diff --git a/tasks/construct-ae-p38-factorial-factors-l5-s5/environment/Dockerfile b/tasks/construct-ae-p38-factorial-factors-l5-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-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-ae-p38-factorial-factors-l5-s5/environment/instance.json b/tasks/construct-ae-p38-factorial-factors-l5-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f5ad53fc3efea9b8b05443996b9f47766ae1d978
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l5-s5/environment/instance.json
@@ -0,0 +1 @@
+{"N": 2400, "t": 764, "family": "ae_p38_factorial_factors", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-ae-p38-factorial-factors-l5-s5/solution/solve.sh b/tasks/construct-ae-p38-factorial-factors-l5-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..1c5ea6edb26b16c3c7a2a8e07fb2a0738ecd725f
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-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'
+{"factors": [764, 764, 764, 764, 764, 764, 764, 764, 764, 764, 764, 764, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 765, 766, 766, 766, 766, 766, 766, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 767, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 768, 769, 769, 769, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 770, 771, 771, 771, 771, 771, 771, 771, 771, 771, 772, 772, 772, 772, 772, 772, 772, 772, 772, 772, 772, 772, 773, 773, 773, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 774, 775, 775, 775, 775, 775, 775, 775, 775, 775, 775, 775, 775, 775, 775, 775, 775, 775, 775, 775, 775, 775, 775, 775, 775, 775, 775, 775, 775, 775, 775, 775, 775, 776, 776, 776, 776, 776, 776, 776, 776, 776, 776, 776, 776, 776, 776, 776, 776, 776, 776, 776, 776, 776, 776, 776, 776, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 777, 778, 778, 778, 778, 778, 778, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 779, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 780, 781, 781, 781, 781, 781, 781, 781, 781, 781, 781, 781, 781, 781, 781, 781, 781, 781, 781, 781, 781, 781, 781, 781, 781, 781, 781, 781, 781, 781, 781, 781, 781, 781, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 782, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 783, 785, 785, 785, 785, 785, 785, 785, 785, 785, 785, 785, 785, 785, 785, 785, 786, 786, 786, 786, 786, 786, 786, 786, 786, 786, 786, 786, 786, 786, 786, 786, 786, 786, 787, 787, 787, 788, 788, 788, 788, 788, 788, 788, 788, 788, 788, 788, 788, 789, 789, 789, 789, 789, 789, 789, 789, 789, 790, 790, 790, 790, 790, 790, 790, 790, 790, 790, 790, 790, 790, 790, 790, 790, 790, 790, 790, 790, 790, 790, 790, 790, 790, 790, 790, 790, 790, 790, 791, 791, 791, 791, 791, 791, 791, 791, 791, 791, 791, 791, 791, 791, 791, 791, 791, 791, 791, 791, 791, 792, 792, 792, 792, 792, 792, 792, 792, 792, 792, 792, 792, 792, 792, 792, 792, 792, 792, 792, 792, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 793, 794, 794, 794, 794, 794, 794, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 795, 796, 796, 796, 796, 796, 796, 796, 796, 796, 796, 796, 796, 797, 797, 797, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 798, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 799, 801, 801, 801, 801, 801, 801, 801, 801, 801, 801, 801, 801, 801, 801, 801, 801, 801, 801, 801, 801, 801, 801, 801, 801, 801, 801, 802, 802, 802, 802, 802, 803, 803, 803, 803, 803, 803, 803, 803, 803, 803, 803, 803, 803, 803, 803, 803, 803, 803, 803, 803, 803, 803, 803, 803, 803, 803, 803, 803, 803, 803, 803, 803, 804, 804, 804, 804, 804, 804, 804, 804, 804, 804, 804, 804, 804, 804, 804, 804, 804, 804, 804, 804, 804, 804, 804, 804, 804, 804, 804, 804, 804, 804, 804, 804, 804, 804, 804, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 806, 807, 807, 807, 807, 807, 807, 807, 807, 808, 808, 808, 808, 808, 808, 808, 808, 808, 808, 808, 808, 808, 808, 808, 808, 808, 808, 808, 808, 808, 808, 808, 809, 809, 811, 811, 813, 813, 813, 813, 813, 813, 813, 813, 815, 815, 815, 815, 815, 815, 815, 815, 815, 815, 815, 815, 815, 815, 818, 818, 818, 818, 818, 821, 821, 822, 822, 822, 822, 822, 822, 822, 822, 822, 822, 822, 822, 822, 822, 822, 822, 822, 823, 823, 824, 824, 824, 824, 824, 824, 824, 824, 824, 824, 824, 824, 824, 824, 824, 824, 824, 824, 824, 824, 824, 824, 824, 827, 827, 829, 829, 830, 830, 830, 830, 830, 830, 830, 830, 830, 830, 830, 830, 830, 830, 830, 830, 830, 830, 830, 830, 830, 830, 830, 830, 830, 830, 830, 830, 831, 831, 831, 831, 831, 831, 831, 831, 834, 834, 834, 834, 834, 834, 834, 834, 834, 834, 834, 834, 834, 834, 834, 834, 834, 835, 835, 835, 835, 835, 835, 835, 835, 835, 835, 835, 835, 835, 835, 838, 838, 838, 838, 838, 839, 839, 842, 842, 842, 842, 842, 843, 843, 843, 843, 843, 843, 843, 843, 844, 844, 844, 844, 844, 844, 844, 844, 844, 844, 844, 849, 849, 849, 849, 849, 849, 849, 849, 853, 853, 856, 856, 856, 856, 856, 856, 856, 856, 856, 856, 856, 856, 856, 856, 856, 856, 856, 856, 856, 856, 856, 856, 857, 857, 859, 859, 862, 862, 862, 862, 862, 863, 863, 864, 865, 865, 865, 865, 865, 865, 865, 865, 865, 865, 865, 865, 865, 866, 866, 866, 866, 866, 867, 872, 872, 872, 872, 872, 872, 872, 872, 872, 872, 872, 872, 872, 872, 872, 872, 872, 872, 872, 872, 872, 872, 877, 877, 878, 878, 878, 878, 878, 879, 879, 879, 879, 879, 879, 879, 879, 881, 881, 883, 883, 886, 886, 886, 886, 886, 887, 887, 889, 889, 889, 889, 889, 889, 889, 889, 889, 889, 889, 889, 889, 889, 889, 889, 889, 889, 892, 892, 892, 892, 892, 892, 892, 892, 892, 892, 894, 894, 894, 894, 894, 894, 894, 894, 894, 894, 894, 894, 894, 894, 894, 894, 895, 895, 895, 895, 895, 895, 895, 895, 895, 895, 895, 895, 895, 898, 898, 898, 898, 898, 905, 905, 905, 905, 905, 905, 905, 905, 905, 905, 905, 905, 905, 906, 906, 906, 906, 906, 906, 906, 906, 906, 906, 906, 906, 906, 906, 906, 907, 907, 908, 908, 908, 908, 908, 908, 908, 908, 908, 908, 911, 911, 914, 914, 914, 914, 914, 916, 916, 916, 916, 916, 916, 916, 916, 916, 916, 919, 919, 921, 921, 921, 921, 921, 921, 921, 922, 922, 922, 922, 922, 926, 926, 926, 926, 926, 929, 929, 932, 932, 932, 932, 932, 932, 932, 932, 932, 932, 933, 933, 933, 933, 933, 933, 933, 934, 934, 934, 934, 934, 937, 937, 939, 939, 939, 939, 939, 939, 939, 941, 941, 947, 947, 951, 951, 951, 951, 951, 951, 951, 953, 953, 956, 956, 956, 956, 956, 956, 956, 956, 956, 956, 958, 958, 958, 958, 958, 964, 964, 964, 964, 964, 964, 964, 964, 964, 967, 967, 971, 971, 974, 974, 974, 974, 977, 977, 982, 982, 982, 982, 983, 983, 991, 991, 993, 993, 993, 993, 993, 993, 993, 997, 997, 998, 998, 998, 998, 1004, 1004, 1004, 1004, 1004, 1004, 1004, 1004, 1004, 1006, 1006, 1006, 1006, 1009, 1009, 1011, 1011, 1011, 1011, 1011, 1011, 1011, 1013, 1013, 1018, 1018, 1018, 1018, 1019, 1019, 1021, 1021, 1031, 1031, 1033, 1033, 1037, 1039, 1039, 1041, 1041, 1041, 1041, 1041, 1041, 1042, 1042, 1042, 1042, 1046, 1046, 1046, 1046, 1047, 1047, 1047, 1047, 1047, 1047, 1049, 1049, 1051, 1051, 1059, 1059, 1059, 1059, 1059, 1059, 1061, 1061, 1063, 1063, 1069, 1069, 1077, 1077, 1077, 1077, 1077, 1077, 1081, 1082, 1082, 1082, 1082, 1087, 1087, 1089, 1091, 1091, 1093, 1093, 1094, 1094, 1094, 1094, 1097, 1097, 1101, 1101, 1101, 1101, 1101, 1101, 1103, 1103, 1109, 1109, 1114, 1114, 1114, 1114, 1117, 1117, 1119, 1119, 1119, 1119, 1119, 1119, 1123, 1123, 1126, 1126, 1126, 1126, 1129, 1129, 1137, 1137, 1137, 1137, 1137, 1137, 1138, 1138, 1138, 1138, 1142, 1142, 1142, 1142, 1151, 1151, 1153, 1153, 1154, 1154, 1154, 1154, 1163, 1163, 1171, 1171, 1174, 1174, 1174, 1174, 1181, 1181, 1186, 1186, 1186, 1186, 1187, 1187, 1193, 1193, 1198, 1198, 1198, 1198, 1201, 1202, 1202, 1202, 1213, 1214, 1214, 1214, 1217, 1223, 1226, 1226, 1226, 1229, 1231, 1234, 1234, 1234, 1237, 1238, 1238, 1238, 1249, 1259, 1262, 1262, 1262, 1277, 1279, 1282, 1282, 1282, 1283, 1286, 1286, 1286, 1289, 1291, 1294, 1294, 1294, 1297, 1301, 1303, 1306, 1306, 1306, 1307, 1318, 1318, 1318, 1319, 1321, 1322, 1322, 1322, 1327, 1346, 1346, 1346, 1354, 1354, 1354, 1361, 1366, 1366, 1366, 1367, 1373, 1381, 1382, 1382, 1382, 1399, 1402, 1402, 1402, 1409, 1418, 1418, 1418, 1423, 1427, 1429, 1433, 1438, 1438, 1438, 1439, 1447, 1451, 1453, 1454, 1454, 1454, 1459, 1466, 1466, 1466, 1471, 1478, 1478, 1478, 1481, 1483, 1486, 1486, 1486, 1487, 1489, 1493, 1499, 1502, 1502, 1502, 1511, 1514, 1514, 1514, 1522, 1522, 1522, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399]}
+RLMATH_EOF
diff --git a/tasks/construct-ae-p38-factorial-factors-l5-s5/tests/check.py b/tasks/construct-ae-p38-factorial-factors-l5-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..93ec67a19039afa8b4ce6e39811691b802132562
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l5-s5/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-l5-s5/tests/instance.json b/tasks/construct-ae-p38-factorial-factors-l5-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f5ad53fc3efea9b8b05443996b9f47766ae1d978
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-l5-s5/tests/instance.json
@@ -0,0 +1 @@
+{"N": 2400, "t": 764, "family": "ae_p38_factorial_factors", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-ae-p38-factorial-factors-l5-s5/tests/run.py b/tasks/construct-ae-p38-factorial-factors-l5-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-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-ae-p38-factorial-factors-l5-s5/tests/test.sh b/tasks/construct-ae-p38-factorial-factors-l5-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-ae-p38-factorial-factors-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-ae-p52-erdos-squarefree-l1-s6/environment/Dockerfile b/tasks/construct-ae-p52-erdos-squarefree-l1-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-ae-p52-erdos-squarefree-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-ae-p52-erdos-squarefree-l1-s6/environment/instance.json b/tasks/construct-ae-p52-erdos-squarefree-l1-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e8e401a76da2da5650de774e44d9dd826963c566
--- /dev/null
+++ b/tasks/construct-ae-p52-erdos-squarefree-l1-s6/environment/instance.json
@@ -0,0 +1 @@
+{"N": 66, "t": 3, "family": "ae_p52_erdos_squarefree", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-ae-p52-erdos-squarefree-l1-s6/solution/solve.sh b/tasks/construct-ae-p52-erdos-squarefree-l1-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..50f3f4974dadd8b33f98b78aaa992c9a3457fa09
--- /dev/null
+++ b/tasks/construct-ae-p52-erdos-squarefree-l1-s6/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"A": [7, 32, 57]}
+RLMATH_EOF
diff --git a/tasks/construct-ae-p52-erdos-squarefree-l1-s6/tests/check.py b/tasks/construct-ae-p52-erdos-squarefree-l1-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..87e15b242f211059193ba122baecdee9add99130
--- /dev/null
+++ b/tasks/construct-ae-p52-erdos-squarefree-l1-s6/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-l1-s6/tests/instance.json b/tasks/construct-ae-p52-erdos-squarefree-l1-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e8e401a76da2da5650de774e44d9dd826963c566
--- /dev/null
+++ b/tasks/construct-ae-p52-erdos-squarefree-l1-s6/tests/instance.json
@@ -0,0 +1 @@
+{"N": 66, "t": 3, "family": "ae_p52_erdos_squarefree", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-ae-p52-erdos-squarefree-l1-s6/tests/run.py b/tasks/construct-ae-p52-erdos-squarefree-l1-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-ae-p52-erdos-squarefree-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-ae-p52-erdos-squarefree-l1-s6/tests/test.sh b/tasks/construct-ae-p52-erdos-squarefree-l1-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-ae-p52-erdos-squarefree-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-ae-p52-erdos-squarefree-l4-s0/environment/Dockerfile b/tasks/construct-ae-p52-erdos-squarefree-l4-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-ae-p52-erdos-squarefree-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-ae-p52-erdos-squarefree-l4-s0/environment/instance.json b/tasks/construct-ae-p52-erdos-squarefree-l4-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..435a39b0a41c19e76c3cba59fbc860254bc5cc4f
--- /dev/null
+++ b/tasks/construct-ae-p52-erdos-squarefree-l4-s0/environment/instance.json
@@ -0,0 +1 @@
+{"N": 2767, "t": 111, "family": "ae_p52_erdos_squarefree", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-ae-p52-erdos-squarefree-l4-s0/solution/solve.sh b/tasks/construct-ae-p52-erdos-squarefree-l4-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..62e770dc0bc03d1c62954304f1205a0d4f03d600
--- /dev/null
+++ b/tasks/construct-ae-p52-erdos-squarefree-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'
+{"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, 1082, 1107, 1132, 1157, 1182, 1207, 1232, 1257, 1282, 1307, 1332, 1357, 1382, 1407, 1432, 1457, 1482, 1507, 1532, 1557, 1582, 1607, 1632, 1657, 1682, 1707, 1732, 1757, 1782, 1807, 1832, 1857, 1882, 1907, 1932, 1957, 1982, 2007, 2032, 2057, 2082, 2107, 2132, 2157, 2182, 2207, 2232, 2257, 2282, 2307, 2332, 2357, 2382, 2407, 2432, 2457, 2482, 2507, 2532, 2557, 2582, 2607, 2632, 2657, 2682, 2707, 2732, 2757]}
+RLMATH_EOF
diff --git a/tasks/construct-ae-p52-erdos-squarefree-l4-s0/tests/check.py b/tasks/construct-ae-p52-erdos-squarefree-l4-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..87e15b242f211059193ba122baecdee9add99130
--- /dev/null
+++ b/tasks/construct-ae-p52-erdos-squarefree-l4-s0/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-l4-s0/tests/instance.json b/tasks/construct-ae-p52-erdos-squarefree-l4-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..435a39b0a41c19e76c3cba59fbc860254bc5cc4f
--- /dev/null
