subset-sum-permutation-l4-s5/environment/Dockerfile b/tasks/construct-rlve-distinct-subset-sum-permutation-l4-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-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-distinct-subset-sum-permutation-l4-s5/environment/instance.json b/tasks/construct-rlve-distinct-subset-sum-permutation-l4-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8d8a3adb30c54bf0b943e23c88a2b5e8b48aa874
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l4-s5/environment/instance.json
@@ -0,0 +1 @@
+{"A": [4, 17, 12, 13, 5, 25, 18, 7, 9, 15, 14, 27, 21, 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-l4-s5/solution/solve.sh b/tasks/construct-rlve-distinct-subset-sum-permutation-l4-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..81c07d615d75918c3c91f8a1540c02f691e19386
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-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'
+{"B": [5, 18, 13, 14, 7, 27, 21, 9, 12, 17, 15, 1, 25, 4]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l4-s5/tests/check.py b/tasks/construct-rlve-distinct-subset-sum-permutation-l4-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..06bbbcf872c2fca05785f40fdfdcf85a5e5a0e7c
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l4-s5/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-l4-s5/tests/instance.json b/tasks/construct-rlve-distinct-subset-sum-permutation-l4-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8d8a3adb30c54bf0b943e23c88a2b5e8b48aa874
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l4-s5/tests/instance.json
@@ -0,0 +1 @@
+{"A": [4, 17, 12, 13, 5, 25, 18, 7, 9, 15, 14, 27, 21, 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-l4-s5/tests/run.py b/tasks/construct-rlve-distinct-subset-sum-permutation-l4-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-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-distinct-subset-sum-permutation-l4-s5/tests/test.sh b/tasks/construct-rlve-distinct-subset-sum-permutation-l4-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-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-even-degree-partition-l1-s3/environment/Dockerfile b/tasks/construct-rlve-even-degree-partition-l1-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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-even-degree-partition-l1-s3/environment/instance.json b/tasks/construct-rlve-even-degree-partition-l1-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..dd0c45640c11eb84bc4f1c07cf5eb65adffe0244
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l1-s3/environment/instance.json
@@ -0,0 +1 @@
+{"n": 5, "edges": [[0, 1], [0, 4], [1, 2], [1, 3], [2, 4], [3, 4]], "family": "rlve_even_degree_partition", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-even-degree-partition-l1-s3/solution/solve.sh b/tasks/construct-rlve-even-degree-partition-l1-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..cd8b2affce0dcdcfb946c73700debe04c103fd9b
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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'
+{"groups": [1, 2, 2, 2, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-even-degree-partition-l1-s3/tests/check.py b/tasks/construct-rlve-even-degree-partition-l1-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..9c7f0f71ea6d10264ececcc933b97a4a0e6eaf8e
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l1-s3/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-l1-s3/tests/instance.json b/tasks/construct-rlve-even-degree-partition-l1-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..dd0c45640c11eb84bc4f1c07cf5eb65adffe0244
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l1-s3/tests/instance.json
@@ -0,0 +1 @@
+{"n": 5, "edges": [[0, 1], [0, 4], [1, 2], [1, 3], [2, 4], [3, 4]], "family": "rlve_even_degree_partition", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-even-degree-partition-l1-s3/tests/run.py b/tasks/construct-rlve-even-degree-partition-l1-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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-even-degree-partition-l1-s3/tests/test.sh b/tasks/construct-rlve-even-degree-partition-l1-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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-even-degree-partition-l1-s4/environment/Dockerfile b/tasks/construct-rlve-even-degree-partition-l1-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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-even-degree-partition-l1-s4/environment/instance.json b/tasks/construct-rlve-even-degree-partition-l1-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..38cac585fac3fd34e5e4f6c4828d771eb6320273
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l1-s4/environment/instance.json
@@ -0,0 +1 @@
+{"n": 5, "edges": [[0, 1], [1, 4], [2, 3], [2, 4], [3, 4]], "family": "rlve_even_degree_partition", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-even-degree-partition-l1-s4/solution/solve.sh b/tasks/construct-rlve-even-degree-partition-l1-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..2ae2ffd201fe81063d975c996c28af1f55ac01d6
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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'
+{"groups": [1, 2, 1, 1, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-even-degree-partition-l1-s4/tests/check.py b/tasks/construct-rlve-even-degree-partition-l1-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..9c7f0f71ea6d10264ececcc933b97a4a0e6eaf8e
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l1-s4/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-l1-s4/tests/instance.json b/tasks/construct-rlve-even-degree-partition-l1-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..38cac585fac3fd34e5e4f6c4828d771eb6320273
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l1-s4/tests/instance.json
@@ -0,0 +1 @@
+{"n": 5, "edges": [[0, 1], [1, 4], [2, 3], [2, 4], [3, 4]], "family": "rlve_even_degree_partition", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-even-degree-partition-l1-s4/tests/run.py b/tasks/construct-rlve-even-degree-partition-l1-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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-even-degree-partition-l1-s4/tests/test.sh b/tasks/construct-rlve-even-degree-partition-l1-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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-even-degree-partition-l5-s2/environment/Dockerfile b/tasks/construct-rlve-even-degree-partition-l5-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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-even-degree-partition-l5-s2/environment/instance.json b/tasks/construct-rlve-even-degree-partition-l5-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..7d8fe05e44495a7f7e8257f308f6dab6d8e1a354
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l5-s2/environment/instance.json
@@ -0,0 +1 @@
+{"n": 28, "edges": [[0, 1], [0, 5], [0, 6], [0, 7], [0, 8], [0, 9], [0, 10], [0, 14], [0, 17], [0, 20], [0, 23], [0, 25], [0, 27], [1, 9], [1, 11], [1, 13], [1, 15], [1, 16], [1, 18], [1, 21], [1, 27], [2, 3], [2, 5], [2, 6], [2, 11], [2, 12], [2, 14], [2, 17], [2, 19], [2, 24], [2, 25], [3, 5], [3, 14], [3, 16], [3, 17], [3, 25], [3, 26], [3, 27], [4, 16], [4, 17], [4, 24], [5, 11], [5, 13], [5, 14], [5, 15], [5, 17], [5, 21], [5, 25], [5, 26], [5, 27], [6, 10], [6, 18], [6, 19], [6, 23], [6, 24], [6, 25], [6, 27], [7, 8], [7, 9], [7, 10], [7, 17], [7, 18], [7, 20], [7, 21], [7, 25], [7, 26], [8, 11], [8, 17], [8, 18], [8, 20], [8, 21], [8, 22], [8, 25], [8, 26], [9, 12], [9, 13], [9, 17], [9, 19], [9, 25], [9, 27], [10, 15], [10, 17], [10, 18], [10, 25], [10, 26], [11, 19], [11, 20], [11, 21], [11, 25], [12, 14], [12, 19], [12, 23], [13, 14], [13, 16], [13, 17], [13, 24], [13, 25], [14, 15], [14, 16], [14, 17], [14, 18], [14, 25], [15, 16], [15, 17], [15, 22], [15, 23], [15, 24], [15, 25], [15, 26], [15, 27], [16, 22], [16, 24], [16, 25], [16, 26], [16, 27], [17, 18], [17, 19], [17, 25], [17, 26], [17, 27], [18, 22], [18, 25], [18, 27], [19, 20], [19, 21], [19, 22], [19, 24], [20, 21], [20, 22], [20, 23], [20, 24], [20, 26], [21, 22], [21, 23], [21, 25], [22, 24], [22, 27], [23, 24], [23, 25], [24, 27], [25, 27]], "family": "rlve_even_degree_partition", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-even-degree-partition-l5-s2/solution/solve.sh b/tasks/construct-rlve-even-degree-partition-l5-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..1df082635a958e669157d33a3cc0c188663c772e
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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'
+{"groups": [2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-even-degree-partition-l5-s2/tests/check.py b/tasks/construct-rlve-even-degree-partition-l5-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..9c7f0f71ea6d10264ececcc933b97a4a0e6eaf8e
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l5-s2/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-l5-s2/tests/instance.json b/tasks/construct-rlve-even-degree-partition-l5-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..7d8fe05e44495a7f7e8257f308f6dab6d8e1a354
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l5-s2/tests/instance.json
@@ -0,0 +1 @@
+{"n": 28, "edges": [[0, 1], [0, 5], [0, 6], [0, 7], [0, 8], [0, 9], [0, 10], [0, 14], [0, 17], [0, 20], [0, 23], [0, 25], [0, 27], [1, 9], [1, 11], [1, 13], [1, 15], [1, 16], [1, 18], [1, 21], [1, 27], [2, 3], [2, 5], [2, 6], [2, 11], [2, 12], [2, 14], [2, 17], [2, 19], [2, 24], [2, 25], [3, 5], [3, 14], [3, 16], [3, 17], [3, 25], [3, 26], [3, 27], [4, 16], [4, 17], [4, 24], [5, 11], [5, 13], [5, 14], [5, 15], [5, 17], [5, 21], [5, 25], [5, 26], [5, 27], [6, 10], [6, 18], [6, 19], [6, 23], [6, 24], [6, 25], [6, 27], [7, 8], [7, 9], [7, 10], [7, 17], [7, 18], [7, 20], [7, 21], [7, 25], [7, 26], [8, 11], [8, 17], [8, 18], [8, 20], [8, 21], [8, 22], [8, 25], [8, 26], [9, 12], [9, 13], [9, 17], [9, 19], [9, 25], [9, 27], [10, 15], [10, 17], [10, 18], [10, 25], [10, 26], [11, 19], [11, 20], [11, 21], [11, 25], [12, 14], [12, 19], [12, 23], [13, 14], [13, 16], [13, 17], [13, 24], [13, 25], [14, 15], [14, 16], [14, 17], [14, 18], [14, 25], [15, 16], [15, 17], [15, 22], [15, 23], [15, 24], [15, 25], [15, 26], [15, 27], [16, 22], [16, 24], [16, 25], [16, 26], [16, 27], [17, 18], [17, 19], [17, 25], [17, 26], [17, 27], [18, 22], [18, 25], [18, 27], [19, 20], [19, 21], [19, 22], [19, 24], [20, 21], [20, 22], [20, 23], [20, 24], [20, 26], [21, 22], [21, 23], [21, 25], [22, 24], [22, 27], [23, 24], [23, 25], [24, 27], [25, 27]], "family": "rlve_even_degree_partition", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-even-degree-partition-l5-s2/tests/run.py b/tasks/construct-rlve-even-degree-partition-l5-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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-even-degree-partition-l5-s2/tests/test.sh b/tasks/construct-rlve-even-degree-partition-l5-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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-even-degree-partition-l5-s5/environment/Dockerfile b/tasks/construct-rlve-even-degree-partition-l5-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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-even-degree-partition-l5-s5/environment/instance.json b/tasks/construct-rlve-even-degree-partition-l5-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..bcc9c49e19e48dcd36c0ec22cf8d9d856650599b
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l5-s5/environment/instance.json
@@ -0,0 +1 @@
+{"n": 28, "edges": [[0, 1], [0, 4], [0, 17], [0, 24], [1, 2], [1, 6], [1, 7], [1, 12], [1, 17], [1, 19], [1, 24], [1, 25], [2, 4], [2, 8], [2, 9], [2, 15], [2, 17], [3, 8], [3, 10], [3, 15], [3, 16], [3, 18], [3, 22], [3, 25], [3, 26], [3, 27], [4, 5], [4, 6], [4, 9], [4, 11], [4, 12], [4, 15], [4, 17], [4, 20], [4, 24], [5, 7], [5, 8], [5, 16], [5, 19], [5, 20], [5, 24], [6, 27], [7, 11], [7, 14], [7, 17], [7, 24], [8, 12], [8, 14], [8, 17], [8, 25], [9, 16], [9, 18], [9, 22], [9, 23], [9, 24], [9, 25], [9, 26], [10, 14], [10, 15], [10, 18], [10, 23], [10, 27], [11, 21], [11, 22], [11, 25], [11, 27], [12, 17], [12, 19], [12, 24], [13, 15], [13, 16], [13, 23], [13, 27], [14, 15], [14, 21], [14, 23], [15, 16], [15, 18], [15, 23], [15, 27], [16, 18], [16, 23], [16, 26], [16, 27], [17, 19], [18, 21], [18, 23], [18, 25], [18, 26], [18, 27], [20, 22], [21, 22], [21, 23], [21, 27], [22, 26], [22, 27], [23, 25], [23, 27], [24, 25], [25, 27], [26, 27]], "family": "rlve_even_degree_partition", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-even-degree-partition-l5-s5/solution/solve.sh b/tasks/construct-rlve-even-degree-partition-l5-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..d2304ad7d6a27b40f75d2851651d9f7ecc28adef
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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'
+{"groups": [2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-even-degree-partition-l5-s5/tests/check.py b/tasks/construct-rlve-even-degree-partition-l5-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..9c7f0f71ea6d10264ececcc933b97a4a0e6eaf8e
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l5-s5/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-l5-s5/tests/instance.json b/tasks/construct-rlve-even-degree-partition-l5-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..bcc9c49e19e48dcd36c0ec22cf8d9d856650599b
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l5-s5/tests/instance.json
@@ -0,0 +1 @@
+{"n": 28, "edges": [[0, 1], [0, 4], [0, 17], [0, 24], [1, 2], [1, 6], [1, 7], [1, 12], [1, 17], [1, 19], [1, 24], [1, 25], [2, 4], [2, 8], [2, 9], [2, 15], [2, 17], [3, 8], [3, 10], [3, 15], [3, 16], [3, 18], [3, 22], [3, 25], [3, 26], [3, 27], [4, 5], [4, 6], [4, 9], [4, 11], [4, 12], [4, 15], [4, 17], [4, 20], [4, 24], [5, 7], [5, 8], [5, 16], [5, 19], [5, 20], [5, 24], [6, 27], [7, 11], [7, 14], [7, 17], [7, 24], [8, 12], [8, 14], [8, 17], [8, 25], [9, 16], [9, 18], [9, 22], [9, 23], [9, 24], [9, 25], [9, 26], [10, 14], [10, 15], [10, 18], [10, 23], [10, 27], [11, 21], [11, 22], [11, 25], [11, 27], [12, 17], [12, 19], [12, 24], [13, 15], [13, 16], [13, 23], [13, 27], [14, 15], [14, 21], [14, 23], [15, 16], [15, 18], [15, 23], [15, 27], [16, 18], [16, 23], [16, 26], [16, 27], [17, 19], [18, 21], [18, 23], [18, 25], [18, 26], [18, 27], [20, 22], [21, 22], [21, 23], [21, 27], [22, 26], [22, 27], [23, 25], [23, 27], [24, 25], [25, 27], [26, 27]], "family": "rlve_even_degree_partition", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-even-degree-partition-l5-s5/tests/run.py b/tasks/construct-rlve-even-degree-partition-l5-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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-even-degree-partition-l5-s5/tests/test.sh b/tasks/construct-rlve-even-degree-partition-l5-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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-graph-isomorphism-l3-s1/environment/Dockerfile b/tasks/construct-rlve-graph-isomorphism-l3-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-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-graph-isomorphism-l3-s1/environment/instance.json b/tasks/construct-rlve-graph-isomorphism-l3-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..564782b413d7e35880aac49b9348dd6d144dd864
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l3-s1/environment/instance.json
@@ -0,0 +1 @@
+{"n": 12, "edges1": [[1, 5], [2, 9], [0, 4], [6, 10], [2, 10], [2, 6], [6, 7], [1, 9], [7, 11], [2, 4], [0, 6], [1, 7], [5, 11], [0, 10], [3, 11], [0, 11], [3, 8], [0, 7], [1, 11], [9, 10], [3, 4], [7, 8], [1, 4], [0, 9], [1, 8], [2, 3], [7, 9], [8, 9], [6, 8], [5, 10], [0, 1], [9, 11], [0, 2], [0, 5], [4, 6], [1, 10], [8, 11], [0, 8], [2, 11], [3, 5], [10, 11], [5, 8]], "edges2": [[3, 11], [0, 1], [9, 10], [6, 8], [0, 4], [4, 10], [3, 4], [5, 9], [2, 11], [5, 7], [1, 5], [10, 11], [7, 9], [3, 5], [1, 6], [4, 6], [0, 5], [4, 5], [2, 7], [4, 8], [1, 10], [1, 7], [6, 10], [0, 3], [0, 6], [0, 2], [1, 2], [1, 9], [4, 11], [3, 8], [3, 6], [2, 3], [5, 6], [3, 9], [5, 8], [8, 9], [5, 11], [0, 11], [0, 8], [4, 7], [0, 10], [5, 10]], "family": "rlve_graph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-graph-isomorphism-l3-s1/solution/solve.sh b/tasks/construct-rlve-graph-isomorphism-l3-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..a169ad565ede52e877f7b8060ac9897fe46835ff
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-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'
+{"p": [5, 4, 1, 2, 7, 11, 9, 8, 3, 6, 10, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-graph-isomorphism-l3-s1/tests/check.py b/tasks/construct-rlve-graph-isomorphism-l3-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..64b755f5ab12e7270811ccfc2f02d104715517b7
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l3-s1/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-l3-s1/tests/instance.json b/tasks/construct-rlve-graph-isomorphism-l3-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..564782b413d7e35880aac49b9348dd6d144dd864
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l3-s1/tests/instance.json
@@ -0,0 +1 @@
+{"n": 12, "edges1": [[1, 5], [2, 9], [0, 4], [6, 10], [2, 10], [2, 6], [6, 7], [1, 9], [7, 11], [2, 4], [0, 6], [1, 7], [5, 11], [0, 10], [3, 11], [0, 11], [3, 8], [0, 7], [1, 11], [9, 10], [3, 4], [7, 8], [1, 4], [0, 9], [1, 8], [2, 3], [7, 9], [8, 9], [6, 8], [5, 10], [0, 1], [9, 11], [0, 2], [0, 5], [4, 6], [1, 10], [8, 11], [0, 8], [2, 11], [3, 5], [10, 11], [5, 8]], "edges2": [[3, 11], [0, 1], [9, 10], [6, 8], [0, 4], [4, 10], [3, 4], [5, 9], [2, 11], [5, 7], [1, 5], [10, 11], [7, 9], [3, 5], [1, 6], [4, 6], [0, 5], [4, 5], [2, 7], [4, 8], [1, 10], [1, 7], [6, 10], [0, 3], [0, 6], [0, 2], [1, 2], [1, 9], [4, 11], [3, 8], [3, 6], [2, 3], [5, 6], [3, 9], [5, 8], [8, 9], [5, 11], [0, 11], [0, 8], [4, 7], [0, 10], [5, 10]], "family": "rlve_graph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-graph-isomorphism-l3-s1/tests/run.py b/tasks/construct-rlve-graph-isomorphism-l3-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-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-graph-isomorphism-l3-s1/tests/test.sh b/tasks/construct-rlve-graph-isomorphism-l3-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-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-graph-isomorphism-l3-s6/environment/Dockerfile b/tasks/construct-rlve-graph-isomorphism-l3-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-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-rlve-graph-isomorphism-l3-s6/environment/instance.json b/tasks/construct-rlve-graph-isomorphism-l3-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e4fd29964aecd8c5426e6e3b19bd4c2acc8ca93a
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l3-s6/environment/instance.json
@@ -0,0 +1 @@
+{"n": 12, "edges1": [[8, 10], [6, 10], [8, 11], [0, 10], [1, 10], [1, 9], [6, 8], [8, 9], [0, 9]], "edges2": [[2, 9], [1, 9], [3, 9], [3, 7], [1, 3], [3, 11], [10, 11], [9, 10], [7, 10]], "family": "rlve_graph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-graph-isomorphism-l3-s6/solution/solve.sh b/tasks/construct-rlve-graph-isomorphism-l3-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..325522eafee3f048c4402ff075cc9e09bc6d86b3
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-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'
+{"p": [11, 7, 0, 5, 4, 8, 1, 6, 9, 10, 3, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-graph-isomorphism-l3-s6/tests/check.py b/tasks/construct-rlve-graph-isomorphism-l3-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..64b755f5ab12e7270811ccfc2f02d104715517b7
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l3-s6/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-l3-s6/tests/instance.json b/tasks/construct-rlve-graph-isomorphism-l3-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e4fd29964aecd8c5426e6e3b19bd4c2acc8ca93a
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l3-s6/tests/instance.json
@@ -0,0 +1 @@
+{"n": 12, "edges1": [[8, 10], [6, 10], [8, 11], [0, 10], [1, 10], [1, 9], [6, 8], [8, 9], [0, 9]], "edges2": [[2, 9], [1, 9], [3, 9], [3, 7], [1, 3], [3, 11], [10, 11], [9, 10], [7, 10]], "family": "rlve_graph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-graph-isomorphism-l3-s6/tests/run.py b/tasks/construct-rlve-graph-isomorphism-l3-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-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-rlve-graph-isomorphism-l3-s6/tests/test.sh b/tasks/construct-rlve-graph-isomorphism-l3-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-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-rlve-graph-isomorphism-l3-s8/environment/Dockerfile b/tasks/construct-rlve-graph-isomorphism-l3-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-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-graph-isomorphism-l3-s8/environment/instance.json b/tasks/construct-rlve-graph-isomorphism-l3-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..597f043181e447b65d3c084d9832982f0a7a5fbd
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l3-s8/environment/instance.json
@@ -0,0 +1 @@
+{"n": 12, "edges1": [[9, 10], [4, 6], [2, 4], [6, 11], [1, 5], [5, 8], [3, 5], [9, 11], [2, 11], [5, 11], [5, 9], [1, 3], [1, 9], [2, 10], [1, 4], [2, 7], [3, 6], [0, 6], [2, 3], [0, 8], [0, 7], [0, 4], [4, 9], [10, 11], [7, 10], [0, 3], [8, 9], [2, 5], [4, 7]], "edges2": [[7, 11], [3, 10], [1, 6], [3, 5], [1, 10], [0, 2], [7, 8], [2, 10], [10, 11], [3, 9], [5, 8], [5, 11], [4, 8], [4, 9], [1, 3], [0, 11], [0, 6], [2, 4], [8, 9], [6, 11], [8, 11], [4, 6], [6, 8], [2, 3], [7, 9], [4, 10], [7, 10], [0, 4], [1, 9]], "family": "rlve_graph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-graph-isomorphism-l3-s8/solution/solve.sh b/tasks/construct-rlve-graph-isomorphism-l3-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..ff780893985d5ea37065b25c608754656fdf934d
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-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'
+{"p": [3, 7, 4, 9, 10, 8, 1, 2, 5, 11, 0, 6]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-graph-isomorphism-l3-s8/tests/check.py b/tasks/construct-rlve-graph-isomorphism-l3-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..64b755f5ab12e7270811ccfc2f02d104715517b7
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l3-s8/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-l3-s8/tests/instance.json b/tasks/construct-rlve-graph-isomorphism-l3-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..597f043181e447b65d3c084d9832982f0a7a5fbd
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l3-s8/tests/instance.json
@@ -0,0 +1 @@
+{"n": 12, "edges1": [[9, 10], [4, 6], [2, 4], [6, 11], [1, 5], [5, 8], [3, 5], [9, 11], [2, 11], [5, 11], [5, 9], [1, 3], [1, 9], [2, 10], [1, 4], [2, 7], [3, 6], [0, 6], [2, 3], [0, 8], [0, 7], [0, 4], [4, 9], [10, 11], [7, 10], [0, 3], [8, 9], [2, 5], [4, 7]], "edges2": [[7, 11], [3, 10], [1, 6], [3, 5], [1, 10], [0, 2], [7, 8], [2, 10], [10, 11], [3, 9], [5, 8], [5, 11], [4, 8], [4, 9], [1, 3], [0, 11], [0, 6], [2, 4], [8, 9], [6, 11], [8, 11], [4, 6], [6, 8], [2, 3], [7, 9], [4, 10], [7, 10], [0, 4], [1, 9]], "family": "rlve_graph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-graph-isomorphism-l3-s8/tests/run.py b/tasks/construct-rlve-graph-isomorphism-l3-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-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-graph-isomorphism-l3-s8/tests/test.sh b/tasks/construct-rlve-graph-isomorphism-l3-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-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-grid-parity-construction-l3-s1/environment/Dockerfile b/tasks/construct-rlve-grid-parity-construction-l3-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-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-grid-parity-construction-l3-s1/environment/instance.json b/tasks/construct-rlve-grid-parity-construction-l3-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..9a0be4a563c410c31e54cf1d6cb8f9a85756a61c
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l3-s1/environment/instance.json
@@ -0,0 +1 @@
+{"parity": ["0001", "1101", "0000", "0000", "1000", "0111"], "family": "rlve_grid_parity_construction", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-grid-parity-construction-l3-s1/solution/solve.sh b/tasks/construct-rlve-grid-parity-construction-l3-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..41b3d03097ac553885a5add6765b7281ebadc1de
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-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'
+{"grid": ["0001", "0010", "1011", "1010", "0000", "0010"]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-grid-parity-construction-l3-s1/tests/check.py b/tasks/construct-rlve-grid-parity-construction-l3-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..89d5d286593348d0000ac10db74e1c6242be6c97
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l3-s1/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-l3-s1/tests/instance.json b/tasks/construct-rlve-grid-parity-construction-l3-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..9a0be4a563c410c31e54cf1d6cb8f9a85756a61c
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l3-s1/tests/instance.json
@@ -0,0 +1 @@
+{"parity": ["0001", "1101", "0000", "0000", "1000", "0111"], "family": "rlve_grid_parity_construction", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-grid-parity-construction-l3-s1/tests/run.py b/tasks/construct-rlve-grid-parity-construction-l3-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-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-grid-parity-construction-l3-s1/tests/test.sh b/tasks/construct-rlve-grid-parity-construction-l3-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-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-grid-parity-construction-l3-s8/environment/Dockerfile b/tasks/construct-rlve-grid-parity-construction-l3-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-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-grid-parity-construction-l3-s8/environment/instance.json b/tasks/construct-rlve-grid-parity-construction-l3-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b6041f4112d39baf710931e4c27f2eba7b45a120
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l3-s8/environment/instance.json
@@ -0,0 +1 @@
+{"parity": ["11", "00", "00", "10", "11", "01"], "family": "rlve_grid_parity_construction", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-grid-parity-construction-l3-s8/solution/solve.sh b/tasks/construct-rlve-grid-parity-construction-l3-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..eb43cf80dd0d68088842c1696e27aa95a3239191
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-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'
+{"grid": ["11", "11", "11", "11", "01", "11"]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-grid-parity-construction-l3-s8/tests/check.py b/tasks/construct-rlve-grid-parity-construction-l3-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..89d5d286593348d0000ac10db74e1c6242be6c97
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l3-s8/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-l3-s8/tests/instance.json b/tasks/construct-rlve-grid-parity-construction-l3-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b6041f4112d39baf710931e4c27f2eba7b45a120
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l3-s8/tests/instance.json
@@ -0,0 +1 @@
+{"parity": ["11", "00", "00", "10", "11", "01"], "family": "rlve_grid_parity_construction", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-grid-parity-construction-l3-s8/tests/run.py b/tasks/construct-rlve-grid-parity-construction-l3-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-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-grid-parity-construction-l3-s8/tests/test.sh b/tasks/construct-rlve-grid-parity-construction-l3-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-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-imp-party-l1-s9/environment/Dockerfile b/tasks/construct-rlve-imp-party-l1-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-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-imp-party-l1-s9/environment/instance.json b/tasks/construct-rlve-imp-party-l1-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..595891a821a91e32c3c8d7c32058b393fe94fdd5
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-l1-s9/environment/instance.json
@@ -0,0 +1 @@
+{"n": 3, "edges": [[0, 1], [0, 5], [0, 6], [0, 7], [0, 8], [1, 2], [1, 5], [1, 6], [1, 7], [1, 8], [2, 5], [3, 5], [3, 6], [3, 7], [4, 5], [4, 6], [4, 7], [4, 8], [5, 6], [5, 7], [5, 8], [6, 7], [6, 8], [7, 8]], "family": "rlve_imp_party", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-imp-party-l1-s9/solution/solve.sh b/tasks/construct-rlve-imp-party-l1-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..da2a43c1fe05b22807800c696916113bf111109c
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-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'
+{"clique": [5, 6, 7]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-imp-party-l1-s9/tests/check.py b/tasks/construct-rlve-imp-party-l1-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..97b5a037a2ba8bf4495cf87cc074d1c60b11311c
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-l1-s9/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-l1-s9/tests/instance.json b/tasks/construct-rlve-imp-party-l1-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..595891a821a91e32c3c8d7c32058b393fe94fdd5
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-l1-s9/tests/instance.json
@@ -0,0 +1 @@
+{"n": 3, "edges": [[0, 1], [0, 5], [0, 6], [0, 7], [0, 8], [1, 2], [1, 5], [1, 6], [1, 7], [1, 8], [2, 5], [3, 5], [3, 6], [3, 7], [4, 5], [4, 6], [4, 7], [4, 8], [5, 6], [5, 7], [5, 8], [6, 7], [6, 8], [7, 8]], "family": "rlve_imp_party", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-imp-party-l1-s9/tests/run.py b/tasks/construct-rlve-imp-party-l1-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-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-imp-party-l1-s9/tests/test.sh b/tasks/construct-rlve-imp-party-l1-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-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-imp-party-l5-s1/environment/Dockerfile b/tasks/construct-rlve-imp-party-l5-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-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-rlve-imp-party-l5-s1/environment/instance.json b/tasks/construct-rlve-imp-party-l5-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..eba63ec41936188a592e7b24177fd44b677dade2
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-l5-s1/environment/instance.json
@@ -0,0 +1 @@
+{"n": 18, "edges": [[0, 1], [0, 4], [0, 5], [0, 6], [0, 7], [0, 8], [0, 9], [0, 10], [0, 12], [0, 13], [0, 14], [0, 15], [0, 16], [0, 18], [0, 19], [0, 20], [0, 21], [0, 22], [0, 24], [0, 25], [0, 26], [0, 30], [0, 33], [0, 35], [0, 36], [0, 37], [0, 38], [0, 39], [0, 40], [0, 41], [0, 43], [0, 45], [0, 48], [0, 50], [0, 51], [1, 4], [1, 5], [1, 6], [1, 7], [1, 8], [1, 9], [1, 10], [1, 12], [1, 13], [1, 14], [1, 15], [1, 16], [1, 18], [1, 19], [1, 20], [1, 21], [1, 22], [1, 24], [1, 25], [1, 26], [1, 30], [1, 33], [1, 35], [1, 36], [1, 37], [1, 38], [1, 39], [1, 40], [1, 41], [1, 43], [1, 45], [1, 48], [1, 50], [1, 51], [3, 24], [4, 5], [4, 6], [4, 7], [4, 8], [4, 9], [4, 10], [4, 12], [4, 13], [4, 14], [4, 15], [4, 16], [4, 18], [4, 19], [4, 20], [4, 21], [4, 22], [4, 24], [4, 25], [4, 26], [4, 30], [4, 33], [4, 35], [4, 36], [4, 37], [4, 38], [4, 39], [4, 40], [4, 41], [4, 43], [4, 45], [4, 48], [4, 50], [4, 51], [5, 6], [5, 7], [5, 8], [5, 9], [5, 10], [5, 12], [5, 13], [5, 14], 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39], [25, 40], [25, 41], [25, 43], [25, 45], [25, 48], [25, 50], [25, 51], [26, 30], [26, 33], [26, 35], [26, 36], [26, 37], [26, 38], [26, 39], [26, 40], [26, 41], [26, 43], [26, 45], [26, 48], [26, 50], [26, 51], [30, 33], [30, 35], [30, 36], [30, 37], [30, 38], [30, 39], [30, 40], [30, 41], [30, 43], [30, 45], [30, 48], [30, 50], [30, 51], [30, 52], [33, 35], [33, 36], [33, 37], [33, 38], [33, 39], [33, 40], [33, 41], [33, 43], [33, 45], [33, 48], [33, 50], [33, 51], [35, 36], [35, 37], [35, 38], [35, 39], [35, 40], [35, 41], [35, 43], [35, 45], [35, 48], [35, 50], [35, 51], [36, 37], [36, 38], [36, 39], [36, 40], [36, 41], [36, 43], [36, 45], [36, 48], [36, 50], [36, 51], [37, 38], [37, 39], [37, 40], [37, 41], [37, 43], [37, 45], [37, 48], [37, 50], [37, 51], [38, 39], [38, 40], [38, 41], [38, 43], [38, 45], [38, 48], [38, 50], [38, 51], [39, 40], [39, 41], [39, 43], [39, 45], [39, 48], [39, 50], [39, 51], [40, 41], [40, 43], [40, 45], [40, 48], [40, 50], [40, 51], [41, 43], [41, 45], [41, 48], [41, 49], [41, 50], [41, 51], [43, 45], [43, 48], [43, 50], [43, 51], [45, 48], [45, 50], [45, 51], [48, 50], [48, 51], [50, 51]], "family": "rlve_imp_party", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-imp-party-l5-s1/solution/solve.sh b/tasks/construct-rlve-imp-party-l5-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..dc9e37a78e3550b62630dc98e5b89aa1dd7b383a
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-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'
+{"clique": [4, 5, 6, 7, 8, 16, 20, 24, 30, 33, 37, 38, 40, 41, 43, 48, 50, 51]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-imp-party-l5-s1/tests/check.py b/tasks/construct-rlve-imp-party-l5-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..97b5a037a2ba8bf4495cf87cc074d1c60b11311c
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-l5-s1/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-l5-s1/tests/instance.json b/tasks/construct-rlve-imp-party-l5-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..eba63ec41936188a592e7b24177fd44b677dade2
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-l5-s1/tests/instance.json
@@ -0,0 +1 @@
+{"n": 18, "edges": [[0, 1], [0, 4], [0, 5], [0, 6], [0, 7], [0, 8], [0, 9], [0, 10], [0, 12], [0, 13], [0, 14], [0, 15], [0, 16], [0, 18], [0, 19], [0, 20], [0, 21], [0, 22], [0, 24], [0, 25], [0, 26], [0, 30], [0, 33], [0, 35], [0, 36], [0, 37], [0, 38], [0, 39], [0, 40], [0, 41], [0, 43], [0, 45], [0, 48], [0, 50], [0, 51], [1, 4], [1, 5], [1, 6], [1, 7], [1, 8], [1, 9], [1, 10], [1, 12], [1, 13], [1, 14], [1, 15], [1, 16], [1, 18], [1, 19], [1, 20], [1, 21], [1, 22], [1, 24], [1, 25], [1, 26], [1, 30], [1, 33], [1, 35], [1, 36], [1, 37], [1, 38], [1, 39], [1, 40], [1, 41], [1, 43], [1, 45], [1, 48], [1, 50], [1, 51], [3, 24], [4, 5], [4, 6], [4, 7], [4, 8], [4, 9], [4, 10], [4, 12], [4, 13], [4, 14], [4, 15], [4, 16], [4, 18], [4, 19], [4, 20], [4, 21], [4, 22], [4, 24], [4, 25], [4, 26], [4, 30], [4, 33], [4, 35], [4, 36], [4, 37], [4, 38], [4, 39], [4, 40], [4, 41], [4, 43], [4, 45], [4, 48], [4, 50], [4, 51], [5, 6], [5, 7], [5, 8], [5, 9], [5, 10], [5, 12], [5, 13], [5, 14], 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37], [13, 38], [13, 39], [13, 40], [13, 41], [13, 43], [13, 45], [13, 48], [13, 50], [13, 51], [14, 15], [14, 16], [14, 18], [14, 19], [14, 20], [14, 21], [14, 22], [14, 24], [14, 25], [14, 26], [14, 30], [14, 33], [14, 35], [14, 36], [14, 37], [14, 38], [14, 39], [14, 40], [14, 41], [14, 43], [14, 45], [14, 48], [14, 50], [14, 51], [15, 16], [15, 18], [15, 19], [15, 20], [15, 21], [15, 22], [15, 24], [15, 25], [15, 26], [15, 30], [15, 33], [15, 35], [15, 36], [15, 37], [15, 38], [15, 39], [15, 40], [15, 41], [15, 43], [15, 45], [15, 48], [15, 50], [15, 51], [16, 18], [16, 19], [16, 20], [16, 21], [16, 22], [16, 24], [16, 25], [16, 26], [16, 30], [16, 33], [16, 35], [16, 36], [16, 37], [16, 38], [16, 39], [16, 40], [16, 41], [16, 43], [16, 45], [16, 48], [16, 50], [16, 51], [18, 19], [18, 20], [18, 21], [18, 22], [18, 23], [18, 24], [18, 25], [18, 26], [18, 30], [18, 33], [18, 35], [18, 36], [18, 37], [18, 38], [18, 39], [18, 40], [18, 41], [18, 43], [18, 45], [18, 48], [18, 50], [18, 51], [19, 20], [19, 21], [19, 22], [19, 24], [19, 25], [19, 26], [19, 30], [19, 33], [19, 35], [19, 36], [19, 37], [19, 38], [19, 39], [19, 40], [19, 41], [19, 43], [19, 45], [19, 48], [19, 50], [19, 51], [20, 21], [20, 22], [20, 24], [20, 25], [20, 26], [20, 30], [20, 33], [20, 35], [20, 36], [20, 37], [20, 38], [20, 39], [20, 40], [20, 41], [20, 43], [20, 45], [20, 48], [20, 50], [20, 51], [21, 22], [21, 24], [21, 25], [21, 26], [21, 30], [21, 33], [21, 34], [21, 35], [21, 36], [21, 37], [21, 38], [21, 39], [21, 40], [21, 41], [21, 43], [21, 45], [21, 48], [21, 50], [21, 51], [22, 24], [22, 25], [22, 26], [22, 30], [22, 33], [22, 35], [22, 36], [22, 37], [22, 38], [22, 39], [22, 40], [22, 41], [22, 43], [22, 45], [22, 48], [22, 50], [22, 51], [23, 30], [24, 25], [24, 26], [24, 30], [24, 33], [24, 35], [24, 36], [24, 37], [24, 38], [24, 39], [24, 40], [24, 41], [24, 43], [24, 45], [24, 48], [24, 50], [24, 51], [25, 26], [25, 30], [25, 33], [25, 35], [25, 36], [25, 37], [25, 38], [25, 39], [25, 40], [25, 41], [25, 43], [25, 45], [25, 48], [25, 50], [25, 51], [26, 30], [26, 33], [26, 35], [26, 36], [26, 37], [26, 38], [26, 39], [26, 40], [26, 41], [26, 43], [26, 45], [26, 48], [26, 50], [26, 51], [30, 33], [30, 35], [30, 36], [30, 37], [30, 38], [30, 39], [30, 40], [30, 41], [30, 43], [30, 45], [30, 48], [30, 50], [30, 51], [30, 52], [33, 35], [33, 36], [33, 37], [33, 38], [33, 39], [33, 40], [33, 41], [33, 43], [33, 45], [33, 48], [33, 50], [33, 51], [35, 36], [35, 37], [35, 38], [35, 39], [35, 40], [35, 41], [35, 43], [35, 45], [35, 48], [35, 50], [35, 51], [36, 37], [36, 38], [36, 39], [36, 40], [36, 41], [36, 43], [36, 45], [36, 48], [36, 50], [36, 51], [37, 38], [37, 39], [37, 40], [37, 41], [37, 43], [37, 45], [37, 48], [37, 50], [37, 51], [38, 39], [38, 40], [38, 41], [38, 43], [38, 45], [38, 48], [38, 50], [38, 51], [39, 40], [39, 41], [39, 43], [39, 45], [39, 48], [39, 50], [39, 51], [40, 41], [40, 43], [40, 45], [40, 48], [40, 50], [40, 51], [41, 43], [41, 45], [41, 48], [41, 49], [41, 50], [41, 51], [43, 45], [43, 48], [43, 50], [43, 51], [45, 48], [45, 50], [45, 51], [48, 50], [48, 51], [50, 51]], "family": "rlve_imp_party", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-imp-party-l5-s1/tests/run.py b/tasks/construct-rlve-imp-party-l5-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-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-rlve-imp-party-l5-s1/tests/test.sh b/tasks/construct-rlve-imp-party-l5-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-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-rlve-imp-party-l5-s6/tests/run.py b/tasks/construct-rlve-imp-party-l5-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-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-integer-linear-system-l2-s1/environment/Dockerfile b/tasks/construct-rlve-integer-linear-system-l2-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-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-rlve-integer-linear-system-l2-s1/environment/instance.json b/tasks/construct-rlve-integer-linear-system-l2-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..91487c44a2bc52caed308c6e7405e7d9e75d25bc
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-l2-s1/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [[1, 1, 0, 1, 0], [0, 0, 0, 1, 1], [1, 1, 1, 1, 0], [0, 1, 0, 1, 1], [0, 0, 0, 0, 1], [0, 1, 1, 0, 1], [0, 0, 1, 1, 1], [0, 0, 0, 1, 0], [1, 0, 1, 0, 1], [1, 1, 0, 1, 1], [1, 0, 1, 1, 0]], "rhs": [12, 6, 16, 9, 1, 8, 10, 5, 9, 13, 13], "family": "rlve_integer_linear_system", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-integer-linear-system-l2-s1/solution/solve.sh b/tasks/construct-rlve-integer-linear-system-l2-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..a3a9edf7f3be13499361324345b714501ffb827d
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-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'
+{"x": [4, 3, 4, 5, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-integer-linear-system-l2-s1/tests/check.py b/tasks/construct-rlve-integer-linear-system-l2-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..ff19002b3fddc9597f84ee31ef5ffbee5d3fac32
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-l2-s1/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-l2-s1/tests/instance.json b/tasks/construct-rlve-integer-linear-system-l2-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..91487c44a2bc52caed308c6e7405e7d9e75d25bc
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-l2-s1/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [[1, 1, 0, 1, 0], [0, 0, 0, 1, 1], [1, 1, 1, 1, 0], [0, 1, 0, 1, 1], [0, 0, 0, 0, 1], [0, 1, 1, 0, 1], [0, 0, 1, 1, 1], [0, 0, 0, 1, 0], [1, 0, 1, 0, 1], [1, 1, 0, 1, 1], [1, 0, 1, 1, 0]], "rhs": [12, 6, 16, 9, 1, 8, 10, 5, 9, 13, 13], "family": "rlve_integer_linear_system", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-integer-linear-system-l2-s1/tests/run.py b/tasks/construct-rlve-integer-linear-system-l2-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-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-rlve-integer-linear-system-l2-s1/tests/test.sh b/tasks/construct-rlve-integer-linear-system-l2-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-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-rlve-integer-linear-system-l2-s6/environment/Dockerfile b/tasks/construct-rlve-integer-linear-system-l2-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-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-integer-linear-system-l2-s6/environment/instance.json b/tasks/construct-rlve-integer-linear-system-l2-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..ea5e630f0d4022896b80c54f206ab5acdcc50af1
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-l2-s6/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [[1, 0, 0, 0, 0], [1, 1, 1, 1, 1], [1, 1, 0, 0, 1], [1, 1, 0, 1, 1], [1, 1, 0, 0, 1], [1, 0, 0, 0, 1], [0, 1, 1, 1, 1], [0, 0, 1, 0, 1], [0, 1, 1, 0, 1], [0, 1, 0, 1, 0], [0, 0, 0, 1, 0], [1, 1, 0, 1, 1], [0, 1, 1, 1, 1], [0, 1, 0, 1, 0]], "rhs": [2, 12, 5, 7, 5, 3, 10, 6, 8, 4, 2, 7, 10, 4], "family": "rlve_integer_linear_system", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-integer-linear-system-l2-s6/solution/solve.sh b/tasks/construct-rlve-integer-linear-system-l2-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..3f6e66ddcf650459fab390403ff5163c980ac0e1
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-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'
+{"x": [2, 2, 5, 2, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-integer-linear-system-l2-s6/tests/check.py b/tasks/construct-rlve-integer-linear-system-l2-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..ff19002b3fddc9597f84ee31ef5ffbee5d3fac32
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-l2-s6/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-l2-s6/tests/instance.json b/tasks/construct-rlve-integer-linear-system-l2-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..ea5e630f0d4022896b80c54f206ab5acdcc50af1
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-l2-s6/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [[1, 0, 0, 0, 0], [1, 1, 1, 1, 1], [1, 1, 0, 0, 1], [1, 1, 0, 1, 1], [1, 1, 0, 0, 1], [1, 0, 0, 0, 1], [0, 1, 1, 1, 1], [0, 0, 1, 0, 1], [0, 1, 1, 0, 1], [0, 1, 0, 1, 0], [0, 0, 0, 1, 0], [1, 1, 0, 1, 1], [0, 1, 1, 1, 1], [0, 1, 0, 1, 0]], "rhs": [2, 12, 5, 7, 5, 3, 10, 6, 8, 4, 2, 7, 10, 4], "family": "rlve_integer_linear_system", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-integer-linear-system-l2-s6/tests/run.py b/tasks/construct-rlve-integer-linear-system-l2-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-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-integer-linear-system-l2-s6/tests/test.sh b/tasks/construct-rlve-integer-linear-system-l2-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-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-integer-linear-system-l6-s9/environment/Dockerfile b/tasks/construct-rlve-integer-linear-system-l6-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-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-integer-linear-system-l6-s9/environment/instance.json b/tasks/construct-rlve-integer-linear-system-l6-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..51265bd02ca653c13b3085c6df0b559f2952a844
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-l6-s9/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [[0, 0, 2, 0, 0, 5, 3, 0, 0, 0, 1, 0, 0, 0, 4, 1, 0, 4, 2, 0, 5, 0, 1, 0, 0, 3], [0, 0, 0, 0, 5, 3, 0, 1, 3, 4, 0, 0, 2, 1, 4, 3, 0, 0, 3, 0, 0, 5, 0, 1, 0, 0], [0, 0, 4, 0, 2, 1, 0, 2, 0, 2, 0, 0, 2, 2, 0, 3, 2, 1, 0, 0, 0, 2, 0, 0, 4, 3], [0, 0, 2, 5, 0, 4, 0, 0, 5, 0, 1, 2, 0, 4, 0, 4, 4, 3, 0, 0, 2, 1, 1, 0, 0, 2], [2, 3, 2, 1, 0, 2, 0, 3, 0, 1, 0, 5, 0, 0, 5, 0, 0, 2, 3, 4, 0, 5, 3, 0, 1, 0], [4, 1, 0, 0, 4, 0, 0, 0, 0, 1, 2, 0, 2, 0, 0, 5, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0], [0, 3, 3, 1, 3, 1, 0, 5, 1, 1, 0, 1, 3, 0, 4, 3, 0, 0, 0, 0, 0, 2, 5, 0, 0, 2], [4, 0, 5, 0, 3, 0, 0, 0, 2, 5, 2, 0, 0, 4, 0, 2, 0, 0, 3, 2, 0, 3, 0, 4, 4, 3], [0, 4, 3, 0, 4, 0, 0, 0, 3, 5, 1, 3, 5, 4, 0, 4, 4, 0, 1, 4, 2, 0, 4, 3, 4, 0], [4, 4, 0, 3, 0, 0, 0, 0, 4, 0, 1, 0, 0, 4, 0, 0, 0, 2, 0, 4, 4, 1, 0, 0, 0, 5], [4, 2, 4, 4, 0, 1, 0, 0, 0, 0, 1, 0, 0, 4, 2, 4, 0, 0, 4, 0, 0, 3, 1, 1, 5, 5], [0, 3, 0, 0, 2, 0, 1, 4, 4, 5, 0, 2, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 3, 2, 1], [0, 5, 4, 4, 2, 2, 0, 0, 2, 1, 3, 4, 5, 0, 5, 0, 4, 5, 1, 0, 4, 3, 0, 0, 0, 0], [1, 4, 0, 4, 0, 0, 0, 0, 5, 0, 0, 0, 5, 4, 1, 2, 0, 3, 0, 3, 0, 4, 3, 3, 0, 0], [3, 0, 1, 5, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 3, 0, 0], [0, 0, 0, 0, 4, 3, 0, 3, 3, 3, 0, 0, 0, 0, 5, 2, 0, 1, 0, 4, 0, 4, 1, 0, 0, 2], [0, 0, 1, 0, 0, 5, 1, 0, 3, 2, 0, 3, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [1, 0, 5, 0, 2, 2, 0, 0, 1, 4, 2, 3, 2, 4, 3, 0, 0, 2, 2, 0, 5, 1, 0, 3, 5, 3], [3, 5, 4, 0, 4, 0, 0, 0, 1, 0, 0, 4, 0, 5, 3, 0, 2, 0, 0, 0, 5, 1, 0, 5, 0, 3], [2, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 2, 0, 4, 0, 0, 0, 2, 4, 3, 5, 0, 0, 0, 3, 0], [0, 3, 0, 0, 0, 5, 0, 5, 0, 1, 3, 0, 2, 0, 0, 4, 0, 0, 0, 4, 5, 3, 0, 0, 1, 4], [1, 0, 0, 4, 3, 3, 5, 4, 3, 0, 0, 0, 4, 5, 3, 0, 0, 3, 0, 4, 5, 0, 1, 0, 0, 3], [2, 0, 3, 5, 3, 3, 3, 1, 5, 3, 0, 5, 0, 1, 1, 5, 0, 0, 4, 0, 0, 0, 4, 0, 5, 1], [5, 0, 4, 0, 0, 5, 3, 0, 0, 0, 0, 0, 4, 0, 0, 0, 5, 1, 0, 0, 0, 1, 0, 0, 0, 0], [1, 1, 5, 0, 0, 0, 0, 1, 0, 4, 3, 0, 0, 0, 0, 0, 0, 5, 0, 2, 0, 5, 4, 0, 2, 3], [4, 0, 0, 4, 0, 2, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 2, 0, 4, 0, 0, 0, 3], [1, 0, 0, 0, 0, 0, 0, 5, 3, 4, 1, 0, 0, 5, 3, 0, 5, 0, 0, 3, 4, 1, 0, 3, 0, 5], [0, 0, 3, 0, 0, 2, 0, 0, 0, 5, 0, 1, 0, 1, 0, 0, 5, 5, 4, 3, 0, 0, 3, 0, 0, 0], [0, 3, 0, 0, 0, 3, 5, 0, 0, 2, 0, 0, 5, 0, 0, 2, 2, 0, 0, 4, 2, 3, 1, 0, 0, 0], [0, 0, 2, 4, 3, 3, 4, 1, 5, 0, 0, 4, 0, 0, 1, 0, 3, 4, 0, 2, 3, 0, 0, 0, 0, 0], [2, 0, 0, 0, 0, 5, 0, 5, 1, 2, 0, 4, 0, 0, 0, 0, 2, 5, 4, 5, 3, 3, 2, 3, 0, 5], [5, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 3, 0, 2, 1, 0, 0, 5, 2, 1, 2, 0, 2, 4], [0, 0, 0, 0, 1, 0, 0, 3, 1, 3, 0, 0, 0, 3, 0, 5, 0, 1, 2, 1, 0, 1, 0, 0, 0, 5], [2, 3, 0, 3, 1, 4, 0, 0, 0, 0, 2, 3, 0, 0, 5, 0, 0, 5, 1, 4, 0, 1, 1, 0, 2, 0], [0, 0, 0, 1, 2, 0, 0, 5, 0, 0, 1, 0, 0, 5, 5, 1, 1, 0, 0, 0, 0, 0, 1, 5, 0, 0], [0, 5, 0, 0, 2, 1, 1, 0, 0, 4, 4, 2, 5, 0, 0, 1, 0, 0, 5, 0, 0, 0, 0, 0, 0, 2], [3, 0, 5, 0, 0, 0, 0, 2, 4, 3, 1, 5, 1, 0, 2, 2, 4, 0, 0, 0, 0, 5, 0, 0, 4, 0], [0, 0, 1, 2, 4, 5, 0, 5, 5, 0, 0, 3, 0, 0, 5, 4, 0, 0, 1, 2, 0, 0, 5, 0, 1, 0], [0, 0, 0, 4, 0, 0, 4, 1, 0, 0, 0, 0, 1, 5, 1, 0, 2, 0, 4, 0, 4, 5, 0, 0, 1, 1], [0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 3, 5, 0, 0, 2, 0, 2, 4, 0], [0, 0, 0, 3, 3, 4, 2, 0, 0, 2, 0, 0, 5, 0, 0, 0, 0, 0, 4, 1, 0, 1, 0, 5, 0, 3], [0, 3, 5, 3, 3, 0, 0, 0, 1, 2, 0, 0, 3, 0, 2, 0, 0, 4, 0, 5, 0, 0, 5, 0, 1, 0], [4, 0, 1, 0, 0, 1, 4, 3, 0, 0, 4, 3, 0, 0, 0, 0, 5, 1, 1, 5, 2, 0, 5, 5, 0, 2], [0, 0, 2, 5, 0, 2, 0, 0, 1, 0, 0, 4, 2, 0, 5, 0, 3, 0, 0, 5, 5, 0, 2, 5, 0, 4], [3, 3, 0, 3, 0, 0, 0, 5, 0, 0, 0, 0, 5, 5, 0, 0, 3, 0, 2, 1, 0, 0, 0, 0, 0, 0], [5, 0, 0, 3, 3, 3, 1, 0, 2, 4, 1, 0, 2, 5, 5, 0, 0, 5, 0, 4, 0, 0, 2, 3, 2, 2], [0, 3, 1, 5, 0, 2, 0, 1, 0, 3, 0, 5, 1, 1, 5, 0, 2, 1, 4, 4, 2, 0, 0, 1, 2, 1], [3, 4, 3, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 2, 2, 5, 1, 0, 3, 0, 1, 5, 0, 0, 1], [0, 0, 0, 0, 2, 0, 5, 5, 5, 0, 0, 0, 5, 0, 0, 4, 3, 4, 0, 0, 0, 5, 2, 5, 4, 0], [4, 5, 0, 4, 3, 1, 2, 0, 1, 0, 0, 0, 2, 0, 5, 0, 0, 0, 3, 5, 1, 0, 0, 4, 2, 0], [0, 0, 3, 3, 0, 1, 3, 4, 0, 5, 5, 0, 0, 0, 1, 5, 0, 2, 3, 0, 0, 2, 0, 0, 0, 0], [5, 1, 5, 0, 4, 0, 3, 5, 3, 0, 0, 4, 1, 1, 5, 5, 0, 3, 3, 3, 1, 0, 5, 2, 5, 1], [0, 5, 0, 0, 0, 4, 0, 2, 0, 0, 1, 5, 0, 3, 4, 3, 3, 2, 0, 0, 0, 3, 1, 0, 0, 5], [0, 1, 5, 4, 1, 2, 3, 0, 2, 0, 2, 1, 0, 0, 1, 0, 0, 5, 0, 1, 5, 0, 0, 0, 0, 0], [1, 4, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 4, 4, 3, 0, 0, 0, 5, 0, 0, 2, 2, 0, 4], [0, 3, 5, 4, 0, 0, 1, 0, 1, 0, 2, 0, 1, 3, 0, 0, 5, 0, 0, 0, 5, 3, 0, 5, 0, 0], [0, 0, 4, 0, 0, 4, 4, 0, 5, 0, 1, 0, 0, 2, 0, 0, 0, 1, 4, 0, 0, 2, 5, 3, 3, 0], [3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 2, 0, 0, 5, 0, 1, 5, 0, 0, 0, 1, 5, 0], [4, 0, 0, 0, 2, 2, 5, 5, 1, 0, 0, 0, 0, 0, 0, 5, 0, 3, 0, 0, 4, 5, 0, 0, 0, 0], [1, 1, 4, 0, 0, 0, 1, 0, 0, 0, 0, 4, 0, 0, 0, 0, 1, 0, 3, 3, 0, 2, 0, 5, 5, 3], [0, 0, 0, 0, 5, 5, 0, 1, 5, 0, 0, 4, 0, 0, 0, 4, 3, 0, 4, 0, 0, 0, 0, 4, 0, 4], [0, 3, 1, 0, 5, 2, 2, 4, 0, 0, 3, 1, 0, 0, 4, 1, 0, 0, 3, 0, 0, 2, 0, 3, 0, 5], [0, 0, 2, 2, 2, 1, 1, 0, 2, 0, 2, 1, 1, 3, 1, 2, 0, 0, 1, 0, 4, 4, 1, 0, 3, 5], [0, 1, 2, 3, 0, 3, 3, 0, 3, 0, 0, 4, 1, 5, 5, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 4], [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 3, 0, 2, 0, 0, 2, 0, 5, 4, 0, 0, 0, 0, 0, 0, 4], [0, 0, 0, 1, 0, 0, 4, 4, 4, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0], [0, 4, 4, 0, 5, 2, 0, 0, 2, 4, 0, 0, 0, 3, 3, 0, 0, 1, 1, 5, 2, 0, 2, 0, 1, 0]], "rhs": [485, 522, 489, 566, 720, 322, 601, 692, 802, 382, 697, 393, 812, 542, 380, 510, 348, 747, 477, 380, 629, 621, 893, 419, 715, 395, 439, 590, 464, 545, 826, 375, 376, 611, 277, 523, 639, 659, 434, 370, 512, 641, 638, 588, 356, 700, 678, 512, 664, 514, 718, 962, 573, 546, 422, 486, 612, 307, 531, 521, 517, 546, 538, 493, 364, 231, 561], "family": "rlve_integer_linear_system", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-integer-linear-system-l6-s9/solution/solve.sh b/tasks/construct-rlve-integer-linear-system-l6-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..1fd8da02660878aba9dd8e4e5b259f114a9d6971
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-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'
+{"x": [9, 10, 26, 21, 11, 24, 10, 17, 2, 23, 20, 19, 13, 3, 6, 17, 5, 22, 21, 14, 8, 21, 22, 8, 18, 10]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-integer-linear-system-l6-s9/tests/check.py b/tasks/construct-rlve-integer-linear-system-l6-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..ff19002b3fddc9597f84ee31ef5ffbee5d3fac32
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-l6-s9/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-l6-s9/tests/instance.json b/tasks/construct-rlve-integer-linear-system-l6-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..51265bd02ca653c13b3085c6df0b559f2952a844
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-l6-s9/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [[0, 0, 2, 0, 0, 5, 3, 0, 0, 0, 1, 0, 0, 0, 4, 1, 0, 4, 2, 0, 5, 0, 1, 0, 0, 3], [0, 0, 0, 0, 5, 3, 0, 1, 3, 4, 0, 0, 2, 1, 4, 3, 0, 0, 3, 0, 0, 5, 0, 1, 0, 0], [0, 0, 4, 0, 2, 1, 0, 2, 0, 2, 0, 0, 2, 2, 0, 3, 2, 1, 0, 0, 0, 2, 0, 0, 4, 3], [0, 0, 2, 5, 0, 4, 0, 0, 5, 0, 1, 2, 0, 4, 0, 4, 4, 3, 0, 0, 2, 1, 1, 0, 0, 2], [2, 3, 2, 1, 0, 2, 0, 3, 0, 1, 0, 5, 0, 0, 5, 0, 0, 2, 3, 4, 0, 5, 3, 0, 1, 0], [4, 1, 0, 0, 4, 0, 0, 0, 0, 1, 2, 0, 2, 0, 0, 5, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0], [0, 3, 3, 1, 3, 1, 0, 5, 1, 1, 0, 1, 3, 0, 4, 3, 0, 0, 0, 0, 0, 2, 5, 0, 0, 2], [4, 0, 5, 0, 3, 0, 0, 0, 2, 5, 2, 0, 0, 4, 0, 2, 0, 0, 3, 2, 0, 3, 0, 4, 4, 3], [0, 4, 3, 0, 4, 0, 0, 0, 3, 5, 1, 3, 5, 4, 0, 4, 4, 0, 1, 4, 2, 0, 4, 3, 4, 0], [4, 4, 0, 3, 0, 0, 0, 0, 4, 0, 1, 0, 0, 4, 0, 0, 0, 2, 0, 4, 4, 1, 0, 0, 0, 5], [4, 2, 4, 4, 0, 1, 0, 0, 0, 0, 1, 0, 0, 4, 2, 4, 0, 0, 4, 0, 0, 3, 1, 1, 5, 5], [0, 3, 0, 0, 2, 0, 1, 4, 4, 5, 0, 2, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 3, 2, 1], [0, 5, 4, 4, 2, 2, 0, 0, 2, 1, 3, 4, 5, 0, 5, 0, 4, 5, 1, 0, 4, 3, 0, 0, 0, 0], [1, 4, 0, 4, 0, 0, 0, 0, 5, 0, 0, 0, 5, 4, 1, 2, 0, 3, 0, 3, 0, 4, 3, 3, 0, 0], [3, 0, 1, 5, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 3, 0, 0], [0, 0, 0, 0, 4, 3, 0, 3, 3, 3, 0, 0, 0, 0, 5, 2, 0, 1, 0, 4, 0, 4, 1, 0, 0, 2], [0, 0, 1, 0, 0, 5, 1, 0, 3, 2, 0, 3, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [1, 0, 5, 0, 2, 2, 0, 0, 1, 4, 2, 3, 2, 4, 3, 0, 0, 2, 2, 0, 5, 1, 0, 3, 5, 3], [3, 5, 4, 0, 4, 0, 0, 0, 1, 0, 0, 4, 0, 5, 3, 0, 2, 0, 0, 0, 5, 1, 0, 5, 0, 3], [2, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 2, 0, 4, 0, 0, 0, 2, 4, 3, 5, 0, 0, 0, 3, 0], [0, 3, 0, 0, 0, 5, 0, 5, 0, 1, 3, 0, 2, 0, 0, 4, 0, 0, 0, 4, 5, 3, 0, 0, 1, 4], [1, 0, 0, 4, 3, 3, 5, 4, 3, 0, 0, 0, 4, 5, 3, 0, 0, 3, 0, 4, 5, 0, 1, 0, 0, 3], [2, 0, 3, 5, 3, 3, 3, 1, 5, 3, 0, 5, 0, 1, 1, 5, 0, 0, 4, 0, 0, 0, 4, 0, 5, 1], [5, 0, 4, 0, 0, 5, 3, 0, 0, 0, 0, 0, 4, 0, 0, 0, 5, 1, 0, 0, 0, 1, 0, 0, 0, 0], [1, 1, 5, 0, 0, 0, 0, 1, 0, 4, 3, 0, 0, 0, 0, 0, 0, 5, 0, 2, 0, 5, 4, 0, 2, 3], [4, 0, 0, 4, 0, 2, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 2, 0, 4, 0, 0, 0, 3], [1, 0, 0, 0, 0, 0, 0, 5, 3, 4, 1, 0, 0, 5, 3, 0, 5, 0, 0, 3, 4, 1, 0, 3, 0, 5], [0, 0, 3, 0, 0, 2, 0, 0, 0, 5, 0, 1, 0, 1, 0, 0, 5, 5, 4, 3, 0, 0, 3, 0, 0, 0], [0, 3, 0, 0, 0, 3, 5, 0, 0, 2, 0, 0, 5, 0, 0, 2, 2, 0, 0, 4, 2, 3, 1, 0, 0, 0], [0, 0, 2, 4, 3, 3, 4, 1, 5, 0, 0, 4, 0, 0, 1, 0, 3, 4, 0, 2, 3, 0, 0, 0, 0, 0], [2, 0, 0, 0, 0, 5, 0, 5, 1, 2, 0, 4, 0, 0, 0, 0, 2, 5, 4, 5, 3, 3, 2, 3, 0, 5], [5, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 3, 0, 2, 1, 0, 0, 5, 2, 1, 2, 0, 2, 4], [0, 0, 0, 0, 1, 0, 0, 3, 1, 3, 0, 0, 0, 3, 0, 5, 0, 1, 2, 1, 0, 1, 0, 0, 0, 5], [2, 3, 0, 3, 1, 4, 0, 0, 0, 0, 2, 3, 0, 0, 5, 0, 0, 5, 1, 4, 0, 1, 1, 0, 2, 0], [0, 0, 0, 1, 2, 0, 0, 5, 0, 0, 1, 0, 0, 5, 5, 1, 1, 0, 0, 0, 0, 0, 1, 5, 0, 0], [0, 5, 0, 0, 2, 1, 1, 0, 0, 4, 4, 2, 5, 0, 0, 1, 0, 0, 5, 0, 0, 0, 0, 0, 0, 2], [3, 0, 5, 0, 0, 0, 0, 2, 4, 3, 1, 5, 1, 0, 2, 2, 4, 0, 0, 0, 0, 5, 0, 0, 4, 0], [0, 0, 1, 2, 4, 5, 0, 5, 5, 0, 0, 3, 0, 0, 5, 4, 0, 0, 1, 2, 0, 0, 5, 0, 1, 0], [0, 0, 0, 4, 0, 0, 4, 1, 0, 0, 0, 0, 1, 5, 1, 0, 2, 0, 4, 0, 4, 5, 0, 0, 1, 1], [0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 3, 5, 0, 0, 2, 0, 2, 4, 0], [0, 0, 0, 3, 3, 4, 2, 0, 0, 2, 0, 0, 5, 0, 0, 0, 0, 0, 4, 1, 0, 1, 0, 5, 0, 3], [0, 3, 5, 3, 3, 0, 0, 0, 1, 2, 0, 0, 3, 0, 2, 0, 0, 4, 0, 5, 0, 0, 5, 0, 1, 0], [4, 0, 1, 0, 0, 1, 4, 3, 0, 0, 4, 3, 0, 0, 0, 0, 5, 1, 1, 5, 2, 0, 5, 5, 0, 2], [0, 0, 2, 5, 0, 2, 0, 0, 1, 0, 0, 4, 2, 0, 5, 0, 3, 0, 0, 5, 5, 0, 2, 5, 0, 4], [3, 3, 0, 3, 0, 0, 0, 5, 0, 0, 0, 0, 5, 5, 0, 0, 3, 0, 2, 1, 0, 0, 0, 0, 0, 0], [5, 0, 0, 3, 3, 3, 1, 0, 2, 4, 1, 0, 2, 5, 5, 0, 0, 5, 0, 4, 0, 0, 2, 3, 2, 2], [0, 3, 1, 5, 0, 2, 0, 1, 0, 3, 0, 5, 1, 1, 5, 0, 2, 1, 4, 4, 2, 0, 0, 1, 2, 1], [3, 4, 3, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 2, 2, 5, 1, 0, 3, 0, 1, 5, 0, 0, 1], [0, 0, 0, 0, 2, 0, 5, 5, 5, 0, 0, 0, 5, 0, 0, 4, 3, 4, 0, 0, 0, 5, 2, 5, 4, 0], [4, 5, 0, 4, 3, 1, 2, 0, 1, 0, 0, 0, 2, 0, 5, 0, 0, 0, 3, 5, 1, 0, 0, 4, 2, 0], [0, 0, 3, 3, 0, 1, 3, 4, 0, 5, 5, 0, 0, 0, 1, 5, 0, 2, 3, 0, 0, 2, 0, 0, 0, 0], [5, 1, 5, 0, 4, 0, 3, 5, 3, 0, 0, 4, 1, 1, 5, 5, 0, 3, 3, 3, 1, 0, 5, 2, 5, 1], [0, 5, 0, 0, 0, 4, 0, 2, 0, 0, 1, 5, 0, 3, 4, 3, 3, 2, 0, 0, 0, 3, 1, 0, 0, 5], [0, 1, 5, 4, 1, 2, 3, 0, 2, 0, 2, 1, 0, 0, 1, 0, 0, 5, 0, 1, 5, 0, 0, 0, 0, 0], [1, 4, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 4, 4, 3, 0, 0, 0, 5, 0, 0, 2, 2, 0, 4], [0, 3, 5, 4, 0, 0, 1, 0, 1, 0, 2, 0, 1, 3, 0, 0, 5, 0, 0, 0, 5, 3, 0, 5, 0, 0], [0, 0, 4, 0, 0, 4, 4, 0, 5, 0, 1, 0, 0, 2, 0, 0, 0, 1, 4, 0, 0, 2, 5, 3, 3, 0], [3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 2, 0, 0, 5, 0, 1, 5, 0, 0, 0, 1, 5, 0], [4, 0, 0, 0, 2, 2, 5, 5, 1, 0, 0, 0, 0, 0, 0, 5, 0, 3, 0, 0, 4, 5, 0, 0, 0, 0], [1, 1, 4, 0, 0, 0, 1, 0, 0, 0, 0, 4, 0, 0, 0, 0, 1, 0, 3, 3, 0, 2, 0, 5, 5, 3], [0, 0, 0, 0, 5, 5, 0, 1, 5, 0, 0, 4, 0, 0, 0, 4, 3, 0, 4, 0, 0, 0, 0, 4, 0, 4], [0, 3, 1, 0, 5, 2, 2, 4, 0, 0, 3, 1, 0, 0, 4, 1, 0, 0, 3, 0, 0, 2, 0, 3, 0, 5], [0, 0, 2, 2, 2, 1, 1, 0, 2, 0, 2, 1, 1, 3, 1, 2, 0, 0, 1, 0, 4, 4, 1, 0, 3, 5], [0, 1, 2, 3, 0, 3, 3, 0, 3, 0, 0, 4, 1, 5, 5, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 4], [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 3, 0, 2, 0, 0, 2, 0, 5, 4, 0, 0, 0, 0, 0, 0, 4], [0, 0, 0, 1, 0, 0, 4, 4, 4, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0], [0, 4, 4, 0, 5, 2, 0, 0, 2, 4, 0, 0, 0, 3, 3, 0, 0, 1, 1, 5, 2, 0, 2, 0, 1, 0]], "rhs": [485, 522, 489, 566, 720, 322, 601, 692, 802, 382, 697, 393, 812, 542, 380, 510, 348, 747, 477, 380, 629, 621, 893, 419, 715, 395, 439, 590, 464, 545, 826, 375, 376, 611, 277, 523, 639, 659, 434, 370, 512, 641, 638, 588, 356, 700, 678, 512, 664, 514, 718, 962, 573, 546, 422, 486, 612, 307, 531, 521, 517, 546, 538, 493, 364, 231, 561], "family": "rlve_integer_linear_system", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-integer-linear-system-l6-s9/tests/run.py b/tasks/construct-rlve-integer-linear-system-l6-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-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-integer-linear-system-l6-s9/tests/test.sh b/tasks/construct-rlve-integer-linear-system-l6-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-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-largest-convex-polygon-l1-s1/environment/Dockerfile b/tasks/construct-rlve-largest-convex-polygon-l1-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l1-s1/environment/instance.json b/tasks/construct-rlve-largest-convex-polygon-l1-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b6bd381fd38dda2bcb53202d6da5103a1af425cd
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l1-s1/environment/instance.json
@@ -0,0 +1 @@
+{"points": [[3, 1], [4, 6], [5, 1], [3, 0], [2, 5], [1, 3]], "k": 5, "family": "rlve_largest_convex_polygon", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-convex-polygon-l1-s1/solution/solve.sh b/tasks/construct-rlve-largest-convex-polygon-l1-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..28d9e05ec79eccf4f4ac0d540414da4f0ad1fd13
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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'
+{"indices": [5, 4, 1, 2, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-largest-convex-polygon-l1-s1/tests/check.py b/tasks/construct-rlve-largest-convex-polygon-l1-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3a09cc858f41ce5745e4e491188cd42360c65dae
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l1-s1/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-l1-s1/tests/instance.json b/tasks/construct-rlve-largest-convex-polygon-l1-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b6bd381fd38dda2bcb53202d6da5103a1af425cd
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l1-s1/tests/instance.json
@@ -0,0 +1 @@
+{"points": [[3, 1], [4, 6], [5, 1], [3, 0], [2, 5], [1, 3]], "k": 5, "family": "rlve_largest_convex_polygon", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-convex-polygon-l1-s1/tests/run.py b/tasks/construct-rlve-largest-convex-polygon-l1-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l1-s1/tests/test.sh b/tasks/construct-rlve-largest-convex-polygon-l1-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l1-s6/environment/Dockerfile b/tasks/construct-rlve-largest-convex-polygon-l1-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l1-s6/environment/instance.json b/tasks/construct-rlve-largest-convex-polygon-l1-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..51b5cef3bad73159fadf3cb148eb6eafa6e2aaf8
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l1-s6/environment/instance.json
@@ -0,0 +1 @@
+{"points": [[6, 2], [3, 4], [5, 3], [4, 6], [0, 5], [6, 6]], "k": 4, "family": "rlve_largest_convex_polygon", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-convex-polygon-l1-s6/solution/solve.sh b/tasks/construct-rlve-largest-convex-polygon-l1-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..601dee1c6c70f39b17522e2a1d5ba185e4baf5f0
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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'
+{"indices": [1, 3, 5, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-largest-convex-polygon-l1-s6/tests/check.py b/tasks/construct-rlve-largest-convex-polygon-l1-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3a09cc858f41ce5745e4e491188cd42360c65dae
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l1-s6/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-l1-s6/tests/instance.json b/tasks/construct-rlve-largest-convex-polygon-l1-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..51b5cef3bad73159fadf3cb148eb6eafa6e2aaf8
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l1-s6/tests/instance.json
@@ -0,0 +1 @@
+{"points": [[6, 2], [3, 4], [5, 3], [4, 6], [0, 5], [6, 6]], "k": 4, "family": "rlve_largest_convex_polygon", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-convex-polygon-l1-s6/tests/run.py b/tasks/construct-rlve-largest-convex-polygon-l1-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l1-s6/tests/test.sh b/tasks/construct-rlve-largest-convex-polygon-l1-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l1-s8/environment/Dockerfile b/tasks/construct-rlve-largest-convex-polygon-l1-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l1-s8/environment/instance.json b/tasks/construct-rlve-largest-convex-polygon-l1-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0ee0bc89a7fdec7c37fb6845bec030f1ef34a713
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l1-s8/environment/instance.json
@@ -0,0 +1 @@
+{"points": [[6, 5], [3, 0], [2, 3], [0, 6], [5, 3], [2, 5]], "k": 5, "family": "rlve_largest_convex_polygon", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-convex-polygon-l1-s8/solution/solve.sh b/tasks/construct-rlve-largest-convex-polygon-l1-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..7cb1d1f2a244eba594970b99c3e4f5e4fe494b45