+++ b/tasks/construct-ae-p52-erdos-squarefree-l4-s0/tests/instance.json
@@ -0,0 +1 @@
+{"N": 2767, "t": 111, "family": "ae_p52_erdos_squarefree", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-ae-p52-erdos-squarefree-l4-s0/tests/run.py b/tasks/construct-ae-p52-erdos-squarefree-l4-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-ae-p52-erdos-squarefree-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-ae-p52-erdos-squarefree-l4-s0/tests/test.sh b/tasks/construct-ae-p52-erdos-squarefree-l4-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-ae-p52-erdos-squarefree-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-all-interval-l4-s2/environment/Dockerfile b/tasks/construct-mc-all-interval-l4-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-all-interval-l4-s2/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-all-interval-l4-s2/environment/instance.json b/tasks/construct-mc-all-interval-l4-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a4c6847f73d4019f24865550da8234643beae5f1
--- /dev/null
+++ b/tasks/construct-mc-all-interval-l4-s2/environment/instance.json
@@ -0,0 +1 @@
+{"n": 24, "fixed": [], "family": "mc_all_interval", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-all-interval-l4-s2/solution/solve.sh b/tasks/construct-mc-all-interval-l4-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..b880d08403ebb792fba8f94351b30f9659144b85
--- /dev/null
+++ b/tasks/construct-mc-all-interval-l4-s2/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": [0, 23, 1, 22, 2, 21, 3, 20, 4, 19, 5, 18, 6, 17, 7, 16, 8, 15, 9, 14, 10, 13, 11, 12]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-all-interval-l4-s2/tests/check.py b/tasks/construct-mc-all-interval-l4-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2e39eec264b6422a47e35921b8cdbe1fa1748307
--- /dev/null
+++ b/tasks/construct-mc-all-interval-l4-s2/tests/check.py
@@ -0,0 +1,17 @@
+"""All-interval series of size n (MathConstraint all_interval, CC-BY-4.0). Accepts any valid series."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n = inst["n"]
+    x = int_list(ans.get("x"), n, 0, n - 1, "x")
+    if sorted(x) != list(range(n)):
+        raise Invalid("x is not a permutation of 0..n-1")
+    d = [abs(x[i + 1] - x[i]) for i in range(n - 1)]
+    if sorted(d) != list(range(1, n)):
+        raise Invalid("consecutive differences are not a permutation of 1..n-1")
+    seqs = {"x": x, "d": d}
+    for name, i, v in inst.get("fixed", []):
+        if seqs[name][i] != v:
+            raise Invalid(f"fixed value {name}[{i}] = {v} not respected")
+    return None, {}
diff --git a/tasks/construct-mc-all-interval-l4-s2/tests/instance.json b/tasks/construct-mc-all-interval-l4-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a4c6847f73d4019f24865550da8234643beae5f1
--- /dev/null
+++ b/tasks/construct-mc-all-interval-l4-s2/tests/instance.json
@@ -0,0 +1 @@
+{"n": 24, "fixed": [], "family": "mc_all_interval", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-all-interval-l4-s2/tests/run.py b/tasks/construct-mc-all-interval-l4-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-all-interval-l4-s2/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-all-interval-l4-s2/tests/test.sh b/tasks/construct-mc-all-interval-l4-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-all-interval-l4-s2/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-debruijn-l3-s1/environment/Dockerfile b/tasks/construct-mc-debruijn-l3-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-debruijn-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-mc-debruijn-l3-s1/environment/instance.json b/tasks/construct-mc-debruijn-l3-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..91f28787a9513d5c45f9ded1d81ff9091303acd8
--- /dev/null
+++ b/tasks/construct-mc-debruijn-l3-s1/environment/instance.json
@@ -0,0 +1 @@
+{"b": 5, "n": 2, "family": "mc_debruijn", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-debruijn-l3-s1/solution/solve.sh b/tasks/construct-mc-debruijn-l3-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..306ac010cd28e431cbc2abdb1a2c82a727509eb6
--- /dev/null
+++ b/tasks/construct-mc-debruijn-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'
+{"seq": [0, 0, 1, 0, 2, 0, 3, 0, 4, 1, 1, 2, 1, 3, 1, 4, 2, 2, 3, 2, 4, 3, 3, 4, 4]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-debruijn-l3-s1/tests/check.py b/tasks/construct-mc-debruijn-l3-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..c9d6f0177d2b3bde8306d74939ee9704c1501a23
--- /dev/null
+++ b/tasks/construct-mc-debruijn-l3-s1/tests/check.py
@@ -0,0 +1,15 @@
+"""De Bruijn sequence B(b, n) (MathConstraint debruijn, CC-BY-4.0). Accepts any valid cyclic sequence."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    b, n = inst["b"], inst["n"]
+    m = b ** n
+    x = int_list(ans.get("seq"), m, 0, b - 1, "seq")
+    seen = set()
+    for i in range(m):
+        w = tuple(x[(i + k) % m] for k in range(n))
+        if w in seen:
+            raise Invalid(f"window {w} occurs twice")
+        seen.add(w)
+    return None, {}
diff --git a/tasks/construct-mc-debruijn-l3-s1/tests/instance.json b/tasks/construct-mc-debruijn-l3-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..91f28787a9513d5c45f9ded1d81ff9091303acd8
--- /dev/null
+++ b/tasks/construct-mc-debruijn-l3-s1/tests/instance.json
@@ -0,0 +1 @@
+{"b": 5, "n": 2, "family": "mc_debruijn", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-debruijn-l3-s1/tests/run.py b/tasks/construct-mc-debruijn-l3-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-debruijn-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-mc-debruijn-l3-s1/tests/test.sh b/tasks/construct-mc-debruijn-l3-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-debruijn-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-mc-golomb-l1-s0/environment/Dockerfile b/tasks/construct-mc-golomb-l1-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-golomb-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-golomb-l1-s0/environment/instance.json b/tasks/construct-mc-golomb-l1-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e9d3b935f3db8fbb833bd24e94ee61b7adbb013a
--- /dev/null
+++ b/tasks/construct-mc-golomb-l1-s0/environment/instance.json
@@ -0,0 +1 @@
+{"n": 5, "max_length": 11, "fixed": [], "family": "mc_golomb", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-golomb-l1-s0/solution/solve.sh b/tasks/construct-mc-golomb-l1-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..bd6efa4f1d02614cf61aa098484e4ba90f64e6b5
--- /dev/null
+++ b/tasks/construct-mc-golomb-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'
+{"marks": [0, 1, 4, 9, 11]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-golomb-l1-s0/tests/check.py b/tasks/construct-mc-golomb-l1-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..84aedd6e33fee7271543bfa49ab05128e23c31ee
--- /dev/null
+++ b/tasks/construct-mc-golomb-l1-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-l1-s0/tests/instance.json b/tasks/construct-mc-golomb-l1-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e9d3b935f3db8fbb833bd24e94ee61b7adbb013a
--- /dev/null
+++ b/tasks/construct-mc-golomb-l1-s0/tests/instance.json
@@ -0,0 +1 @@
+{"n": 5, "max_length": 11, "fixed": [], "family": "mc_golomb", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-golomb-l1-s0/tests/run.py b/tasks/construct-mc-golomb-l1-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-golomb-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-golomb-l1-s0/tests/test.sh b/tasks/construct-mc-golomb-l1-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-golomb-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-golomb-l5-s1/environment/Dockerfile b/tasks/construct-mc-golomb-l5-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-golomb-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-golomb-l5-s1/environment/instance.json b/tasks/construct-mc-golomb-l5-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3dea60364421e60309fc40182edc119d5102d041
--- /dev/null
+++ b/tasks/construct-mc-golomb-l5-s1/environment/instance.json
@@ -0,0 +1 @@
+{"n": 14, "max_length": 127, "fixed": [], "family": "mc_golomb", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-golomb-l5-s1/solution/solve.sh b/tasks/construct-mc-golomb-l5-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..1b9c99c2026919425a3613ed19bad39337275d7c
--- /dev/null
+++ b/tasks/construct-mc-golomb-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'
+{"marks": [0, 4, 6, 20, 35, 52, 59, 77, 78, 86, 89, 99, 122, 127]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-golomb-l5-s1/tests/check.py b/tasks/construct-mc-golomb-l5-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..84aedd6e33fee7271543bfa49ab05128e23c31ee
--- /dev/null
+++ b/tasks/construct-mc-golomb-l5-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-l5-s1/tests/instance.json b/tasks/construct-mc-golomb-l5-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3dea60364421e60309fc40182edc119d5102d041
--- /dev/null
+++ b/tasks/construct-mc-golomb-l5-s1/tests/instance.json
@@ -0,0 +1 @@
+{"n": 14, "max_length": 127, "fixed": [], "family": "mc_golomb", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-golomb-l5-s1/tests/run.py b/tasks/construct-mc-golomb-l5-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-golomb-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-golomb-l5-s1/tests/test.sh b/tasks/construct-mc-golomb-l5-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-golomb-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-graceful-graph-l2-s2/environment/Dockerfile b/tasks/construct-mc-graceful-graph-l2-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-graceful-graph-l2-s2/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-graceful-graph-l2-s2/environment/instance.json b/tasks/construct-mc-graceful-graph-l2-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..13ae486d53521f54d1916e3c208055f21bc32cd8
--- /dev/null
+++ b/tasks/construct-mc-graceful-graph-l2-s2/environment/instance.json
@@ -0,0 +1 @@
+{"k": 2, "p": 4, "num_vertices": 8, "edges": [[0, 1], [2, 3], [4, 5], [6, 7], [0, 2], [1, 3], [2, 4], [3, 5], [4, 6], [5, 7]], "fixed": [], "family": "mc_graceful_graph", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-graceful-graph-l2-s2/solution/solve.sh b/tasks/construct-mc-graceful-graph-l2-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..90b9fb78b7e8b595c18a098e4d2fe73dad6d6fe9
--- /dev/null
+++ b/tasks/construct-mc-graceful-graph-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'
+{"labels": [0, 2, 10, 3, 1, 7, 9, 4]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-graceful-graph-l2-s2/tests/instance.json b/tasks/construct-mc-graceful-graph-l2-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..13ae486d53521f54d1916e3c208055f21bc32cd8
--- /dev/null
+++ b/tasks/construct-mc-graceful-graph-l2-s2/tests/instance.json
@@ -0,0 +1 @@
+{"k": 2, "p": 4, "num_vertices": 8, "edges": [[0, 1], [2, 3], [4, 5], [6, 7], [0, 2], [1, 3], [2, 4], [3, 5], [4, 6], [5, 7]], "fixed": [], "family": "mc_graceful_graph", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-graceful-graph-l2-s2/tests/run.py b/tasks/construct-mc-graceful-graph-l2-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-graceful-graph-l2-s2/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-graceful-graph-l2-s2/tests/test.sh b/tasks/construct-mc-graceful-graph-l2-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-graceful-graph-l2-s2/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-graph-k-coloring-l3-s3/environment/Dockerfile b/tasks/construct-mc-graph-k-coloring-l3-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l3-s3/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-graph-k-coloring-l3-s3/environment/instance.json b/tasks/construct-mc-graph-k-coloring-l3-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e8d799fda5932b63b93f5fadfa51cb50fcefd362
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l3-s3/environment/instance.json
@@ -0,0 +1 @@
+{"n": 45, "k": 4, "edges": [[0, 4], [0, 5], [0, 31], [0, 44], [1, 3], [1, 8], [1, 18], [1, 21], [1, 33], [2, 11], [2, 13], [2, 22], [2, 23], [2, 27], [2, 30], [2, 31], [2, 41], [3, 6], [3, 9], [3, 17], [3, 21], [3, 25], [3, 28], [3, 35], [3, 42], [3, 43], [4, 6], [4, 7], [4, 8], [4, 11], [4, 18], [4, 22], [4, 27], [4, 35], [5, 30], [5, 38], [6, 9], [6, 15], [6, 17], [6, 27], [6, 33], [6, 37], [6, 41], [7, 11], [7, 25], [7, 39], [8, 15], [8, 17], [8, 24], [8, 25], [8, 26], [8, 34], [8, 44], [9, 15], [9, 25], [9, 31], [10, 12], [10, 23], [10, 29], [11, 12], [11, 16], [11, 17], [11, 24], [11, 25], [11, 28], [11, 35], [11, 36], [12, 14], [12, 27], [12, 28], [12, 39], [12, 41], [13, 16], [13, 25], [13, 27], [13, 36], [13, 42], [13, 43], [14, 15], [14, 17], [14, 31], [14, 37], [15, 22], [15, 26], [15, 29], [15, 30], [15, 36], [15, 39], [16, 17], [16, 26], [16, 30], [16, 33], [16, 40], [16, 43], [17, 18], [17, 38], [18, 19], [18, 20], [18, 35], [18, 36], [18, 38], [19, 32], [20, 23], [20, 26], [20, 34], [21, 40], [21, 42], [21, 43], [22, 26], [22, 35], [22, 42], [23, 24], [23, 37], [23, 39], [24, 28], [24, 41], [25, 37], [25, 39], [25, 43], [26, 28], [26, 31], [26, 34], [27, 29], [27, 38], [27, 40], [28, 39], [29, 30], [30, 31], [30, 36], [30, 42], [31, 33], [31, 38], [33, 41], [34, 39], [35, 36], [35, 44], [36, 37], [37, 42], [37, 44], [39, 44], [42, 44]], "family": "mc_graph_k_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-graph-k-coloring-l3-s3/solution/solve.sh b/tasks/construct-mc-graph-k-coloring-l3-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..6169b2851c56949f507199f6031f3cec5d24961b
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-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'
+{"coloring": [0, 0, 0, 1, 1, 1, 0, 0, 2, 2, 0, 2, 1, 1, 0, 1, 0, 3, 2, 0, 0, 3, 2, 1, 3, 3, 3, 2, 0, 3, 2, 1, 1, 3, 1, 0, 3, 2, 0, 2, 1, 2, 0, 2, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-graph-k-coloring-l3-s3/tests/check.py b/tasks/construct-mc-graph-k-coloring-l3-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2b5f38cd0c22ac1b4ba37c0d3ff0a34635f08854
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l3-s3/tests/check.py
@@ -0,0 +1,11 @@
+"""Proper k-colouring of a given graph (MathConstraint graph_k_coloring, CC-BY-4.0). Any valid colouring."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    c = int_list(ans.get("coloring"), n, 0, k - 1, "coloring")
+    for u, v in inst["edges"]:
+        if c[u] == c[v]:
+            raise Invalid(f"edge ({u},{v}) is monochromatic")
+    return None, {}
diff --git a/tasks/construct-mc-graph-k-coloring-l3-s3/tests/instance.json b/tasks/construct-mc-graph-k-coloring-l3-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e8d799fda5932b63b93f5fadfa51cb50fcefd362
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l3-s3/tests/instance.json
@@ -0,0 +1 @@
+{"n": 45, "k": 4, "edges": [[0, 4], [0, 5], [0, 31], [0, 44], [1, 3], [1, 8], [1, 18], [1, 21], [1, 33], [2, 11], [2, 13], [2, 22], [2, 23], [2, 27], [2, 30], [2, 31], [2, 41], [3, 6], [3, 9], [3, 17], [3, 21], [3, 25], [3, 28], [3, 35], [3, 42], [3, 43], [4, 6], [4, 7], [4, 8], [4, 11], [4, 18], [4, 22], [4, 27], [4, 35], [5, 30], [5, 38], [6, 9], [6, 15], [6, 17], [6, 27], [6, 33], [6, 37], [6, 41], [7, 11], [7, 25], [7, 39], [8, 15], [8, 17], [8, 24], [8, 25], [8, 26], [8, 34], [8, 44], [9, 15], [9, 25], [9, 31], [10, 12], [10, 23], [10, 29], [11, 12], [11, 16], [11, 17], [11, 24], [11, 25], [11, 28], [11, 35], [11, 36], [12, 14], [12, 27], [12, 28], [12, 39], [12, 41], [13, 16], [13, 25], [13, 27], [13, 36], [13, 42], [13, 43], [14, 15], [14, 17], [14, 31], [14, 37], [15, 22], [15, 26], [15, 29], [15, 30], [15, 36], [15, 39], [16, 17], [16, 26], [16, 30], [16, 33], [16, 40], [16, 43], [17, 18], [17, 38], [18, 19], [18, 20], [18, 35], [18, 36], [18, 38], [19, 32], [20, 23], [20, 26], [20, 34], [21, 40], [21, 42], [21, 43], [22, 26], [22, 35], [22, 42], [23, 24], [23, 37], [23, 39], [24, 28], [24, 41], [25, 37], [25, 39], [25, 43], [26, 28], [26, 31], [26, 34], [27, 29], [27, 38], [27, 40], [28, 39], [29, 30], [30, 31], [30, 36], [30, 42], [31, 33], [31, 38], [33, 41], [34, 39], [35, 36], [35, 44], [36, 37], [37, 42], [37, 44], [39, 44], [42, 44]], "family": "mc_graph_k_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-graph-k-coloring-l3-s3/tests/run.py b/tasks/construct-mc-graph-k-coloring-l3-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l3-s3/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-graph-k-coloring-l3-s3/tests/test.sh b/tasks/construct-mc-graph-k-coloring-l3-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l3-s3/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-graph-k-coloring-l3-s4/environment/Dockerfile b/tasks/construct-mc-graph-k-coloring-l3-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l3-s4/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-graph-k-coloring-l3-s4/environment/instance.json b/tasks/construct-mc-graph-k-coloring-l3-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e96fa02a86b0394725b9d449801fd2951121379b
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l3-s4/environment/instance.json
@@ -0,0 +1 @@
+{"n": 50, "k": 4, "edges": [[0, 7], [0, 11], [0, 12], [0, 39], [0, 42], [0, 46], [1, 3], [1, 8], [1, 9], [1, 20], [1, 26], [1, 30], [1, 33], [1, 48], [2, 8], [2, 17], [2, 24], [2, 27], [2, 32], [2, 39], [3, 10], [3, 16], [3, 22], [3, 25], [3, 27], [3, 30], [3, 33], [3, 43], [3, 49], [4, 21], [4, 23], [5, 11], [5, 17], [5, 23], [5, 28], [5, 32], [5, 37], [5, 39], [6, 15], [6, 16], [6, 20], [6, 21], [6, 33], [6, 41], [6, 48], [7, 16], [7, 18], [7, 27], [7, 28], [7, 30], [7, 45], [8, 10], [8, 13], [8, 17], [8, 25], [8, 33], [9, 20], [9, 24], [9, 31], [9, 33], [9, 35], [9, 38], [9, 44], [10, 19], [10, 24], [10, 28], [10, 34], [10, 38], [10, 40], [10, 48], [11, 15], [11, 23], [11, 25], [11, 29], [11, 30], [12, 29], [12, 36], [12, 37], [13, 19], [13, 43], [14, 21], [14, 39], [14, 44], [14, 47], [15, 20], [15, 34], [15, 45], [16, 23], [17, 29], [17, 38], [17, 41], [17, 42], [17, 46], [18, 35], [18, 40], [18, 45], [18, 47], [19, 28], [19, 44], [20, 24], [20, 25], [20, 43], [20, 47], [21, 27], [21, 29], [21, 30], [21, 35], [22, 24], [22, 47], [23, 24], [24, 35], [25, 27], [25, 47], [26, 30], [26, 31], [26, 38], [26, 39], [26, 46], [26, 47], [26, 49], [27, 32], [27, 35], [29, 48], [30, 31], [30, 46], [31, 34], [31, 37], [31, 40], [31, 43], [32, 46], [34, 37], [34, 40], [34, 43], [36, 37], [36, 38], [37, 39], [37, 47], [38, 47], [38, 49], [40, 42], [40, 49], [44, 46], [44, 47], [47, 48]], "family": "mc_graph_k_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-graph-k-coloring-l3-s4/solution/solve.sh b/tasks/construct-mc-graph-k-coloring-l3-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..266f293bc140cbeabc6fb729ccb1839656df31bf
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-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'
+{"coloring": [0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 2, 2, 2, 0, 1, 3, 1, 0, 3, 2, 0, 2, 2, 2, 0, 3, 0, 3, 2, 1, 3, 0, 2, 3, 1, 2, 1, 1, 2, 2, 3, 1, 1, 2, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-graph-k-coloring-l3-s4/tests/check.py b/tasks/construct-mc-graph-k-coloring-l3-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2b5f38cd0c22ac1b4ba37c0d3ff0a34635f08854
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l3-s4/tests/check.py
@@ -0,0 +1,11 @@
+"""Proper k-colouring of a given graph (MathConstraint graph_k_coloring, CC-BY-4.0). Any valid colouring."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    c = int_list(ans.get("coloring"), n, 0, k - 1, "coloring")
+    for u, v in inst["edges"]:
+        if c[u] == c[v]:
+            raise Invalid(f"edge ({u},{v}) is monochromatic")
+    return None, {}
diff --git a/tasks/construct-mc-graph-k-coloring-l3-s4/tests/instance.json b/tasks/construct-mc-graph-k-coloring-l3-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e96fa02a86b0394725b9d449801fd2951121379b
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l3-s4/tests/instance.json
@@ -0,0 +1 @@
+{"n": 50, "k": 4, "edges": [[0, 7], [0, 11], [0, 12], [0, 39], [0, 42], [0, 46], [1, 3], [1, 8], [1, 9], [1, 20], [1, 26], [1, 30], [1, 33], [1, 48], [2, 8], [2, 17], [2, 24], [2, 27], [2, 32], [2, 39], [3, 10], [3, 16], [3, 22], [3, 25], [3, 27], [3, 30], [3, 33], [3, 43], [3, 49], [4, 21], [4, 23], [5, 11], [5, 17], [5, 23], [5, 28], [5, 32], [5, 37], [5, 39], [6, 15], [6, 16], [6, 20], [6, 21], [6, 33], [6, 41], [6, 48], [7, 16], [7, 18], [7, 27], [7, 28], [7, 30], [7, 45], [8, 10], [8, 13], [8, 17], [8, 25], [8, 33], [9, 20], [9, 24], [9, 31], [9, 33], [9, 35], [9, 38], [9, 44], [10, 19], [10, 24], [10, 28], [10, 34], [10, 38], [10, 40], [10, 48], [11, 15], [11, 23], [11, 25], [11, 29], [11, 30], [12, 29], [12, 36], [12, 37], [13, 19], [13, 43], [14, 21], [14, 39], [14, 44], [14, 47], [15, 20], [15, 34], [15, 45], [16, 23], [17, 29], [17, 38], [17, 41], [17, 42], [17, 46], [18, 35], [18, 40], [18, 45], [18, 47], [19, 28], [19, 44], [20, 24], [20, 25], [20, 43], [20, 47], [21, 27], [21, 29], [21, 30], [21, 35], [22, 24], [22, 47], [23, 24], [24, 35], [25, 27], [25, 47], [26, 30], [26, 31], [26, 38], [26, 39], [26, 46], [26, 47], [26, 49], [27, 32], [27, 35], [29, 48], [30, 31], [30, 46], [31, 34], [31, 37], [31, 40], [31, 43], [32, 46], [34, 37], [34, 40], [34, 43], [36, 37], [36, 38], [37, 39], [37, 47], [38, 47], [38, 49], [40, 42], [40, 49], [44, 46], [44, 47], [47, 48]], "family": "mc_graph_k_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-graph-k-coloring-l3-s4/tests/run.py b/tasks/construct-mc-graph-k-coloring-l3-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l3-s4/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-graph-k-coloring-l3-s4/tests/test.sh b/tasks/construct-mc-graph-k-coloring-l3-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-graph-k-coloring-l3-s4/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-langford-l3-s3/environment/Dockerfile b/tasks/construct-mc-langford-l3-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-langford-l3-s3/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-langford-l3-s3/environment/instance.json b/tasks/construct-mc-langford-l3-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f0b5141d74a7fc9247fd6d82d614cb3d267cb72a
--- /dev/null
+++ b/tasks/construct-mc-langford-l3-s3/environment/instance.json
@@ -0,0 +1 @@
+{"n": 20, "fixed": [], "family": "mc_langford", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-langford-l3-s3/solution/solve.sh b/tasks/construct-mc-langford-l3-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..cbd33a93268d223cdc959c30327bbf2719eb28f0
--- /dev/null
+++ b/tasks/construct-mc-langford-l3-s3/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"seq": [13, 7, 9, 1, 10, 1, 19, 14, 16, 7, 2, 11, 9, 2, 13, 10, 18, 12, 20, 8, 17, 15, 14, 11, 6, 16, 19, 5, 8, 4, 12, 6, 3, 5, 4, 18, 3, 15, 17, 20]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-langford-l3-s3/tests/check.py b/tasks/construct-mc-langford-l3-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..4672d7eacf321991933cb280d6c940f1c4a67935
--- /dev/null
+++ b/tasks/construct-mc-langford-l3-s3/tests/check.py
@@ -0,0 +1,17 @@
+"""Langford sequence L(2, n) (MathConstraint langford, CC-BY-4.0). Accepts any valid sequence."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n = inst["n"]
+    seq = int_list(ans.get("seq"), 2 * n, 1, n, "seq")
+    for v in range(1, n + 1):
+        pos = [i for i, s in enumerate(seq) if s == v]
+        if len(pos) != 2:
+            raise Invalid(f"value {v} occurs {len(pos)} times (need exactly 2)")
+        if pos[1] - pos[0] != v + 1:
+            raise Invalid(f"the two copies of {v} are {pos[1] - pos[0]} positions apart (need {v + 1})")
+    for name, i, v in inst.get("fixed", []):
+        if seq[i] != v:
+            raise Invalid(f"fixed value {name}[{i}] = {v} not respected")
+    return None, {}
diff --git a/tasks/construct-mc-langford-l3-s3/tests/instance.json b/tasks/construct-mc-langford-l3-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f0b5141d74a7fc9247fd6d82d614cb3d267cb72a
--- /dev/null
+++ b/tasks/construct-mc-langford-l3-s3/tests/instance.json
@@ -0,0 +1 @@
+{"n": 20, "fixed": [], "family": "mc_langford", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-langford-l3-s3/tests/run.py b/tasks/construct-mc-langford-l3-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-langford-l3-s3/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-langford-l3-s3/tests/test.sh b/tasks/construct-mc-langford-l3-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-langford-l3-s3/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-langford-l6-s5/environment/Dockerfile b/tasks/construct-mc-langford-l6-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-langford-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-mc-langford-l6-s5/environment/instance.json b/tasks/construct-mc-langford-l6-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e00094a8f7ed9867c0c089042e45eda214443c8b
--- /dev/null
+++ b/tasks/construct-mc-langford-l6-s5/environment/instance.json
@@ -0,0 +1 @@
+{"n": 48, "fixed": [], "family": "mc_langford", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-langford-l6-s5/solution/solve.sh b/tasks/construct-mc-langford-l6-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..7b791e4e45cc59af03b6fd9c928d7eadac19dba0
--- /dev/null
+++ b/tasks/construct-mc-langford-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'
+{"seq": [44, 21, 32, 43, 26, 27, 29, 36, 46, 20, 23, 9, 6, 11, 3, 1, 5, 1, 3, 6, 35, 9, 5, 21, 38, 11, 34, 39, 24, 30, 20, 26, 7, 27, 23, 32, 29, 45, 48, 40, 7, 37, 47, 41, 36, 44, 19, 43, 42, 18, 31, 17, 25, 24, 16, 46, 35, 15, 33, 14, 30, 34, 13, 38, 12, 28, 19, 39, 18, 17, 10, 16, 22, 15, 14, 8, 13, 12, 25, 37, 40, 10, 31, 45, 8, 41, 2, 48, 4, 2, 47, 42, 33, 4, 28, 22]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-langford-l6-s5/tests/check.py b/tasks/construct-mc-langford-l6-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..4672d7eacf321991933cb280d6c940f1c4a67935
--- /dev/null
+++ b/tasks/construct-mc-langford-l6-s5/tests/check.py
@@ -0,0 +1,17 @@
+"""Langford sequence L(2, n) (MathConstraint langford, CC-BY-4.0). Accepts any valid sequence."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n = inst["n"]
+    seq = int_list(ans.get("seq"), 2 * n, 1, n, "seq")
+    for v in range(1, n + 1):
+        pos = [i for i, s in enumerate(seq) if s == v]
+        if len(pos) != 2:
+            raise Invalid(f"value {v} occurs {len(pos)} times (need exactly 2)")
+        if pos[1] - pos[0] != v + 1:
+            raise Invalid(f"the two copies of {v} are {pos[1] - pos[0]} positions apart (need {v + 1})")
+    for name, i, v in inst.get("fixed", []):
+        if seq[i] != v:
+            raise Invalid(f"fixed value {name}[{i}] = {v} not respected")
+    return None, {}
diff --git a/tasks/construct-mc-langford-l6-s5/tests/instance.json b/tasks/construct-mc-langford-l6-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e00094a8f7ed9867c0c089042e45eda214443c8b
--- /dev/null