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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'
+{"indices": [2, 5, 0, 4, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-largest-convex-polygon-l1-s8/tests/check.py b/tasks/construct-rlve-largest-convex-polygon-l1-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3a09cc858f41ce5745e4e491188cd42360c65dae
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l1-s8/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-l1-s8/tests/instance.json b/tasks/construct-rlve-largest-convex-polygon-l1-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0ee0bc89a7fdec7c37fb6845bec030f1ef34a713
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l1-s8/tests/instance.json
@@ -0,0 +1 @@
+{"points": [[6, 5], [3, 0], [2, 3], [0, 6], [5, 3], [2, 5]], "k": 5, "family": "rlve_largest_convex_polygon", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-convex-polygon-l1-s8/tests/run.py b/tasks/construct-rlve-largest-convex-polygon-l1-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l1-s8/tests/test.sh b/tasks/construct-rlve-largest-convex-polygon-l1-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l5-s0/environment/Dockerfile b/tasks/construct-rlve-largest-convex-polygon-l5-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l5-s0/environment/instance.json b/tasks/construct-rlve-largest-convex-polygon-l5-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0271f4e3d95f53f3a8f977e430bbbbaf40c1dd59
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l5-s0/environment/instance.json
@@ -0,0 +1 @@
+{"points": [[4, 0], [14, 7], [3, 10], [22, 17], [21, 25], [4, 24], [19, 18], [9, 20], [16, 7], [1, 12], [16, 10], [17, 2], [14, 21], [5, 15], [2, 1], [10, 14], [1, 11], [0, 22], [21, 2], [20, 21], [23, 17], [5, 14], [9, 12], [24, 6], [10, 22]], "k": 11, "family": "rlve_largest_convex_polygon", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-convex-polygon-l5-s0/solution/solve.sh b/tasks/construct-rlve-largest-convex-polygon-l5-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..4ead44da3d9c0615299ca286941413e0106b1bdb
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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'
+{"indices": [16, 9, 13, 12, 4, 19, 6, 10, 1, 0, 14]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-largest-convex-polygon-l5-s0/tests/check.py b/tasks/construct-rlve-largest-convex-polygon-l5-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3a09cc858f41ce5745e4e491188cd42360c65dae
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l5-s0/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-l5-s0/tests/instance.json b/tasks/construct-rlve-largest-convex-polygon-l5-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0271f4e3d95f53f3a8f977e430bbbbaf40c1dd59
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l5-s0/tests/instance.json
@@ -0,0 +1 @@
+{"points": [[4, 0], [14, 7], [3, 10], [22, 17], [21, 25], [4, 24], [19, 18], [9, 20], [16, 7], [1, 12], [16, 10], [17, 2], [14, 21], [5, 15], [2, 1], [10, 14], [1, 11], [0, 22], [21, 2], [20, 21], [23, 17], [5, 14], [9, 12], [24, 6], [10, 22]], "k": 11, "family": "rlve_largest_convex_polygon", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-convex-polygon-l5-s0/tests/run.py b/tasks/construct-rlve-largest-convex-polygon-l5-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l5-s0/tests/test.sh b/tasks/construct-rlve-largest-convex-polygon-l5-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l5-s7/environment/Dockerfile b/tasks/construct-rlve-largest-convex-polygon-l5-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l5-s7/environment/instance.json b/tasks/construct-rlve-largest-convex-polygon-l5-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..19313410518e364433529783b2848d8623f09026
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l5-s7/environment/instance.json
@@ -0,0 +1 @@
+{"points": [[12, 4], [17, 0], [0, 2], [19, 18], [1, 3], [1, 12], [18, 4], [7, 1], [21, 3], [6, 17], [9, 1], [9, 10], [25, 9], [8, 23], [10, 20], [24, 13], [6, 13], [20, 6], [22, 6], [3, 17], [3, 20], [8, 16], [25, 8], [0, 15], [13, 18]], "k": 12, "family": "rlve_largest_convex_polygon", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-convex-polygon-l5-s7/solution/solve.sh b/tasks/construct-rlve-largest-convex-polygon-l5-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..479cb5ea5cb9d9308bd1f7eb6e933451c8dd0936
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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'
+{"indices": [2, 16, 21, 24, 3, 15, 12, 22, 18, 6, 10, 7]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-largest-convex-polygon-l5-s7/tests/check.py b/tasks/construct-rlve-largest-convex-polygon-l5-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3a09cc858f41ce5745e4e491188cd42360c65dae
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l5-s7/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-l5-s7/tests/instance.json b/tasks/construct-rlve-largest-convex-polygon-l5-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..19313410518e364433529783b2848d8623f09026
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l5-s7/tests/instance.json
@@ -0,0 +1 @@
+{"points": [[12, 4], [17, 0], [0, 2], [19, 18], [1, 3], [1, 12], [18, 4], [7, 1], [21, 3], [6, 17], [9, 1], [9, 10], [25, 9], [8, 23], [10, 20], [24, 13], [6, 13], [20, 6], [22, 6], [3, 17], [3, 20], [8, 16], [25, 8], [0, 15], [13, 18]], "k": 12, "family": "rlve_largest_convex_polygon", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-convex-polygon-l5-s7/tests/run.py b/tasks/construct-rlve-largest-convex-polygon-l5-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l5-s7/tests/test.sh b/tasks/construct-rlve-largest-convex-polygon-l5-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l5-s9/environment/Dockerfile b/tasks/construct-rlve-largest-convex-polygon-l5-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l5-s9/environment/instance.json b/tasks/construct-rlve-largest-convex-polygon-l5-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..813bc75a7b5a952373dbbf57352e22ac3e18543c
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l5-s9/environment/instance.json
@@ -0,0 +1 @@
+{"points": [[18, 23], [14, 4], [6, 24], [21, 25], [14, 13], [2, 11], [0, 23], [25, 16], [6, 2], [7, 22], [21, 21], [17, 5], [9, 19], [0, 22], [11, 25], [7, 12], [18, 18], [20, 3], [4, 1], [23, 8], [4, 16], [8, 4], [1, 13], [15, 3], [13, 24]], "k": 13, "family": "rlve_largest_convex_polygon", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-convex-polygon-l5-s9/solution/solve.sh b/tasks/construct-rlve-largest-convex-polygon-l5-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..cf98108bb51f2c7654f9eff760e11c35ceaf53c2
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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'
+{"indices": [13, 6, 2, 24, 0, 10, 7, 19, 17, 23, 1, 15, 20]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-largest-convex-polygon-l5-s9/tests/check.py b/tasks/construct-rlve-largest-convex-polygon-l5-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3a09cc858f41ce5745e4e491188cd42360c65dae
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l5-s9/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-l5-s9/tests/instance.json b/tasks/construct-rlve-largest-convex-polygon-l5-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..813bc75a7b5a952373dbbf57352e22ac3e18543c
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l5-s9/tests/instance.json
@@ -0,0 +1 @@
+{"points": [[18, 23], [14, 4], [6, 24], [21, 25], [14, 13], [2, 11], [0, 23], [25, 16], [6, 2], [7, 22], [21, 21], [17, 5], [9, 19], [0, 22], [11, 25], [7, 12], [18, 18], [20, 3], [4, 1], [23, 8], [4, 16], [8, 4], [1, 13], [15, 3], [13, 24]], "k": 13, "family": "rlve_largest_convex_polygon", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-convex-polygon-l5-s9/tests/run.py b/tasks/construct-rlve-largest-convex-polygon-l5-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l5-s9/tests/test.sh b/tasks/construct-rlve-largest-convex-polygon-l5-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-rectangle-l3-s1/environment/Dockerfile b/tasks/construct-rlve-largest-rectangle-l3-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-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-largest-rectangle-l3-s1/environment/instance.json b/tasks/construct-rlve-largest-rectangle-l3-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..ce0b8c7929fa65d55ac5765975209f67416b160d
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l3-s1/environment/instance.json
@@ -0,0 +1 @@
+{"points": [[11, 12], [1, 3], [2, 3], [1, 11], [1, -1], [16, -15], [-10, 13], [14, -9], [8, -14], [3, 0], [-3, -1], [-3, 3], [15, 7], [-3, -15], [-6, -15], [13, 5]], "area": 16, "family": "rlve_largest_rectangle", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-rectangle-l3-s1/solution/solve.sh b/tasks/construct-rlve-largest-rectangle-l3-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..d82f282dc81572e2f21e90b942b313c1d69460bb
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-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'
+{"indices": [1, 4, 10, 11]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-largest-rectangle-l3-s1/tests/check.py b/tasks/construct-rlve-largest-rectangle-l3-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2251b2af23a481e598fa1e48d0a8249f70b6cc44
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l3-s1/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-l3-s1/tests/instance.json b/tasks/construct-rlve-largest-rectangle-l3-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..ce0b8c7929fa65d55ac5765975209f67416b160d
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l3-s1/tests/instance.json
@@ -0,0 +1 @@
+{"points": [[11, 12], [1, 3], [2, 3], [1, 11], [1, -1], [16, -15], [-10, 13], [14, -9], [8, -14], [3, 0], [-3, -1], [-3, 3], [15, 7], [-3, -15], [-6, -15], [13, 5]], "area": 16, "family": "rlve_largest_rectangle", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-rectangle-l3-s1/tests/run.py b/tasks/construct-rlve-largest-rectangle-l3-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-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-largest-rectangle-l3-s1/tests/test.sh b/tasks/construct-rlve-largest-rectangle-l3-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-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-largest-rectangle-l3-s6/environment/Dockerfile b/tasks/construct-rlve-largest-rectangle-l3-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-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-rlve-largest-rectangle-l3-s6/environment/instance.json b/tasks/construct-rlve-largest-rectangle-l3-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..07ea7318e4724b8711c4d6cf628fd49374f86efb
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l3-s6/environment/instance.json
@@ -0,0 +1 @@
+{"points": [[-16, -5], [-10, 4], [15, 10], [-6, 2], [7, 11], [14, -12], [0, 2], [-2, 2], [13, -6], [-4, 16], [-16, 15], [13, 9], [-6, -4], [9, 2], [0, -4], [-16, -13]], "area": 36, "family": "rlve_largest_rectangle", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-rectangle-l3-s6/solution/solve.sh b/tasks/construct-rlve-largest-rectangle-l3-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..0188fb09534f13c0f4933ad4a7e5c0ffda690e87
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-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'
+{"indices": [3, 6, 14, 12]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-largest-rectangle-l3-s6/tests/check.py b/tasks/construct-rlve-largest-rectangle-l3-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2251b2af23a481e598fa1e48d0a8249f70b6cc44
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l3-s6/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-l3-s6/tests/instance.json b/tasks/construct-rlve-largest-rectangle-l3-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..07ea7318e4724b8711c4d6cf628fd49374f86efb
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l3-s6/tests/instance.json
@@ -0,0 +1 @@
+{"points": [[-16, -5], [-10, 4], [15, 10], [-6, 2], [7, 11], [14, -12], [0, 2], [-2, 2], [13, -6], [-4, 16], [-16, 15], [13, 9], [-6, -4], [9, 2], [0, -4], [-16, -13]], "area": 36, "family": "rlve_largest_rectangle", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-rectangle-l3-s6/tests/run.py b/tasks/construct-rlve-largest-rectangle-l3-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-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-rlve-largest-rectangle-l3-s6/tests/test.sh b/tasks/construct-rlve-largest-rectangle-l3-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-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-rlve-largest-rectangle-l3-s8/environment/Dockerfile b/tasks/construct-rlve-largest-rectangle-l3-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-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-largest-rectangle-l3-s8/environment/instance.json b/tasks/construct-rlve-largest-rectangle-l3-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2936bdfdcd99e5f9d256bf57e78fdb5d3b286b48
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l3-s8/environment/instance.json
@@ -0,0 +1 @@
+{"points": [[0, 6], [-13, -1], [-3, 5], [2, -7], [-2, 8], [4, -3], [-13, -10], [-10, 7], [10, -6], [16, 13], [-9, -1], [3, 11], [-14, -5], [5, 6], [15, -4], [-1, 0]], "area": 65, "family": "rlve_largest_rectangle", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-rectangle-l3-s8/solution/solve.sh b/tasks/construct-rlve-largest-rectangle-l3-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..bad7b699aacb21f571cd1a41c671e9af898c6e3c
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-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'
+{"indices": [4, 7, 10, 15]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-largest-rectangle-l3-s8/tests/check.py b/tasks/construct-rlve-largest-rectangle-l3-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2251b2af23a481e598fa1e48d0a8249f70b6cc44
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l3-s8/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-l3-s8/tests/instance.json b/tasks/construct-rlve-largest-rectangle-l3-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2936bdfdcd99e5f9d256bf57e78fdb5d3b286b48
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l3-s8/tests/instance.json
@@ -0,0 +1 @@
+{"points": [[0, 6], [-13, -1], [-3, 5], [2, -7], [-2, 8], [4, -3], [-13, -10], [-10, 7], [10, -6], [16, 13], [-9, -1], [3, 11], [-14, -5], [5, 6], [15, -4], [-1, 0]], "area": 65, "family": "rlve_largest_rectangle", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-rectangle-l3-s8/tests/run.py b/tasks/construct-rlve-largest-rectangle-l3-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-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-largest-rectangle-l3-s8/tests/test.sh b/tasks/construct-rlve-largest-rectangle-l3-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-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-matrix-permutation-equivalence-l4-s3/environment/Dockerfile b/tasks/construct-rlve-matrix-permutation-equivalence-l4-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-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-matrix-permutation-equivalence-l4-s3/environment/instance.json b/tasks/construct-rlve-matrix-permutation-equivalence-l4-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..ce7878556c7f60a6bfc1ffa420487e34f85a918b
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l4-s3/environment/instance.json
@@ -0,0 +1 @@
+{"A": ["100101000001", "101111111001", "101110001001", "110110101010", "111010111001", "100100010000", "010001111110", "001100011111", "000011001101", "100100100101"], "B": ["011011100110", "000100100010", "111010010011", "001101011111", "101101101111", "001110111001", "001101101100", "100101100000", "010101100001", "111001001000"], "family": "rlve_matrix_permutation_equivalence", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l4-s3/solution/solve.sh b/tasks/construct-rlve-matrix-permutation-equivalence-l4-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..de7e4c0c34788589e6caaa710aa8d3fbab67f363
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-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'
+{"a": [7, 5, 6, 4, 1, 3, 2, 0, 9, 8], "b": [5, 9, 8, 0, 10, 11, 3, 1, 4, 2, 7, 6]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l4-s3/tests/check.py b/tasks/construct-rlve-matrix-permutation-equivalence-l4-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..310a1d2c8b6afd7a82a35ba2a4c802c1ce289d03
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l4-s3/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-l4-s3/tests/instance.json b/tasks/construct-rlve-matrix-permutation-equivalence-l4-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..ce7878556c7f60a6bfc1ffa420487e34f85a918b
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l4-s3/tests/instance.json
@@ -0,0 +1 @@
+{"A": ["100101000001", "101111111001", "101110001001", "110110101010", "111010111001", "100100010000", "010001111110", "001100011111", "000011001101", "100100100101"], "B": ["011011100110", "000100100010", "111010010011", "001101011111", "101101101111", "001110111001", "001101101100", "100101100000", "010101100001", "111001001000"], "family": "rlve_matrix_permutation_equivalence", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l4-s3/tests/run.py b/tasks/construct-rlve-matrix-permutation-equivalence-l4-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-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-matrix-permutation-equivalence-l4-s3/tests/test.sh b/tasks/construct-rlve-matrix-permutation-equivalence-l4-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-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-matrix-permutation-equivalence-l4-s4/environment/Dockerfile b/tasks/construct-rlve-matrix-permutation-equivalence-l4-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-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-matrix-permutation-equivalence-l4-s4/environment/instance.json b/tasks/construct-rlve-matrix-permutation-equivalence-l4-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..bd5b560e15264c72c35389a85ae965785be06f59
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l4-s4/environment/instance.json
@@ -0,0 +1 @@
+{"A": ["1101111", "1111111", "1110111", "1110111", "1111111", "1111111", "1111111"], "B": ["1011111", "1111110", "1011111", "1111111", "1111111", "1111111", "1111111"], "family": "rlve_matrix_permutation_equivalence", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l4-s4/solution/solve.sh b/tasks/construct-rlve-matrix-permutation-equivalence-l4-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..7ebad3efac061f2734a4edc22f12a1d2bf371237
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-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'
+{"a": [2, 0, 3, 1, 6, 4, 5], "b": [5, 3, 4, 1, 6, 0, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l4-s4/tests/check.py b/tasks/construct-rlve-matrix-permutation-equivalence-l4-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..310a1d2c8b6afd7a82a35ba2a4c802c1ce289d03
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l4-s4/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-l4-s4/tests/instance.json b/tasks/construct-rlve-matrix-permutation-equivalence-l4-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..bd5b560e15264c72c35389a85ae965785be06f59
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-l4-s4/tests/instance.json
@@ -0,0 +1 @@
+{"A": ["1101111", "1111111", "1110111", "1110111", "1111111", "1111111", "1111111"], "B": ["1011111", "1111110", "1011111", "1111111", "1111111", "1111111", "1111111"], "family": "rlve_matrix_permutation_equivalence", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-matrix-permutation-equivalence-l4-s4/tests/run.py b/tasks/construct-rlve-matrix-permutation-equivalence-l4-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-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-matrix-permutation-equivalence-l4-s4/tests/test.sh b/tasks/construct-rlve-matrix-permutation-equivalence-l4-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-matrix-permutation-equivalence-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-max-achromatic-coloring-l1-s0/environment/Dockerfile b/tasks/construct-rlve-max-achromatic-coloring-l1-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-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-rlve-max-achromatic-coloring-l1-s0/environment/instance.json b/tasks/construct-rlve-max-achromatic-coloring-l1-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..539b06342d6907831d09effc3c32444d4386f757
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l1-s0/environment/instance.json
@@ -0,0 +1 @@
+{"n": 6, "k": 4, "edges": [[0, 2], [0, 3], [0, 4], [1, 2], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]], "family": "rlve_max_achromatic_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l1-s0/solution/solve.sh b/tasks/construct-rlve-max-achromatic-coloring-l1-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..9724d48b7abcf18981860acd0757df0cbf16483a
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-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'
+{"colors": [0, 0, 1, 2, 3, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l1-s0/tests/check.py b/tasks/construct-rlve-max-achromatic-coloring-l1-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..bb676a518264e5db4fb0964e27ccffccf6980807
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l1-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("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-l1-s0/tests/instance.json b/tasks/construct-rlve-max-achromatic-coloring-l1-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..539b06342d6907831d09effc3c32444d4386f757
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l1-s0/tests/instance.json
@@ -0,0 +1 @@
+{"n": 6, "k": 4, "edges": [[0, 2], [0, 3], [0, 4], [1, 2], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]], "family": "rlve_max_achromatic_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l1-s0/tests/run.py b/tasks/construct-rlve-max-achromatic-coloring-l1-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-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-rlve-max-achromatic-coloring-l1-s0/tests/test.sh b/tasks/construct-rlve-max-achromatic-coloring-l1-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-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-rlve-max-achromatic-coloring-l1-s7/environment/Dockerfile b/tasks/construct-rlve-max-achromatic-coloring-l1-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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-max-achromatic-coloring-l1-s7/environment/instance.json b/tasks/construct-rlve-max-achromatic-coloring-l1-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..11356a79469d2e4b2c44240a293c8ee53b78d853
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l1-s7/environment/instance.json
@@ -0,0 +1 @@
+{"n": 6, "k": 4, "edges": [[0, 3], [1, 3], [1, 4], [1, 5], [2, 5], [3, 5], [4, 5]], "family": "rlve_max_achromatic_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l1-s7/solution/solve.sh b/tasks/construct-rlve-max-achromatic-coloring-l1-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..41038d6e4789a1a5901f6c511bb604b3e2b71401
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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'
+{"colors": [0, 1, 0, 2, 0, 3]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l1-s7/tests/check.py b/tasks/construct-rlve-max-achromatic-coloring-l1-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..bb676a518264e5db4fb0964e27ccffccf6980807
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l1-s7/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-l1-s7/tests/instance.json b/tasks/construct-rlve-max-achromatic-coloring-l1-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..11356a79469d2e4b2c44240a293c8ee53b78d853
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l1-s7/tests/instance.json
@@ -0,0 +1 @@
+{"n": 6, "k": 4, "edges": [[0, 3], [1, 3], [1, 4], [1, 5], [2, 5], [3, 5], [4, 5]], "family": "rlve_max_achromatic_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l1-s7/tests/run.py b/tasks/construct-rlve-max-achromatic-coloring-l1-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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-max-achromatic-coloring-l1-s7/tests/test.sh b/tasks/construct-rlve-max-achromatic-coloring-l1-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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-max-achromatic-coloring-l1-s9/environment/Dockerfile b/tasks/construct-rlve-max-achromatic-coloring-l1-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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-max-achromatic-coloring-l1-s9/environment/instance.json b/tasks/construct-rlve-max-achromatic-coloring-l1-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8d55f51fb454ea7207ac904939026e204d7caac5
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l1-s9/environment/instance.json
@@ -0,0 +1 @@
+{"n": 6, "k": 4, "edges": [[0, 1], [0, 2], [0, 3], [1, 2], [1, 5], [3, 5], [4, 5]], "family": "rlve_max_achromatic_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l1-s9/solution/solve.sh b/tasks/construct-rlve-max-achromatic-coloring-l1-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..167203314e16a61cd80e98c3ad21c96d79753324
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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'
+{"colors": [0, 1, 2, 2, 0, 3]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l1-s9/tests/check.py b/tasks/construct-rlve-max-achromatic-coloring-l1-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..bb676a518264e5db4fb0964e27ccffccf6980807
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l1-s9/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-l1-s9/tests/instance.json b/tasks/construct-rlve-max-achromatic-coloring-l1-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8d55f51fb454ea7207ac904939026e204d7caac5
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l1-s9/tests/instance.json
@@ -0,0 +1 @@
+{"n": 6, "k": 4, "edges": [[0, 1], [0, 2], [0, 3], [1, 2], [1, 5], [3, 5], [4, 5]], "family": "rlve_max_achromatic_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l1-s9/tests/run.py b/tasks/construct-rlve-max-achromatic-coloring-l1-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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-max-achromatic-coloring-l1-s9/tests/test.sh b/tasks/construct-rlve-max-achromatic-coloring-l1-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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-max-achromatic-coloring-l5-s1/environment/Dockerfile b/tasks/construct-rlve-max-achromatic-coloring-l5-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-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-rlve-max-achromatic-coloring-l5-s1/environment/instance.json b/tasks/construct-rlve-max-achromatic-coloring-l5-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c1ac88b14e19b886785bcfd75898769bd1ff0366
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l5-s1/environment/instance.json
@@ -0,0 +1 @@
+{"n": 13, "k": 7, "edges": [[0, 4], [0, 5], [0, 12], [1, 2], [1, 7], [1, 10], [1, 12], [2, 6], [2, 7], [2, 9], [3, 5], [3, 6], [3, 8], [3, 9], [3, 10], [3, 11], [3, 12], [4, 8], [4, 9], [4, 11], [4, 12], [5, 7], [5, 9], [5, 11], [6, 7], [6, 8], [6, 9], [7, 10], [9, 10], [10, 12], [11, 12]], "family": "rlve_max_achromatic_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l5-s1/solution/solve.sh b/tasks/construct-rlve-max-achromatic-coloring-l5-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..2cf3ab208f1ae361afb73ad608300cb45122f8f7
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-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'
+{"colors": [0, 1, 2, 0, 3, 1, 4, 3, 5, 6, 5, 4, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l5-s1/tests/check.py b/tasks/construct-rlve-max-achromatic-coloring-l5-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..bb676a518264e5db4fb0964e27ccffccf6980807
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l5-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("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-l5-s1/tests/instance.json b/tasks/construct-rlve-max-achromatic-coloring-l5-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c1ac88b14e19b886785bcfd75898769bd1ff0366
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l5-s1/tests/instance.json
@@ -0,0 +1 @@
+{"n": 13, "k": 7, "edges": [[0, 4], [0, 5], [0, 12], [1, 2], [1, 7], [1, 10], [1, 12], [2, 6], [2, 7], [2, 9], [3, 5], [3, 6], [3, 8], [3, 9], [3, 10], [3, 11], [3, 12], [4, 8], [4, 9], [4, 11], [4, 12], [5, 7], [5, 9], [5, 11], [6, 7], [6, 8], [6, 9], [7, 10], [9, 10], [10, 12], [11, 12]], "family": "rlve_max_achromatic_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l5-s1/tests/run.py b/tasks/construct-rlve-max-achromatic-coloring-l5-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-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-rlve-max-achromatic-coloring-l5-s1/tests/test.sh b/tasks/construct-rlve-max-achromatic-coloring-l5-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-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-rlve-max-achromatic-coloring-l5-s6/environment/Dockerfile b/tasks/construct-rlve-max-achromatic-coloring-l5-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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-max-achromatic-coloring-l5-s6/environment/instance.json b/tasks/construct-rlve-max-achromatic-coloring-l5-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2a074a0796bebea44b0b17f812c434999c3635c1
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l5-s6/environment/instance.json
@@ -0,0 +1 @@
+{"n": 13, "k": 7, "edges": [[0, 4], [0, 6], [0, 7], [0, 10], [1, 7], [1, 8], [1, 9], [1, 10], [1, 11], [2, 4], [2, 7], [2, 9], [2, 10], [2, 11], [3, 5], [3, 7], [3, 9], [3, 10], [4, 6], [4, 9], [5, 8], [5, 11], [6, 10], [6, 11], [7, 11], [7, 12], [8, 9], [8, 11], [8, 12], [9, 10], [10, 12]], "family": "rlve_max_achromatic_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l5-s6/solution/solve.sh b/tasks/construct-rlve-max-achromatic-coloring-l5-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..f7d7807eafcb102d6de6b305524eee11f1988b1e
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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'
+{"colors": [0, 1, 2, 3, 1, 0, 4, 5, 2, 5, 6, 3, 4]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l5-s6/tests/check.py b/tasks/construct-rlve-max-achromatic-coloring-l5-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..bb676a518264e5db4fb0964e27ccffccf6980807
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l5-s6/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-l5-s6/tests/instance.json b/tasks/construct-rlve-max-achromatic-coloring-l5-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2a074a0796bebea44b0b17f812c434999c3635c1