+++ b/tasks/construct-mc-langford-l6-s5/tests/instance.json
@@ -0,0 +1 @@
+{"n": 48, "fixed": [], "family": "mc_langford", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-langford-l6-s5/tests/run.py b/tasks/construct-mc-langford-l6-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-langford-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-mc-langford-l6-s5/tests/test.sh b/tasks/construct-mc-langford-l6-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-langford-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-mc-low-autocorrelation-l1-s1/environment/Dockerfile b/tasks/construct-mc-low-autocorrelation-l1-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-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-low-autocorrelation-l1-s1/environment/instance.json b/tasks/construct-mc-low-autocorrelation-l1-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0223d7e0736631a049afd966d0e35e73d9fcc073
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-l1-s1/environment/instance.json
@@ -0,0 +1 @@
+{"n": 12, "bound": 25, "fixed": [], "family": "mc_low_autocorrelation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-low-autocorrelation-l1-s1/solution/solve.sh b/tasks/construct-mc-low-autocorrelation-l1-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..90be1e0093037ff225095ffe6f71be37918a9653
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-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'
+{"seq": [-1, 1, 1, -1, 1, 1, 1, 1, -1, -1, -1, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-low-autocorrelation-l1-s1/tests/check.py b/tasks/construct-mc-low-autocorrelation-l1-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..8456f7b1c0ff0c9fb4bc9ae5a49074c8afa7d1aa
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-l1-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-l1-s1/tests/instance.json b/tasks/construct-mc-low-autocorrelation-l1-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0223d7e0736631a049afd966d0e35e73d9fcc073
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-l1-s1/tests/instance.json
@@ -0,0 +1 @@
+{"n": 12, "bound": 25, "fixed": [], "family": "mc_low_autocorrelation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-low-autocorrelation-l1-s1/tests/run.py b/tasks/construct-mc-low-autocorrelation-l1-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-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-low-autocorrelation-l1-s1/tests/test.sh b/tasks/construct-mc-low-autocorrelation-l1-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-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-low-autocorrelation-l5-s0/environment/Dockerfile b/tasks/construct-mc-low-autocorrelation-l5-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-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-low-autocorrelation-l5-s0/environment/instance.json b/tasks/construct-mc-low-autocorrelation-l5-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..6ce14db4fcc9a997ed88f7a55de6a373d196f847
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-l5-s0/environment/instance.json
@@ -0,0 +1 @@
+{"n": 36, "bound": 82, "fixed": [], "family": "mc_low_autocorrelation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-low-autocorrelation-l5-s0/solution/solve.sh b/tasks/construct-mc-low-autocorrelation-l5-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..c34f8c453fbe7905054b9b68e4da60dee25888a3
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-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'
+{"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]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-low-autocorrelation-l5-s0/tests/check.py b/tasks/construct-mc-low-autocorrelation-l5-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..8456f7b1c0ff0c9fb4bc9ae5a49074c8afa7d1aa
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-l5-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-l5-s0/tests/instance.json b/tasks/construct-mc-low-autocorrelation-l5-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..6ce14db4fcc9a997ed88f7a55de6a373d196f847
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-l5-s0/tests/instance.json
@@ -0,0 +1 @@
+{"n": 36, "bound": 82, "fixed": [], "family": "mc_low_autocorrelation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-low-autocorrelation-l5-s0/tests/run.py b/tasks/construct-mc-low-autocorrelation-l5-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-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-low-autocorrelation-l5-s0/tests/test.sh b/tasks/construct-mc-low-autocorrelation-l5-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-low-autocorrelation-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-magic-sequence-l2-s1/environment/Dockerfile b/tasks/construct-mc-magic-sequence-l2-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-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-magic-sequence-l2-s1/environment/instance.json b/tasks/construct-mc-magic-sequence-l2-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..658d44b26538ace79c899ff279699b6071261bff
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-l2-s1/environment/instance.json
@@ -0,0 +1 @@
+{"n": 9, "fixed": [], "family": "mc_magic_sequence", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-magic-sequence-l2-s1/solution/solve.sh b/tasks/construct-mc-magic-sequence-l2-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..ddb4d9051e8932e2715ae167552403c3aceac21e
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-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": [5, 2, 1, 0, 0, 1, 0, 0, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-magic-sequence-l2-s1/tests/check.py b/tasks/construct-mc-magic-sequence-l2-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..27f636ddd6cfe523924ae01c6b9f37ee392f0332
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-l2-s1/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-l2-s1/tests/instance.json b/tasks/construct-mc-magic-sequence-l2-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..658d44b26538ace79c899ff279699b6071261bff
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-l2-s1/tests/instance.json
@@ -0,0 +1 @@
+{"n": 9, "fixed": [], "family": "mc_magic_sequence", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-magic-sequence-l2-s1/tests/run.py b/tasks/construct-mc-magic-sequence-l2-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-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-magic-sequence-l2-s1/tests/test.sh b/tasks/construct-mc-magic-sequence-l2-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-magic-sequence-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-l1-s1/environment/Dockerfile b/tasks/construct-mc-max-clique-l1-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-max-clique-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-max-clique-l1-s1/environment/instance.json b/tasks/construct-mc-max-clique-l1-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..fad5841dbd91b77831a8ef6838a5fee589c6b3a4
--- /dev/null
+++ b/tasks/construct-mc-max-clique-l1-s1/environment/instance.json
@@ -0,0 +1 @@
+{"n": 25, "k": 5, "edges": [[0, 2], [0, 4], [0, 5], [0, 7], [0, 9], [0, 13], [0, 14], [0, 17], [1, 2], [1, 6], [1, 7], [1, 11], [1, 14], [1, 15], [1, 17], [1, 20], [1, 21], [1, 22], [1, 24], [2, 13], [2, 15], [2, 23], [3, 6], [3, 7], [3, 8], [3, 10], [3, 13], [3, 14], [4, 5], [4, 6], [4, 7], [4, 8], [4, 10], [4, 13], [4, 14], [4, 22], [4, 24], [5, 6], [5, 7], [5, 9], [5, 11], [5, 12], [5, 15], [5, 17], [5, 18], [5, 20], [5, 22], [5, 24], [6, 9], [6, 10], [6, 12], [6, 14], [6, 16], [6, 17], [6, 21], [6, 22], [6, 24], [7, 8], [7, 11], [7, 14], [7, 16], [7, 20], [7, 21], [7, 22], [8, 12], [8, 15], [8, 20], [9, 10], [9, 17], [9, 18], [9, 24], [10, 14], [10, 15], [10, 16], [10, 20], [11, 16], [11, 23], [12, 16], [12, 19], [12, 24], [13, 19], [13, 20], [13, 22], [13, 24], [14, 16], [14, 24], [15, 17], [15, 18], [15, 19], [15, 20], [15, 22], [16, 17], [16, 18], [17, 20], [18, 21], [19, 24], [20, 23], [20, 24], [21, 22], [22, 24], [23, 24]], "family": "mc_max_clique", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-max-clique-l1-s1/solution/solve.sh b/tasks/construct-mc-max-clique-l1-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..85bb9fe9b22b9e4425d6d7434f2f7e91c777c905
--- /dev/null
+++ b/tasks/construct-mc-max-clique-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'
+{"clique": [4, 5, 6, 22, 24]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-max-clique-l1-s1/tests/check.py b/tasks/construct-mc-max-clique-l1-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..e3f70dbd5b155a3514c2102aa4ad460201495210
--- /dev/null
+++ b/tasks/construct-mc-max-clique-l1-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-l1-s1/tests/instance.json b/tasks/construct-mc-max-clique-l1-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..fad5841dbd91b77831a8ef6838a5fee589c6b3a4
--- /dev/null
+++ b/tasks/construct-mc-max-clique-l1-s1/tests/instance.json
@@ -0,0 +1 @@
+{"n": 25, "k": 5, "edges": [[0, 2], [0, 4], [0, 5], [0, 7], [0, 9], [0, 13], [0, 14], [0, 17], [1, 2], [1, 6], [1, 7], [1, 11], [1, 14], [1, 15], [1, 17], [1, 20], [1, 21], [1, 22], [1, 24], [2, 13], [2, 15], [2, 23], [3, 6], [3, 7], [3, 8], [3, 10], [3, 13], [3, 14], [4, 5], [4, 6], [4, 7], [4, 8], [4, 10], [4, 13], [4, 14], [4, 22], [4, 24], [5, 6], [5, 7], [5, 9], [5, 11], [5, 12], [5, 15], [5, 17], [5, 18], [5, 20], [5, 22], [5, 24], [6, 9], [6, 10], [6, 12], [6, 14], [6, 16], [6, 17], [6, 21], [6, 22], [6, 24], [7, 8], [7, 11], [7, 14], [7, 16], [7, 20], [7, 21], [7, 22], [8, 12], [8, 15], [8, 20], [9, 10], [9, 17], [9, 18], [9, 24], [10, 14], [10, 15], [10, 16], [10, 20], [11, 16], [11, 23], [12, 16], [12, 19], [12, 24], [13, 19], [13, 20], [13, 22], [13, 24], [14, 16], [14, 24], [15, 17], [15, 18], [15, 19], [15, 20], [15, 22], [16, 17], [16, 18], [17, 20], [18, 21], [19, 24], [20, 23], [20, 24], [21, 22], [22, 24], [23, 24]], "family": "mc_max_clique", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-max-clique-l1-s1/tests/run.py b/tasks/construct-mc-max-clique-l1-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-max-clique-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-max-clique-l1-s1/tests/test.sh b/tasks/construct-mc-max-clique-l1-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-max-clique-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-max-independent-set-l3-s3/environment/Dockerfile b/tasks/construct-mc-max-independent-set-l3-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-l3-s3/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-max-independent-set-l3-s3/environment/instance.json b/tasks/construct-mc-max-independent-set-l3-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..180c73574320f51936f7017b815ba864a99cd30e
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-l3-s3/environment/instance.json
@@ -0,0 +1 @@
+{"n": 45, "k": 11, "edges": [[0, 2], [0, 6], [0, 16], [0, 21], [0, 22], [0, 23], [0, 29], [0, 30], [0, 32], [0, 33], [0, 35], [0, 36], [0, 39], [1, 12], [1, 16], [1, 24], [1, 30], [1, 37], [1, 42], [1, 44], [2, 3], [2, 7], [2, 8], [2, 12], [2, 16], [2, 21], [2, 23], [2, 29], [2, 31], [2, 32], [2, 33], [2, 37], [2, 39], [2, 40], [2, 41], [2, 43], [2, 44], [3, 6], [3, 15], [3, 23], [3, 25], [3, 29], [3, 30], [3, 32], [3, 34], [3, 35], [3, 37], [3, 40], [3, 41], [3, 42], [4, 8], [4, 17], [4, 21], [4, 23], [4, 27], [4, 28], [4, 30], [4, 36], [4, 40], [5, 10], [5, 13], [5, 14], [5, 18], [5, 19], [5, 20], [5, 22], [5, 27], [5, 28], [5, 30], [6, 16], [6, 17], [6, 20], [6, 21], [6, 24], [6, 31], [6, 34], [6, 36], [6, 38], [6, 40], [6, 44], [7, 14], [7, 17], [7, 22], [7, 25], [7, 28], [7, 29], [7, 30], [7, 37], [7, 39], [7, 40], [7, 42], [8, 9], [8, 11], [8, 18], [8, 26], [8, 27], [8, 33], [8, 34], [8, 36], [8, 38], [8, 39], [8, 43], [8, 44], [9, 12], [9, 31], [9, 39], [10, 11], [10, 14], [10, 19], [10, 21], [10, 23], [10, 24], [10, 28], [10, 31], [10, 34], [10, 39], [10, 42], [10, 44], [11, 14], [11, 21], [11, 22], [11, 32], [11, 41], [11, 43], [12, 15], [12, 21], [12, 24], [12, 25], [12, 29], [12, 34], [12, 35], [12, 37], [12, 41], [13, 20], [13, 25], [13, 29], [13, 32], [13, 34], [13, 36], [13, 38], [14, 21], [14, 25], [14, 26], [14, 28], [14, 36], [14, 37], [14, 38], [14, 43], [15, 16], [15, 21], [15, 23], [15, 28], [15, 30], [15, 31], [15, 33], [15, 35], [15, 38], [15, 39], [15, 41], [15, 42], [15, 44], [16, 18], [16, 20], [16, 25], [16, 28], [16, 29], [16, 32], [16, 33], [17, 18], [17, 19], [17, 27], [17, 31], [17, 34], [17, 35], [17, 36], [17, 37], [17, 39], [17, 41], [17, 42], [17, 44], [18, 22], [18, 23], [18, 28], [18, 30], [18, 31], [18, 32], [18, 42], [18, 44], [19, 21], [19, 22], [19, 29], [19, 30], [19, 32], [19, 33], [19, 42], [19, 43], [20, 21], [20, 36], [20, 41], [20, 43], [20, 44], [21, 22], [21, 23], [21, 24], [21, 29], [21, 35], [21, 43], [22, 25], [22, 26], [22, 37], [22, 41], [22, 43], [23, 27], [23, 29], [23, 34], [23, 38], [23, 40], [23, 41], [23, 43], [23, 44], [24, 36], [24, 44], [25, 30], [25, 35], [25, 42], [25, 44], [26, 35], [26, 36], [26, 42], [27, 28], [27, 31], [27, 35], [27, 37], [27, 40], [27, 44], [28, 30], [28, 33], [28, 36], [28, 37], [28, 40], [28, 43], [29, 30], [29, 31], [29, 34], [29, 40], [29, 41], [30, 37], [30, 39], [30, 40], [30, 44], [31, 33], [31, 36], [31, 37], [31, 38], [31, 39], [31, 43], [32, 37], [32, 39], [32, 41], [32, 42], [32, 43], [33, 34], [33, 36], [33, 38], [33, 39], [34, 38], [34, 40], [34, 42], [35, 36], [35, 43], [36, 42], [37, 43], [38, 40], [38, 41], [39, 40], [39, 41], [39, 42], [40, 42], [42, 44]], "family": "mc_max_independent_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-max-independent-set-l3-s3/solution/solve.sh b/tasks/construct-mc-max-independent-set-l3-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..cbf83fdb0f4e4e3d95470c056dfa40e78460fef3
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-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'
+{"independent_set": [1, 6, 7, 9, 10, 13, 18, 26, 33, 41, 43]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-max-independent-set-l3-s3/tests/check.py b/tasks/construct-mc-max-independent-set-l3-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..adce413e8578d996549583b990e4ff559f38ed41
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-l3-s3/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-l3-s3/tests/instance.json b/tasks/construct-mc-max-independent-set-l3-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..180c73574320f51936f7017b815ba864a99cd30e
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-l3-s3/tests/instance.json
@@ -0,0 +1 @@
+{"n": 45, "k": 11, "edges": [[0, 2], [0, 6], [0, 16], [0, 21], [0, 22], [0, 23], [0, 29], [0, 30], [0, 32], [0, 33], [0, 35], [0, 36], [0, 39], [1, 12], [1, 16], [1, 24], [1, 30], [1, 37], [1, 42], [1, 44], [2, 3], [2, 7], [2, 8], [2, 12], [2, 16], [2, 21], [2, 23], [2, 29], [2, 31], [2, 32], [2, 33], [2, 37], [2, 39], [2, 40], [2, 41], [2, 43], [2, 44], [3, 6], [3, 15], [3, 23], [3, 25], [3, 29], [3, 30], [3, 32], [3, 34], [3, 35], [3, 37], [3, 40], [3, 41], [3, 42], [4, 8], [4, 17], [4, 21], [4, 23], [4, 27], [4, 28], [4, 30], [4, 36], [4, 40], [5, 10], [5, 13], [5, 14], [5, 18], [5, 19], [5, 20], [5, 22], [5, 27], [5, 28], [5, 30], [6, 16], [6, 17], [6, 20], [6, 21], [6, 24], [6, 31], [6, 34], [6, 36], [6, 38], [6, 40], [6, 44], [7, 14], [7, 17], [7, 22], [7, 25], [7, 28], [7, 29], [7, 30], [7, 37], [7, 39], [7, 40], [7, 42], [8, 9], [8, 11], [8, 18], [8, 26], [8, 27], [8, 33], [8, 34], [8, 36], [8, 38], [8, 39], [8, 43], [8, 44], [9, 12], [9, 31], [9, 39], [10, 11], [10, 14], [10, 19], [10, 21], [10, 23], [10, 24], [10, 28], [10, 31], [10, 34], [10, 39], [10, 42], [10, 44], [11, 14], [11, 21], [11, 22], [11, 32], [11, 41], [11, 43], [12, 15], [12, 21], [12, 24], [12, 25], [12, 29], [12, 34], [12, 35], [12, 37], [12, 41], [13, 20], [13, 25], [13, 29], [13, 32], [13, 34], [13, 36], [13, 38], [14, 21], [14, 25], [14, 26], [14, 28], [14, 36], [14, 37], [14, 38], [14, 43], [15, 16], [15, 21], [15, 23], [15, 28], [15, 30], [15, 31], [15, 33], [15, 35], [15, 38], [15, 39], [15, 41], [15, 42], [15, 44], [16, 18], [16, 20], [16, 25], [16, 28], [16, 29], [16, 32], [16, 33], [17, 18], [17, 19], [17, 27], [17, 31], [17, 34], [17, 35], [17, 36], [17, 37], [17, 39], [17, 41], [17, 42], [17, 44], [18, 22], [18, 23], [18, 28], [18, 30], [18, 31], [18, 32], [18, 42], [18, 44], [19, 21], [19, 22], [19, 29], [19, 30], [19, 32], [19, 33], [19, 42], [19, 43], [20, 21], [20, 36], [20, 41], [20, 43], [20, 44], [21, 22], [21, 23], [21, 24], [21, 29], [21, 35], [21, 43], [22, 25], [22, 26], [22, 37], [22, 41], [22, 43], [23, 27], [23, 29], [23, 34], [23, 38], [23, 40], [23, 41], [23, 43], [23, 44], [24, 36], [24, 44], [25, 30], [25, 35], [25, 42], [25, 44], [26, 35], [26, 36], [26, 42], [27, 28], [27, 31], [27, 35], [27, 37], [27, 40], [27, 44], [28, 30], [28, 33], [28, 36], [28, 37], [28, 40], [28, 43], [29, 30], [29, 31], [29, 34], [29, 40], [29, 41], [30, 37], [30, 39], [30, 40], [30, 44], [31, 33], [31, 36], [31, 37], [31, 38], [31, 39], [31, 43], [32, 37], [32, 39], [32, 41], [32, 42], [32, 43], [33, 34], [33, 36], [33, 38], [33, 39], [34, 38], [34, 40], [34, 42], [35, 36], [35, 43], [36, 42], [37, 43], [38, 40], [38, 41], [39, 40], [39, 41], [39, 42], [40, 42], [42, 44]], "family": "mc_max_independent_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-max-independent-set-l3-s3/tests/run.py b/tasks/construct-mc-max-independent-set-l3-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-l3-s3/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-max-independent-set-l3-s3/tests/test.sh b/tasks/construct-mc-max-independent-set-l3-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-l3-s3/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-max-independent-set-l6-s5/environment/Dockerfile b/tasks/construct-mc-max-independent-set-l6-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-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-mc-max-independent-set-l6-s5/environment/instance.json b/tasks/construct-mc-max-independent-set-l6-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e56a2d1d258f6c627d425c9827e3b598872a7e6a
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-l6-s5/environment/instance.json
@@ -0,0 +1 @@
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\ No newline at end of file
diff --git a/tasks/construct-mc-max-independent-set-l6-s5/solution/solve.sh b/tasks/construct-mc-max-independent-set-l6-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..5cbd738530899f5830e491d82d53bdd50ce25c58
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-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'
+{"independent_set": [8, 16, 20, 24, 41, 54, 59, 66, 67, 69, 70, 76, 83, 92, 119, 120, 126, 127]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-max-independent-set-l6-s5/tests/check.py b/tasks/construct-mc-max-independent-set-l6-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..adce413e8578d996549583b990e4ff559f38ed41
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-l6-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-l6-s5/tests/instance.json b/tasks/construct-mc-max-independent-set-l6-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e56a2d1d258f6c627d425c9827e3b598872a7e6a
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-l6-s5/tests/instance.json
@@ -0,0 +1 @@
+{"n": 140, "k": 18, "edges": [[0, 4], [0, 5], [0, 6], [0, 7], [0, 12], [0, 16], [0, 23], [0, 26], [0, 29], [0, 31], [0, 37], [0, 38], [0, 47], [0, 58], [0, 63], [0, 71], [0, 79], [0, 81], [0, 86], [0, 90], [0, 95], [0, 101], [0, 102], [0, 103], [0, 104], [0, 106], [0, 110], [0, 111], [0, 112], [0, 113], [0, 117], [0, 119], [0, 121], [0, 123], [0, 126], [0, 127], [0, 135], [0, 137], [1, 3], [1, 9], [1, 11], [1, 12], [1, 17], [1, 20], [1, 27], [1, 30], [1, 35], [1, 39], [1, 49], [1, 50], [1, 53], [1, 55], [1, 57], [1, 58], [1, 64], [1, 69], [1, 77], [1, 83], [1, 86], [1, 87], [1, 90], [1, 92], [1, 98], [1, 107], [1, 112], [1, 118], [1, 121], [1, 122], [1, 123], [1, 124], [1, 126], [1, 127], [1, 131], [2, 6], [2, 8], [2, 9], [2, 10], [2, 19], [2, 25], [2, 29], [2, 30], [2, 35], [2, 40], [2, 42], [2, 45], [2, 47], [2, 48], [2, 49], [2, 70], [2, 71], [2, 76], [2, 79], [2, 81], [2, 82], [2, 87], [2, 90], [2, 101], [2, 108], [2, 112], [2, 115], [2, 116], [2, 118], [2, 120], [2, 127], [2, 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[98, 112], [98, 117], [98, 119], [98, 120], [98, 121], [98, 124], [98, 125], [98, 129], [98, 131], [98, 137], [99, 103], [99, 106], [99, 109], [99, 117], [99, 118], [99, 121], [99, 128], [99, 133], [99, 134], [99, 137], [100, 102], [100, 104], [100, 109], [100, 122], [100, 124], [100, 126], [100, 133], [100, 135], [101, 102], [101, 108], [101, 116], [101, 117], [101, 119], [101, 120], [101, 132], [102, 110], [102, 114], [102, 117], [102, 124], [102, 125], [102, 126], [102, 129], [102, 134], [102, 136], [102, 138], [103, 110], [103, 118], [103, 119], [103, 121], [103, 126], [103, 127], [103, 128], [103, 133], [103, 134], [103, 135], [104, 105], [104, 113], [104, 114], [104, 115], [104, 116], [104, 117], [104, 118], [104, 119], [104, 120], [104, 127], [104, 130], [104, 131], [104, 134], [104, 136], [104, 137], [104, 138], [105, 108], [105, 110], [105, 111], [105, 129], [105, 130], [105, 133], [105, 136], [106, 107], [106, 108], [106, 113], [106, 116], [106, 120], [106, 128], [106, 133], [106, 134], [106, 137], [106, 139], [107, 112], [107, 125], [107, 128], [107, 132], [107, 134], [107, 139], [108, 109], [108, 110], [108, 111], [108, 124], [108, 128], [108, 130], [108, 132], [108, 134], [108, 136], [108, 139], [109, 112], [109, 113], [109, 117], [109, 126], [109, 128], [109, 130], [109, 132], [109, 136], [110, 113], [110, 118], [110, 120], [110, 123], [110, 124], [110, 126], [110, 129], [110, 132], [110, 139], [111, 113], [111, 119], [111, 120], [111, 124], [111, 128], [111, 129], [111, 131], [111, 135], [112, 122], [112, 128], [112, 131], [112, 132], [112, 134], [112, 136], [112, 138], [113, 114], [113, 116], [113, 121], [113, 128], [113, 129], [113, 132], [113, 135], [113, 136], [114, 117], [114, 127], [114, 131], [114, 135], [114, 136], [115, 119], [115, 120], [115, 122], [115, 130], [115, 132], [115, 135], [115, 136], [115, 138], [116, 119], [116, 128], [116, 131], [116, 136], [116, 137], [116, 138], [117, 122], [117, 130], [117, 132], [118, 119], [118, 122], [118, 130], [118, 132], [119, 121], [119, 122], [119, 129], [119, 130], [119, 133], [119, 134], [119, 139], [120, 124], [120, 139], [121, 122], [121, 127], [121, 128], [121, 130], [121, 131], [121, 132], [121, 138], [122, 124], [122, 125], [122, 126], [122, 131], [122, 132], [122, 133], [122, 136], [123, 124], [123, 127], [123, 130], [123, 131], [123, 133], [123, 134], [123, 138], [123, 139], [124, 127], [124, 132], [124, 134], [124, 135], [124, 137], [124, 138], [125, 127], [125, 129], [125, 131], [125, 133], [125, 134], [125, 137], [125, 139], [126, 128], [126, 130], [126, 134], [126, 136], [127, 128], [127, 129], [127, 130], [127, 135], [127, 139], [128, 130], [128, 132], [128, 134], [128, 135], [129, 134], [129, 135], [129, 139], [130, 131], [130, 132], [130, 138], [131, 132], [131, 138], [131, 139], [132, 136], [133, 139], [135, 136], [135, 137], [135, 138], [136, 137]], "family": "mc_max_independent_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-max-independent-set-l6-s5/tests/run.py b/tasks/construct-mc-max-independent-set-l6-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-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-mc-max-independent-set-l6-s5/tests/test.sh b/tasks/construct-mc-max-independent-set-l6-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-max-independent-set-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-mc-non-transitive-dice-l6-s2/environment/Dockerfile b/tasks/construct-mc-non-transitive-dice-l6-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-non-transitive-dice-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-mc-non-transitive-dice-l6-s2/environment/instance.json b/tasks/construct-mc-non-transitive-dice-l6-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1354eed8e81b74bc2b5a82dbdc4f76dcaf1cf1d0
--- /dev/null
+++ b/tasks/construct-mc-non-transitive-dice-l6-s2/environment/instance.json
@@ -0,0 +1 @@
+{"n": 12, "m": 12, "family": "mc_non_transitive_dice", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-non-transitive-dice-l6-s2/solution/solve.sh b/tasks/construct-mc-non-transitive-dice-l6-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..ab2924c5ba066417a80d1e9907faaab364eb1d59
--- /dev/null
+++ b/tasks/construct-mc-non-transitive-dice-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'
+{"dice": [[0, 1, 6, 7, 7, 12, 14, 15, 15, 21, 22, 23], [2, 3, 5, 6, 10, 11, 12, 13, 19, 20, 21, 22], [0, 3, 6, 7, 9, 12, 13, 18, 18, 18, 18, 20], [0, 1, 2, 10, 11, 11, 13, 16, 16, 17, 19, 21], [0, 0, 6, 8, 10, 11, 11, 12, 12, 15, 20, 23], [3, 4, 6, 7, 7, 9, 9, 9, 15, 19, 20, 23], [1, 3, 5, 6, 6, 12, 13, 15, 15, 17, 17, 18], [1, 9, 10, 10, 10, 11, 11, 12, 12, 13, 14, 15], [4, 4, 5, 6, 8, 8, 10, 12, 21, 22, 23, 23], [3, 3, 3, 8, 8, 9, 18, 19, 20, 21, 21, 22], [0, 6, 10, 10, 14, 15, 16, 16, 17, 19, 20, 20], [1, 6, 9, 9, 11, 13, 14, 15, 18, 19, 22, 22]]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-non-transitive-dice-l6-s2/tests/check.py b/tasks/construct-mc-non-transitive-dice-l6-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..e3d5c04fa6e2819bf32f34fafe13a383437dc62a
--- /dev/null
+++ b/tasks/construct-mc-non-transitive-dice-l6-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-l6-s2/tests/instance.json b/tasks/construct-mc-non-transitive-dice-l6-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1354eed8e81b74bc2b5a82dbdc4f76dcaf1cf1d0
--- /dev/null
+++ b/tasks/construct-mc-non-transitive-dice-l6-s2/tests/instance.json
@@ -0,0 +1 @@
+{"n": 12, "m": 12, "family": "mc_non_transitive_dice", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-non-transitive-dice-l6-s2/tests/run.py b/tasks/construct-mc-non-transitive-dice-l6-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-non-transitive-dice-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-mc-non-transitive-dice-l6-s2/tests/test.sh b/tasks/construct-mc-non-transitive-dice-l6-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-non-transitive-dice-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-mc-number-partitioning-l2-s0/environment/Dockerfile b/tasks/construct-mc-number-partitioning-l2-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-number-partitioning-l2-s0/environment/instance.json b/tasks/construct-mc-number-partitioning-l2-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..76c24ca49e7bceb8424210465813bf6c73a3c438
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l2-s0/environment/instance.json
@@ -0,0 +1 @@
+{"n": 16, "k": 8, "family": "mc_number_partitioning", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-number-partitioning-l2-s0/solution/solve.sh b/tasks/construct-mc-number-partitioning-l2-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..1951f91e58a24a660b1d91c5dd7c22b851cb9eb6