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l5-s6/tests/instance.json
@@ -0,0 +1 @@
+{"n": 13, "k": 7, "edges": [[0, 4], [0, 6], [0, 7], [0, 10], [1, 7], [1, 8], [1, 9], [1, 10], [1, 11], [2, 4], [2, 7], [2, 9], [2, 10], [2, 11], [3, 5], [3, 7], [3, 9], [3, 10], [4, 6], [4, 9], [5, 8], [5, 11], [6, 10], [6, 11], [7, 11], [7, 12], [8, 9], [8, 11], [8, 12], [9, 10], [10, 12]], "family": "rlve_max_achromatic_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l5-s6/tests/run.py b/tasks/construct-rlve-max-achromatic-coloring-l5-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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-max-achromatic-coloring-l5-s6/tests/test.sh b/tasks/construct-rlve-max-achromatic-coloring-l5-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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-maximum-clique-l4-s4/environment/instance.json b/tasks/construct-rlve-maximum-clique-l4-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1dcd19662fb858f53d4f007df7b4c910cba2d2fd
--- /dev/null
+++ b/tasks/construct-rlve-maximum-clique-l4-s4/environment/instance.json
@@ -0,0 +1 @@
+{"n": 32, "k": 8, "edges": [[0, 2], [0, 3], [0, 4], [0, 8], [0, 9], [0, 10], [0, 11], [0, 13], [0, 14], [0, 15], [0, 21], [0, 22], [0, 23], [0, 26], [0, 27], [0, 29], [0, 30], [0, 31], [1, 3], [1, 4], [1, 5], [1, 6], [1, 7], [1, 9], [1, 11], [1, 12], [1, 13], [1, 16], [1, 17], [1, 18], [1, 19], [1, 20], [1, 21], [1, 22], [1, 24], [1, 25], [1, 27], [1, 28], [1, 29], [1, 31], [2, 4], [2, 5], [2, 6], [2, 7], [2, 8], [2, 9], [2, 10], [2, 11], [2, 13], [2, 14], [2, 15], [2, 16], [2, 17], [2, 18], [2, 19], [2, 20], [2, 22], [2, 25], [2, 26], [2, 28], [2, 29], [2, 31], [3, 5], [3, 6], [3, 7], [3, 9], [3, 10], [3, 13], [3, 15], [3, 16], [3, 18], [3, 19], [3, 21], [3, 22], [3, 23], [3, 24], [3, 25], [3, 26], [3, 28], [3, 29], [4, 6], [4, 8], [4, 9], [4, 10], [4, 11], [4, 13], [4, 16], [4, 17], [4, 18], [4, 19], [4, 20], [4, 21], [4, 22], [4, 28], [4, 31], [5, 6], [5, 8], [5, 9], [5, 10], [5, 12], [5, 15], [5, 16], [5, 17], [5, 19], [5, 20], [5, 21], [5, 22], [5, 24], [5, 27], [5, 28], [5, 29], [5, 30], [6, 8], [6, 9], [6, 10], [6, 12], [6, 14], [6, 21], [6, 22], [6, 23], [6, 25], [6, 27], [6, 28], [6, 30], [6, 31], [7, 10], [7, 11], [7, 12], [7, 13], [7, 14], [7, 15], [7, 16], [7, 17], [7, 18], [7, 20], [7, 21], [7, 22], [7, 23], [7, 24], [7, 26], [7, 29], [7, 30], [8, 10], [8, 11], [8, 14], [8, 15], [8, 16], [8, 17], [8, 18], [8, 20], [8, 23], [8, 24], [8, 25], [8, 26], [8, 27], [8, 28], [8, 29], [8, 30], [9, 11], [9, 12], [9, 14], [9, 15], [9, 16], [9, 18], [9, 20], [9, 21], [9, 22], [9, 23], [9, 24], [9, 25], [9, 26], [9, 28], [9, 30], [9, 31], [10, 11], [10, 13], [10, 14], [10, 15], [10, 16], [10, 17], [10, 19], [10, 20], [10, 22], [10, 24], [10, 25], [10, 26], [10, 28], [10, 29], [10, 30], [11, 12], [11, 14], [11, 15], [11, 16], [11, 18], [11, 19], [11, 21], [11, 23], [11, 25], [11, 27], [11, 28], [11, 29], [11, 30], [11, 31], [12, 14], [12, 16], [12, 17], [12, 18], [12, 21], [12, 23], [12, 27], [12, 30], [13, 14], [13, 15], [13, 18], [13, 21], [13, 23], [13, 24], [13, 25], [13, 28], [13, 31], [14, 15], [14, 16], [14, 17], [14, 20], [14, 21], [14, 22], [14, 23], [14, 24], [14, 25], [14, 26], [14, 29], [14, 30], [15, 16], [15, 18], [15, 19], [15, 21], [15, 22], [15, 23], [15, 24], [15, 26], [15, 27], [15, 28], [15, 30], [15, 31], [16, 18], [16, 20], [16, 22], [16, 23], [16, 24], [16, 26], [16, 27], [16, 28], [16, 31], [17, 18], [17, 19], [17, 20], [17, 21], [17, 26], [17, 27], [17, 31], [18, 20], [18, 21], [18, 22], [18, 24], [18, 27], [18, 29], [18, 31], [19, 20], [19, 21], [19, 22], [19, 23], [19, 25], [19, 26], [19, 27], [19, 29], [19, 30], [19, 31], [20, 21], [20, 22], [20, 23], [20, 25], [20, 26], [20, 27], [20, 28], [20, 29], [21, 22], [21, 23], [21, 25], [21, 26], [21, 28], [21, 29], [21, 30], [21, 31], [22, 23], [22, 24], [22, 25], [22, 26], [22, 28], [22, 30], [23, 26], [23, 27], [23, 29], [23, 31], [24, 25], [24, 26], [24, 30], [24, 31], [25, 28], [25, 29], [26, 27], [26, 28], [26, 30], [26, 31], [27, 28], [27, 29], [27, 30], [27, 31], [28, 29], [28, 30], [28, 31], [29, 30], [30, 31]], "family": "rlve_maximum_clique", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-chromatic-number-l4-s1/environment/Dockerfile b/tasks/construct-rlve-min-chromatic-number-l4-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l4-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-min-chromatic-number-l4-s1/environment/instance.json b/tasks/construct-rlve-min-chromatic-number-l4-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d757096e139c85012355c96ab14648e2364c3c5a
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l4-s1/environment/instance.json
@@ -0,0 +1 @@
+{"n": 18, "k": 5, "edges": [[0, 1], [0, 6], [0, 7], [0, 9], [0, 10], [0, 11], [0, 13], [0, 15], [0, 16], [1, 3], [1, 7], [1, 9], [1, 10], [1, 12], [1, 16], [1, 17], [2, 5], [2, 9], [2, 12], [2, 13], [2, 15], [2, 17], [3, 4], [3, 5], [3, 6], [3, 9], [3, 10], [3, 13], [3, 14], [3, 16], [4, 5], [4, 7], [4, 10], [4, 16], [5, 7], [5, 8], [5, 9], [5, 13], [5, 14], [5, 15], [5, 16], [6, 9], [6, 10], [6, 15], [6, 16], [7, 8], [7, 9], [7, 10], [7, 11], [7, 13], [7, 14], [7, 15], [7, 17], [8, 10], [8, 17], [9, 10], [9, 11], [9, 12], [9, 14], [9, 15], [10, 15], [11, 16], [11, 17], [12, 15], [12, 17], [14, 16], [15, 16], [16, 17]], "family": "rlve_min_chromatic_number", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-chromatic-number-l4-s1/solution/solve.sh b/tasks/construct-rlve-min-chromatic-number-l4-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..4a5ceec0aee6708259e68b0d6c4712c68d05e754
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l4-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'
+{"colors": [0, 1, 0, 0, 1, 2, 3, 3, 0, 4, 2, 1, 3, 1, 1, 1, 4, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-min-chromatic-number-l4-s1/tests/check.py b/tasks/construct-rlve-min-chromatic-number-l4-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..d2b77793fb15e0d97174b3c28e69a86e2a26996f
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l4-s1/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-l4-s1/tests/instance.json b/tasks/construct-rlve-min-chromatic-number-l4-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d757096e139c85012355c96ab14648e2364c3c5a
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l4-s1/tests/instance.json
@@ -0,0 +1 @@
+{"n": 18, "k": 5, "edges": [[0, 1], [0, 6], [0, 7], [0, 9], [0, 10], [0, 11], [0, 13], [0, 15], [0, 16], [1, 3], [1, 7], [1, 9], [1, 10], [1, 12], [1, 16], [1, 17], [2, 5], [2, 9], [2, 12], [2, 13], [2, 15], [2, 17], [3, 4], [3, 5], [3, 6], [3, 9], [3, 10], [3, 13], [3, 14], [3, 16], [4, 5], [4, 7], [4, 10], [4, 16], [5, 7], [5, 8], [5, 9], [5, 13], [5, 14], [5, 15], [5, 16], [6, 9], [6, 10], [6, 15], [6, 16], [7, 8], [7, 9], [7, 10], [7, 11], [7, 13], [7, 14], [7, 15], [7, 17], [8, 10], [8, 17], [9, 10], [9, 11], [9, 12], [9, 14], [9, 15], [10, 15], [11, 16], [11, 17], [12, 15], [12, 17], [14, 16], [15, 16], [16, 17]], "family": "rlve_min_chromatic_number", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-chromatic-number-l4-s1/tests/run.py b/tasks/construct-rlve-min-chromatic-number-l4-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l4-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-min-chromatic-number-l4-s1/tests/test.sh b/tasks/construct-rlve-min-chromatic-number-l4-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l4-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-l4-s6/environment/Dockerfile b/tasks/construct-rlve-min-chromatic-number-l4-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l4-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-chromatic-number-l4-s6/environment/instance.json b/tasks/construct-rlve-min-chromatic-number-l4-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..89916d8368bd2509876415fc45097c3c536ea6df
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l4-s6/environment/instance.json
@@ -0,0 +1 @@
+{"n": 18, "k": 7, "edges": [[0, 2], [0, 3], [0, 4], [0, 7], [0, 10], [0, 11], [0, 12], [0, 13], [0, 14], [0, 15], [0, 16], [0, 17], [1, 2], [1, 3], [1, 5], [1, 6], [1, 8], [1, 9], [1, 10], [1, 12], [1, 14], [1, 15], [1, 16], [1, 17], [2, 3], [2, 4], [2, 5], [2, 7], [2, 8], [2, 10], [2, 11], [2, 13], [2, 14], [2, 15], [2, 16], [2, 17], [3, 7], [3, 8], [3, 10], [3, 11], [3, 12], [3, 13], [3, 14], [4, 5], [4, 6], [4, 7], [4, 9], [4, 10], [4, 11], [4, 12], [4, 15], [4, 16], [4, 17], [5, 6], [5, 9], [5, 13], [5, 14], [5, 15], [5, 16], [5, 17], [6, 7], [6, 8], [6, 12], [6, 13], [6, 15], [7, 9], [7, 12], [7, 13], [7, 15], [7, 16], [8, 11], [8, 12], [8, 13], [8, 14], [8, 17], [9, 11], [9, 14], [9, 16], [9, 17], [10, 11], [10, 16], [10, 17], [11, 12], [11, 13], [11, 14], [11, 15], [11, 16], [11, 17], [12, 14], [12, 15], [12, 17], [13, 15], [13, 16], [14, 15], [14, 16], [14, 17], [15, 16], [15, 17], [16, 17]], "family": "rlve_min_chromatic_number", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-chromatic-number-l4-s6/solution/solve.sh b/tasks/construct-rlve-min-chromatic-number-l4-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..179440c7326d24893ec524571d20c47abd241aca
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l4-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, 1, 2, 3, 4, 1, 4, 5, 1, 5, 4, 6, 3, 3, 5, 6, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-min-chromatic-number-l4-s6/tests/check.py b/tasks/construct-rlve-min-chromatic-number-l4-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..d2b77793fb15e0d97174b3c28e69a86e2a26996f
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l4-s6/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-l4-s6/tests/instance.json b/tasks/construct-rlve-min-chromatic-number-l4-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..89916d8368bd2509876415fc45097c3c536ea6df
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l4-s6/tests/instance.json
@@ -0,0 +1 @@
+{"n": 18, "k": 7, "edges": [[0, 2], [0, 3], [0, 4], [0, 7], [0, 10], [0, 11], [0, 12], [0, 13], [0, 14], [0, 15], [0, 16], [0, 17], [1, 2], [1, 3], [1, 5], [1, 6], [1, 8], [1, 9], [1, 10], [1, 12], [1, 14], [1, 15], [1, 16], [1, 17], [2, 3], [2, 4], [2, 5], [2, 7], [2, 8], [2, 10], [2, 11], [2, 13], [2, 14], [2, 15], [2, 16], [2, 17], [3, 7], [3, 8], [3, 10], [3, 11], [3, 12], [3, 13], [3, 14], [4, 5], [4, 6], [4, 7], [4, 9], [4, 10], [4, 11], [4, 12], [4, 15], [4, 16], [4, 17], [5, 6], [5, 9], [5, 13], [5, 14], [5, 15], [5, 16], [5, 17], [6, 7], [6, 8], [6, 12], [6, 13], [6, 15], [7, 9], [7, 12], [7, 13], [7, 15], [7, 16], [8, 11], [8, 12], [8, 13], [8, 14], [8, 17], [9, 11], [9, 14], [9, 16], [9, 17], [10, 11], [10, 16], [10, 17], [11, 12], [11, 13], [11, 14], [11, 15], [11, 16], [11, 17], [12, 14], [12, 15], [12, 17], [13, 15], [13, 16], [14, 15], [14, 16], [14, 17], [15, 16], [15, 17], [16, 17]], "family": "rlve_min_chromatic_number", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-chromatic-number-l4-s6/tests/run.py b/tasks/construct-rlve-min-chromatic-number-l4-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l4-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-chromatic-number-l4-s6/tests/test.sh b/tasks/construct-rlve-min-chromatic-number-l4-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l4-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-chromatic-number-l4-s8/environment/Dockerfile b/tasks/construct-rlve-min-chromatic-number-l4-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l4-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-chromatic-number-l4-s8/environment/instance.json b/tasks/construct-rlve-min-chromatic-number-l4-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a77064e5795cc7e55ae3a22269f702b385035b1f
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l4-s8/environment/instance.json
@@ -0,0 +1 @@
+{"n": 18, "k": 5, "edges": [[0, 1], [0, 8], [0, 11], [0, 16], [0, 17], [1, 5], [1, 8], [1, 11], [1, 13], [1, 14], [1, 15], [1, 16], [2, 3], [2, 8], [2, 9], [2, 12], [2, 13], [2, 15], [2, 17], [3, 7], [3, 8], [3, 9], [3, 10], [3, 12], [3, 13], [3, 17], [4, 6], [4, 7], [4, 10], [4, 11], [4, 13], [4, 14], [4, 16], [4, 17], [5, 6], [5, 7], [5, 8], [5, 9], [5, 15], [5, 17], [6, 8], [6, 10], [6, 12], [7, 10], [7, 11], [7, 17], [8, 9], [8, 10], [8, 12], [8, 13], [8, 17], [9, 12], [9, 14], [9, 16], [9, 17], [10, 11], [10, 12], [10, 13], [10, 16], [11, 14], [11, 15], [11, 16], [11, 17], [12, 13], [12, 16], [13, 17], [14, 17], [15, 16]], "family": "rlve_min_chromatic_number", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-chromatic-number-l4-s8/solution/solve.sh b/tasks/construct-rlve-min-chromatic-number-l4-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..20483b681505b4040a1899ac6bdee81dcc449772
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l4-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, 1, 0, 1, 1, 0, 2, 3, 3, 2, 0, 2, 4, 2, 0, 4, 3, 4]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-min-chromatic-number-l4-s8/tests/check.py b/tasks/construct-rlve-min-chromatic-number-l4-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..d2b77793fb15e0d97174b3c28e69a86e2a26996f
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l4-s8/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-l4-s8/tests/instance.json b/tasks/construct-rlve-min-chromatic-number-l4-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a77064e5795cc7e55ae3a22269f702b385035b1f
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l4-s8/tests/instance.json
@@ -0,0 +1 @@
+{"n": 18, "k": 5, "edges": [[0, 1], [0, 8], [0, 11], [0, 16], [0, 17], [1, 5], [1, 8], [1, 11], [1, 13], [1, 14], [1, 15], [1, 16], [2, 3], [2, 8], [2, 9], [2, 12], [2, 13], [2, 15], [2, 17], [3, 7], [3, 8], [3, 9], [3, 10], [3, 12], [3, 13], [3, 17], [4, 6], [4, 7], [4, 10], [4, 11], [4, 13], [4, 14], [4, 16], [4, 17], [5, 6], [5, 7], [5, 8], [5, 9], [5, 15], [5, 17], [6, 8], [6, 10], [6, 12], [7, 10], [7, 11], [7, 17], [8, 9], [8, 10], [8, 12], [8, 13], [8, 17], [9, 12], [9, 14], [9, 16], [9, 17], [10, 11], [10, 12], [10, 13], [10, 16], [11, 14], [11, 15], [11, 16], [11, 17], [12, 13], [12, 16], [13, 17], [14, 17], [15, 16]], "family": "rlve_min_chromatic_number", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-chromatic-number-l4-s8/tests/run.py b/tasks/construct-rlve-min-chromatic-number-l4-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l4-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-chromatic-number-l4-s8/tests/test.sh b/tasks/construct-rlve-min-chromatic-number-l4-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l4-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-l4-s2/environment/Dockerfile b/tasks/construct-rlve-min-harmonious-coloring-l4-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-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-min-harmonious-coloring-l4-s2/environment/instance.json b/tasks/construct-rlve-min-harmonious-coloring-l4-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..eaf099d47f8771acb0313205b78767515d52c26c
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l4-s2/environment/instance.json
@@ -0,0 +1 @@
+{"n": 12, "k": 5, "edges": [[0, 6], [0, 10], [1, 3], [1, 8], [1, 11], [4, 8], [5, 7], [5, 11], [7, 10]], "family": "rlve_min_harmonious_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l4-s2/solution/solve.sh b/tasks/construct-rlve-min-harmonious-coloring-l4-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..32db8ba3973a608142046ea59a20f7fc4a37b295
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-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, 0, 0, 2, 2, 3, 3, 0, 4, 4]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l4-s2/tests/check.py b/tasks/construct-rlve-min-harmonious-coloring-l4-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..91282a31c765b6103e6575180adb47abddb6a506
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l4-s2/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-l4-s2/tests/instance.json b/tasks/construct-rlve-min-harmonious-coloring-l4-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..eaf099d47f8771acb0313205b78767515d52c26c
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l4-s2/tests/instance.json
@@ -0,0 +1 @@
+{"n": 12, "k": 5, "edges": [[0, 6], [0, 10], [1, 3], [1, 8], [1, 11], [4, 8], [5, 7], [5, 11], [7, 10]], "family": "rlve_min_harmonious_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l4-s2/tests/run.py b/tasks/construct-rlve-min-harmonious-coloring-l4-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-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-min-harmonious-coloring-l4-s2/tests/test.sh b/tasks/construct-rlve-min-harmonious-coloring-l4-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-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-min-harmonious-coloring-l4-s5/environment/Dockerfile b/tasks/construct-rlve-min-harmonious-coloring-l4-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-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-min-harmonious-coloring-l4-s5/environment/instance.json b/tasks/construct-rlve-min-harmonious-coloring-l4-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..631b2e1f142437e43ecd65bb7f3cc36f2fe80cdc
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l4-s5/environment/instance.json
@@ -0,0 +1 @@
+{"n": 12, "k": 5, "edges": [[0, 4], [0, 11], [2, 9], [3, 9], [4, 11], [5, 7], [6, 9], [7, 11], [8, 11]], "family": "rlve_min_harmonious_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l4-s5/solution/solve.sh b/tasks/construct-rlve-min-harmonious-coloring-l4-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..af1a1df689f76301840c3340cdde32bda12857dc
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-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, 0, 0, 1, 1, 0, 2, 2, 3, 3, 0, 4]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l4-s5/tests/check.py b/tasks/construct-rlve-min-harmonious-coloring-l4-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..91282a31c765b6103e6575180adb47abddb6a506
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l4-s5/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-l4-s5/tests/instance.json b/tasks/construct-rlve-min-harmonious-coloring-l4-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..631b2e1f142437e43ecd65bb7f3cc36f2fe80cdc
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l4-s5/tests/instance.json
@@ -0,0 +1 @@
+{"n": 12, "k": 5, "edges": [[0, 4], [0, 11], [2, 9], [3, 9], [4, 11], [5, 7], [6, 9], [7, 11], [8, 11]], "family": "rlve_min_harmonious_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l4-s5/tests/run.py b/tasks/construct-rlve-min-harmonious-coloring-l4-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-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-min-harmonious-coloring-l4-s5/tests/test.sh b/tasks/construct-rlve-min-harmonious-coloring-l4-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-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-pairwise-sums-recovery-l2-s3/environment/Dockerfile b/tasks/construct-rlve-pairwise-sums-recovery-l2-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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-pairwise-sums-recovery-l2-s3/environment/instance.json b/tasks/construct-rlve-pairwise-sums-recovery-l2-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..41809b5dc2c8df44fd4b95102b799435d10f108b
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l2-s3/environment/instance.json
@@ -0,0 +1 @@
+{"n": 5, "sums": [15, 12, 13, 8, 9, 12, 11, 10, 7, 11], "family": "rlve_pairwise_sums_recovery", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l2-s3/solution/solve.sh b/tasks/construct-rlve-pairwise-sums-recovery-l2-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..a7bb46f3f90c5524d48bee13e2c2011c13d9c803
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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'
+{"numbers": [3, 4, 5, 7, 8]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l2-s3/tests/check.py b/tasks/construct-rlve-pairwise-sums-recovery-l2-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..01b5b475e927688c0209fb93478a95f6dd1d3407
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l2-s3/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-l2-s3/tests/instance.json b/tasks/construct-rlve-pairwise-sums-recovery-l2-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..41809b5dc2c8df44fd4b95102b799435d10f108b
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l2-s3/tests/instance.json
@@ -0,0 +1 @@
+{"n": 5, "sums": [15, 12, 13, 8, 9, 12, 11, 10, 7, 11], "family": "rlve_pairwise_sums_recovery", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l2-s3/tests/run.py b/tasks/construct-rlve-pairwise-sums-recovery-l2-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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-pairwise-sums-recovery-l2-s3/tests/test.sh b/tasks/construct-rlve-pairwise-sums-recovery-l2-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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-pairwise-sums-recovery-l2-s4/environment/Dockerfile b/tasks/construct-rlve-pairwise-sums-recovery-l2-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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-pairwise-sums-recovery-l2-s4/environment/instance.json b/tasks/construct-rlve-pairwise-sums-recovery-l2-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e5550b68a5f0c15bc4547763aed86a3531093b39
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l2-s4/environment/instance.json
@@ -0,0 +1 @@
+{"n": 5, "sums": [7, 10, 13, 17, 8, 15, 12, 5, 9, 12], "family": "rlve_pairwise_sums_recovery", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l2-s4/solution/solve.sh b/tasks/construct-rlve-pairwise-sums-recovery-l2-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..53634c1d6248e4000d57178733ec23b0d6c59197
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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'
+{"numbers": [2, 3, 5, 7, 10]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l2-s4/tests/check.py b/tasks/construct-rlve-pairwise-sums-recovery-l2-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..01b5b475e927688c0209fb93478a95f6dd1d3407
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l2-s4/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-l2-s4/tests/instance.json b/tasks/construct-rlve-pairwise-sums-recovery-l2-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e5550b68a5f0c15bc4547763aed86a3531093b39
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l2-s4/tests/instance.json
@@ -0,0 +1 @@
+{"n": 5, "sums": [7, 10, 13, 17, 8, 15, 12, 5, 9, 12], "family": "rlve_pairwise_sums_recovery", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l2-s4/tests/run.py b/tasks/construct-rlve-pairwise-sums-recovery-l2-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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-pairwise-sums-recovery-l2-s4/tests/test.sh b/tasks/construct-rlve-pairwise-sums-recovery-l2-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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-pairwise-sums-recovery-l6-s2/environment/Dockerfile b/tasks/construct-rlve-pairwise-sums-recovery-l6-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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-pairwise-sums-recovery-l6-s2/environment/instance.json b/tasks/construct-rlve-pairwise-sums-recovery-l6-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2772249ddfe0723f117387b3058fe02f72878819
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l6-s2/environment/instance.json
@@ -0,0 +1 @@
+{"n": 26, "sums": [18, 77, 60, 94, 64, 61, 58, 39, 61, 39, 69, 27, 60, 46, 58, 18, 39, 45, 63, 26, 47, 57, 87, 59, 43, 25, 64, 62, 34, 55, 30, 14, 33, 29, 32, 31, 39, 65, 61, 44, 6, 21, 73, 71, 13, 60, 81, 32, 39, 54, 48, 23, 46, 48, 47, 85, 12, 45, 14, 61, 43, 53, 75, 54, 13, 36, 24, 79, 34, 68, 36, 62, 80, 59, 50, 29, 86, 5, 51, 23, 46, 74, 44, 47, 50, 22, 56, 28, 60, 17, 3, 57, 52, 36, 31, 34, 96, 78, 56, 38, 39, 87, 11, 25, 62, 21, 83, 82, 53, 91, 49, 49, 33, 40, 78, 36, 59, 17, 27, 50, 7, 80, 75, 33, 82, 36, 23, 57, 23, 51, 80, 76, 16, 73, 16, 67, 37, 56, 53, 20, 70, 81, 50, 72, 57, 15, 62, 72, 19, 19, 77, 71, 75, 43, 53, 65, 42, 73, 17, 50, 75, 43, 69, 59, 57, 47, 63, 21, 9, 48, 35, 57, 32, 68, 33, 59, 68, 64, 48, 60, 40, 74, 42, 83, 49, 71, 38, 15, 63, 44, 37, 26, 69, 4, 51, 58, 35, 41, 54, 46, 51, 36, 46, 10, 51, 26, 89, 46, 50, 30, 37, 31, 50, 55, 95, 35, 54, 55, 42, 70, 59, 29, 25, 57, 27, 97, 31, 49, 22, 38, 19, 52, 67, 13, 41, 15, 18, 72, 79, 41, 47, 35, 47, 37, 25, 26, 84, 52, 25, 35, 60, 15, 64, 71, 40, 66, 30, 58, 27, 56, 92, 44, 65, 28, 14, 24, 37, 59, 39, 24, 30, 56, 90, 26, 74, 62, 46, 45, 16, 12, 49, 52, 44, 49, 50, 10, 12, 41, 45, 16, 66, 42, 44, 28, 41, 20, 93, 62, 43, 80, 52, 29, 28, 53, 41, 65, 35, 28, 55, 11, 41, 20, 34, 34, 49, 38, 51, 48, 77, 5, 62, 40, 38, 17, 56], "family": "rlve_pairwise_sums_recovery", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l6-s2/solution/solve.sh b/tasks/construct-rlve-pairwise-sums-recovery-l6-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..51eaf71d3fba148a814b7d10c74753e99f962cc7
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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'
+{"numbers": [1, 2, 3, 4, 8, 9, 11, 12, 13, 14, 15, 17, 22, 24, 26, 27, 30, 32, 33, 35, 38, 42, 45, 47, 48, 49]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l6-s2/tests/check.py b/tasks/construct-rlve-pairwise-sums-recovery-l6-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..01b5b475e927688c0209fb93478a95f6dd1d3407
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l6-s2/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-l6-s2/tests/instance.json b/tasks/construct-rlve-pairwise-sums-recovery-l6-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2772249ddfe0723f117387b3058fe02f72878819
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l6-s2/tests/instance.json
@@ -0,0 +1 @@
+{"n": 26, "sums": [18, 77, 60, 94, 64, 61, 58, 39, 61, 39, 69, 27, 60, 46, 58, 18, 39, 45, 63, 26, 47, 57, 87, 59, 43, 25, 64, 62, 34, 55, 30, 14, 33, 29, 32, 31, 39, 65, 61, 44, 6, 21, 73, 71, 13, 60, 81, 32, 39, 54, 48, 23, 46, 48, 47, 85, 12, 45, 14, 61, 43, 53, 75, 54, 13, 36, 24, 79, 34, 68, 36, 62, 80, 59, 50, 29, 86, 5, 51, 23, 46, 74, 44, 47, 50, 22, 56, 28, 60, 17, 3, 57, 52, 36, 31, 34, 96, 78, 56, 38, 39, 87, 11, 25, 62, 21, 83, 82, 53, 91, 49, 49, 33, 40, 78, 36, 59, 17, 27, 50, 7, 80, 75, 33, 82, 36, 23, 57, 23, 51, 80, 76, 16, 73, 16, 67, 37, 56, 53, 20, 70, 81, 50, 72, 57, 15, 62, 72, 19, 19, 77, 71, 75, 43, 53, 65, 42, 73, 17, 50, 75, 43, 69, 59, 57, 47, 63, 21, 9, 48, 35, 57, 32, 68, 33, 59, 68, 64, 48, 60, 40, 74, 42, 83, 49, 71, 38, 15, 63, 44, 37, 26, 69, 4, 51, 58, 35, 41, 54, 46, 51, 36, 46, 10, 51, 26, 89, 46, 50, 30, 37, 31, 50, 55, 95, 35, 54, 55, 42, 70, 59, 29, 25, 57, 27, 97, 31, 49, 22, 38, 19, 52, 67, 13, 41, 15, 18, 72, 79, 41, 47, 35, 47, 37, 25, 26, 84, 52, 25, 35, 60, 15, 64, 71, 40, 66, 30, 58, 27, 56, 92, 44, 65, 28, 14, 24, 37, 59, 39, 24, 30, 56, 90, 26, 74, 62, 46, 45, 16, 12, 49, 52, 44, 49, 50, 10, 12, 41, 45, 16, 66, 42, 44, 28, 41, 20, 93, 62, 43, 80, 52, 29, 28, 53, 41, 65, 35, 28, 55, 11, 41, 20, 34, 34, 49, 38, 51, 48, 77, 5, 62, 40, 38, 17, 56], "family": "rlve_pairwise_sums_recovery", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l6-s2/tests/run.py b/tasks/construct-rlve-pairwise-sums-recovery-l6-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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-pairwise-sums-recovery-l6-s2/tests/test.sh b/tasks/construct-rlve-pairwise-sums-recovery-l6-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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-pairwise-sums-recovery-l6-s5/environment/Dockerfile b/tasks/construct-rlve-pairwise-sums-recovery-l6-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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-pairwise-sums-recovery-l6-s5/environment/instance.json b/tasks/construct-rlve-pairwise-sums-recovery-l6-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..ddf34964934ee8e08d7bc5e592054e469af462d0
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l6-s5/environment/instance.json
@@ -0,0 +1 @@
+{"n": 26, "sums": [8, 52, 30, 51, 30, 39, 43, 46, 59, 83, 38, 86, 66, 56, 47, 74, 63, 41, 77, 36, 39, 43, 36, 53, 27, 40, 48, 46, 59, 46, 68, 59, 36, 53, 52, 38, 49, 75, 55, 56, 58, 47, 61, 29, 46, 50, 43, 65, 60, 41, 62, 54, 56, 60, 37, 31, 18, 35, 34, 41, 36, 49, 43, 69, 49, 68, 87, 28, 64, 81, 46, 76, 67, 51, 71, 61, 76, 13, 40, 45, 21, 31, 74, 42, 90, 72, 4, 63, 25, 46, 37, 28, 26, 38, 57, 70, 54, 57, 31, 50, 60, 74, 39, 50, 31, 65, 38, 58, 44, 45, 52, 53, 51, 94, 41, 41, 11, 66, 58, 66, 44, 24, 84, 23, 14, 54, 40, 58, 28, 25, 45, 29, 55, 75, 54, 60, 68, 42, 47, 63, 40, 43, 76, 17, 76, 33, 41, 35, 61, 75, 56, 38, 59, 65, 31, 62, 71, 24, 69, 78, 49, 64, 54, 55, 47, 68, 23, 42, 41, 35, 36, 79, 46, 71, 48, 21, 61, 16, 44, 26, 27, 73, 75, 97, 61, 52, 34, 26, 55, 45, 65, 50, 49, 35, 31, 56, 57, 38, 57, 60, 50, 16, 34, 42, 71, 44, 51, 19, 25, 29, 54, 24, 69, 44, 33, 58, 48, 43, 64, 61, 32, 20, 14, 19, 34, 21, 79, 64, 22, 50, 49, 11, 51, 60, 29, 6, 27, 79, 73, 65, 33, 56, 32, 81, 37, 25, 63, 63, 17, 36, 57, 53, 45, 56, 89, 18, 86, 67, 49, 67, 45, 38, 15, 45, 72, 39, 69, 64, 62, 39, 53, 15, 37, 66, 52, 16, 62, 47, 72, 35, 59, 40, 48, 46, 56, 7, 53, 91, 9, 47, 79, 37, 33, 30, 82, 34, 24, 29, 36, 70, 66, 85, 72, 54, 53, 83, 39, 49, 34, 26, 33, 48, 27, 12, 82, 35, 26, 67, 16, 57, 78, 51, 44, 71, 20], "family": "rlve_pairwise_sums_recovery", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l6-s5/solution/solve.sh b/tasks/construct-rlve-pairwise-sums-recovery-l6-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..3b9b5e2c121c8bd2cd8d053e69b9b6997278946a
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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'
+{"numbers": [1, 3, 5, 6, 10, 11, 13, 14, 15, 20, 21, 23, 24, 25, 26, 28, 30, 32, 33, 35, 36, 39, 40, 43, 46, 51]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l6-s5/tests/check.py b/tasks/construct-rlve-pairwise-sums-recovery-l6-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..01b5b475e927688c0209fb93478a95f6dd1d3407
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l6-s5/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-l6-s5/tests/instance.json b/tasks/construct-rlve-pairwise-sums-recovery-l6-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..ddf34964934ee8e08d7bc5e592054e469af462d0
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l6-s5/tests/instance.json
@@ -0,0 +1 @@