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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'
+{"x": [0, 1, 2, 3, 4, 5, 6, 7, 7, 6, 5, 4, 3, 2, 1, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-number-partitioning-l2-s0/tests/check.py b/tasks/construct-mc-number-partitioning-l2-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..67793721db8ba7d3c3979ea0e1e07091491ffa7e
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l2-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-l2-s0/tests/instance.json b/tasks/construct-mc-number-partitioning-l2-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..76c24ca49e7bceb8424210465813bf6c73a3c438
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l2-s0/tests/instance.json
@@ -0,0 +1 @@
+{"n": 16, "k": 8, "family": "mc_number_partitioning", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-number-partitioning-l2-s0/tests/run.py b/tasks/construct-mc-number-partitioning-l2-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-number-partitioning-l2-s0/tests/test.sh b/tasks/construct-mc-number-partitioning-l2-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-number-partitioning-l2-s9/environment/Dockerfile b/tasks/construct-mc-number-partitioning-l2-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-mc-number-partitioning-l2-s9/environment/instance.json b/tasks/construct-mc-number-partitioning-l2-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..cc5cf2c9c344748011fb372f77313ffbbbcdfb95
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l2-s9/environment/instance.json
@@ -0,0 +1 @@
+{"n": 14, "k": 5, "family": "mc_number_partitioning", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-number-partitioning-l2-s9/solution/solve.sh b/tasks/construct-mc-number-partitioning-l2-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..57fa89f71ad87cdc1b11d367b2906e100897e89f
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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'
+{"x": [2, 4, 1, 3, 1, 3, 0, 2, 4, 4, 3, 2, 1, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-number-partitioning-l2-s9/tests/check.py b/tasks/construct-mc-number-partitioning-l2-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..67793721db8ba7d3c3979ea0e1e07091491ffa7e
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l2-s9/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-l2-s9/tests/instance.json b/tasks/construct-mc-number-partitioning-l2-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..cc5cf2c9c344748011fb372f77313ffbbbcdfb95
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l2-s9/tests/instance.json
@@ -0,0 +1 @@
+{"n": 14, "k": 5, "family": "mc_number_partitioning", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-number-partitioning-l2-s9/tests/run.py b/tasks/construct-mc-number-partitioning-l2-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-mc-number-partitioning-l2-s9/tests/test.sh b/tasks/construct-mc-number-partitioning-l2-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-mc-number-partitioning-l6-s1/environment/Dockerfile b/tasks/construct-mc-number-partitioning-l6-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-number-partitioning-l6-s1/environment/instance.json b/tasks/construct-mc-number-partitioning-l6-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a05f89f73775933f69e9113841fa066819590a02
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l6-s1/environment/instance.json
@@ -0,0 +1 @@
+{"n": 107, "k": 6, "family": "mc_number_partitioning", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-number-partitioning-l6-s1/solution/solve.sh b/tasks/construct-mc-number-partitioning-l6-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..18e3cfb6462dde79876bfcd9397f0be3b4a093d1
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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'
+{"x": [1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-number-partitioning-l6-s1/tests/check.py b/tasks/construct-mc-number-partitioning-l6-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..67793721db8ba7d3c3979ea0e1e07091491ffa7e
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l6-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-l6-s1/tests/instance.json b/tasks/construct-mc-number-partitioning-l6-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a05f89f73775933f69e9113841fa066819590a02
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l6-s1/tests/instance.json
@@ -0,0 +1 @@
+{"n": 107, "k": 6, "family": "mc_number_partitioning", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-number-partitioning-l6-s1/tests/run.py b/tasks/construct-mc-number-partitioning-l6-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-number-partitioning-l6-s1/tests/test.sh b/tasks/construct-mc-number-partitioning-l6-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-number-partitioning-l6-s6/environment/Dockerfile b/tasks/construct-mc-number-partitioning-l6-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-mc-number-partitioning-l6-s6/environment/instance.json b/tasks/construct-mc-number-partitioning-l6-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1f826194c7f8810a822920625b674ba351d9cbba
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l6-s6/environment/instance.json
@@ -0,0 +1 @@
+{"n": 110, "k": 3, "family": "mc_number_partitioning", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-number-partitioning-l6-s6/solution/solve.sh b/tasks/construct-mc-number-partitioning-l6-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..b09c8c50f231c468bf14706da9dd654b96920dcc
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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'
+{"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, 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-l6-s6/tests/check.py b/tasks/construct-mc-number-partitioning-l6-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..67793721db8ba7d3c3979ea0e1e07091491ffa7e
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l6-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-l6-s6/tests/instance.json b/tasks/construct-mc-number-partitioning-l6-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1f826194c7f8810a822920625b674ba351d9cbba
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l6-s6/tests/instance.json
@@ -0,0 +1 @@
+{"n": 110, "k": 3, "family": "mc_number_partitioning", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-number-partitioning-l6-s6/tests/run.py b/tasks/construct-mc-number-partitioning-l6-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-mc-number-partitioning-l6-s6/tests/test.sh b/tasks/construct-mc-number-partitioning-l6-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-mc-number-partitioning-l6-s8/environment/Dockerfile b/tasks/construct-mc-number-partitioning-l6-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-mc-number-partitioning-l6-s8/environment/instance.json b/tasks/construct-mc-number-partitioning-l6-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b3a746930ef3dd16a2f82608594a5e717089b932
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l6-s8/environment/instance.json
@@ -0,0 +1 @@
+{"n": 119, "k": 5, "family": "mc_number_partitioning", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-number-partitioning-l6-s8/solution/solve.sh b/tasks/construct-mc-number-partitioning-l6-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..a953fb8e6cc96708d7e90a3a5c76daf98a7991c1
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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'
+{"x": [1, 2, 3, 4, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 4, 3, 2, 1, 0, 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, 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-l6-s8/tests/check.py b/tasks/construct-mc-number-partitioning-l6-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..67793721db8ba7d3c3979ea0e1e07091491ffa7e
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l6-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-l6-s8/tests/instance.json b/tasks/construct-mc-number-partitioning-l6-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b3a746930ef3dd16a2f82608594a5e717089b932
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-l6-s8/tests/instance.json
@@ -0,0 +1 @@
+{"n": 119, "k": 5, "family": "mc_number_partitioning", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-number-partitioning-l6-s8/tests/run.py b/tasks/construct-mc-number-partitioning-l6-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-mc-number-partitioning-l6-s8/tests/test.sh b/tasks/construct-mc-number-partitioning-l6-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-number-partitioning-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-mc-ortholatin-l4-s1/environment/Dockerfile b/tasks/construct-mc-ortholatin-l4-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-ortholatin-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-ortholatin-l4-s1/environment/instance.json b/tasks/construct-mc-ortholatin-l4-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d505145346e1c8f7e549f176a10e8b70d0916898
--- /dev/null
+++ b/tasks/construct-mc-ortholatin-l4-s1/environment/instance.json
@@ -0,0 +1 @@
+{"n": 15, "family": "mc_ortholatin", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-ortholatin-l4-s1/solution/solve.sh b/tasks/construct-mc-ortholatin-l4-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..56b3ffd06e5c011a720d92e47420bc975dd0d34e
--- /dev/null
+++ b/tasks/construct-mc-ortholatin-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'
+{"X": [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]], "Y": [[0, 2, 4, 6, 8, 10, 12, 14, 1, 3, 5, 7, 9, 11, 13], [1, 3, 5, 7, 9, 11, 13, 0, 2, 4, 6, 8, 10, 12, 14], [2, 4, 6, 8, 10, 12, 14, 1, 3, 5, 7, 9, 11, 13, 0], [3, 5, 7, 9, 11, 13, 0, 2, 4, 6, 8, 10, 12, 14, 1], [4, 6, 8, 10, 12, 14, 1, 3, 5, 7, 9, 11, 13, 0, 2], [5, 7, 9, 11, 13, 0, 2, 4, 6, 8, 10, 12, 14, 1, 3], [6, 8, 10, 12, 14, 1, 3, 5, 7, 9, 11, 13, 0, 2, 4], [7, 9, 11, 13, 0, 2, 4, 6, 8, 10, 12, 14, 1, 3, 5], [8, 10, 12, 14, 1, 3, 5, 7, 9, 11, 13, 0, 2, 4, 6], [9, 11, 13, 0, 2, 4, 6, 8, 10, 12, 14, 1, 3, 5, 7], [10, 12, 14, 1, 3, 5, 7, 9, 11, 13, 0, 2, 4, 6, 8], [11, 13, 0, 2, 4, 6, 8, 10, 12, 14, 1, 3, 5, 7, 9], [12, 14, 1, 3, 5, 7, 9, 11, 13, 0, 2, 4, 6, 8, 10], [13, 0, 2, 4, 6, 8, 10, 12, 14, 1, 3, 5, 7, 9, 11], [14, 1, 3, 5, 7, 9, 11, 13, 0, 2, 4, 6, 8, 10, 12]]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-ortholatin-l4-s1/tests/check.py b/tasks/construct-mc-ortholatin-l4-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..a5418a59fae1c86318eee903e90d75b36d8a3306
--- /dev/null
+++ b/tasks/construct-mc-ortholatin-l4-s1/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-l4-s1/tests/instance.json b/tasks/construct-mc-ortholatin-l4-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d505145346e1c8f7e549f176a10e8b70d0916898
--- /dev/null
+++ b/tasks/construct-mc-ortholatin-l4-s1/tests/instance.json
@@ -0,0 +1 @@
+{"n": 15, "family": "mc_ortholatin", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-ortholatin-l4-s1/tests/run.py b/tasks/construct-mc-ortholatin-l4-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-ortholatin-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-ortholatin-l4-s1/tests/test.sh b/tasks/construct-mc-ortholatin-l4-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-ortholatin-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-pysms-chromatic-girth-l4-s1/environment/Dockerfile b/tasks/construct-mc-pysms-chromatic-girth-l4-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-pysms-chromatic-girth-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-pysms-chromatic-girth-l4-s1/environment/instance.json b/tasks/construct-mc-pysms-chromatic-girth-l4-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f4013625555b6e5009040089c6712c63964ce427
--- /dev/null
+++ b/tasks/construct-mc-pysms-chromatic-girth-l4-s1/environment/instance.json
@@ -0,0 +1 @@
+{"vertices": 24, "required_edges": [], "max_chromatic_number": 3, "min_edges": 29, "min_girth": 7, "family": "mc_pysms_chromatic_girth", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-chromatic-girth-l4-s1/solution/solve.sh b/tasks/construct-mc-pysms-chromatic-girth-l4-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..a1f64629f3adc6ee97610aaf0e580f5bc7c6dac6
--- /dev/null
+++ b/tasks/construct-mc-pysms-chromatic-girth-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'
+{"edges": [[0, 22], [0, 23], [1, 20], [1, 21], [2, 18], [2, 19], [3, 16], [3, 17], [4, 14], [4, 15], [4, 23], [5, 13], [5, 15], [5, 21], [6, 12], [6, 14], [6, 20], [7, 11], [7, 13], [7, 19], [8, 10], [8, 12], [8, 18], [9, 10], [9, 11], [9, 23], [10, 21], [11, 17], [12, 22], [13, 22], [14, 19], [15, 16], [16, 18], [17, 20]], "coloring": [2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 2, 1, 2, 0, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-pysms-chromatic-girth-l4-s1/tests/check.py b/tasks/construct-mc-pysms-chromatic-girth-l4-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..4ab0b12508cf59b733f007a9d1f50831aec9efdf
--- /dev/null
+++ b/tasks/construct-mc-pysms-chromatic-girth-l4-s1/tests/check.py
@@ -0,0 +1,151 @@
+"""Graph-existence constraints (MathConstraint pysms_* problem types, CC-BY-4.0). The answer is a graph on
+the stated vertex set, plus a proper colouring when a chromatic-number upper bound is asked for. Every
+constraint present in the instance is checked exactly (no stored answer is compared).
+
+Upstream note: upstream verifies by fixing the edge variables in the pysms CNF and calling CaDiCaL; the
+min-chromatic-number bound of pysms_graph_builder is only a raw smsg solver parameter, not a clause, so the
+upstream verifier ignored it. It is enforced here. Edge endpoints are also type-checked (upstream accepted
+floats/bools that compare in range)."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def _has_clique(adj, n, k):
+    def rec(cand, size):
+        if size == k:
+            return True
+        while cand:
+            if size + bin(cand).count("1") < k:
+                return False
+            v = cand.bit_length() - 1
+            cand &= ~(1 << v)
+            if rec(cand & adj[v], size + 1):
+                return True
+        return False
+    return rec((1 << n) - 1, 0)
+
+
+def _connected(adj, n, removed):
+    alive = [v for v in range(n) if not removed >> v & 1]
+    if len(alive) <= 1:
+        return True
+    mask = sum(1 << v for v in alive)
+    seen = 1 << alive[0]
+    stack = [alive[0]]
+    while stack:
+        v = stack.pop()
+        new = adj[v] & mask & ~seen
+        seen |= new
+        while new:
+            u = new.bit_length() - 1
+            new &= ~(1 << u)
+            stack.append(u)
+    return seen == mask
+
+
+def _girth(adj, n):
+    best = None
+    for r in range(n):
+        dist, par, q = {r: 0}, {r: -1}, [r]
+        for v in q:
+            nb = adj[v]
+            while nb:
+                u = nb.bit_length() - 1
+                nb &= ~(1 << u)
+                if u not in dist:
+                    dist[u], par[u] = dist[v] + 1, v
+                    q.append(u)
+                elif par[v] != u:
+                    c = dist[u] + dist[v] + 1
+                    best = c if best is None else min(best, c)
+    return best  # None = acyclic
+
+
+def _colourable(adj, n, k):
+    col = [-1] * n
+    order = sorted(range(n), key=lambda v: -bin(adj[v]).count("1"))
+
+    def rec(i):
+        if i == n:
+            return True
+        v = order[i]
+        used = {col[u] for u in range(n) if adj[v] >> u & 1 and col[u] >= 0}
+        for c in range(k):
+            if c not in used:
+                col[v] = c
+                if rec(i + 1):
+                    return True
+        col[v] = -1
+        return False
+    return rec(0)
+
+
+def check(inst, ans):
+    n = inst["vertices"]
+    E = ans.get("edges")
+    if not isinstance(E, list):
+        raise Invalid("edges must be a list of [u, v] pairs")
+    edges = set()
+    for e in E:
+        u, v = int_list(e, 2, 0, n - 1, "edge")
+        if u == v:
+            raise Invalid(f"self-loop at {u}")
+        e2 = (min(u, v), max(u, v))
+        if e2 in edges:
+            raise Invalid(f"duplicate edge {e2}")
+        edges.add(e2)
+    adj = [0] * n
+    for u, v in edges:
+        adj[u] |= 1 << v
+        adj[v] |= 1 << u
+    deg = [bin(a).count("1") for a in adj]
+    for u, v in inst.get("required_edges", []):
+        if (min(u, v), max(u, v)) not in edges:
+            raise Invalid(f"required edge ({u},{v}) is missing")
+    c = inst.get
+    if c("min_degree") is not None and min(deg) < c("min_degree"):
+        raise Invalid(f"minimum degree {min(deg)} < {c('min_degree')}")
+    if c("max_degree") is not None and max(deg) > c("max_degree"):
+        raise Invalid(f"maximum degree {max(deg)} > {c('max_degree')}")
+    if c("min_edges") is not None and len(edges) < c("min_edges"):
+        raise Invalid(f"{len(edges)} edges < {c('min_edges')}")
+    if c("max_edges") is not None and len(edges) > c("max_edges"):
+        raise Invalid(f"{len(edges)} edges > {c('max_edges')}")
+    if c("max_clique") is not None and _has_clique(adj, n, c("max_clique") + 1):
+        raise Invalid(f"graph contains a clique of size {c('max_clique') + 1}")
+    if c("max_independent_set") is not None:
+        full = (1 << n) - 1
+        comp = [full & ~adj[v] & ~(1 << v) for v in range(n)]
+        if _has_clique(comp, n, c("max_independent_set") + 1):
+            raise Invalid(f"graph contains an independent set of size {c('max_independent_set') + 1}")
+    if c("min_girth") is not None:
+        g = _girth(adj, n)
+        if g is not None and g < c("min_girth"):
+            raise Invalid(f"graph has a cycle of length {g} < {c('min_girth')}")
+    if c("min_connectivity") is not None:
+        k = c("min_connectivity")
+        if n <= k:
+            raise Invalid("too few vertices for this connectivity")
+        for s in range(k):
+            for I in itertools.combinations(range(n), s):
+                if not _connected(adj, n, sum(1 << v for v in I)):
+                    raise Invalid(f"deleting vertices {list(I)} disconnects the graph")
+    if c("mtf"):
+        for u, v in itertools.combinations(range(n), 2):
+            if adj[u] >> v & 1:
+                if adj[u] & adj[v]:
+                    raise Invalid(f"triangle on edge ({u},{v})")
+            elif not adj[u] & adj[v]:
+                raise Invalid(f"non-adjacent {u}, {v} have no common neighbour (edge could be added)")
+    if c("max_chromatic_number") is not None:
+        k = c("max_chromatic_number")
+        col = int_list(ans.get("coloring"), n, 0, k - 1, "coloring")
+        for u, v in edges:
+            if col[u] == col[v]:
+                raise Invalid(f"coloring: edge ({u},{v}) is monochromatic")
+    if c("min_chromatic_number") is not None and c("min_chromatic_number") >= 2:
+        m = c("min_chromatic_number")
+        if not edges or _colourable(adj, n, m - 1):
+            raise Invalid(f"graph is {m - 1}-colourable, chromatic number must be at least {m}")
+    return None, {"num_edges": len(edges)}
diff --git a/tasks/construct-mc-pysms-chromatic-girth-l4-s1/tests/instance.json b/tasks/construct-mc-pysms-chromatic-girth-l4-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f4013625555b6e5009040089c6712c63964ce427
--- /dev/null
+++ b/tasks/construct-mc-pysms-chromatic-girth-l4-s1/tests/instance.json
@@ -0,0 +1 @@
+{"vertices": 24, "required_edges": [], "max_chromatic_number": 3, "min_edges": 29, "min_girth": 7, "family": "mc_pysms_chromatic_girth", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-chromatic-girth-l4-s1/tests/run.py b/tasks/construct-mc-pysms-chromatic-girth-l4-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-pysms-chromatic-girth-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-pysms-chromatic-girth-l4-s1/tests/test.sh b/tasks/construct-mc-pysms-chromatic-girth-l4-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-pysms-chromatic-girth-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-pysms-clique-coloring-l3-s2/environment/Dockerfile b/tasks/construct-mc-pysms-clique-coloring-l3-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-pysms-clique-coloring-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-mc-pysms-clique-coloring-l3-s2/environment/instance.json b/tasks/construct-mc-pysms-clique-coloring-l3-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..57d27e69d28c4a43039ccd23f524e94f6ac59bb3
--- /dev/null
+++ b/tasks/construct-mc-pysms-clique-coloring-l3-s2/environment/instance.json
@@ -0,0 +1 @@
+{"vertices": 15, "required_edges": [], "max_chromatic_number": 5, "max_clique": 2, "min_degree": 3, "family": "mc_pysms_clique_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-clique-coloring-l3-s2/solution/solve.sh b/tasks/construct-mc-pysms-clique-coloring-l3-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..1930d6d1e7db64f41591ba00202053aba83d8895
--- /dev/null
+++ b/tasks/construct-mc-pysms-clique-coloring-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'
+{"edges": [[0, 3], [0, 4], [0, 5], [0, 6], [0, 7], [0, 8], [0, 9], [0, 10], [0, 11], [0, 12], [0, 13], [0, 14], [1, 3], [1, 4], [1, 5], [1, 6], [1, 7], [1, 8], [1, 9], [1, 10], [1, 11], [1, 12], [1, 13], [1, 14], [2, 3], [2, 4], [2, 5], [2, 6], [2, 7], [2, 8], [2, 9], [2, 10], [2, 11], [2, 12], [2, 13], [2, 14]], "coloring": [0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-pysms-clique-coloring-l3-s2/tests/check.py b/tasks/construct-mc-pysms-clique-coloring-l3-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..4ab0b12508cf59b733f007a9d1f50831aec9efdf
--- /dev/null
+++ b/tasks/construct-mc-pysms-clique-coloring-l3-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-clique-coloring-l3-s2/tests/instance.json b/tasks/construct-mc-pysms-clique-coloring-l3-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..57d27e69d28c4a43039ccd23f524e94f6ac59bb3
--- /dev/null
+++ b/tasks/construct-mc-pysms-clique-coloring-l3-s2/tests/instance.json
@@ -0,0 +1 @@
+{"vertices": 15, "required_edges": [], "max_chromatic_number": 5, "max_clique": 2, "min_degree": 3, "family": "mc_pysms_clique_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-clique-coloring-l3-s2/tests/run.py b/tasks/construct-mc-pysms-clique-coloring-l3-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-pysms-clique-coloring-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-mc-pysms-clique-coloring-l3-s2/tests/test.sh b/tasks/construct-mc-pysms-clique-coloring-l3-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-pysms-clique-coloring-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-mc-pysms-degree-bounds-l5-s1/environment/Dockerfile b/tasks/construct-mc-pysms-degree-bounds-l5-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-pysms-degree-bounds-l5-s1/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-pysms-degree-bounds-l5-s1/environment/instance.json b/tasks/construct-mc-pysms-degree-bounds-l5-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2f0c12257c8984c9c15ca80f12d02a922a30b686
--- /dev/null
+++ b/tasks/construct-mc-pysms-degree-bounds-l5-s1/environment/instance.json
@@ -0,0 +1 @@
+{"vertices": 15, "required_edges": [], "max_degree": 5, "min_degree": 1, "family": "mc_pysms_degree_bounds", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-degree-bounds-l5-s1/solution/solve.sh b/tasks/construct-mc-pysms-degree-bounds-l5-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..d5f59035f07c114c678f52ea1328cafaeeede017
--- /dev/null
+++ b/tasks/construct-mc-pysms-degree-bounds-l5-s1/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"edges": [[0, 10], [0, 11], [0, 12], [0, 13], [0, 14], [1, 10], [1, 11], [1, 12], [1, 13], [1, 14], [2, 10], [2, 11], [2, 12], [2, 13], [2, 14], [3, 10], [3, 11], [3, 12], [3, 13], [3, 14], [4, 10], [4, 11], [4, 12], [4, 13], [4, 14], [5, 6], [5, 7], [5, 8], [5, 9], [6, 7], [6, 8], [6, 9], [7, 8], [7, 9], [8, 9]]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-pysms-degree-bounds-l5-s1/tests/check.py b/tasks/construct-mc-pysms-degree-bounds-l5-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..4ab0b12508cf59b733f007a9d1f50831aec9efdf
--- /dev/null
+++ b/tasks/construct-mc-pysms-degree-bounds-l5-s1/tests/check.py
@@ -0,0 +1,151 @@
+"""Graph-existence constraints (MathConstraint pysms_* problem types, CC-BY-4.0). The answer is a graph on
+the stated vertex set, plus a proper colouring when a chromatic-number upper bound is asked for. Every
+constraint present in the instance is checked exactly (no stored answer is compared).
+
+Upstream note: upstream verifies by fixing the edge variables in the pysms CNF and calling CaDiCaL; the
+min-chromatic-number bound of pysms_graph_builder is only a raw smsg solver parameter, not a clause, so the
+upstream verifier ignored it. It is enforced here. Edge endpoints are also type-checked (upstream accepted
+floats/bools that compare in range)."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def _has_clique(adj, n, k):
+    def rec(cand, size):
+        if size == k:
+            return True
+        while cand:
+            if size + bin(cand).count("1") < k:
+                return False
+            v = cand.bit_length() - 1
+            cand &= ~(1 << v)
+            if rec(cand & adj[v], size + 1):
+                return True
+        return False
+    return rec((1 << n) - 1, 0)
+
+
+def _connected(adj, n, removed):
+    alive = [v for v in range(n) if not removed >> v & 1]
+    if len(alive) <= 1:
+        return True
+    mask = sum(1 << v for v in alive)
+    seen = 1 << alive[0]
+    stack = [alive[0]]
+    while stack:
+        v = stack.pop()
+        new = adj[v] & mask & ~seen
+        seen |= new
+        while new:
+            u = new.bit_length() - 1
+            new &= ~(1 << u)
+            stack.append(u)
+    return seen == mask
+
+
+def _girth(adj, n):
+    best = None
+    for r in range(n):
+        dist, par, q = {r: 0}, {r: -1}, [r]
+        for v in q:
+            nb = adj[v]
+            while nb:
+                u = nb.bit_length() - 1
+                nb &= ~(1 << u)
+                if u not in dist:
+                    dist[u], par[u] = dist[v] + 1, v
+                    q.append(u)
+                elif par[v] != u:
+                    c = dist[u] + dist[v] + 1
+                    best = c if best is None else min(best, c)
+    return best  # None = acyclic
+
+
+def _colourable(adj, n, k):
+    col = [-1] * n
+    order = sorted(range(n), key=lambda v: -bin(adj[v]).count("1"))
+
+    def rec(i):
+        if i == n:
+            return True
+        v = order[i]
+        used = {col[u] for u in range(n) if adj[v] >> u & 1 and col[u] >= 0}
+        for c in range(k):
+            if c not in used:
+                col[v] = c
+                if rec(i + 1):
+                    return True
+        col[v] = -1
+        return False
+    return rec(0)
+
+
+def check(inst, ans):
+    n = inst["vertices"]
+    E = ans.get("edges")
+    if not isinstance(E, list):
+        raise Invalid("edges must be a list of [u, v] pairs")
+    edges = set()
+    for e in E:
+        u, v = int_list(e, 2, 0, n - 1, "edge")
+        if u == v:
+            raise Invalid(f"self-loop at {u}")
+        e2 = (min(u, v), max(u, v))
+        if e2 in edges:
+            raise Invalid(f"duplicate edge {e2}")
+        edges.add(e2)
+    adj = [0] * n
+    for u, v in edges:
+        adj[u] |= 1 << v
+        adj[v] |= 1 << u
+    deg = [bin(a).count("1") for a in adj]
+    for u, v in inst.get("required_edges", []):
+        if (min(u, v), max(u, v)) not in edges:
+            raise Invalid(f"required edge ({u},{v}) is missing")
+    c = inst.get
+    if c("min_degree") is not None and min(deg) < c("min_degree"):
+        raise Invalid(f"minimum degree {min(deg)} < {c('min_degree')}")
+    if c("max_degree") is not None and max(deg) > c("max_degree"):
+        raise Invalid(f"maximum degree {max(deg)} > {c('max_degree')}")
+    if c("min_edges") is not None and len(edges) < c("min_edges"):
+        raise Invalid(f"{len(edges)} edges < {c('min_edges')}")
+    if c("max_edges") is not None and len(edges) > c("max_edges"):
+        raise Invalid(f"{len(edges)} edges > {c('max_edges')}")
+    if c("max_clique") is not None and _has_clique(adj, n, c("max_clique") + 1):
+        raise Invalid(f"graph contains a clique of size {c('max_clique') + 1}")
+    if c("max_independent_set") is not None:
+        full = (1 << n) - 1
+        comp = [full & ~adj[v] & ~(1 << v) for v in range(n)]
+        if _has_clique(comp, n, c("max_independent_set") + 1):
+            raise Invalid(f"graph contains an independent set of size {c('max_independent_set') + 1}")
+    if c("min_girth") is not None:
+        g = _girth(adj, n)
+        if g is not None and g < c("min_girth"):