+{"n": 26, "sums": [8, 52, 30, 51, 30, 39, 43, 46, 59, 83, 38, 86, 66, 56, 47, 74, 63, 41, 77, 36, 39, 43, 36, 53, 27, 40, 48, 46, 59, 46, 68, 59, 36, 53, 52, 38, 49, 75, 55, 56, 58, 47, 61, 29, 46, 50, 43, 65, 60, 41, 62, 54, 56, 60, 37, 31, 18, 35, 34, 41, 36, 49, 43, 69, 49, 68, 87, 28, 64, 81, 46, 76, 67, 51, 71, 61, 76, 13, 40, 45, 21, 31, 74, 42, 90, 72, 4, 63, 25, 46, 37, 28, 26, 38, 57, 70, 54, 57, 31, 50, 60, 74, 39, 50, 31, 65, 38, 58, 44, 45, 52, 53, 51, 94, 41, 41, 11, 66, 58, 66, 44, 24, 84, 23, 14, 54, 40, 58, 28, 25, 45, 29, 55, 75, 54, 60, 68, 42, 47, 63, 40, 43, 76, 17, 76, 33, 41, 35, 61, 75, 56, 38, 59, 65, 31, 62, 71, 24, 69, 78, 49, 64, 54, 55, 47, 68, 23, 42, 41, 35, 36, 79, 46, 71, 48, 21, 61, 16, 44, 26, 27, 73, 75, 97, 61, 52, 34, 26, 55, 45, 65, 50, 49, 35, 31, 56, 57, 38, 57, 60, 50, 16, 34, 42, 71, 44, 51, 19, 25, 29, 54, 24, 69, 44, 33, 58, 48, 43, 64, 61, 32, 20, 14, 19, 34, 21, 79, 64, 22, 50, 49, 11, 51, 60, 29, 6, 27, 79, 73, 65, 33, 56, 32, 81, 37, 25, 63, 63, 17, 36, 57, 53, 45, 56, 89, 18, 86, 67, 49, 67, 45, 38, 15, 45, 72, 39, 69, 64, 62, 39, 53, 15, 37, 66, 52, 16, 62, 47, 72, 35, 59, 40, 48, 46, 56, 7, 53, 91, 9, 47, 79, 37, 33, 30, 82, 34, 24, 29, 36, 70, 66, 85, 72, 54, 53, 83, 39, 49, 34, 26, 33, 48, 27, 12, 82, 35, 26, 67, 16, 57, 78, 51, 44, 71, 20], "family": "rlve_pairwise_sums_recovery", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l6-s5/tests/run.py b/tasks/construct-rlve-pairwise-sums-recovery-l6-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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-pairwise-sums-recovery-l6-s5/tests/test.sh b/tasks/construct-rlve-pairwise-sums-recovery-l6-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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-pan-solar-panels-l3-s1/environment/Dockerfile b/tasks/construct-rlve-pan-solar-panels-l3-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-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-pan-solar-panels-l3-s1/environment/instance.json b/tasks/construct-rlve-pan-solar-panels-l3-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..dab88c42ec4fd20cd2ef331bed55de6836030c07
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l3-s1/environment/instance.json
@@ -0,0 +1 @@
+{"A": 84820, "B": 88690, "C": 15258, "D": 22353, "target": 22172, "family": "rlve_pan_solar_panels", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pan-solar-panels-l3-s1/solution/solve.sh b/tasks/construct-rlve-pan-solar-panels-l3-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..7b645d85885382261abd060ad3a5605bf2a79ae3
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-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'
+{"X": 88688, "Y": 22172}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-pan-solar-panels-l3-s1/tests/check.py b/tasks/construct-rlve-pan-solar-panels-l3-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..8bb2e7cc0b0280ab877755cc5d97b510680f7aa3
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l3-s1/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-l3-s1/tests/instance.json b/tasks/construct-rlve-pan-solar-panels-l3-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..dab88c42ec4fd20cd2ef331bed55de6836030c07
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l3-s1/tests/instance.json
@@ -0,0 +1 @@
+{"A": 84820, "B": 88690, "C": 15258, "D": 22353, "target": 22172, "family": "rlve_pan_solar_panels", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pan-solar-panels-l3-s1/tests/run.py b/tasks/construct-rlve-pan-solar-panels-l3-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-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-pan-solar-panels-l3-s1/tests/test.sh b/tasks/construct-rlve-pan-solar-panels-l3-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-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-pan-solar-panels-l3-s6/environment/Dockerfile b/tasks/construct-rlve-pan-solar-panels-l3-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-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-rlve-pan-solar-panels-l3-s6/environment/instance.json b/tasks/construct-rlve-pan-solar-panels-l3-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..72a5421c627f468cda460690b15b88e5f835b7fd
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l3-s6/environment/instance.json
@@ -0,0 +1 @@
+{"A": 26804, "B": 63517, "C": 74163, "D": 80659, "target": 40329, "family": "rlve_pan_solar_panels", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pan-solar-panels-l3-s6/solution/solve.sh b/tasks/construct-rlve-pan-solar-panels-l3-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..cd755425982be4c39d921177c2416f6054305391
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-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'
+{"X": 40329, "Y": 80658}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-pan-solar-panels-l3-s6/tests/check.py b/tasks/construct-rlve-pan-solar-panels-l3-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..8bb2e7cc0b0280ab877755cc5d97b510680f7aa3
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l3-s6/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-l3-s6/tests/instance.json b/tasks/construct-rlve-pan-solar-panels-l3-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..72a5421c627f468cda460690b15b88e5f835b7fd
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l3-s6/tests/instance.json
@@ -0,0 +1 @@
+{"A": 26804, "B": 63517, "C": 74163, "D": 80659, "target": 40329, "family": "rlve_pan_solar_panels", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pan-solar-panels-l3-s6/tests/run.py b/tasks/construct-rlve-pan-solar-panels-l3-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-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-rlve-pan-solar-panels-l3-s6/tests/test.sh b/tasks/construct-rlve-pan-solar-panels-l3-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-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-rlve-pan-solar-panels-l3-s8/environment/Dockerfile b/tasks/construct-rlve-pan-solar-panels-l3-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-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-pan-solar-panels-l3-s8/environment/instance.json b/tasks/construct-rlve-pan-solar-panels-l3-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f2d784e3f9ea6a4aefe816159c017bf441235ce3
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l3-s8/environment/instance.json
@@ -0,0 +1 @@
+{"A": 2862, "B": 20567, "C": 29606, "D": 64020, "target": 20567, "family": "rlve_pan_solar_panels", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pan-solar-panels-l3-s8/solution/solve.sh b/tasks/construct-rlve-pan-solar-panels-l3-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..8cbeab293edf84059decbc4199835d82f6f62b70
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-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'
+{"X": 20567, "Y": 61701}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-pan-solar-panels-l3-s8/tests/check.py b/tasks/construct-rlve-pan-solar-panels-l3-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..8bb2e7cc0b0280ab877755cc5d97b510680f7aa3
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l3-s8/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-l3-s8/tests/instance.json b/tasks/construct-rlve-pan-solar-panels-l3-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f2d784e3f9ea6a4aefe816159c017bf441235ce3
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l3-s8/tests/instance.json
@@ -0,0 +1 @@
+{"A": 2862, "B": 20567, "C": 29606, "D": 64020, "target": 20567, "family": "rlve_pan_solar_panels", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pan-solar-panels-l3-s8/tests/run.py b/tasks/construct-rlve-pan-solar-panels-l3-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-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-pan-solar-panels-l3-s8/tests/test.sh b/tasks/construct-rlve-pan-solar-panels-l3-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-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-polynomial-global-minimum-l2-s2/environment/Dockerfile b/tasks/construct-rlve-polynomial-global-minimum-l2-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-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-rlve-polynomial-global-minimum-l2-s2/environment/instance.json b/tasks/construct-rlve-polynomial-global-minimum-l2-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c8f3ee20b094f79cb5e700c0b195cb694fa0da07
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l2-s2/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [36, 68, 49, 16, 2], "fmin": "0", "family": "rlve_polynomial_global_minimum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l2-s2/solution/solve.sh b/tasks/construct-rlve-polynomial-global-minimum-l2-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..f1950410fa3b584939153d3ef59e69446e7b3733
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-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'
+{"x0": -2.0}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l2-s2/tests/check.py b/tasks/construct-rlve-polynomial-global-minimum-l2-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3be068f4391ad0838c29a4f7d521b8072cf39991
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l2-s2/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-l2-s2/tests/instance.json b/tasks/construct-rlve-polynomial-global-minimum-l2-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c8f3ee20b094f79cb5e700c0b195cb694fa0da07
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l2-s2/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [36, 68, 49, 16, 2], "fmin": "0", "family": "rlve_polynomial_global_minimum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l2-s2/tests/run.py b/tasks/construct-rlve-polynomial-global-minimum-l2-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-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-rlve-polynomial-global-minimum-l2-s2/tests/test.sh b/tasks/construct-rlve-polynomial-global-minimum-l2-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-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-rlve-polynomial-global-minimum-l2-s5/environment/Dockerfile b/tasks/construct-rlve-polynomial-global-minimum-l2-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l2-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-global-minimum-l2-s5/environment/instance.json b/tasks/construct-rlve-polynomial-global-minimum-l2-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..502b74d3be6358a65a56bc57affd53cf51133166
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l2-s5/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [1, -2, 1, 0, 1], "fmin": "0.289273423937777937935889154385276791977392087", "family": "rlve_polynomial_global_minimum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l2-s5/solution/solve.sh b/tasks/construct-rlve-polynomial-global-minimum-l2-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..6f66cc4eb57262ab9aeea25923f3d8454a3baa4e
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l2-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'
+{"x0": 0.5897545123014584}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l2-s5/tests/check.py b/tasks/construct-rlve-polynomial-global-minimum-l2-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3be068f4391ad0838c29a4f7d521b8072cf39991
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l2-s5/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-l2-s5/tests/instance.json b/tasks/construct-rlve-polynomial-global-minimum-l2-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..502b74d3be6358a65a56bc57affd53cf51133166
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l2-s5/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [1, -2, 1, 0, 1], "fmin": "0.289273423937777937935889154385276791977392087", "family": "rlve_polynomial_global_minimum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l2-s5/tests/run.py b/tasks/construct-rlve-polynomial-global-minimum-l2-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l2-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-global-minimum-l2-s5/tests/test.sh b/tasks/construct-rlve-polynomial-global-minimum-l2-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l2-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-l6-s3/environment/Dockerfile b/tasks/construct-rlve-polynomial-global-minimum-l6-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-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-rlve-polynomial-global-minimum-l6-s3/environment/instance.json b/tasks/construct-rlve-polynomial-global-minimum-l6-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..276c149ef7507f5787ad2c651a3118c324fb21d8
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l6-s3/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [2069, 10246, 23227, 31192, 27901, 17724, 8570, 3504, 1352, 480, 134, 24, 2, 0, 2], "fmin": "74.2472391815217440684469375737734650173534310", "family": "rlve_polynomial_global_minimum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l6-s3/solution/solve.sh b/tasks/construct-rlve-polynomial-global-minimum-l6-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..3671fa4ffca041520bea9f7505faffb4b7cc03fe
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-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'
+{"x0": -0.812132799110263}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l6-s3/tests/check.py b/tasks/construct-rlve-polynomial-global-minimum-l6-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3be068f4391ad0838c29a4f7d521b8072cf39991
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l6-s3/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-l6-s3/tests/instance.json b/tasks/construct-rlve-polynomial-global-minimum-l6-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..276c149ef7507f5787ad2c651a3118c324fb21d8
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l6-s3/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [2069, 10246, 23227, 31192, 27901, 17724, 8570, 3504, 1352, 480, 134, 24, 2, 0, 2], "fmin": "74.2472391815217440684469375737734650173534310", "family": "rlve_polynomial_global_minimum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l6-s3/tests/run.py b/tasks/construct-rlve-polynomial-global-minimum-l6-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-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-rlve-polynomial-global-minimum-l6-s3/tests/test.sh b/tasks/construct-rlve-polynomial-global-minimum-l6-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-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-rlve-polynomial-integer-roots-l1-s1/environment/Dockerfile b/tasks/construct-rlve-polynomial-integer-roots-l1-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l1-s1/environment/instance.json b/tasks/construct-rlve-polynomial-integer-roots-l1-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4ba04c9bbed002694529eef45b8f949704062afe
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l1-s1/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [0, 2, -3, 1], "family": "rlve_polynomial_integer_roots", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l1-s1/solution/solve.sh b/tasks/construct-rlve-polynomial-integer-roots-l1-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..1668283e1ed43b3d19ab7dac3e8b19e8bd4c67ae
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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'
+{"roots": [0, 1, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l1-s1/tests/check.py b/tasks/construct-rlve-polynomial-integer-roots-l1-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2d30553cb884c5e03c8afe73ef652efc483ab791
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l1-s1/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-l1-s1/tests/instance.json b/tasks/construct-rlve-polynomial-integer-roots-l1-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4ba04c9bbed002694529eef45b8f949704062afe
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l1-s1/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [0, 2, -3, 1], "family": "rlve_polynomial_integer_roots", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l1-s1/tests/run.py b/tasks/construct-rlve-polynomial-integer-roots-l1-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l1-s1/tests/test.sh b/tasks/construct-rlve-polynomial-integer-roots-l1-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l1-s6/environment/Dockerfile b/tasks/construct-rlve-polynomial-integer-roots-l1-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l1-s6/environment/instance.json b/tasks/construct-rlve-polynomial-integer-roots-l1-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b26c3695db032e7b0cd99320bb3baafe046e6144
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l1-s6/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [-27, -9, 3, 1], "family": "rlve_polynomial_integer_roots", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l1-s6/solution/solve.sh b/tasks/construct-rlve-polynomial-integer-roots-l1-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..876a2af9549c338d793b3904c6619f43826cde7f
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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'
+{"roots": [-3, -3, 3]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l1-s6/tests/check.py b/tasks/construct-rlve-polynomial-integer-roots-l1-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2d30553cb884c5e03c8afe73ef652efc483ab791
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l1-s6/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-l1-s6/tests/instance.json b/tasks/construct-rlve-polynomial-integer-roots-l1-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b26c3695db032e7b0cd99320bb3baafe046e6144
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l1-s6/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [-27, -9, 3, 1], "family": "rlve_polynomial_integer_roots", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l1-s6/tests/run.py b/tasks/construct-rlve-polynomial-integer-roots-l1-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l1-s6/tests/test.sh b/tasks/construct-rlve-polynomial-integer-roots-l1-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l1-s8/environment/Dockerfile b/tasks/construct-rlve-polynomial-integer-roots-l1-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l1-s8/environment/instance.json b/tasks/construct-rlve-polynomial-integer-roots-l1-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..da8fc2129b6b3efdf2407de1d1d0501b35d1aa24
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l1-s8/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [3, 7, 5, 1], "family": "rlve_polynomial_integer_roots", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l1-s8/solution/solve.sh b/tasks/construct-rlve-polynomial-integer-roots-l1-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..c171033ddb27dd05009d746534e91515c4c5af55
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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'
+{"roots": [-3, -1, -1]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l1-s8/tests/check.py b/tasks/construct-rlve-polynomial-integer-roots-l1-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2d30553cb884c5e03c8afe73ef652efc483ab791
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l1-s8/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-l1-s8/tests/instance.json b/tasks/construct-rlve-polynomial-integer-roots-l1-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..da8fc2129b6b3efdf2407de1d1d0501b35d1aa24
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l1-s8/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [3, 7, 5, 1], "family": "rlve_polynomial_integer_roots", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l1-s8/tests/run.py b/tasks/construct-rlve-polynomial-integer-roots-l1-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l1-s8/tests/test.sh b/tasks/construct-rlve-polynomial-integer-roots-l1-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l5-s0/environment/Dockerfile b/tasks/construct-rlve-polynomial-integer-roots-l5-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l5-s0/environment/instance.json b/tasks/construct-rlve-polynomial-integer-roots-l5-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c54c864e03e57abef87d8ebae9635488b6c85076
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l5-s0/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [-1565650944000, -1009613721600, 1285219128960, 521911678464, -320491747448, -94618547484, 26729556402, 7765018167, -604310943, -248952012, -3717344, 2615154, 114990, -10704, -618, 15, 1], "family": "rlve_polynomial_integer_roots", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l5-s0/solution/solve.sh b/tasks/construct-rlve-polynomial-integer-roots-l5-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..0a0f181b9d6b53a70ebb4468b2d345c514be70b1
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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'
+{"roots": [-15, -13, -11, -11, -9, -5, -4, -2, -1, 2, 2, 3, 5, 12, 16, 16]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l5-s0/tests/check.py b/tasks/construct-rlve-polynomial-integer-roots-l5-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2d30553cb884c5e03c8afe73ef652efc483ab791
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l5-s0/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-l5-s0/tests/instance.json b/tasks/construct-rlve-polynomial-integer-roots-l5-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c54c864e03e57abef87d8ebae9635488b6c85076
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l5-s0/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [-1565650944000, -1009613721600, 1285219128960, 521911678464, -320491747448, -94618547484, 26729556402, 7765018167, -604310943, -248952012, -3717344, 2615154, 114990, -10704, -618, 15, 1], "family": "rlve_polynomial_integer_roots", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l5-s0/tests/run.py b/tasks/construct-rlve-polynomial-integer-roots-l5-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l5-s0/tests/test.sh b/tasks/construct-rlve-polynomial-integer-roots-l5-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l5-s7/environment/Dockerfile b/tasks/construct-rlve-polynomial-integer-roots-l5-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l5-s7/environment/instance.json b/tasks/construct-rlve-polynomial-integer-roots-l5-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..234ffc743b5ed1bd54f67b5a6ab7ffbbdabdedf4
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l5-s7/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [-3662382366720, 4873861988352, -1126426411008, -1012991840256, 497829860864, 18500979712, -46987583456, 6840202672, 972794944, -350882696, 26985762, 1373423, -314179, 12122, 512, -49, 1], "family": "rlve_polynomial_integer_roots", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l5-s7/solution/solve.sh b/tasks/construct-rlve-polynomial-integer-roots-l5-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..8b1368e8c3bd86fa3632eefe58bda8681734ef97
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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'
+{"roots": [-16, -12, -4, -3, -3, 2, 2, 2, 6, 7, 7, 8, 10, 11, 16, 16]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l5-s7/tests/check.py b/tasks/construct-rlve-polynomial-integer-roots-l5-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2d30553cb884c5e03c8afe73ef652efc483ab791
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l5-s7/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-l5-s7/tests/instance.json b/tasks/construct-rlve-polynomial-integer-roots-l5-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..234ffc743b5ed1bd54f67b5a6ab7ffbbdabdedf4
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l5-s7/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [-3662382366720, 4873861988352, -1126426411008, -1012991840256, 497829860864, 18500979712, -46987583456, 6840202672, 972794944, -350882696, 26985762, 1373423, -314179, 12122, 512, -49, 1], "family": "rlve_polynomial_integer_roots", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l5-s7/tests/run.py b/tasks/construct-rlve-polynomial-integer-roots-l5-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l5-s7/tests/test.sh b/tasks/construct-rlve-polynomial-integer-roots-l5-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l5-s9/environment/Dockerfile b/tasks/construct-rlve-polynomial-integer-roots-l5-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l5-s9/environment/instance.json b/tasks/construct-rlve-polynomial-integer-roots-l5-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..de3fce71551215eb899e63021a6d780100c213f5
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l5-s9/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [-768592281600000, 752654592000000, -152350626816000, -61959249388800, 30607940473920, -3395661872112, -415182717876, 122036613848, -5357232657, -964839954, 109445834, 43686, -527184, 21946, 522, -54, 1], "family": "rlve_polynomial_integer_roots", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l5-s9/solution/solve.sh b/tasks/construct-rlve-polynomial-integer-roots-l5-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..06615568556f615ab9ba0bd463b2a3d23250bf73
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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'
+{"roots": [-16, -13, -12, -8, -3, 2, 4, 5, 8, 9, 10, 11, 12, 15, 15, 15]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l5-s9/tests/check.py b/tasks/construct-rlve-polynomial-integer-roots-l5-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2d30553cb884c5e03c8afe73ef652efc483ab791
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l5-s9/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-l5-s9/tests/instance.json b/tasks/construct-rlve-polynomial-integer-roots-l5-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..de3fce71551215eb899e63021a6d780100c213f5
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l5-s9/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [-768592281600000, 752654592000000, -152350626816000, -61959249388800, 30607940473920, -3395661872112, -415182717876, 122036613848, -5357232657, -964839954, 109445834, 43686, -527184, 21946, 522, -54, 1], "family": "rlve_polynomial_integer_roots", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l5-s9/tests/run.py b/tasks/construct-rlve-polynomial-integer-roots-l5-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l5-s9/tests/test.sh b/tasks/construct-rlve-polynomial-integer-roots-l5-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-interpolation-l2-s0/environment/Dockerfile b/tasks/construct-rlve-polynomial-interpolation-l2-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l2-s0/environment/instance.json b/tasks/construct-rlve-polynomial-interpolation-l2-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..27b5e41e9690a187bb182c0bc4159b598432b8fa
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l2-s0/environment/instance.json
@@ -0,0 +1 @@
+{"X": [2, -1, 1, -4, 3], "Y": [42, 6, 2, 492, 198], "family": "rlve_polynomial_interpolation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-interpolation-l2-s0/solution/solve.sh b/tasks/construct-rlve-polynomial-interpolation-l2-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..36b49a115a674cded12b626e6dab30ead7d4116b
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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'
+{"coeffs": [0, -3, 2, 1, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-interpolation-l2-s0/tests/instance.json b/tasks/construct-rlve-polynomial-interpolation-l2-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..27b5e41e9690a187bb182c0bc4159b598432b8fa
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l2-s0/tests/instance.json
@@ -0,0 +1 @@
+{"X": [2, -1, 1, -4, 3], "Y": [42, 6, 2, 492, 198], "family": "rlve_polynomial_interpolation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-interpolation-l2-s0/tests/test.sh b/tasks/construct-rlve-polynomial-interpolation-l2-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l2-s7/environment/Dockerfile b/tasks/construct-rlve-polynomial-interpolation-l2-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l2-s7/environment/instance.json b/tasks/construct-rlve-polynomial-interpolation-l2-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..25a1e2da20a34327672f23a5edeb777907b8c936
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l2-s7/environment/instance.json
@@ -0,0 +1 @@
+{"X": [4, 2, -2, -4, 3], "Y": [1661, 137, 65, 1037, 575], "family": "rlve_polynomial_interpolation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-interpolation-l2-s7/solution/solve.sh b/tasks/construct-rlve-polynomial-interpolation-l2-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..bd6c5267f40b3733a5b9ab3040231c44f8a7617d
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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'
+{"coeffs": [5, -2, 4, 5, 5]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-interpolation-l2-s7/tests/check.py b/tasks/construct-rlve-polynomial-interpolation-l2-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..f05264e3fd4584db7cb3768800e4a2497564b4f5
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l2-s7/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-l2-s7/tests/instance.json b/tasks/construct-rlve-polynomial-interpolation-l2-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..25a1e2da20a34327672f23a5edeb777907b8c936
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l2-s7/tests/instance.json
@@ -0,0 +1 @@
+{"X": [4, 2, -2, -4, 3], "Y": [1661, 137, 65, 1037, 575], "family": "rlve_polynomial_interpolation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-interpolation-l2-s7/tests/run.py b/tasks/construct-rlve-polynomial-interpolation-l2-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l2-s7/tests/test.sh b/tasks/construct-rlve-polynomial-interpolation-l2-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l2-s9/environment/Dockerfile b/tasks/construct-rlve-polynomial-interpolation-l2-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-rlve-polynomial-interpolation-l2-s9/environment/instance.json b/tasks/construct-rlve-polynomial-interpolation-l2-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3d4a078ffe3a5ec7d0bfdc0d7547452a11dc0810
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l2-s9/environment/instance.json
@@ -0,0 +1 @@
+{"X": [4, 0, -3, -4, -1], "Y": [1327, 3, 375, 1207, 7], "family": "rlve_polynomial_interpolation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-interpolation-l2-s9/solution/solve.sh b/tasks/construct-rlve-polynomial-interpolation-l2-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..36c723a81149d030446177d9a16636ebf97fafc7
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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'
+{"coeffs": [3, -1, -1, 1, 5]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-interpolation-l2-s9/tests/check.py b/tasks/construct-rlve-polynomial-interpolation-l2-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..f05264e3fd4584db7cb3768800e4a2497564b4f5
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l2-s9/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-l2-s9/tests/instance.json b/tasks/construct-rlve-polynomial-interpolation-l2-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3d4a078ffe3a5ec7d0bfdc0d7547452a11dc0810
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l2-s9/tests/instance.json
@@ -0,0 +1 @@
+{"X": [4, 0, -3, -4, -1], "Y": [1327, 3, 375, 1207, 7], "family": "rlve_polynomial_interpolation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-interpolation-l2-s9/tests/run.py b/tasks/construct-rlve-polynomial-interpolation-l2-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-rlve-polynomial-interpolation-l2-s9/tests/test.sh b/tasks/construct-rlve-polynomial-interpolation-l2-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-rlve-polynomial-interpolation-l6-s1/environment/Dockerfile b/tasks/construct-rlve-polynomial-interpolation-l6-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l6-s1/environment/instance.json b/tasks/construct-rlve-polynomial-interpolation-l6-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..33daa6e578b57976fa27f3ea6b47991cf3904e07
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l6-s1/environment/instance.json
@@ -0,0 +1 @@
+{"X": [0, -13, 3, -5, -18, 12, -2, 13, 17, 18, 9, -7, -1, 7, 16, 11, -10, 14, -17], "Y": [3, 485513006071948042958, 1215189630, 18483476208358, 166339859780774143440813, 98071143210280824183, 1570525, 416823369214381662640, 53055714045136763278028, 148920159160575235375545, 538504296172885812, 7489305830982020, 2, 5676334414488538, 17751833939710160270323, 20332287485320779158, 4415368630452681413, 1590730566594875451917, 59644952208302055193810], "family": "rlve_polynomial_interpolation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-interpolation-l6-s1/solution/solve.sh b/tasks/construct-rlve-polynomial-interpolation-l6-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..1f625cb7c16998568d7b07ae22699efabee821b3
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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'
+{"coeffs": [3, -1, 4, -1, -2, 3, 3, 5, -5, -1, 3, 4, -1, 3, 4, 5, 2, -4, 4]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-interpolation-l6-s1/tests/check.py b/tasks/construct-rlve-polynomial-interpolation-l6-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..f05264e3fd4584db7cb3768800e4a2497564b4f5
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l6-s1/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-l6-s1/tests/instance.json b/tasks/construct-rlve-polynomial-interpolation-l6-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..33daa6e578b57976fa27f3ea6b47991cf3904e07
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l6-s1/tests/instance.json
@@ -0,0 +1 @@
+{"X": [0, -13, 3, -5, -18, 12, -2, 13, 17, 18, 9, -7, -1, 7, 16, 11, -10, 14, -17], "Y": [3, 485513006071948042958, 1215189630, 18483476208358, 166339859780774143440813, 98071143210280824183, 1570525, 416823369214381662640, 53055714045136763278028, 148920159160575235375545, 538504296172885812, 7489305830982020, 2, 5676334414488538, 17751833939710160270323, 20332287485320779158, 4415368630452681413, 1590730566594875451917, 59644952208302055193810], "family": "rlve_polynomial_interpolation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-interpolation-l6-s1/tests/run.py b/tasks/construct-rlve-polynomial-interpolation-l6-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l6-s1/tests/test.sh b/tasks/construct-rlve-polynomial-interpolation-l6-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l6-s6/environment/Dockerfile b/tasks/construct-rlve-polynomial-interpolation-l6-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l6-s6/environment/instance.json b/tasks/construct-rlve-polynomial-interpolation-l6-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..7516908058fd3d1e229ade99d6af96d3b8bb4ef9
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l6-s6/environment/instance.json
@@ -0,0 +1 @@
+{"X": [12, -12, -18, -8, 5, 6, 4, -13, -15, 8, -5, 17, 1, 3, -6, -3, 16, -14, 14], "Y": [128850990831703782512, 137757317747849775848, 201232907979463747385762, 94904064881322212, 17676827354546, 476398010158922, 312733797656, 580303949907394010762, 7593586581418457031746, 85821016504391828, 20801729674346, 68707030106966828544962, 2, 1711555658, 545174899192646, 2272401362, 23038953169452567216812, 2197736901841749467342, 2075452449684711137666], "family": "rlve_polynomial_interpolation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-interpolation-l6-s6/solution/solve.sh b/tasks/construct-rlve-polynomial-interpolation-l6-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..25c32fc20956d2cef7e9fcd36d810fb3411c8ee8
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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'
+{"coeffs": [-4, -5, 3, -4, -2, 4, 0, 0, 3, 0, 2, 1, 4, -5, 2, -1, 1, -2, 5]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-interpolation-l6-s6/tests/check.py b/tasks/construct-rlve-polynomial-interpolation-l6-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..f05264e3fd4584db7cb3768800e4a2497564b4f5
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l6-s6/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-l6-s6/tests/instance.json b/tasks/construct-rlve-polynomial-interpolation-l6-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..7516908058fd3d1e229ade99d6af96d3b8bb4ef9
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l6-s6/tests/instance.json
@@ -0,0 +1 @@
+{"X": [12, -12, -18, -8, 5, 6, 4, -13, -15, 8, -5, 17, 1, 3, -6, -3, 16, -14, 14], "Y": [128850990831703782512, 137757317747849775848, 201232907979463747385762, 94904064881322212, 17676827354546, 476398010158922, 312733797656, 580303949907394010762, 7593586581418457031746, 85821016504391828, 20801729674346, 68707030106966828544962, 2, 1711555658, 545174899192646, 2272401362, 23038953169452567216812, 2197736901841749467342, 2075452449684711137666], "family": "rlve_polynomial_interpolation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-interpolation-l6-s6/tests/run.py b/tasks/construct-rlve-polynomial-interpolation-l6-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l6-s6/tests/test.sh b/tasks/construct-rlve-polynomial-interpolation-l6-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l6-s8/environment/Dockerfile b/tasks/construct-rlve-polynomial-interpolation-l6-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l6-s8/environment/instance.json b/tasks/construct-rlve-polynomial-interpolation-l6-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..bab68ae7345c70eb29895ae8e954ef74eae7e408
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l6-s8/environment/instance.json
@@ -0,0 +1 @@
+{"X": [1, 10, -16, 16, -4, 12, 3, 17, -5, -15, -8, -11, 14, -6, -9, 0, -18, 4, -17], "Y": [13, 3393016632331515346, 12964592541221740762188, 15332711185621724541868, 129757757328, 88605965538995490368, 1679952599, 45458158411536500623773, 8145241173871, 4031684803366693572221, 44647699867347748, 14599405106326960993, 1400892102561487729518, 232710532272098, 381098689811724443, -4, 109153853684984495793038, 273875199208, 38823022433522958765619], "family": "rlve_polynomial_interpolation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-interpolation-l6-s8/solution/solve.sh b/tasks/construct-rlve-polynomial-interpolation-l6-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..d4d6e2731f0fe07e9ff07cc1adfcf3edee30a7a5
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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'
+{"coeffs": [-4, -5, 4, -5, 2, 5, 1, 3, 3, 2, 3, -4, -3, 2, 0, 3, -1, 4, 3]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-interpolation-l6-s8/tests/check.py b/tasks/construct-rlve-polynomial-interpolation-l6-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..f05264e3fd4584db7cb3768800e4a2497564b4f5
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l6-s8/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-l6-s8/tests/instance.json b/tasks/construct-rlve-polynomial-interpolation-l6-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..bab68ae7345c70eb29895ae8e954ef74eae7e408
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l6-s8/tests/instance.json
@@ -0,0 +1 @@
+{"X": [1, 10, -16, 16, -4, 12, 3, 17, -5, -15, -8, -11, 14, -6, -9, 0, -18, 4, -17], "Y": [13, 3393016632331515346, 12964592541221740762188, 15332711185621724541868, 129757757328, 88605965538995490368, 1679952599, 45458158411536500623773, 8145241173871, 4031684803366693572221, 44647699867347748, 14599405106326960993, 1400892102561487729518, 232710532272098, 381098689811724443, -4, 109153853684984495793038, 273875199208, 38823022433522958765619], "family": "rlve_polynomial_interpolation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-interpolation-l6-s8/tests/run.py b/tasks/construct-rlve-polynomial-interpolation-l6-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l6-s8/tests/test.sh b/tasks/construct-rlve-polynomial-interpolation-l6-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-product-permutation-l2-s2/environment/Dockerfile b/tasks/construct-rlve-prefix-product-permutation-l2-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-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-rlve-prefix-product-permutation-l2-s2/environment/instance.json b/tasks/construct-rlve-prefix-product-permutation-l2-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..84d45b38e4d105354ec9e2d2d0562c57a6dfda13
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l2-s2/environment/instance.json
@@ -0,0 +1 @@
+{"n": 97, "family": "rlve_prefix_product_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-product-permutation-l2-s2/solution/solve.sh b/tasks/construct-rlve-prefix-product-permutation-l2-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..95ef3f3364fca9537feb71e152fcf4e870ff1281
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-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'
+{"perm": [1, 2, 50, 66, 74, 40, 82, 15, 86, 55, 69, 54, 90, 16, 8, 14, 92, 41, 28, 47, 35, 38, 76, 39, 94, 67, 57, 19, 53, 88, 56, 73, 95, 51, 21, 62, 63, 22, 24, 6, 18, 72, 68, 89, 87, 70, 20, 65, 96, 3, 34, 79, 29, 12, 10, 31, 27, 81, 93, 75, 77, 36, 37, 78, 48, 4, 26, 43, 11, 46, 80, 42, 32, 5, 60, 23, 61, 64, 52, 71, 58, 7, 85, 91, 83, 9, 45, 30, 44, 13, 84, 17, 59, 25, 33, 49, 97]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-prefix-product-permutation-l2-s2/tests/check.py b/tasks/construct-rlve-prefix-product-permutation-l2-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3d93850acd79cc0f8fff05cdbb0c63aabc484815
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l2-s2/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-l2-s2/tests/instance.json b/tasks/construct-rlve-prefix-product-permutation-l2-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..84d45b38e4d105354ec9e2d2d0562c57a6dfda13
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l2-s2/tests/instance.json
@@ -0,0 +1 @@
+{"n": 97, "family": "rlve_prefix_product_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-product-permutation-l2-s2/tests/run.py b/tasks/construct-rlve-prefix-product-permutation-l2-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-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-rlve-prefix-product-permutation-l2-s2/tests/test.sh b/tasks/construct-rlve-prefix-product-permutation-l2-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-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-rlve-prefix-product-permutation-l2-s5/environment/Dockerfile b/tasks/construct-rlve-prefix-product-permutation-l2-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l2-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-prefix-product-permutation-l2-s5/environment/instance.json b/tasks/construct-rlve-prefix-product-permutation-l2-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b28b746e2651e0f2e0671c88b5d2a8e36db7f5f4
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l2-s5/environment/instance.json
@@ -0,0 +1 @@
+{"n": 73, "family": "rlve_prefix_product_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-product-permutation-l2-s5/solution/solve.sh b/tasks/construct-rlve-prefix-product-permutation-l2-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..687cffb3c20446a6df0aad7bdf6f78787c3e55e2
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l2-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'
+{"perm": [1, 2, 38, 50, 56, 45, 62, 22, 65, 66, 23, 21, 68, 46, 48, 40, 33, 44, 70, 51, 12, 8, 11, 55, 71, 39, 60, 47, 61, 69, 57, 34, 17, 32, 59, 49, 72, 3, 26, 16, 43, 58, 41, 18, 6, 14, 28, 15, 36, 4, 20, 64, 67, 63, 24, 5, 31, 42, 35, 27, 29, 7, 54, 52, 9, 10, 53, 13, 30, 19, 25, 37, 73]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-prefix-product-permutation-l2-s5/tests/check.py b/tasks/construct-rlve-prefix-product-permutation-l2-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3d93850acd79cc0f8fff05cdbb0c63aabc484815
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l2-s5/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-l2-s5/tests/instance.json b/tasks/construct-rlve-prefix-product-permutation-l2-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b28b746e2651e0f2e0671c88b5d2a8e36db7f5f4
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l2-s5/tests/instance.json
@@ -0,0 +1 @@
+{"n": 73, "family": "rlve_prefix_product_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-product-permutation-l2-s5/tests/run.py b/tasks/construct-rlve-prefix-product-permutation-l2-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l2-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-prefix-product-permutation-l2-s5/tests/test.sh b/tasks/construct-rlve-prefix-product-permutation-l2-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l2-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-prefix-product-permutation-l6-s3/environment/Dockerfile b/tasks/construct-rlve-prefix-product-permutation-l6-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-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-rlve-prefix-product-permutation-l6-s3/environment/instance.json b/tasks/construct-rlve-prefix-product-permutation-l6-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d014c8cc1f4027ecaa4996df061f7fe708346758
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l6-s3/environment/instance.json
@@ -0,0 +1 @@
+{"n": 12011, "family": "rlve_prefix_product_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-product-permutation-l6-s3/solution/solve.sh b/tasks/construct-rlve-prefix-product-permutation-l6-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..961cad6e9b1b932e64dbdfdee117f3690d886c97
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-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'
+{"perm": [1, 2, 6007, 4005, 3004, 9610, 2003, 1717, 7508, 9343, 10811, 1093, 1002, 925, 859, 3204, 9760, 10599, 4672, 3794, 5406, 573, 547, 4701, 6507, 4325, 463, 3115, 430, 9527, 7608, 4263, 10886, 365, 5300, 9953, 8342, 2598, 7903, 309, 8709, 6153, 287, 839, 274, 9076, 2351, 2301, 3254, 1962, 2163, 11541, 232, 1814, 1558, 7426, 6221, 5269, 4764, 5294, 9810, 10043, 2132, 8199, 11449, 2588, 183, 4662, 8656, 9575, 4977, 10997, 10177, 9544, 7305, 5446, 3952, 157, 155, 3954, 4355, 1039, 3077, 6513, 144, 6925, 420, 7180, 6143, 9043, 10544, 133, 1176, 9429, 1151, 5564, 7633, 4954, 6987, 4126, 1082, 4520, 5771, 9913, 6122, 7322, 6913, 11563, 6785, 9147, 9719, 8874, 3111, 9461, 2635, 941, 8388, 8111, 8653, 1515, 10911, 3376, 5022, 10059, 7072, 3268, 4100, 3784, 5725, 4284, 7300, 1468, 92, 10838, 8337, 3026, 10334, 5875, 4788, 3371, 2489, 8775, 5499, 85, 5089, 11515, 10778, 8662, 3653, 1452, 8729, 9228, 7982, 11855, 79, 8060, 78, 11859, 7983, 4609, 2178, 5820, 520, 1180, 1539, 2476, 3257, 6474, 6078, 996, 3463, 1757, 6216, 487, 9596, 9198, 3072, 5769, 4522, 11341, 11278, 9424, 67, 7352, 6594, 7727, 4715, 6424, 576, 10741, 8788, 10377, 3817, 1868, 8483, 4867, 3494, 2013, 8069, 846, 6547, 9562, 8266, 11657, 2886, 8438, 4957, 7196, 9067, 10173, 9667, 3758, 3457, 7670, 5782, 4973, 3393, 10905, 4574, 3182, 4860, 3588, 10443, 1940, 1556, 1816, 4731, 10213, 1318, 1050, 471, 53, 10200, 8352, 4056, 461, 4327, 9326, 758, 2363, 5456, 3390, 7694, 347, 8517, 10002, 5030, 9532, 9542, 10179, 7640, 4547, 8056, 3704, 7898, 2309, 2863, 10423, 8148, 372, 9656, 2394, 6740, 5938, 6052, 5168, 11425, 3015, 4169, 3707, 7519, 8422, 11173, 45, 2938, 1486, 8400, 1562, 1686, 11151, 1245, 5173, 4388, 6919, 2750, 5859, 43, 11175, 2545, 2744, 5758, 1652, 11395, 6027, 10337, 5864, 1827, 1376, 6732, 6830, 4365, 8700, 10620, 5511, 9997, 9216, 5928, 4109, 40, 3305, 10036, 1121, 6045, 4107, 5930, 6445, 3992, 5836, 2305, 867, 7095, 3855, 8916, 9409, 6266, 7725, 6596, 11057, 770, 7193, 7244, 3920, 1629, 10966, 9243, 10542, 9045, 1391, 6504, 7158, 1732, 5848, 879, 1997, 9114, 4318, 244, 3497, 10804, 4888, 10605, 6708, 7542, 6670, 2885, 11809, 8267, 8513, 5671, 9302, 11645, 9683, 10718, 1126, 34, 6714, 9682, 11652, 9303, 11361, 3864, 260, 2358, 5926, 9218, 1090, 6294, 9973, 5371, 7290, 10400, 1262, 5189, 7747, 1909, 9641, 6940, 9436, 4242, 4756, 2434, 984, 7753, 490, 1007, 5596, 4035, 8563, 6429, 7617, 3274, 5692, 10787, 10492, 10139, 5013, 5829, 1329, 7449, 11131, 10225, 1959, 2479, 2473, 9604, 8510, 4534, 5128, 5087, 87, 4834, 5764, 7885, 10933, 1729, 3788, 9841, 8299, 8897, 29, 2487, 3373, 1697, 639, 5453, 3839, 8293, 2777, 7597, 685, 8436, 2888, 7800, 1682, 5222, 9016, 6976, 8492, 6784, 11905, 6914, 3836, 2366, 11084, 5107, 7234, 6665, 8753, 6531, 7956, 236, 7687, 27, 8899, 11106, 6691, 10182, 2110, 8034, 10962, 231, 11961, 2164, 2261, 10669, 5918, 6385, 1537, 1182, 7674, 8734, 9440, 7701, 9948, 9853, 5796, 174, 2418, 4259, 394, 11007, 7144, 8521, 4094, 10772, 826, 10777, 11867, 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11707, 9998, 3946, 1460, 7332, 3067, 6358, 3699, 1119, 10038, 9097, 814, 11349, 7967, 10127, 2992, 3354, 3071, 11837, 9597, 2445, 7130, 2699, 2681, 4043, 10079, 11497, 5213, 2336, 7586, 5998, 6408, 4881, 9316, 3095, 2606, 6367, 5750, 9592, 717, 7222, 11233, 8385, 6034, 1477, 2897, 7697, 9794, 1779, 5464, 2453, 8561, 4037, 8770, 2989, 5680, 11275, 4653, 1192, 1322, 1065, 8077, 5519, 11260, 9614, 9703, 257, 1591, 9718, 11903, 6786, 11459, 10207, 9969, 8325, 6770, 2665, 11364, 11289, 4843, 2678, 8860, 6524, 8712, 7462, 2344, 8257, 5342, 355, 205, 3576, 442, 4214, 7102, 8683, 2070, 5288, 9276, 8671, 10535, 2842, 4317, 11668, 1998, 4180, 6700, 6081, 5570, 1643, 8080, 10305, 5050, 6878, 7656, 10976, 1441, 7512, 10841, 813, 9206, 10039, 4028, 7406, 5881, 8347, 3821, 10413, 8548, 10258, 7532, 7591, 1983, 8718, 571, 5408, 3349, 8362, 3445, 6072, 2350, 11967, 275, 10528, 2780, 5839, 9921, 10628, 2564, 10172, 11804, 7197, 4256, 11299, 4135, 8995, 7254, 1421, 8632, 4559, 6541, 10342, 6630, 2685, 11257, 4988, 10142, 7296, 5591, 3798, 11137, 1390, 11676, 10543, 11923, 6144, 8891, 7813, 5956, 8252, 542, 1797, 8397, 7523, 1771, 11484, 7318, 9986, 8211, 2243, 4061, 5379, 3242, 2851, 6334, 1885, 2812, 8660, 10780, 8237, 6975, 11567, 5223, 8851, 11211, 7644, 5148, 8007, 5, 2404, 1336, 3697, 6360, 2289, 5286, 2072, 1457, 10389, 587, 268, 7845, 7253, 9062, 4136, 10349, 2264, 2140, 625, 4153, 3675, 136, 1680, 7802, 6066, 5263, 552, 1808, 8419, 1490, 1058, 5122, 7316, 11486, 11075, 10024, 5805, 11384, 11400, 11266, 9901, 3981, 1945, 11062, 1660, 6883, 6928, 7469, 8049, 2212, 9727, 4656, 6655, 7584, 2338, 7261, 4197, 9383, 1268, 1546, 3972, 4680, 2802, 5656, 9573, 8658, 2814, 177, 6245, 8678, 11422, 10973, 83, 5501, 10775, 828, 1206, 4414, 11330, 3406, 6875, 4747, 7828, 5561, 11378, 5258, 7538, 2655, 10027, 2696, 2831, 9408, 11690, 3856, 5045, 2528, 7010, 1957, 10227, 10754, 8038, 8488, 9889, 8498, 11042, 3138, 113, 2553, 11105, 11549, 28, 11584, 8300, 10987, 6723, 2739, 7812, 9041, 6145, 1724, 11170, 1573, 4362, 11432, 4628, 1214, 5216, 6346, 5205, 8577, 8260, 7085, 970, 3110, 11901, 9720, 1795, 544, 6427, 8565, 7569, 1162, 9832, 4081, 5438, 6489, 7852, 6523, 9135, 2679, 2701, 7535, 9504, 10844, 1587, 7419, 11210, 9014, 5224, 5435, 10634, 10661, 6310, 8246, 3508, 1464, 6590, 5016, 1580, 1606, 2298, 10722, 4074, 742, 5878, 4618, 7438, 220, 7154, 5665, 6810, 4466, 1656, 604, 698, 3564, 4892, 10066, 953, 2258, 7082, 1313, 4274, 8503, 10551, 4683, 7917, 1243, 11153, 16, 2254, 3876, 11229, 6499, 11222, 1981, 7593, 2582, 10864, 6285, 797, 9225, 4352, 8312, 10896, 2792, 6898, 6206, 631, 10376, 11822, 10742, 3646, 1099, 10319, 2735, 3344, 2369, 4271, 9482, 8584, 9465, 5498, 11871, 2490, 8593, 6633, 2988, 9163, 4038, 6299, 534, 10506, 4144, 6735, 6415, 7980, 9230, 9711, 49, 10052, 9536, 167, 5540, 6530, 11555, 6666, 6303, 6603, 8666, 932, 917, 4940, 126, 7914, 3734, 5911, 9471, 4395, 401, 6322, 1501, 8012, 9439, 11532, 7675, 1279, 4350, 9227, 11862, 1453, 3416, 7973, 5716, 9254, 10449, 10329, 6793, 4631, 570, 9084, 1984, 9319, 8573, 1913, 7461, 9133, 6525, 6152, 11972, 310, 1978, 506, 4156, 9779, 10404, 1594, 10619, 11711, 4366, 3339, 6646, 6481, 8123, 1525, 10170, 2566, 7936, 5334, 11065, 1132, 4278, 4497, 6113, 2069, 9122, 7103, 6935, 913, 11421, 8939, 6246, 5146, 7646, 3315, 5368, 10534, 9118, 9277, 3231, 9645, 931, 8749, 6604, 2932, 3652, 11865, 10779, 9020, 2813, 8943, 9574, 11944, 4663, 1514, 11894, 8112, 6237, 6228, 5994, 9402, 9260, 10367, 6497, 11231, 7224, 140, 9525, 432, 10317, 1101, 122, 6992, 5834, 3994, 4558, 9059, 1422, 8457, 7116, 7373, 6686, 10398, 7292, 6556, 242, 4320, 7039, 217, 1109, 5688, 11124, 6583, 9590, 5752, 4669, 10984, 1230, 8605, 5140, 682, 3084, 5139, 8610, 1231, 3778, 4189, 3480, 2525, 1743, 10559, 3286, 4041, 2683, 6632, 8773, 2491, 9930, 11482, 1773, 1113, 7923, 11111, 9464, 8778, 9483, 5005, 4804, 6010, 5340, 8259, 8879, 5206, 6169, 1912, 8715, 9320, 6099, 11178, 3650, 2934, 5942, 7568, 8869, 6428, 11615, 4036, 9165, 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1798, 10697, 2534, 10482, 4676, 9637, 4176, 8110, 11896, 942, 6033, 9174, 11234, 8133, 6119, 3600, 3642, 1740, 5048, 10307, 2462, 1890, 10939, 1044, 8412, 3633, 10274, 1544, 1270, 3227, 10915, 3739, 834, 3444, 9080, 3350, 149, 10562, 11325, 1000, 1095, 1768, 1055, 4055, 11779, 10201, 9352, 10825, 3820, 9092, 5882, 7053, 4772, 2597, 11976, 9954, 11505, 7859, 3025, 11878, 10839, 7514, 5902, 2381, 10540, 9245, 1283, 3859, 4347, 4235, 6769, 9142, 9970, 2762, 1137, 1576, 5756, 2746, 4914, 10676, 3007, 5654, 2804, 10895, 8795, 4353, 3956, 254, 4116, 7843, 270, 4495, 4280, 6319, 1228, 10986, 8896, 11585, 9842, 11055, 6598, 8283, 2776, 11576, 3840, 9420, 11415, 9373, 817, 4383, 1068, 5254, 2775, 8295, 6599, 7328, 4098, 3270, 6103, 2281, 1766, 1097, 3648, 11180, 4526, 10503, 2495, 4532, 8512, 11656, 11810, 9563, 5247, 4230, 1311, 7084, 8878, 8578, 5341, 9130, 2345, 212, 8557, 541, 9038, 5957, 9908, 9500, 3910, 3507, 8845, 6311, 1496, 6706, 10607, 10687, 9990, 5642, 6974, 9018, 10781, 3410, 7825, 9927, 538, 10160, 7912, 128, 6289, 4853, 10283, 426, 2173, 4645, 1931, 10713, 7340, 20, 6608, 5853, 6421, 2965, 877, 5850, 2242, 9029, 9987, 4969, 3490, 7921, 1115, 9223, 799, 6932, 10937, 1892, 11448, 11949, 2133, 1635, 193, 10146, 1187, 3665, 2922, 1601, 1720, 5386, 1650, 5760, 2277, 10267, 2075, 7909, 5580, 5450, 8467, 7719, 5098, 452, 9648, 6559, 436, 3721, 2594, 11249, 6638, 923, 1004, 4871, 5232, 11322, 5685, 2719, 965, 4433, 2324, 4917, 322, 3098, 6969, 10729, 3684, 7667, 1896, 9346, 651, 371, 11754, 10424, 4596, 5362, 8028, 5620, 6950, 4739, 7238, 767, 9758, 3206, 785, 4837, 6118, 8383, 11235, 8128, 3884, 3883, 8131, 11236, 4016, 4222, 1524, 8695, 6482, 4514, 4411, 7128, 2447, 4238, 7810, 2741, 6518, 6236, 8652, 11895, 8389, 4177, 3536, 6534, 5945, 6752, 2512, 3765, 8507, 6957, 7557, 2518, 894, 1762, 6212, 7069, 4768, 332, 10385, 7411, 6031, 944, 4926, 11045, 10981, 10602, 5398, 7347, 1251, 10304, 9107, 1644, 5518, 9155, 1066, 4385, 5976, 10588, 4906, 10287, 845, 11814, 2014, 2799, 10554, 1626, 4604, 7398, 6566, 77, 11857, 80, 7659, 3703, 11760, 4548, 6401, 11026, 7611, 6851, 2211, 8960, 7470, 10873, 7654, 6880, 1348, 5660, 10466, 3065, 7334, 8487, 8908, 10755, 4487, 10961, 11544, 2111, 3046, 10069, 4251, 5619, 8144, 5363, 6573, 5526, 5112, 11193, 5567, 317, 6178, 3379, 7456, 10626, 9923, 7808, 4240, 9438, 8736, 1502, 7207, 6005, 4, 9010, 5149, 8409, 2773, 5256, 11380, 1639, 962, 1288, 776, 3887, 7792, 2103, 2515, 7051, 5884, 694, 10265, 2279, 6105, 10323, 1973, 2918, 4608, 11854, 11860, 9229, 8762, 6416, 397, 3451, 2849, 3244, 5715, 8726, 3417, 9331, 2821, 1935, 10126, 9203, 11350, 9887, 8490, 6978, 10111, 4484, 10521, 10957, 3660, 235, 11553, 6532, 3538, 9769, 2986, 6635, 3570, 1553, 4248, 7769, 7259, 2340, 5602, 10480, 2536, 774, 1290, 3177, 11272, 5333, 8691, 2567, 5818, 2180, 3148, 6576, 1381, 2272, 10168, 1527, 4572, 10907, 11110, 8587, 1114, 8207, 3491, 494, 1242, 8812, 4684, 3733, 8744, 127, 8230, 10161, 5579, 8183, 2076, 5967, 314, 6084, 308, 11974, 2599, 2724, 11162, 2308, 11758, 3705, 4171, 5194, 5989, 730, 2651, 10613, 11403, 5000, 5025, 100, 10932, 11590, 5765, 6869, 9857, 3554, 4899, 713, 2950, 3019, 1665, 9820, 7452, 5474, 9369, 7435, 1506, 3248, 5279, 5185, 873, 7459, 1915, 855, 2053, 11387, 3024, 8339, 11506, 3308, 2235, 706, 2673, 6522, 8862, 6490, 5741, 3544, 4296, 5535, 7252, 8997, 269, 8307, 4117, 6820, 9919, 5841, 2375, 3623, 3904, 8478, 10014, 2901, 5314, 5126, 4536, 5560, 8926, 4748, 9926, 8234, 3411, 8534, 4814, 7759, 5677, 10130, 7805, 4751, 3061, 2631, 5955, 9040, 8892, 2740, 8116, 4239, 8015, 9924, 4750, 7818, 10131, 6065, 8985, 1681, 11570, 2889, 4912, 2748, 6921, 10642, 6240, 2102, 7996, 3888, 10490, 10789, 5634, 2443, 9599, 9807, 6765, 3750, 10703, 2766, 10732, 7665, 3686, 5245, 9565, 3896, 4204, 3999, 2576, 389, 7258, 7947, 4249, 10071, 10459, 4065, 4245, 1943, 3983, 6395, 5676, 7821, 4815, 2948, 715, 9594, 489, 11620, 985, 4404, 32, 1128, 1908, 11629, 5190, 2388, 6938, 9643, 3233, 2532, 10699, 3197, 3511, 1993, 10880, 3326, 7517, 3709, 5695, 795, 6287, 130, 4714, 11827, 6595, 11687, 6267, 1334, 2406, 5059, 5097, 8179, 8468, 4165, 6479, 6648, 5442, 8528, 5054, 4458, 6805, 5846, 1734, 9554, 2676, 4845, 1777, 9796, 9947, 11530, 9441, 10653, 9793, 9170, 2898, 346, 11770, 3391, 4975, 9577, 11031, 5505, 26, 11551, 237, 2688, 4877, 6388, 2467, 4710, 10817, 10152, 5552, 5789, 1278, 8733, 11533, 1183, 10155, 5781, 11799, 3458, 1895, 8153, 3685, 7779, 10733, 3282, 2787, 3217, 3702, 8058, 81, 10975, 9103, 6879, 8046, 10874, 10439, 3127, 582, 5182, 301, 3314, 8675, 5147, 9012, 11212, 5083, 4546, 11762, 10180, 6693, 11419, 915, 934, 4953, 11916, 5565, 11195, 3726, 10946, 3938, 6038, 6839, 283, 5095, 5061, 7276, 6544, 9816, 2541, 3273, 11613, 6430, 7401, 5575, 11455, 6850, 8052, 11027, 4262, 11982, 9528, 2648, 7019, 9388, 7498, 3893, 4886, 10806, 3082, 684, 11574, 2778, 10530, 2581, 8802, 1982, 9086, 7533, 2703, 1144, 5997, 9187, 2337, 8955, 6656, 10979, 11047, 3852, 9690, 6895, 10926, 4585, 5794, 9855, 6871, 11333, 9586, 1161, 8868, 8566, 5943, 6536, 2051, 857, 927, 6908, 5738, 7029, 7049, 2517, 8100, 6958, 4302, 5209, 6622, 6704, 1498, 892, 2520, 5202, 3186, 10358, 1012, 2079, 6669, 11660, 6709, 702, 2654, 8922, 5259, 9503, 8857, 2702, 7590, 9087, 10259, 1901, 4052, 1493, 1257, 3977, 1053, 1770, 9034, 8398, 1488, 8421, 11742, 3708, 7734, 3327, 5901, 8335, 10840, 9100, 1442, 11305, 9342, 12004, 1718, 1603, 1568, 10075, 2465, 6390, 11357, 5530, 3892, 7603, 9389, 8472, 7231, 10930, 102, 6243, 179, 673, 1751, 832, 3741, 1511, 10765, 10943, 6761, 9517, 3744, 3502, 417, 6886, 4183, 6454, 11464, 7314, 5124, 5316, 10872, 8048, 8961, 6929, 4372, 252, 3958, 5613, 2343, 9132, 8713, 1914, 7865, 874, 10625, 8018, 3380, 2955, 5473, 7874, 9821, 11130, 11604, 1330, 7326, 6601, 6305, 2328, 1297, 10485, 4087, 1107, 219, 8832, 4619, 1505, 7871, 9370, 2708, 654, 6485, 6903, 1086, 4437, 6220, 11957, 1559, 10580, 514, 6802, 5393, 11209, 8853, 1588, 3867, 6652, 9765, 4831, 5872, 6030, 8091, 10386, 3949, 4616, 5880, 9094, 4029, 160, 9836, 5574, 7615, 6431, 6565, 8063, 4605, 6134, 3180, 4576, 10509, 4808, 669, 7363, 3550, 8430, 4634, 580, 3129, 10800, 5219, 3293, 11444, 3582, 4626, 11434, 1022, 3610, 1436, 6685, 8628, 7117, 9895, 9453, 9839, 3790, 10083, 1927, 11371, 3549, 7390, 670, 737, 2854, 10822, 2285, 3056, 5359, 2250, 5826, 6593, 11829, 68, 3358, 10500, 1250, 8083, 5399, 6260, 3400, 1030, 1413, 19, 8220, 10714, 1530, 3621, 2377, 8486, 8040, 3066, 9212, 1461, 3200, 4097, 8281, 6600, 7447, 1331, 1879, 6912, 11907, 6123, 11093, 9985, 9031, 11485, 8976, 5123, 7474, 11465, 24, 5507, 4857, 6350, 647, 726, 5445, 11938, 9545, 4332, 1197, 1467, 11882, 4285, 187, 5590, 9050, 10143, 3532, 6555, 8625, 10399, 11633, 5372, 6126, 5235, 9285, 870, 10094, 10196, 227, 1801, 6471, 5485, 10340, 6543, 7622, 5062, 3872, 4776, 3470, 6859, 5814, 3483, 6961, 5137, 3086, 4186, 2088, 4207, 4196, 8953, 2339, 7946, 7770, 390, 9580, 1420, 9061, 8996, 7846, 5536, 6743, 59, 6720, 1952, 4943, 3919, 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11011, 10920, 1202, 2670, 4505, 10296, 10010, 2403, 9009, 8008, 6006, 12011]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-prefix-product-permutation-l6-s3/tests/check.py b/tasks/construct-rlve-prefix-product-permutation-l6-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3d93850acd79cc0f8fff05cdbb0c63aabc484815
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l6-s3/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-l6-s3/tests/instance.json b/tasks/construct-rlve-prefix-product-permutation-l6-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d014c8cc1f4027ecaa4996df061f7fe708346758
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l6-s3/tests/instance.json
@@ -0,0 +1 @@
+{"n": 12011, "family": "rlve_prefix_product_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-product-permutation-l6-s3/tests/run.py b/tasks/construct-rlve-prefix-product-permutation-l6-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-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-rlve-prefix-product-permutation-l6-s3/tests/test.sh b/tasks/construct-rlve-prefix-product-permutation-l6-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-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-rlve-prefix-product-permutation-l6-s4/environment/Dockerfile b/tasks/construct-rlve-prefix-product-permutation-l6-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-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-prefix-product-permutation-l6-s4/environment/instance.json b/tasks/construct-rlve-prefix-product-permutation-l6-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2c3575eae12645e2bb58660e7639a35ff4af57b1
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l6-s4/environment/instance.json
@@ -0,0 +1 @@
+{"n": 22391, "family": "rlve_prefix_product_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-product-permutation-l6-s4/solution/solve.sh b/tasks/construct-rlve-prefix-product-permutation-l6-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..889b536933a04dd3232a40a05e76d7f447a711a6
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-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'
+{"perm": [1, 2, 11197, 7465, 5599, 17914, 3733, 12796, 2800, 2489, 20153, 18321, 1867, 8613, 17594, 5972, 12596, 10538, 1245, 2358, 10077, 4266, 9161, 20445, 934, 12540, 4307, 8294, 19993, 14671, 14182, 17336, 17494, 21035, 16465, 2560, 623, 3632, 12375, 17799, 5039, 4370, 13329, 9374, 4581, 9455, 10223, 20010, 11663, 11425, 17466, 10977, 2154, 15210, 15343, 3665, 9997, 15714, 7336, 21633, 18287, 5507, 19864, 16350, 19943, 15158, 10518, 12032, 8233, 21743, 12476, 9462, 312, 3375, 13012, 19108, 6188, 15413, 8900, 20408, 2520, 10229, 13381, 9443, 6665, 15543, 15883, 4891, 2291, 19373, 4728, 4430, 5112, 13243, 21201, 13907, 5832, 21007, 5713, 14476, 19929, 19510, 5489, 3914, 12273, 854, 18801, 8790, 7672, 9245, 1833, 8675, 4999, 18429, 19053, 8568, 14864, 20861, 10817, 4705, 9144, 3702, 2754, 8921, 21128, 6987, 19371, 2293, 9972, 10589, 18775, 20170, 16455, 19530, 17212, 10616, 4117, 19777, 10872, 13049, 17434, 21598, 15927, 4855, 11352, 2935, 1688, 3809, 17702, 16381, 20750, 1039, 14290, 18587, 7707, 3468, 15646, 17543, 21400, 19998, 12456, 9319, 5115, 2611, 6691, 8686, 4722, 10325, 3333, 7553, 7772, 20166, 7942, 907, 2446, 4991, 1146, 22139, 9687, 15387, 13560, 5073, 13411, 9300, 13752, 18640, 6622, 5030, 10601, 20378, 6954, 1525, 14112, 7426, 10504, 12517, 2857, 5684, 18434, 9565, 9965, 11475, 20951, 21289, 2745, 5353, 13153, 14712, 6137, 10393, 11623, 14327, 9401, 18082, 15591, 10832, 15032, 18471, 4623, 16053, 917, 16313, 4338, 4921, 2500, 6370, 9215, 3552, 9527, 881, 15480, 12602, 18628, 7785, 10431, 8481, 5409, 14267, 2353, 4779, 15768, 1023, 13047, 10874, 12573, 6764, 4461, 12239, 21760, 10612, 3494, 9903, 9686, 22215, 1147, 20109, 16182, 698, 5295, 19712, 9388, 1631, 21281, 14304, 8228, 7521, 20961, 13922, 19802, 1749, 16504, 7685, 2059, 8941, 9889, 5212, 16632, 21907, 6525, 4415, 19913, 4782, 21995, 20414, 7964, 12100, 2428, 13420, 16872, 13792, 1468, 7003, 12040, 16202, 1905, 17762, 20047, 4826, 8191, 3296, 21571, 1340, 520, 6504, 18341, 19015, 9294, 19037, 3854, 8769, 12930, 73, 19019, 9730, 8772, 7749, 21896, 2897, 21195, 17619, 17424, 17858, 4660, 4091, 2558, 16467, 1306, 10546, 3346, 18853, 15539, 7848, 13557, 10356, 5163, 20320, 1667, 16080, 3777, 21071, 15082, 5648, 21279, 1633, 15167, 17784, 454, 9422, 12419, 11806, 2496, 14418, 11769, 17825, 11070, 19806, 4844, 1569, 7694, 20708, 17976, 2482, 2537, 16162, 6706, 5154, 15846, 16779, 18072, 10438, 20516, 21366, 14507, 20351, 13711, 17257, 5301, 8019, 21385, 14180, 14673, 765, 763, 14675, 18252, 16518, 14909, 10994, 16448, 5239, 6259, 17009, 1429, 6724, 14038, 17517, 20413, 22110, 4783, 21438, 4983, 20326, 5738, 8224, 10476, 11003, 10645, 18651, 1373, 3231, 2677, 6593, 6577, 6290, 18552, 15324, 3069, 11814, 5197, 10101, 5812, 20956, 7164, 7200, 4701, 16544, 20237, 7185, 7796, 1619, 16612, 20262, 18712, 5017, 9236, 979, 2312, 1077, 8027, 11171, 459, 8734, 8157, 16074, 13365, 17310, 2461, 5461, 12446, 6833, 14381, 14647, 4608, 347, 12972, 5365, 4764, 11221, 441, 13660, 18936, 7578, 17497, 19442, 20510, 16084, 3893, 19707, 5216, 20912, 4241, 13312, 2705, 19173, 7134, 20695, 1177, 21594, 2390, 17437, 19080, 12337, 512, 3107, 6524, 22115, 16633, 14254, 6287, 871, 14578, 15962, 2231, 12718, 6120, 2896, 22076, 7750, 16502, 1751, 12943, 10906, 4952, 5966, 16039, 8381, 11108, 9982, 574, 4488, 10055, 483, 19287, 21650, 11545, 9740, 2648, 1820, 21052, 303, 15890, 13066, 816, 21707, 10641, 9128, 18348, 15509, 15310, 6731, 3761, 14844, 10481, 8949, 18157, 6237, 21097, 20057, 875, 21395, 19448, 16101, 3843, 16619, 1030, 6328, 4471, 3112, 4945, 18028, 13802, 5487, 19512, 6115, 10954, 6214, 3263, 11297, 2208, 722, 9957, 9141, 13587, 20483, 10998, 12643, 21403, 14257, 15178, 14443, 17246, 4589, 12410, 509, 17906, 15149, 19632, 15795, 18092, 2476, 11930, 10946, 3502, 17668, 17216, 4173, 19297, 13962, 953, 17412, 20077, 9359, 10024, 5325, 13609, 5420, 4096, 3189, 12844, 5795, 10786, 5962, 11866, 18753, 11456, 18654, 14448, 13502, 9171, 7097, 9508, 18764, 15843, 9498, 9519, 1785, 13123, 5880, 4385, 19462, 17661, 19832, 37, 18762, 9510, 19311, 16061, 3465, 15582, 16020, 3875, 10611, 22144, 12240, 1449, 13894, 10598, 19826, 8810, 6028, 19908, 3564, 20125, 17482, 13526, 11075, 2046, 17720, 12475, 22323, 8234, 21085, 11849, 1304, 16469, 8513, 12869, 20279, 9427, 1496, 7770, 7555, 15120, 12902, 6779, 8385, 16374, 8427, 2582, 1641, 21356, 20857, 834, 5291, 19236, 2124, 1889, 5822, 10536, 12598, 18737, 16112, 14020, 1968, 10640, 21869, 817, 7758, 7584, 14950, 20088, 15490, 11423, 11665, 15907, 11567, 6210, 258, 17099, 17523, 12444, 5463, 18405, 4906, 5885, 17755, 8913, 2788, 16731, 20055, 21099, 12220, 13618, 14384, 785, 5450, 15043, 16521, 21550, 9001, 20184, 560, 12437, 15269, 1269, 5066, 19277, 7516, 14549, 11089, 13773, 18992, 19119, 17199, 8390, 9719, 20232, 13308, 16415, 3243, 21454, 11544, 21879, 19288, 7254, 3277, 10176, 3538, 6856, 7654, 8629, 10794, 2651, 18229, 4010, 9165, 10693, 13548, 18286, 22333, 7337, 21627, 383, 7719, 382, 21631, 7338, 1665, 20322, 19655, 19455, 7697, 7455, 12920, 16693, 14129, 19420, 14035, 2620, 20265, 3130, 19181, 8505, 8008, 715, 16944, 14558, 5236, 18215, 19696, 8759, 9031, 10207, 15926, 22251, 17435, 2392, 2389, 21915, 1178, 2492, 12105, 21359, 12521, 14065, 10821, 15308, 15511, 16434, 859, 5502, 13916, 5323, 10026, 9326, 523, 687, 14636, 1616, 2981, 1339, 22092, 3297, 14094, 3289, 1738, 14341, 1084, 20472, 13992, 18858, 7303, 1535, 673, 17103, 6544, 2599, 8936, 5051, 14786, 14102, 9000, 21674, 16522, 14778, 14946, 14796, 19722, 2351, 14269, 19468, 6000, 10119, 106, 3593, 4034, 15094, 5958, 810, 16892, 19502, 4474, 21327, 5242, 20552, 18095, 2509, 4598, 15814, 16105, 490, 7816, 12352, 2335, 539, 999, 4014, 14656, 5586, 12865, 230, 6914, 15563, 12401, 4079, 13385, 19233, 4873, 6683, 16322, 19851, 9073, 1231, 5768, 2731, 20991, 17419, 8563, 3417, 9864, 7191, 4847, 7324, 7911, 13500, 14450, 174, 19948, 17682, 17096, 2683, 8456, 13578, 6966, 5611, 6339, 221, 6081, 18026, 4947, 20664, 13810, 14985, 19675, 8749, 8207, 20917, 17851, 21451, 2122, 19238, 12387, 1947, 25, 9854, 5485, 13804, 11543, 21652, 3244, 2121, 21464, 17852, 2138, 1353, 17511, 9587, 5250, 14763, 15762, 10348, 8430, 589, 4982, 21993, 4784, 12391, 20801, 8719, 15621, 20736, 15006, 6169, 13417, 11452, 1486, 1554, 15376, 14458, 8828, 11058, 10424, 8317, 1865, 18323, 20417, 3144, 13156, 436, 20082, 18485, 3516, 19177, 16743, 1116, 10094, 17555, 6285, 14256, 21826, 12644, 19997, 22234, 17544, 15071, 6392, 19447, 21853, 876, 18380, 6472, 16688, 16649, 18396, 13672, 6774, 14179, 22014, 8020, 15779, 4191, 20288, 16750, 3626, 16187, 13215, 11483, 14488, 13440, 3319, 5028, 6624, 242, 9347, 9644, 14506, 22021, 20517, 5773, 545, 16066, 19699, 12520, 21590, 12106, 20856, 21722, 1642, 152, 8104, 19141, 10605, 17729, 2341, 11604, 15514, 10854, 3936, 5321, 13918, 15760, 14765, 20370, 19734, 7755, 20779, 18851, 3348, 3366, 13406, 1881, 6311, 18618, 6257, 5241, 21530, 4475, 18195, 9079, 19690, 3119, 15451, 10549, 20973, 10029, 20080, 438, 14367, 10698, 12784, 20920, 2818, 8051, 828, 1922, 3695, 8310, 3750, 11711, 5388, 14360, 20196, 2236, 16615, 12752, 16646, 2473, 20555, 20210, 2507, 18097, 20521, 2744, 22189, 20952, 16698, 3058, 3480, 16673, 4564, 14303, 22130, 1632, 22050, 5649, 985, 12300, 4391, 11557, 14248, 4979, 2318, 4571, 9412, 6794, 5753, 10242, 14127, 16695, 4503, 6322, 17601, 10702, 12570, 7129, 18624, 18785, 12191, 7222, 15197, 19819, 9970, 2295, 15183, 17401, 177, 255, 2285, 20149, 12356, 7575, 6907, 21012, 10410, 7898, 20434, 20242, 18715, 12434, 13254, 17161, 11129, 16669, 8607, 12947, 6036, 20030, 1747, 19804, 11072, 2087, 10881, 9649, 13134, 18177, 1697, 477, 800, 19902, 2242, 10039, 20060, 4680, 10676, 16208, 20767, 2663, 7603, 6805, 10189, 13906, 22298, 13244, 2516, 1595, 9913, 17618, 22074, 2898, 6948, 16589, 13760, 14177, 6776, 17129, 13640, 9377, 5826, 16924, 2742, 20523, 16124, 18420, 20583, 17947, 11132, 4586, 4987, 3549, 11371, 15950, 21145, 20578, 12414, 7922, 10310, 15945, 16005, 4760, 13319, 893, 16626, 6562, 9662, 14136, 11405, 2193, 8599, 20927, 9561, 8831, 17322, 21112, 11854, 19, 20036, 20577, 21171, 15951, 12550, 9656, 18866, 8031, 2199, 1733, 15416, 18987, 18295, 19206, 3967, 1938, 6784, 5306, 6986, 22268, 8922, 17316, 7123, 725, 17328, 18143, 10254, 16495, 10062, 21109, 12979, 15601, 7979, 14210, 11853, 21150, 17323, 12978, 21118, 10063, 13575, 19937, 8004, 17959, 4170, 5538, 19114, 12219, 21682, 20056, 21856, 6238, 18922, 11162, 1673, 15313, 11033, 10543, 654, 5925, 327, 11848, 21741, 8235, 16713, 4257, 15287, 6435, 11836, 10140, 11818, 4714, 11188, 11944, 6299, 15081, 22053, 3778, 2912, 18756, 1505, 17647, 11863, 3390, 20286, 4193, 2373, 19383, 11860, 4214, 20698, 12487, 19411, 821, 302, 21874, 1821, 10429, 7787, 11613, 9307, 2646, 9742, 20814, 20533, 12258, 20254, 945, 4883, 14107, 6016, 16464, 22359, 17495, 7580, 9369, 16239, 19252, 8412, 18206, 1789, 12180, 17560, 16516, 18254, 10935, 3741, 409, 19163, 15075, 18956, 14988, 4554, 18671, 10409, 21240, 6908, 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13692, 4769, 14801, 20148, 21245, 256, 6212, 10956, 12483, 9541, 14794, 14948, 7586, 16841, 16297, 13211, 13000, 2949, 1885, 11833, 18828, 643, 4912, 4410, 5846, 19394, 13592, 18609, 8036, 16316, 20141, 2878, 14045, 1550, 15329, 3867, 6076, 20134, 19516, 11988, 7800, 19841, 6491, 7591, 20107, 1149, 10038, 21213, 19903, 11, 4073, 20260, 16614, 21300, 20197, 10279, 5047, 12717, 21900, 15963, 19568, 14620, 172, 14452, 10381, 3617, 132, 5939, 15241, 2786, 8915, 6384, 14056, 10221, 9457, 16301, 9668, 3029, 2848, 13391, 721, 21834, 11298, 11689, 9825, 2420, 12132, 7641, 13344, 1732, 21139, 8032, 1092, 20158, 12115, 8598, 21156, 11406, 2874, 19487, 4821, 18696, 14085, 4264, 10079, 1837, 1099, 19887, 4017, 14829, 8503, 19183, 16537, 13240, 9322, 8368, 7042, 4374, 20545, 9839, 10236, 5591, 10783, 3476, 18367, 14742, 9788, 12673, 737, 9086, 4805, 19392, 5848, 427, 15209, 22340, 10978, 1958, 1157, 3679, 17378, 3745, 11906, 13112, 17118, 15705, 14667, 12760, 20833, 10134, 1352, 21449, 17853, 16477, 13295, 18319, 20155, 5780, 432, 3682, 19772, 780, 19264, 10127, 1888, 21717, 19237, 21463, 21452, 3245, 4494, 9922, 9514, 16474, 9523, 658, 12967, 16484, 12003, 20757, 18515, 19002, 1330, 18201, 1012, 5644, 15010, 1898, 3317, 13442, 4238, 1624, 19732, 20372, 9229, 2729, 5770, 7379, 3440, 11920, 5395, 10880, 21222, 11073, 13528, 13427, 17090, 14376, 12822, 18504, 7145, 14781, 11121, 10371, 10376, 17229, 336, 20727, 769, 2739, 5144, 17409, 402, 16656, 4937, 16683, 12543, 4879, 3164, 8940, 22120, 7686, 19506, 7639, 12134, 1517, 11382, 4084, 3944, 6809, 6375, 10841, 17719, 21746, 11076, 9558, 7885, 374, 8683, 20624, 20733, 18193, 4477, 18662, 3953, 18394, 16651, 12988, 15785, 3422, 14299, 9291, 9384, 6757, 2822, 7627, 1053, 2660, 20297, 13165, 14503, 7794, 7187, 11791, 190, 15440, 15303, 2406, 7810, 20475, 4361, 6918, 1999, 18327, 14068, 17451, 2853, 4656, 7961, 18326, 2007, 6919, 10759, 4670, 11466, 12584, 12669, 3611, 8001, 10628, 14857, 20589, 2425, 13492, 80, 19874, 12394, 6681, 4875, 397, 284, 14430, 4069, 976, 19250, 16241, 13615, 8169, 6150, 8780, 10639, 21709, 14021, 12091, 12804, 17145, 9439, 14719, 13150, 14494, 1156, 2152, 10979, 10034, 4619, 15364, 9625, 10089, 12184, 6469, 13231, 24, 21460, 12388, 18234, 13484, 5985, 9022, 5668, 3269, 6783, 21132, 3968, 11237, 3509, 4507, 10420, 16142, 10454, 16565, 5471, 17535, 15684, 5654, 19308, 6277, 3694, 21308, 829, 8402, 14582, 20383, 18033, 14979, 11992, 5034, 16197, 8320, 8805, 564, 11396, 4134, 8706, 17761, 22098, 16203, 6982, 17233, 6023, 3385, 3316, 2102, 15011, 15078, 8572, 3928, 17599, 6324, 13637, 5821, 21716, 2125, 10128, 11832, 2271, 2950, 465, 6310, 21332, 13407, 9575, 11954, 371, 1028, 16621, 15016, 4295, 1102, 19650, 1211, 6270, 8612, 22380, 18322, 21419, 8318, 16199, 13284, 1578, 1350, 10136, 15905, 11667, 18119, 13467, 19214, 11934, 5505, 18289, 9156, 18018, 20222, 12555, 6870, 14900, 17751, 4292, 17150, 865, 4299, 19919, 1098, 2184, 10080, 19352, 8674, 22282, 9246, 19223, 13749, 19587, 12563, 7714, 12405, 4408, 4914, 6969, 10428, 21051, 21875, 2649, 10796, 2356, 1247, 1223, 9980, 11110, 17941, 3972, 1214, 4447, 11054, 18437, 3983, 7535, 20405, 19969, 19259, 18797, 15895, 6265, 12997, 20684, 4816, 18111, 11977, 13683, 6402, 11897, 12179, 21027, 18207, 8313, 13122, 21777, 9520, 17856, 17426, 12467, 5690, 3036, 16810, 19051, 18431, 19956, 11809, 12776, 3225, 15700, 13709, 20353, 1661, 16192, 14349, 14611, 2503, 7740, 2811, 11082, 7104, 15960, 14580, 8404, 6074, 3869, 5008, 9190, 12942, 21893, 16503, 22123, 19803, 21225, 20031, 5021, 11280, 7289, 9531, 10952, 6117, 14340, 21567, 3290, 2595, 8898, 15415, 21138, 2200, 13345, 17445, 921, 8584, 8576, 9879, 11729, 8396, 8960, 16173, 1446, 14759, 2751, 5439, 10792, 8631, 8218, 13399, 3650, 6204, 9180, 9395, 20596, 17578, 11144, 7828, 17333, 13951, 8197, 17640, 1463, 14511, 15258, 476, 21217, 18178, 1336, 9907, 3799, 17837, 3724, 8101, 3808, 22246, 2936, 14698, 360, 4418, 6697, 11363, 4146, 8616, 3804, 14688, 19867, 13850, 6729, 15312, 21093, 11163, 7925, 5260, 13841, 16079, 22056, 20321, 21625, 7339, 18741, 16191, 1768, 20354, 4201, 6771, 961, 7388, 10655, 5636, 15222, 10159, 10328, 17205, 20748, 16383, 4150, 5187, 20744, 6011, 151, 21355, 21723, 2583, 20902, 6799, 10389, 20283, 3879, 15166, 22049, 21280, 22131, 9389, 6534, 2836, 6184, 1186, 19731, 2098, 4239, 20914, 8446, 16611, 21963, 7797, 2980, 21574, 14637, 1056, 3543, 8243, 16982, 8128, 15360, 11301, 16132, 5386, 11713, 3074, 16340, 4868, 9951, 4482, 10608, 3394, 11435, 9912, 21198, 2517, 10001, 958, 13675, 18686, 11267, 12685, 3234, 3900, 17289, 12321, 17877, 9759, 18013, 12650, 1349, 1861, 13285, 11886, 8579, 11773, 3918, 18444, 2725, 7693, 22035, 4845, 7193, 15968, 18140, 18037, 12802, 12093, 9632, 20252, 12260, 7364, 9887, 8943, 15375, 21426, 1487, 14322, 15328, 2256, 14046, 16863, 4737, 11682, 18665, 7039, 18544, 3372, 12665, 7712, 12565, 10286, 1036, 672, 21560, 7304, 7528, 118, 11577, 6957, 11006, 5969, 18533, 14111, 22201, 6955, 11579, 19065, 18724, 13356, 11226, 11381, 2054, 12135, 4423, 15984, 17046, 8418, 17810, 12940, 9192, 12452, 13278, 17646, 21067, 18757, 5800, 18961, 18078, 17866, 7370, 14345, 7769, 21733, 9428, 5595, 6109, 19809, 20754, 15595, 7570, 14321, 1553, 21427, 11453, 2915, 5425, 17176, 470, 18153, 20771, 13948, 14185, 927, 4543, 9608, 1081, 19576, 17533, 5473, 7002, 22102, 13793, 1239, 12833, 14510, 1701, 17641, 15506, 9869, 16848, 2967, 15118, 7557, 7439, 16644, 12754, 16991, 8664, 13893, 21758, 12241, 14758, 1721, 16174, 18280, 10917, 203, 1105, 5696, 17892, 16580, 422, 15230, 14132, 14970, 14871, 267, 8472, 6723, 22000, 17010, 4005, 15494, 4388, 6059, 5087, 6556, 11843, 1074, 12365, 13069, 14027, 9353, 9803, 17955, 17936, 15251, 13515, 16099, 19450, 7958, 8744, 3523, 13190, 17285, 19661, 896, 4975, 3202, 13631, 16077, 13843, 17031, 11068, 17827, 8092, 13104, 16609, 8448, 14439, 18679, 16560, 98, 16681, 4939, 3452, 15485, 1153, 11984, 3722, 17839, 7405, 3437, 7318, 3230, 21984, 18652, 11458, 4139, 5877, 4833, 10213, 20604, 4187, 13981, 3141, 6154, 13024, 14813, 4898, 34, 5929, 16377, 8286, 17510, 21448, 2139, 10135, 1860, 1579, 12651, 19747, 13086, 10780, 14606, 11964, 20572, 519, 22091, 21572, 2982, 9906, 1695, 18179, 10533, 3010, 20020, 18200, 2107, 19003, 10530, 4746, 20888, 3637, 19481, 18615, 340, 7312, 16094, 10449, 11205, 17679, 10575, 12253, 10557, 15958, 7106, 18136, 5680, 14158, 652, 10545, 22066, 16468, 21739, 11850, 11360, 7080, 20720, 11231, 3471, 16155, 537, 2337, 711, 10174, 3279, 16855, 18223, 4434, 14389, 2456, 8818, 12330, 1275, 9415, 5070, 1243, 10540, 8183, 14414, 6792, 9414, 1284, 12331, 5898, 12139, 4250, 5065, 21668, 15270, 5077, 13471, 125, 15407, 17087, 15609, 20455, 18426, 3187, 4098, 3406, 6977, 20660, 20194, 14362, 3527, 12737, 9843, 6442, 1222, 1816, 2357, 22374, 10539, 1281, 5071, 13562, 12832, 1466, 13794, 20200, 10988, 8257, 12731, 15831, 5767, 21500, 9074, 17633, 6388, 6448, 12083, 14471, 9979, 1815, 1248, 6443, 11022, 18844, 17406, 17807, 11261, 4446, 1810, 3973, 6269, 1870, 19651, 5469, 16567, 13016, 8753, 5264, 15617, 8216, 8633, 5804, 15445, 19495, 319, 4775, 12480, 20798, 19877, 9149, 95, 8487, 12204, 15588, 14790, 19730, 1626, 6185, 11717, 17713, 2333, 12354, 20151, 2491, 21593, 21916, 20696, 4216, 9259, 12744, 11512, 20306, 11321, 2589, 20646, 2363, 16357, 9446, 13786, 5724, 11264, 5232, 9139, 9959, 3678, 2151, 1959, 14495, 11983, 1381, 15486, 14818, 10037, 2244, 20108, 22138, 22216, 4992, 7210, 20098, 12423, 2574, 7196, 15171, 10202, 3608, 3769, 15264, 9823, 11691, 4792, 16071, 17890, 5698, 8266, 12151, 16640, 15599, 12981, 17822, 20075, 17414, 8145, 10836, 18002, 10093, 21408, 16744, 343, 20761, 263, 8090, 17829, 5720, 18913, 19335, 5695, 1441, 204, 19649, 1872, 4296, 19886, 2183, 1838, 19920, 5747, 9641, 5778, 20157, 2197, 8033, 17005, 10682, 18643, 14083, 18698, 20471, 21565, 14342, 19575, 1473, 9609, 11695, 8026, 21955, 2313, 12364, 1420, 11844, 6942, 19274, 2703, 13314, 4198, 17918, 863, 17152, 16136, 3775, 16082, 20512, 8987, 19027, 19045, 3542, 1614, 14638, 2659, 2023, 7628, 6633, 8475, 17072, 18457, 11539, 6879, 10789, 20052, 4664, 11788, 3252, 14289, 22241, 20751, 671, 1537, 10287, 803, 9873, 2694, 6327, 21848, 16620, 1876, 372, 7887, 12749, 13046, 22151, 15769, 17365, 19074, 8953, 7905, 10910, 9178, 6206, 18767, 5643, 2105, 18202, 6614, 14373, 379, 8214, 15619, 8721, 3997, 5744, 5705, 15921, 4013, 21517, 540, 2946, 16001, 7322, 4849, 159, 2396, 9749, 567, 8137, 16108, 4838, 12299, 21277, 5650, 3216, 18877, 3908, 2311, 21957, 9237, 19249, 1976, 4070, 20528, 14076, 11969, 11335, 13565, 7935, 7017, 20839, 20907, 10941, 8976, 16224, 7387, 1657, 6772, 13674, 1592, 10002, 17609, 400, 17411, 21804, 13963, 12045, 6631, 7630, 17143, 12806, 4882, 21040, 20255, 4541, 929, 20272, 19149, 741, 10850, 8589, 16908, 12539, 22368, 20446, 10006, 3155, 20271, 942, 4542, 1476, 14186, 13644, 2718, 7408, 8583, 1729, 17446, 4367, 16312, 22172, 16054, 16782, 15427, 8815, 13937, 19710, 5297, 4711, 2445, 22219, 7943, 8893, 14482, 15069, 17546, 15202, 12529, 18976, 13830, 4974, 1402, 19662, 16625, 21162, 13320, 2542, 6071, 15710, 17520, 3160, 9008, 18314, 9992, 6830, 15479, 22163, 9528, 16807, 7737, 18379, 21394, 21854, 20058, 10041, 14577, 21903, 6288, 6579, 17795, 19884, 4298, 1841, 17151, 1066, 17919, 2891, 5501, 21583, 16435, 7299, 18359, 18800, 22287, 12274, 16393, 2925, 8124, 20042, 2671, 7597, 7447, 7615, 5871, 719, 13393, 8291, 7607, 17342, 13457, 19794, 15849, 5290, 21720, 20858, 15090, 3535, 8401, 1921, 21309, 8052, 20655, 19104, 8299, 19096, 301, 21054, 19412, 20777, 7757, 21706, 21870, 13067, 12367, 17070, 8477, 16891, 21534, 5959, 6882, 7085, 11572, 8328, 9872, 1034, 10288, 19901, 21215, 478, 20004, 20001, 4958, 142, 6467, 12186, 13362, 13634, 2697, 4178, 17157, 7835, 5449, 21678, 14385, 13888, 3212, 19263, 2128, 19773, 8358, 2973, 8264, 5700, 9473, 14938, 14696, 2938, 2738, 2071, 20728, 15055, 762, 22011, 14674, 22010, 766, 15056, 60, 4107, 8845, 11700, 13228, 18383, 4164, 19742, 11599, 13764, 14739, 15537, 18855, 12217, 19116, 15139, 3105, 514, 10849, 939, 19150, 5978, 9085, 2161, 12674, 14003, 5194, 3274, 3401, 8620, 11304, 7844, 14877, 3116, 17327, 21124, 7124, 9956, 21833, 2209, 13392, 843, 5872, 7350, 16943, 21608, 8009, 8775, 10173, 1294, 2338, 5662, 19605, 13480, 4638, 16508, 17487, 3988, 16930, 9949, 4870, 5294, 22135, 16183, 10826, 6486, 10728, 10970, 6903, 2305, 7443, 14809, 14635, 21576, 524, 11753, 20425, 8373, 6281, 3656, 9795, 11857, 16571, 20504, 20269, 3157, 17102, 21559, 1536, 1037, 20752, 19811, 13966, 6019, 14008, 15614, 9491, 7273, 14838, 14623, 20897, 12966, 2114, 9524, 13880, 5924, 21089, 10544, 1308, 14159, 70, 9918, 4673, 20347, 11318, 8867, 4911, 2268, 18829, 2485, 16365, 13583, 2567, 11795, 8499, 20944, 10153, 249, 11782, 18518, 6373, 6811, 18928, 6332, 3082, 12883, 3631, 22356, 2561, 4732, 2931, 18008, 16513, 9270, 20608, 12874, 12895, 6550, 3629, 12885, 15296, 13222, 8891, 7945, 3739, 10937, 3640, 10527, 16431, 11607, 16598, 9549, 19204, 18297, 16973, 8784, 17068, 12369, 13034, 2316, 4981, 21440, 8431, 3096, 18220, 5177, 4725, 18891, 11447, 10463, 19917, 4301, 6598, 2761, 7244, 4487, 21884, 9983, 17804, 5147, 7950, 7215, 8136, 990, 9750, 11395, 1910, 8806, 13252, 12436, 21671, 20185, 11096, 19130, 16179, 11439, 16278, 2881, 16906, 8591, 4365, 17448, 19281, 17922, 16065, 21363, 5774, 18550, 6292, 2945, 998, 21518, 2336, 1296, 16156, 4236, 13444, 11912, 7549, 18632, 15662, 7083, 6884, 4045, 13265, 11752, 686, 21577, 9327, 6503, 22090, 1341, 20573, 19745, 12653, 10848, 743, 3106, 21910, 12338, 17905, 21819, 12411, 11285, 14012, 6354, 16427, 17441, 11487, 9450, 20642, 5891, 14643, 317, 19497, 16273, 9675, 20162, 6431, 7815, 21522, 16106, 8139, 5760, 278, 15869, 19286, 21881, 10056, 3313, 4956, 20003, 799, 21216, 1698, 15259, 3220, 19688, 9081, 18152, 1481, 17177, 2686, 18500, 6309, 1883, 2951, 4896, 14815, 3457, 8733, 21952, 11172, 17629, 17028, 9421, 22046, 17785, 7746, 8012, 15560, 9947, 16932, 19932, 5083, 9028, 6319, 14236, 13659, 21934, 11222, 14366, 21316, 20081, 21414, 13157, 17376, 3681, 2131, 5781, 20774, 14597, 15208, 2156, 5849, 17692, 15193, 15229, 1437, 16581, 12292, 17196, 10579, 19324, 7069, 3841, 16103, 15816, 15800, 19716, 19162, 21020, 3742, 11748, 11390, 11917, 14169, 16655, 2067, 17410, 955, 17610, 283, 1980, 4876, 8355, 15669, 20964, 5384, 16134, 17154, 5945, 11399, 7484, 5875, 4141, 7718, 21630, 21628, 7720, 8213, 1008, 14374, 17092, 5136, 8682, 2042, 7886, 1027, 1877, 11955, 4321, 5614, 6547, 17239, 15687, 6231, 19856, 19911, 4417, 1685, 14699, 20824, 17549, 2587, 11323, 4568, 10624, 7975, 19897, 10587, 9974, 12971, 21939, 4609, 7226, 20760, 1114, 16745, 7311, 1322, 18616, 6313, 20726, 2073, 17230, 12037, 8836, 14545, 6854, 3540, 19047, 11847, 21087, 5926, 19835, 18302, 17733, 4535, 4969, 4774, 1198, 19496, 497, 14644, 13621, 12663, 3374, 22320, 9463, 13624, 18539, 3356, 13099, 3378, 4052, 15889, 21873, 21053, 822, 19097, 14202, 17567, 2346, 4631, 20488, 6191, 10353, 15390, 20925, 8601, 5521, 8973, 19965, 10293, 14429, 1979, 398, 17611, 2480, 17978, 15868, 486, 5761, 17181, 12504, 13452, 20334, 14708, 5889, 20644, 2591, 8471, 1432, 14872, 14165, 8089, 1112, 20762, 13005, 2681, 17098, 21695, 6211, 2284, 21246, 178, 12707, 12490, 18899, 11781, 633, 10154, 17932, 15629, 9820, 11519, 9346, 21370, 6625, 17614, 20040, 8126, 16984, 13912, 11962, 14608, 3765, 9791, 6913, 21512, 12866, 18841, 13178, 16023, 19893, 17472, 18055, 6080, 21476, 6340, 17770, 3922, 7361, 11561, 6802, 4311, 12992, 8066, 10770, 12000, 16256, 7681, 9240, 17040, 19648, 1104, 1442, 10918, 12428, 12828, 3959, 16709, 19536, 9876, 11889, 14967, 8281, 20868, 15439, 2015, 11792, 17363, 15771, 3753, 8641, 13093, 8982, 17320, 8833, 7006, 12706, 254, 21247, 17402, 19947, 21486, 14451, 2227, 14621, 14840, 19060, 12068, 17671, 13707, 15702, 19782, 17278, 13074, 9937, 2395, 993, 4850, 6747, 18925, 14686, 3806, 8103, 21354, 1643, 6012, 4691, 18584, 20705, 19458, 11041, 17538, 6466, 795, 4959, 9344, 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+RLMATH_EOF
diff --git a/tasks/construct-rlve-prefix-product-permutation-l6-s4/tests/check.py b/tasks/construct-rlve-prefix-product-permutation-l6-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3d93850acd79cc0f8fff05cdbb0c63aabc484815
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l6-s4/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-l6-s4/tests/instance.json b/tasks/construct-rlve-prefix-product-permutation-l6-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2c3575eae12645e2bb58660e7639a35ff4af57b1
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l6-s4/tests/instance.json
@@ -0,0 +1 @@
+{"n": 22391, "family": "rlve_prefix_product_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-product-permutation-l6-s4/tests/run.py b/tasks/construct-rlve-prefix-product-permutation-l6-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-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-prefix-product-permutation-l6-s4/tests/test.sh b/tasks/construct-rlve-prefix-product-permutation-l6-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-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-prefix-sum-permutation-l3-s3/environment/Dockerfile b/tasks/construct-rlve-prefix-sum-permutation-l3-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-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-prefix-sum-permutation-l3-s3/environment/instance.json b/tasks/construct-rlve-prefix-sum-permutation-l3-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0a7010d9a33393a05d5d1d9787c9b5019b076f9d
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l3-s3/environment/instance.json
@@ -0,0 +1 @@
+{"n": 80, "family": "rlve_prefix_sum_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l3-s3/solution/solve.sh b/tasks/construct-rlve-prefix-sum-permutation-l3-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..eb1dbe03da349a026209c011d1b990358299cd12
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-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'
+{"perm": [80, 1, 78, 3, 76, 5, 74, 7, 72, 9, 70, 11, 68, 13, 66, 15, 64, 17, 62, 19, 60, 21, 58, 23, 56, 25, 54, 27, 52, 29, 50, 31, 48, 33, 46, 35, 44, 37, 42, 39, 40, 41, 38, 43, 36, 45, 34, 47, 32, 49, 30, 51, 28, 53, 26, 55, 24, 57, 22, 59, 20, 61, 18, 63, 16, 65, 14, 67, 12, 69, 10, 71, 8, 73, 6, 75, 4, 77, 2, 79]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l3-s3/tests/check.py b/tasks/construct-rlve-prefix-sum-permutation-l3-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..60c3e0996cdfa8673fd7f34ad70cccb789b17654
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l3-s3/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-l3-s3/tests/instance.json b/tasks/construct-rlve-prefix-sum-permutation-l3-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0a7010d9a33393a05d5d1d9787c9b5019b076f9d
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l3-s3/tests/instance.json
@@ -0,0 +1 @@
+{"n": 80, "family": "rlve_prefix_sum_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l3-s3/tests/run.py b/tasks/construct-rlve-prefix-sum-permutation-l3-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-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-prefix-sum-permutation-l3-s3/tests/test.sh b/tasks/construct-rlve-prefix-sum-permutation-l3-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-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-prefix-sum-permutation-l3-s4/environment/Dockerfile b/tasks/construct-rlve-prefix-sum-permutation-l3-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-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-prefix-sum-permutation-l3-s4/environment/instance.json b/tasks/construct-rlve-prefix-sum-permutation-l3-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..09307b8083f935e396bf0e3d057bda2d06542d45
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l3-s4/environment/instance.json
@@ -0,0 +1 @@
+{"n": 282, "family": "rlve_prefix_sum_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l3-s4/solution/solve.sh b/tasks/construct-rlve-prefix-sum-permutation-l3-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..a479d4c052323f583a174f88be46cf4d445d1702
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-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'
+{"perm": [282, 1, 280, 3, 278, 5, 276, 7, 274, 9, 272, 11, 270, 13, 268, 15, 266, 17, 264, 19, 262, 21, 260, 23, 258, 25, 256, 27, 254, 29, 252, 31, 250, 33, 248, 35, 246, 37, 244, 39, 242, 41, 240, 43, 238, 45, 236, 47, 234, 49, 232, 51, 230, 53, 228, 55, 226, 57, 224, 59, 222, 61, 220, 63, 218, 65, 216, 67, 214, 69, 212, 71, 210, 73, 208, 75, 206, 77, 204, 79, 202, 81, 200, 83, 198, 85, 196, 87, 194, 89, 192, 91, 190, 93, 188, 95, 186, 97, 184, 99, 182, 101, 180, 103, 178, 105, 176, 107, 174, 109, 172, 111, 170, 113, 168, 115, 166, 117, 164, 119, 162, 121, 160, 123, 158, 125, 156, 127, 154, 129, 152, 131, 150, 133, 148, 135, 146, 137, 144, 139, 142, 141, 140, 143, 138, 145, 136, 147, 134, 149, 132, 151, 130, 153, 128, 155, 126, 157, 124, 159, 122, 161, 120, 163, 118, 165, 116, 167, 114, 169, 112, 171, 110, 173, 108, 175, 106, 177, 104, 179, 102, 181, 100, 183, 98, 185, 96, 187, 94, 189, 92, 191, 90, 193, 88, 195, 86, 197, 84, 199, 82, 201, 80, 203, 78, 205, 76, 207, 74, 209, 72, 211, 70, 213, 68, 215, 66, 217, 64, 219, 62, 221, 60, 223, 58, 225, 56, 227, 54, 229, 52, 231, 50, 233, 48, 235, 46, 237, 44, 239, 42, 241, 40, 243, 38, 245, 36, 247, 34, 249, 32, 251, 30, 253, 28, 255, 26, 257, 24, 259, 22, 261, 20, 263, 18, 265, 16, 267, 14, 269, 12, 271, 10, 273, 8, 275, 6, 277, 4, 279, 2, 281]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l3-s4/tests/check.py b/tasks/construct-rlve-prefix-sum-permutation-l3-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..60c3e0996cdfa8673fd7f34ad70cccb789b17654
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l3-s4/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-l3-s4/tests/instance.json b/tasks/construct-rlve-prefix-sum-permutation-l3-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..09307b8083f935e396bf0e3d057bda2d06542d45
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l3-s4/tests/instance.json
@@ -0,0 +1 @@
+{"n": 282, "family": "rlve_prefix_sum_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l3-s4/tests/run.py b/tasks/construct-rlve-prefix-sum-permutation-l3-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-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-prefix-sum-permutation-l3-s4/tests/test.sh b/tasks/construct-rlve-prefix-sum-permutation-l3-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-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-prime-graph-coloring-l2-s2/environment/Dockerfile b/tasks/construct-rlve-prime-graph-coloring-l2-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-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-rlve-prime-graph-coloring-l2-s2/environment/instance.json b/tasks/construct-rlve-prime-graph-coloring-l2-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..23e579ec87084f4a9be8e3a43bf1cd17f4c631ef
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-l2-s2/environment/instance.json
@@ -0,0 +1 @@
+{"n": 18, "k": 4, "family": "rlve_prime_graph_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prime-graph-coloring-l2-s2/solution/solve.sh b/tasks/construct-rlve-prime-graph-coloring-l2-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..24a7efb57c34db028474656fca84a7fed20de014
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-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'
+{"colors": [1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-prime-graph-coloring-l2-s2/tests/check.py b/tasks/construct-rlve-prime-graph-coloring-l2-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..b3736f1375f1862241999142c975ced361b8fcb5
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-l2-s2/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-l2-s2/tests/instance.json b/tasks/construct-rlve-prime-graph-coloring-l2-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..23e579ec87084f4a9be8e3a43bf1cd17f4c631ef
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-l2-s2/tests/instance.json
@@ -0,0 +1 @@
+{"n": 18, "k": 4, "family": "rlve_prime_graph_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prime-graph-coloring-l2-s2/tests/run.py b/tasks/construct-rlve-prime-graph-coloring-l2-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-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-rlve-prime-graph-coloring-l2-s2/tests/test.sh b/tasks/construct-rlve-prime-graph-coloring-l2-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-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-rlve-prime-graph-coloring-l2-s5/environment/Dockerfile b/tasks/construct-rlve-prime-graph-coloring-l2-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-l2-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-prime-graph-coloring-l2-s5/environment/instance.json b/tasks/construct-rlve-prime-graph-coloring-l2-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..bce0ba3f979f40c7891ece24aaf25e842a3747e8
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-l2-s5/environment/instance.json
@@ -0,0 +1 @@
+{"n": 13, "k": 4, "family": "rlve_prime_graph_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prime-graph-coloring-l2-s5/solution/solve.sh b/tasks/construct-rlve-prime-graph-coloring-l2-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..8ae3ac5c121cc5b8a44bd953a8b894a7ea445d88
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-l2-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": [1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-prime-graph-coloring-l2-s5/tests/check.py b/tasks/construct-rlve-prime-graph-coloring-l2-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..b3736f1375f1862241999142c975ced361b8fcb5
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-l2-s5/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-l2-s5/tests/instance.json b/tasks/construct-rlve-prime-graph-coloring-l2-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..bce0ba3f979f40c7891ece24aaf25e842a3747e8
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-l2-s5/tests/instance.json
@@ -0,0 +1 @@
+{"n": 13, "k": 4, "family": "rlve_prime_graph_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prime-graph-coloring-l2-s5/tests/run.py b/tasks/construct-rlve-prime-graph-coloring-l2-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-l2-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-prime-graph-coloring-l2-s5/tests/test.sh b/tasks/construct-rlve-prime-graph-coloring-l2-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-l2-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-recursive-sequence-sum-l4-s0/environment/Dockerfile b/tasks/construct-rlve-recursive-sequence-sum-l4-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-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-recursive-sequence-sum-l4-s0/environment/instance.json b/tasks/construct-rlve-recursive-sequence-sum-l4-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1943bcaf2d1c859f03d46ed54d8c4f39adf41f98
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l4-s0/environment/instance.json
@@ -0,0 +1 @@
+{"F0": 31, "A": 1, "B": 3523, "S": 204520, "family": "rlve_recursive_sequence_sum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l4-s0/solution/solve.sh b/tasks/construct-rlve-recursive-sequence-sum-l4-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..16eb8363c71f7eac4a46ae0ed657fe62b4966426
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-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'
+{"indices": [2, 4, 7, 11, 14, 20]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l4-s0/tests/check.py b/tasks/construct-rlve-recursive-sequence-sum-l4-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..feda076c3f95e5e12198151dee30e1b8223924bc
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l4-s0/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-l4-s0/tests/instance.json b/tasks/construct-rlve-recursive-sequence-sum-l4-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1943bcaf2d1c859f03d46ed54d8c4f39adf41f98
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l4-s0/tests/instance.json
@@ -0,0 +1 @@
+{"F0": 31, "A": 1, "B": 3523, "S": 204520, "family": "rlve_recursive_sequence_sum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l4-s0/tests/run.py b/tasks/construct-rlve-recursive-sequence-sum-l4-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-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-recursive-sequence-sum-l4-s0/tests/test.sh b/tasks/construct-rlve-recursive-sequence-sum-l4-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-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-recursive-sequence-sum-l4-s7/environment/Dockerfile b/tasks/construct-rlve-recursive-sequence-sum-l4-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-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-recursive-sequence-sum-l4-s7/environment/instance.json b/tasks/construct-rlve-recursive-sequence-sum-l4-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2daba70af127c25d64fea315ad523c1553313f2f
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l4-s7/environment/instance.json
@@ -0,0 +1 @@
+{"F0": 9, "A": 7, "B": 3633, "S": 57181065149912192853, "family": "rlve_recursive_sequence_sum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l4-s7/solution/solve.sh b/tasks/construct-rlve-recursive-sequence-sum-l4-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..8a7fa9a221254fae2dfb20a6c05d26ab6b357d4b
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-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'
+{"indices": [1, 2, 3, 5, 12, 13, 14, 17, 18, 19, 20]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l4-s7/tests/check.py b/tasks/construct-rlve-recursive-sequence-sum-l4-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..feda076c3f95e5e12198151dee30e1b8223924bc
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l4-s7/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-l4-s7/tests/instance.json b/tasks/construct-rlve-recursive-sequence-sum-l4-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2daba70af127c25d64fea315ad523c1553313f2f
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l4-s7/tests/instance.json
@@ -0,0 +1 @@
+{"F0": 9, "A": 7, "B": 3633, "S": 57181065149912192853, "family": "rlve_recursive_sequence_sum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l4-s7/tests/run.py b/tasks/construct-rlve-recursive-sequence-sum-l4-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-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-recursive-sequence-sum-l4-s7/tests/test.sh b/tasks/construct-rlve-recursive-sequence-sum-l4-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-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-recursive-sequence-sum-l4-s9/environment/Dockerfile b/tasks/construct-rlve-recursive-sequence-sum-l4-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-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-recursive-sequence-sum-l4-s9/environment/instance.json b/tasks/construct-rlve-recursive-sequence-sum-l4-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..15a8ce7db16bea1e3aaa61ea6e751ffd101504b0
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l4-s9/environment/instance.json
@@ -0,0 +1 @@
+{"F0": 10, "A": 4, "B": 2254, "S": 1116125671820130, "family": "rlve_recursive_sequence_sum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l4-s9/solution/solve.sh b/tasks/construct-rlve-recursive-sequence-sum-l4-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..905ef10e755c34d06f2c4d89f409b859affe0cd8
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-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'
+{"indices": [1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l4-s9/tests/check.py b/tasks/construct-rlve-recursive-sequence-sum-l4-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..feda076c3f95e5e12198151dee30e1b8223924bc
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l4-s9/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-l4-s9/tests/instance.json b/tasks/construct-rlve-recursive-sequence-sum-l4-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..15a8ce7db16bea1e3aaa61ea6e751ffd101504b0
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l4-s9/tests/instance.json
@@ -0,0 +1 @@
+{"F0": 10, "A": 4, "B": 2254, "S": 1116125671820130, "family": "rlve_recursive_sequence_sum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l4-s9/tests/run.py b/tasks/construct-rlve-recursive-sequence-sum-l4-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-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-recursive-sequence-sum-l4-s9/tests/test.sh b/tasks/construct-rlve-recursive-sequence-sum-l4-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-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-shortest-path-count-l3-s8/environment/Dockerfile b/tasks/construct-rlve-shortest-path-count-l3-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-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-shortest-path-count-l3-s8/environment/instance.json b/tasks/construct-rlve-shortest-path-count-l3-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..ef3f79b80214f358cbfe08c19fb8a87379484ac4
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-l3-s8/environment/instance.json
@@ -0,0 +1 @@
+{"K": 97002, "n_max": 105, "family": "rlve_shortest_path_count", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-shortest-path-count-l3-s8/solution/solve.sh b/tasks/construct-rlve-shortest-path-count-l3-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..f2ee015dec23d3718eef69dea52538c070887777
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-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'
+{"n": 51, "adj": ["NNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNY", "YNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNN", "YNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNN", "NNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNN", "NNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNN", "NNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNN", "NNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNN", "NNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNN", "NNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNN", "NNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNN", "NNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNN", "NNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNYNNNNNNN", "NNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNYNNNNNNN", "NNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNYNNNNN", "NNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNYNNNNN", "NNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNYNNNN", "NNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNYNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNYNNN", "NNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNYNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNYNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNYNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNY", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNY", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNN", "NNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNN", "NNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNN", "NNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNN", "NNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNN", "NNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNN", "NNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNN", "NNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNYNYNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNYNYNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNYNYNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNYNYN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNY", "NYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNYN"]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-shortest-path-count-l3-s8/tests/check.py b/tasks/construct-rlve-shortest-path-count-l3-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..c59d4c4f4bb4e824a2019380643a5b3ade0ca974
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-l3-s8/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-l3-s8/tests/instance.json b/tasks/construct-rlve-shortest-path-count-l3-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..ef3f79b80214f358cbfe08c19fb8a87379484ac4
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-l3-s8/tests/instance.json
@@ -0,0 +1 @@
+{"K": 97002, "n_max": 105, "family": "rlve_shortest_path_count", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-shortest-path-count-l3-s8/tests/run.py b/tasks/construct-rlve-shortest-path-count-l3-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-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-shortest-path-count-l3-s8/tests/test.sh b/tasks/construct-rlve-shortest-path-count-l3-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-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-subgraph-isomorphism-l4-s0/environment/Dockerfile b/tasks/construct-rlve-subgraph-isomorphism-l4-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-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-subgraph-isomorphism-l4-s0/environment/instance.json b/tasks/construct-rlve-subgraph-isomorphism-l4-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b2c035d2bd344796e52436acf588f1ac0dc93bca
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l4-s0/environment/instance.json
@@ -0,0 +1 @@
+{"n1": 9, "n2": 18, "edges1": [[3, 7], [1, 4], [0, 6], [0, 3], [5, 6], [6, 7], [0, 1], [0, 7], [1, 2], [1, 3], [2, 4], [1, 5], [4, 7], [1, 7], [0, 4], [2, 6], [0, 8]], "edges2": [[12, 16], [4, 10], [3, 7], [5, 8], [3, 5], [0, 16], [7, 12], [8, 16], [7, 9], [2, 3], [8, 13], [0, 14], [10, 15], [2, 12], [5, 12], [5, 9], [12, 13], [12, 17], [9, 15], [0, 7], [7, 13], [0, 8], [3, 9], [11, 14], [0, 10], [11, 13], [5, 14], [15, 17], [8, 11], [14, 15], [4, 13], [2, 13], [8, 10], [6, 11], [1, 14], [2, 7], [1, 10], [6, 7], [13, 14], [4, 6], [16, 17], [6, 16], [9, 16], [1, 9], [6, 14], [10, 11], [4, 17], [4, 11], [3, 8], [10, 13], [6, 17], [5, 15], [6, 13]], "family": "rlve_subgraph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l4-s0/solution/solve.sh b/tasks/construct-rlve-subgraph-isomorphism-l4-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..b949cfb6af421f1467d433d8cd604745ca1be05c
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-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'
+{"p": [13, 10, 0, 4, 8, 15, 14, 11, 12]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l4-s0/tests/check.py b/tasks/construct-rlve-subgraph-isomorphism-l4-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..b8467dd00c27c8902c06206c376ff7a11ac78ae0
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l4-s0/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-l4-s0/tests/instance.json b/tasks/construct-rlve-subgraph-isomorphism-l4-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b2c035d2bd344796e52436acf588f1ac0dc93bca
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l4-s0/tests/instance.json
@@ -0,0 +1 @@
+{"n1": 9, "n2": 18, "edges1": [[3, 7], [1, 4], [0, 6], [0, 3], [5, 6], [6, 7], [0, 1], [0, 7], [1, 2], [1, 3], [2, 4], [1, 5], [4, 7], [1, 7], [0, 4], [2, 6], [0, 8]], "edges2": [[12, 16], [4, 10], [3, 7], [5, 8], [3, 5], [0, 16], [7, 12], [8, 16], [7, 9], [2, 3], [8, 13], [0, 14], [10, 15], [2, 12], [5, 12], [5, 9], [12, 13], [12, 17], [9, 15], [0, 7], [7, 13], [0, 8], [3, 9], [11, 14], [0, 10], [11, 13], [5, 14], [15, 17], [8, 11], [14, 15], [4, 13], [2, 13], [8, 10], [6, 11], [1, 14], [2, 7], [1, 10], [6, 7], [13, 14], [4, 6], [16, 17], [6, 16], [9, 16], [1, 9], [6, 14], [10, 11], [4, 17], [4, 11], [3, 8], [10, 13], [6, 17], [5, 15], [6, 13]], "family": "rlve_subgraph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l4-s0/tests/run.py b/tasks/construct-rlve-subgraph-isomorphism-l4-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-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-subgraph-isomorphism-l4-s0/tests/test.sh b/tasks/construct-rlve-subgraph-isomorphism-l4-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-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-subgraph-isomorphism-l4-s7/environment/Dockerfile b/tasks/construct-rlve-subgraph-isomorphism-l4-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-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-subgraph-isomorphism-l4-s7/environment/instance.json b/tasks/construct-rlve-subgraph-isomorphism-l4-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..687fe2a728ac980709bbc9513233c83ae8d69fc3
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l4-s7/environment/instance.json
@@ -0,0 +1 @@
+{"n1": 15, "n2": 18, "edges1": [[6, 7], [3, 4], [2, 4], [6, 8], [0, 11], [2, 14], [4, 13], [8, 9], [2, 3], [0, 9], [10, 14], [1, 12], [8, 11], [1, 9], [3, 10], [5, 13], [3, 7], [8, 13], [0, 1], [1, 7], [1, 3], [4, 9], [10, 11], [1, 5], [5, 8], [2, 12], [3, 9], [1, 6], [5, 12], [2, 5], [4, 10], [12, 13]], "edges2": [[7, 13], [5, 14], [2, 16], [5, 13], [14, 15], [10, 13], [5, 16], [1, 9], [3, 12], [4, 14], [11, 16], [8, 11], [0, 13], [1, 5], [4, 15], [1, 7], [13, 17], [10, 17], [5, 11], [4, 11], [3, 11], [0, 7], [2, 15], [10, 14], [0, 4], [7, 10], [1, 11], [9, 16], [0, 1], [8, 12], [2, 10], [4, 7], [11, 14], [3, 6], [4, 6], [3, 4], [6, 10], [4, 17]], "family": "rlve_subgraph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l4-s7/solution/solve.sh b/tasks/construct-rlve-subgraph-isomorphism-l4-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..41ba8b36536a07bba3ab7aae5f50eacc4b04c28d
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-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'
+{"p": [15, 4, 1, 11, 5, 7, 6, 3, 10, 14, 16, 2, 0, 13, 9]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l4-s7/tests/check.py b/tasks/construct-rlve-subgraph-isomorphism-l4-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..b8467dd00c27c8902c06206c376ff7a11ac78ae0
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l4-s7/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-l4-s7/tests/instance.json b/tasks/construct-rlve-subgraph-isomorphism-l4-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..687fe2a728ac980709bbc9513233c83ae8d69fc3
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l4-s7/tests/instance.json
@@ -0,0 +1 @@
+{"n1": 15, "n2": 18, "edges1": [[6, 7], [3, 4], [2, 4], [6, 8], [0, 11], [2, 14], [4, 13], [8, 9], [2, 3], [0, 9], [10, 14], [1, 12], [8, 11], [1, 9], [3, 10], [5, 13], [3, 7], [8, 13], [0, 1], [1, 7], [1, 3], [4, 9], [10, 11], [1, 5], [5, 8], [2, 12], [3, 9], [1, 6], [5, 12], [2, 5], [4, 10], [12, 13]], "edges2": [[7, 13], [5, 14], [2, 16], [5, 13], [14, 15], [10, 13], [5, 16], [1, 9], [3, 12], [4, 14], [11, 16], [8, 11], [0, 13], [1, 5], [4, 15], [1, 7], [13, 17], [10, 17], [5, 11], [4, 11], [3, 11], [0, 7], [2, 15], [10, 14], [0, 4], [7, 10], [1, 11], [9, 16], [0, 1], [8, 12], [2, 10], [4, 7], [11, 14], [3, 6], [4, 6], [3, 4], [6, 10], [4, 17]], "family": "rlve_subgraph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l4-s7/tests/run.py b/tasks/construct-rlve-subgraph-isomorphism-l4-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-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-subgraph-isomorphism-l4-s7/tests/test.sh b/tasks/construct-rlve-subgraph-isomorphism-l4-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-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/optimize-ea-circles-rectangle-n21/environment/Dockerfile b/tasks/optimize-ea-circles-rectangle-n21/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/optimize-ea-circles-rectangle-n21/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-ea-circles-rectangle-n21/environment/instance.json b/tasks/optimize-ea-circles-rectangle-n21/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..26c6f9e3f2c7448f2165854f1979a97fdcb145cb
--- /dev/null
+++ b/tasks/optimize-ea-circles-rectangle-n21/environment/instance.json
@@ -0,0 +1 @@
+{"n": 21, "direction": "maximize", "family": "ea_circles_rectangle_21", "subset": "optimize"}
\ No newline at end of file
diff --git a/tasks/optimize-ea-circles-rectangle-n21/solution/solve.sh b/tasks/optimize-ea-circles-rectangle-n21/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..50be8e8adb6e3153669f9d0588f9cace570ae91a
--- /dev/null
+++ b/tasks/optimize-ea-circles-rectangle-n21/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'
+{"circles": [[0.12396161, 0.12396161, 0.12396159760383901], [0.0747415, 0.31647214, 0.07474149252585001], [0.12740019, 0.51163445, 0.127400177259981], [0.0747415, 0.70679675, 0.07474149252585001], [0.12396161, 0.89930729, 0.12396159760383901], [0.36284141, 0.11508312, 0.115083108491688], [0.25916864, 0.31452627, 0.10969588903041], [0.37244262, 0.51163445, 0.117642228235776], [0.25916864, 0.70874263, 0.10969588903041], [0.3628414, 0.90818577, 0.11508311849168701], [0.59083599, 0.11292169, 0.11292167870783101], [0.48083152, 0.30914434, 0.112032298796769], [0.62891008, 0.51163445, 0.138825206117478], [0.48083152, 0.71412455, 0.112032298796769], [0.59083598, 0.9103472, 0.11292168870783001], [0.67852787, 0.2919303, 0.086412051358794], [0.85766825, 0.39257159, 0.11906284809371401], [0.85766825, 0.6306973, 0.11906284809371401], [0.67852787, 0.73133859, 0.086412051358794], [0.83965981, 0.8861976, 0.13707128629287002], [0.83965981, 0.1370713, 0.13707128629287002]]}
+RLMATH_EOF
diff --git a/tasks/optimize-ea-circles-rectangle-n21/tests/check.py b/tasks/optimize-ea-circles-rectangle-n21/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..75f56da763151ed4490b1e352b0fa9409811ddf7
--- /dev/null
+++ b/tasks/optimize-ea-circles-rectangle-n21/tests/check.py
@@ -0,0 +1,27 @@
+"""21 disjoint circles in a rectangle of perimeter 4, maximize the sum of radii
+(EinsteinArena circles-rectangle, MIT).
+
+Upstream bug fixed: the perimeter test is exact (Fractions) upstream but the disjointness test compares
+float sqrt distances; both are exact here."""
+from fractions import Fraction
+
+from run import Invalid, float_matrix
+
+
+def check(inst, ans):
+    n = inst["n"]
+    c = float_matrix(ans.get("circles"), n, 3, "circles")
+    F = [tuple(Fraction(v) for v in row) for row in c]
+    if any(r <= 0 for _, _, r in F):
+        raise Invalid("radii must be positive")
+    w = max(x + r for x, _, r in F) - min(x - r for x, _, r in F)
+    h = max(y + r for _, y, r in F) - min(y - r for _, y, r in F)
+    if w + h > 2:
+        raise Invalid("bounding rectangle violates width + height <= 2")
+    for i in range(n):
+        xi, yi, ri = F[i]
+        for j in range(i + 1, n):
+            xj, yj, rj = F[j]
+            if (xi - xj) ** 2 + (yi - yj) ** 2 < (ri + rj) ** 2:
+                raise Invalid(f"circles {i} and {j} overlap")
+    return sum(r for _, _, r in c), {}
diff --git a/tasks/optimize-ea-circles-rectangle-n21/tests/instance.json b/tasks/optimize-ea-circles-rectangle-n21/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..043fd07e65e5d8f0449348aa0085e041f7f79465
--- /dev/null
+++ b/tasks/optimize-ea-circles-rectangle-n21/tests/instance.json
@@ -0,0 +1 @@
+{"n": 21, "direction": "maximize", "best_known": 2.3658323759, "baseline": 1.96875, "family": "ea_circles_rectangle_21", "subset": "optimize"}
\ No newline at end of file
diff --git a/tasks/optimize-ea-circles-rectangle-n21/tests/run.py b/tasks/optimize-ea-circles-rectangle-n21/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/optimize-ea-circles-rectangle-n21/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-ea-circles-rectangle-n21/tests/test.sh b/tasks/optimize-ea-circles-rectangle-n21/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/optimize-ea-circles-rectangle-n21/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-ea-uncertainty-principle/environment/Dockerfile b/tasks/optimize-ea-uncertainty-principle/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/optimize-ea-uncertainty-principle/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-ea-uncertainty-principle/environment/instance.json b/tasks/optimize-ea-uncertainty-principle/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8aff289b37d435fc83d9e042aace96ce9ac2aaa9
--- /dev/null
+++ b/tasks/optimize-ea-uncertainty-principle/environment/instance.json
@@ -0,0 +1 @@
+{"max_roots": 25, "max_root": 300, "direction": "minimize", "family": "ea_uncertainty_principle", "subset": "optimize"}
\ No newline at end of file
diff --git a/tasks/optimize-ea-uncertainty-principle/solution/solve.sh b/tasks/optimize-ea-uncertainty-principle/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..6ea2123a2d48992dc90d2de2cd88cea3e908e419
--- /dev/null
+++ b/tasks/optimize-ea-uncertainty-principle/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'
+{"laguerre_double_roots": [3.64273649, 5.68246114, 33.00463486, 40.97185579, 50.1028231, 53.76768016]}
+RLMATH_EOF
diff --git a/tasks/optimize-ea-uncertainty-principle/tests/check.py b/tasks/optimize-ea-uncertainty-principle/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..8333ef5642f3cf52e0eefca66545e021b9c9fb8d
--- /dev/null
+++ b/tasks/optimize-ea-uncertainty-principle/tests/check.py
@@ -0,0 +1,151 @@
+"""Uncertainty principle upper bound via Laguerre double roots (EinsteinArena uncertainty-principle, MIT).
+
+Given double-root positions z_1..z_m, g = sum_{j=0}^{2m+1} c_j L_{2j}^{(-1/2)} with g(0)=0, g'(0)=1,
+g(z_i)=g'(z_i)=0; h = g / (x prod (x-z_i)^2) is a polynomial and the score is r/(2 pi), r = largest
+positive root of h of odd multiplicity (last sign change) (minimize).
+
+Changes vs the upstream verifier (float64 / mpmath linear solve, 200k-point float grid scan + brentq):
+* the linear system is solved in 400- and 600-digit decimal arithmetic, h is formed by exact synthetic
+  division, and the last sign change is found by exact real-root isolation (sympy) of h with
+  coefficients rounded to 60 significant digits -- no grid, so close root pairs and roots beyond the
+  upstream scan window (1.5 max z + 100) cannot be missed; the two precisions must agree;
+* NEW validity check (upstream omission): the test function -g(2 pi x^2) e^{-pi x^2} must be eventually
+  non-negative, i.e. the leading coefficient of h must be negative; otherwise the bound is not valid;
+* mpmath is not used (not in the allowed runtime set)."""
+import math
+from decimal import Decimal, localcontext
+from fractions import Fraction
+
+import sympy as sp
+
+from run import Invalid, num
+
+SIG = 60
+
+
+def _laguerre(nmax):
+    al = Fraction(-1, 2)
+    L = [[Fraction(1)], [1 + al, Fraction(-1)]]
+    for k in range(1, nmax):
+        a, b = L[k], L[k - 1]
+        nxt = [Fraction(0)] * (k + 2)
+        for i, c in enumerate(a):
+            nxt[i] += (2 * k + 1 + al) * c
+            nxt[i + 1] -= c
+        for i, c in enumerate(b):
+            nxt[i] -= (k + al) * c
+        L.append([c / (k + 1) for c in nxt])
+    return L
+
+
+def _h_coeffs(zs, prec):
+    """Coefficients (high -> low) of h, as exact rationals rounded to SIG significant digits."""
+    m = len(zs)
+    degs = list(range(0, 4 * m + 4, 2))
+    L = _laguerre(degs[-1])
+    with localcontext() as ctx:
+        ctx.prec = prec
+
+        def D(q):
+            return Decimal(q.numerator) / Decimal(q.denominator)
+
+        P = [[D(c) for c in L[d]] for d in degs]
+        Pd = [[i * c for i, c in enumerate(p)][1:] for p in P]
+        Pd[0] = [Decimal(0)]
+
+        def ev(p, x):
+            r = Decimal(0)
+            for c in reversed(p):
+                r = r * x + c
+            return r
+
+        zd = [D(z) for z in zs]
+        A = [[ev(p, Decimal(0)) for p in P] + [Decimal(0)], [ev(p, Decimal(0)) for p in Pd] + [Decimal(1)]]
+        for z in zd:
+            A.append([ev(p, z) for p in P] + [Decimal(0)])
+            A.append([ev(p, z) for p in Pd] + [Decimal(0)])
+        n = len(A)
+        for col in range(n):
+            piv = max(range(col, n), key=lambda r: abs(A[r][col]))
+            if A[piv][col] == 0:
+                raise Invalid("singular interpolation system")
+            A[col], A[piv] = A[piv], A[col]
+            pv = A[col][col]
+            for r in range(col + 1, n):
+                f = A[r][col] / pv
+                if f:
+                    A[r] = [x - f * y for x, y in zip(A[r], A[col])]
+        c = [Decimal(0)] * n
+        for r in range(n - 1, -1, -1):
+            c[r] = (A[r][n] - sum(A[r][k] * c[k] for k in range(r + 1, n))) / A[r][r]
+        g = [Decimal(0)] * (degs[-1] + 1)
+        for cj, p in zip(c, P):
+            for i, v in enumerate(p):
+                g[i] += cj * v
+        poly = list(reversed(g))
+        rems = []
+
+        def synth(pl, z):
+            out = [pl[0]]
+            for co in pl[1:]:
+                out.append(co + out[-1] * z)
+            return out[:-1], out[-1]
+
+        poly, rm = synth(poly, Decimal(0))
+        rems.append(rm)
+        for z in zd:
+            for _ in range(2):
+                poly, rm = synth(poly, z)
+                rems.append(rm)
+        scale = max(abs(x) for x in poly)
+        if scale == 0 or max(abs(r) for r in rems) > scale * Decimal(10) ** (-(prec // 3)):
+            raise Invalid("numerically unstable root configuration")
+        out = []
+        for x in poly:
+            if x == 0:
+                out.append(sp.Integer(0))
+                continue
+            q = x.adjusted() - SIG
+            out.append(sp.Integer(int(x.scaleb(-q).to_integral_value())) * sp.Rational(10) ** q)
+    return out
+
+
+def _last_sign_change(coeffs):
+    x = sp.Symbol("x")
+    h = sp.Poly(coeffs, x, domain="QQ")
+    if not h.LC() < 0:
+        raise Invalid("the test function is not eventually of the right sign (leading coefficient of h >= 0)")
+    best = None
+    for fac, mult in h.sqf_list()[1]:
+        if mult % 2 == 0:
+            continue
+        for (lo, hi), _ in fac.intervals():
+            if best is None or hi > best[2]:
+                best = (fac, lo, hi)
+    if best is None:
+        raise Invalid("no sign change")
+    fac, lo, hi = best
+    lo, hi = fac.refine_root(lo, hi, eps=sp.Rational(1, 10 ** 16))
+    r = (lo + hi) / 2
+    if not r > 0:
+        raise Invalid("no positive sign change")
+    return float(r)
+
+
+def check(inst, ans):
+    zs = ans.get("laguerre_double_roots")
+    if not isinstance(zs, list) or not 1 <= len(zs) <= inst["max_roots"]:
+        raise Invalid(f"laguerre_double_roots must be a list of 1..{inst['max_roots']} numbers")
+    Z = []
+    for z in zs:
+        v = num(z, "root")
+        if not 0 < v <= inst["max_root"]:
+            raise Invalid(f"roots must lie in (0, {inst['max_root']}]")
+        Z.append(Fraction(repr(v)))
+    if len(set(Z)) != len(Z):
+        raise Invalid("duplicate roots")
+    r1 = _last_sign_change(_h_coeffs(Z, 400))
+    r2 = _last_sign_change(_h_coeffs(Z, 600))
+    if abs(r1 - r2) > 1e-12 * max(1.0, abs(r2)):
+        raise Invalid("numerically unstable root configuration")
+    return r2 / (2 * math.pi), {"r": r2}
diff --git a/tasks/optimize-ea-uncertainty-principle/tests/instance.json b/tasks/optimize-ea-uncertainty-principle/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d97d564504d7019f5283c494869fb92aba6d897e
--- /dev/null
+++ b/tasks/optimize-ea-uncertainty-principle/tests/instance.json
@@ -0,0 +1 @@
+{"max_roots": 25, "max_root": 300, "direction": "minimize", "best_known": 0.31885, "baseline": 0.6132901966611566, "family": "ea_uncertainty_principle", "subset": "optimize"}
\ No newline at end of file
diff --git a/tasks/optimize-ea-uncertainty-principle/tests/run.py b/tasks/optimize-ea-uncertainty-principle/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/optimize-ea-uncertainty-principle/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-ea-uncertainty-principle/tests/test.sh b/tasks/optimize-ea-uncertainty-principle/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/optimize-ea-uncertainty-principle/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