+            raise Invalid(f"graph has a cycle of length {g} < {c('min_girth')}")
+    if c("min_connectivity") is not None:
+        k = c("min_connectivity")
+        if n <= k:
+            raise Invalid("too few vertices for this connectivity")
+        for s in range(k):
+            for I in itertools.combinations(range(n), s):
+                if not _connected(adj, n, sum(1 << v for v in I)):
+                    raise Invalid(f"deleting vertices {list(I)} disconnects the graph")
+    if c("mtf"):
+        for u, v in itertools.combinations(range(n), 2):
+            if adj[u] >> v & 1:
+                if adj[u] & adj[v]:
+                    raise Invalid(f"triangle on edge ({u},{v})")
+            elif not adj[u] & adj[v]:
+                raise Invalid(f"non-adjacent {u}, {v} have no common neighbour (edge could be added)")
+    if c("max_chromatic_number") is not None:
+        k = c("max_chromatic_number")
+        col = int_list(ans.get("coloring"), n, 0, k - 1, "coloring")
+        for u, v in edges:
+            if col[u] == col[v]:
+                raise Invalid(f"coloring: edge ({u},{v}) is monochromatic")
+    if c("min_chromatic_number") is not None and c("min_chromatic_number") >= 2:
+        m = c("min_chromatic_number")
+        if not edges or _colourable(adj, n, m - 1):
+            raise Invalid(f"graph is {m - 1}-colourable, chromatic number must be at least {m}")
+    return None, {"num_edges": len(edges)}
diff --git a/tasks/construct-mc-pysms-degree-bounds-l5-s1/tests/instance.json b/tasks/construct-mc-pysms-degree-bounds-l5-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2f0c12257c8984c9c15ca80f12d02a922a30b686
--- /dev/null
+++ b/tasks/construct-mc-pysms-degree-bounds-l5-s1/tests/instance.json
@@ -0,0 +1 @@
+{"vertices": 15, "required_edges": [], "max_degree": 5, "min_degree": 1, "family": "mc_pysms_degree_bounds", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-degree-bounds-l5-s1/tests/run.py b/tasks/construct-mc-pysms-degree-bounds-l5-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-pysms-degree-bounds-l5-s1/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-pysms-degree-bounds-l5-s1/tests/test.sh b/tasks/construct-mc-pysms-degree-bounds-l5-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-pysms-degree-bounds-l5-s1/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-pysms-girth-degree-l3-s2/environment/Dockerfile b/tasks/construct-mc-pysms-girth-degree-l3-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-pysms-girth-degree-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-mc-pysms-girth-degree-l3-s2/environment/instance.json b/tasks/construct-mc-pysms-girth-degree-l3-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..48617bd27011637304ae763bf874628e02bf03c7
--- /dev/null
+++ b/tasks/construct-mc-pysms-girth-degree-l3-s2/environment/instance.json
@@ -0,0 +1 @@
+{"vertices": 21, "required_edges": [], "max_degree": 6, "min_degree": 4, "min_girth": 5, "family": "mc_pysms_girth_degree", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-girth-degree-l3-s2/solution/solve.sh b/tasks/construct-mc-pysms-girth-degree-l3-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..0ee3efcdb0f0c8ddf22358cd742e0375a8783735
--- /dev/null
+++ b/tasks/construct-mc-pysms-girth-degree-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'
+{"edges": [[0, 17], [0, 18], [0, 19], [0, 20], [1, 14], [1, 15], [1, 16], [1, 20], [2, 12], [2, 13], [2, 16], [2, 19], [3, 11], [3, 13], [3, 15], [3, 18], [4, 10], [4, 13], [4, 14], [4, 17], [5, 9], [5, 12], [5, 15], [5, 17], [6, 9], [6, 10], [6, 11], [6, 20], [7, 8], [7, 11], [7, 16], [7, 17], [8, 10], [8, 12], [8, 18], [9, 16], [9, 18], [10, 19], [11, 14], [12, 14], [13, 20], [15, 19]]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-pysms-girth-degree-l3-s2/tests/check.py b/tasks/construct-mc-pysms-girth-degree-l3-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..4ab0b12508cf59b733f007a9d1f50831aec9efdf
--- /dev/null
+++ b/tasks/construct-mc-pysms-girth-degree-l3-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-girth-degree-l3-s2/tests/instance.json b/tasks/construct-mc-pysms-girth-degree-l3-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..48617bd27011637304ae763bf874628e02bf03c7
--- /dev/null
+++ b/tasks/construct-mc-pysms-girth-degree-l3-s2/tests/instance.json
@@ -0,0 +1 @@
+{"vertices": 21, "required_edges": [], "max_degree": 6, "min_degree": 4, "min_girth": 5, "family": "mc_pysms_girth_degree", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-girth-degree-l3-s2/tests/run.py b/tasks/construct-mc-pysms-girth-degree-l3-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-pysms-girth-degree-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-mc-pysms-girth-degree-l3-s2/tests/test.sh b/tasks/construct-mc-pysms-girth-degree-l3-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-pysms-girth-degree-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-mc-pysms-independent-connectivity-l2-s2/environment/Dockerfile b/tasks/construct-mc-pysms-independent-connectivity-l2-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-pysms-independent-connectivity-l2-s2/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-pysms-independent-connectivity-l2-s2/environment/instance.json b/tasks/construct-mc-pysms-independent-connectivity-l2-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2903adda5b66fea5828b3460f7f487ef935ba400
--- /dev/null
+++ b/tasks/construct-mc-pysms-independent-connectivity-l2-s2/environment/instance.json
@@ -0,0 +1 @@
+{"vertices": 11, "required_edges": [], "max_independent_set": 5, "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-l2-s2/solution/solve.sh b/tasks/construct-mc-pysms-independent-connectivity-l2-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..24220ce9a34ebb047de09ce7f88464235e6a9a4b
--- /dev/null
+++ b/tasks/construct-mc-pysms-independent-connectivity-l2-s2/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"edges": [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [0, 6], [0, 7], [0, 8], [0, 9], [0, 10], [1, 2], [1, 3], [1, 4], [1, 5], [1, 6], [1, 7], [1, 8], [1, 9], [1, 10], [2, 3], [2, 4], [2, 5], [2, 6], [2, 7], [2, 8], [2, 9], [2, 10], [3, 4], [3, 5], [3, 6], [3, 7], [3, 8], [3, 9], [3, 10], [4, 5], [4, 6], [4, 7], [4, 8], [4, 9], [4, 10], [5, 6], [5, 7], [5, 8], [5, 9], [5, 10], [6, 7], [6, 8], [6, 9], [6, 10], [7, 8], [7, 9], [7, 10], [8, 9], [8, 10], [9, 10]]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-pysms-independent-connectivity-l2-s2/tests/check.py b/tasks/construct-mc-pysms-independent-connectivity-l2-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..4ab0b12508cf59b733f007a9d1f50831aec9efdf
--- /dev/null
+++ b/tasks/construct-mc-pysms-independent-connectivity-l2-s2/tests/check.py
@@ -0,0 +1,151 @@
+"""Graph-existence constraints (MathConstraint pysms_* problem types, CC-BY-4.0). The answer is a graph on
+the stated vertex set, plus a proper colouring when a chromatic-number upper bound is asked for. Every
+constraint present in the instance is checked exactly (no stored answer is compared).
+
+Upstream note: upstream verifies by fixing the edge variables in the pysms CNF and calling CaDiCaL; the
+min-chromatic-number bound of pysms_graph_builder is only a raw smsg solver parameter, not a clause, so the
+upstream verifier ignored it. It is enforced here. Edge endpoints are also type-checked (upstream accepted
+floats/bools that compare in range)."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def _has_clique(adj, n, k):
+    def rec(cand, size):
+        if size == k:
+            return True
+        while cand:
+            if size + bin(cand).count("1") < k:
+                return False
+            v = cand.bit_length() - 1
+            cand &= ~(1 << v)
+            if rec(cand & adj[v], size + 1):
+                return True
+        return False
+    return rec((1 << n) - 1, 0)
+
+
+def _connected(adj, n, removed):
+    alive = [v for v in range(n) if not removed >> v & 1]
+    if len(alive) <= 1:
+        return True
+    mask = sum(1 << v for v in alive)
+    seen = 1 << alive[0]
+    stack = [alive[0]]
+    while stack:
+        v = stack.pop()
+        new = adj[v] & mask & ~seen
+        seen |= new
+        while new:
+            u = new.bit_length() - 1
+            new &= ~(1 << u)
+            stack.append(u)
+    return seen == mask
+
+
+def _girth(adj, n):
+    best = None
+    for r in range(n):
+        dist, par, q = {r: 0}, {r: -1}, [r]
+        for v in q:
+            nb = adj[v]
+            while nb:
+                u = nb.bit_length() - 1
+                nb &= ~(1 << u)
+                if u not in dist:
+                    dist[u], par[u] = dist[v] + 1, v
+                    q.append(u)
+                elif par[v] != u:
+                    c = dist[u] + dist[v] + 1
+                    best = c if best is None else min(best, c)
+    return best  # None = acyclic
+
+
+def _colourable(adj, n, k):
+    col = [-1] * n
+    order = sorted(range(n), key=lambda v: -bin(adj[v]).count("1"))
+
+    def rec(i):
+        if i == n:
+            return True
+        v = order[i]
+        used = {col[u] for u in range(n) if adj[v] >> u & 1 and col[u] >= 0}
+        for c in range(k):
+            if c not in used:
+                col[v] = c
+                if rec(i + 1):
+                    return True
+        col[v] = -1
+        return False
+    return rec(0)
+
+
+def check(inst, ans):
+    n = inst["vertices"]
+    E = ans.get("edges")
+    if not isinstance(E, list):
+        raise Invalid("edges must be a list of [u, v] pairs")
+    edges = set()
+    for e in E:
+        u, v = int_list(e, 2, 0, n - 1, "edge")
+        if u == v:
+            raise Invalid(f"self-loop at {u}")
+        e2 = (min(u, v), max(u, v))
+        if e2 in edges:
+            raise Invalid(f"duplicate edge {e2}")
+        edges.add(e2)
+    adj = [0] * n
+    for u, v in edges:
+        adj[u] |= 1 << v
+        adj[v] |= 1 << u
+    deg = [bin(a).count("1") for a in adj]
+    for u, v in inst.get("required_edges", []):
+        if (min(u, v), max(u, v)) not in edges:
+            raise Invalid(f"required edge ({u},{v}) is missing")
+    c = inst.get
+    if c("min_degree") is not None and min(deg) < c("min_degree"):
+        raise Invalid(f"minimum degree {min(deg)} < {c('min_degree')}")
+    if c("max_degree") is not None and max(deg) > c("max_degree"):
+        raise Invalid(f"maximum degree {max(deg)} > {c('max_degree')}")
+    if c("min_edges") is not None and len(edges) < c("min_edges"):
+        raise Invalid(f"{len(edges)} edges < {c('min_edges')}")
+    if c("max_edges") is not None and len(edges) > c("max_edges"):
+        raise Invalid(f"{len(edges)} edges > {c('max_edges')}")
+    if c("max_clique") is not None and _has_clique(adj, n, c("max_clique") + 1):
+        raise Invalid(f"graph contains a clique of size {c('max_clique') + 1}")
+    if c("max_independent_set") is not None:
+        full = (1 << n) - 1
+        comp = [full & ~adj[v] & ~(1 << v) for v in range(n)]
+        if _has_clique(comp, n, c("max_independent_set") + 1):
+            raise Invalid(f"graph contains an independent set of size {c('max_independent_set') + 1}")
+    if c("min_girth") is not None:
+        g = _girth(adj, n)
+        if g is not None and g < c("min_girth"):
+            raise Invalid(f"graph has a cycle of length {g} < {c('min_girth')}")
+    if c("min_connectivity") is not None:
+        k = c("min_connectivity")
+        if n <= k:
+            raise Invalid("too few vertices for this connectivity")
+        for s in range(k):
+            for I in itertools.combinations(range(n), s):
+                if not _connected(adj, n, sum(1 << v for v in I)):
+                    raise Invalid(f"deleting vertices {list(I)} disconnects the graph")
+    if c("mtf"):
+        for u, v in itertools.combinations(range(n), 2):
+            if adj[u] >> v & 1:
+                if adj[u] & adj[v]:
+                    raise Invalid(f"triangle on edge ({u},{v})")
+            elif not adj[u] & adj[v]:
+                raise Invalid(f"non-adjacent {u}, {v} have no common neighbour (edge could be added)")
+    if c("max_chromatic_number") is not None:
+        k = c("max_chromatic_number")
+        col = int_list(ans.get("coloring"), n, 0, k - 1, "coloring")
+        for u, v in edges:
+            if col[u] == col[v]:
+                raise Invalid(f"coloring: edge ({u},{v}) is monochromatic")
+    if c("min_chromatic_number") is not None and c("min_chromatic_number") >= 2:
+        m = c("min_chromatic_number")
+        if not edges or _colourable(adj, n, m - 1):
+            raise Invalid(f"graph is {m - 1}-colourable, chromatic number must be at least {m}")
+    return None, {"num_edges": len(edges)}
diff --git a/tasks/construct-mc-pysms-independent-connectivity-l2-s2/tests/instance.json b/tasks/construct-mc-pysms-independent-connectivity-l2-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2903adda5b66fea5828b3460f7f487ef935ba400
--- /dev/null
+++ b/tasks/construct-mc-pysms-independent-connectivity-l2-s2/tests/instance.json
@@ -0,0 +1 @@
+{"vertices": 11, "required_edges": [], "max_independent_set": 5, "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-l2-s2/tests/run.py b/tasks/construct-mc-pysms-independent-connectivity-l2-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-pysms-independent-connectivity-l2-s2/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-pysms-independent-connectivity-l2-s2/tests/test.sh b/tasks/construct-mc-pysms-independent-connectivity-l2-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-pysms-independent-connectivity-l2-s2/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-pysms-min-degree-l2-s1/environment/Dockerfile b/tasks/construct-mc-pysms-min-degree-l2-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-degree-l2-s1/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-pysms-min-degree-l2-s1/environment/instance.json b/tasks/construct-mc-pysms-min-degree-l2-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3f43f356871b98ae8554c9f6305455e6fdaf6d73
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-degree-l2-s1/environment/instance.json
@@ -0,0 +1 @@
+{"vertices": 10, "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-l2-s1/solution/solve.sh b/tasks/construct-mc-pysms-min-degree-l2-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..2dfbd8e10be3814720762c87be2212bdd75316c7
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-degree-l2-s1/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"edges": [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [0, 6], [0, 7], [0, 8], [0, 9], [1, 2], [1, 3], [1, 4], [1, 5], [1, 6], [1, 7], [1, 8], [1, 9], [2, 3], [2, 4], [2, 5], [2, 6], [2, 7], [2, 8], [2, 9], [3, 4], [3, 5], [3, 6], [3, 7], [3, 8], [3, 9], [4, 5], [4, 6], [4, 7], [4, 8], [4, 9], [5, 6], [5, 7], [5, 8], [5, 9], [6, 7], [6, 8], [6, 9], [7, 8], [7, 9], [8, 9]]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-pysms-min-degree-l2-s1/tests/check.py b/tasks/construct-mc-pysms-min-degree-l2-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..4ab0b12508cf59b733f007a9d1f50831aec9efdf
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-degree-l2-s1/tests/check.py
@@ -0,0 +1,151 @@
+"""Graph-existence constraints (MathConstraint pysms_* problem types, CC-BY-4.0). The answer is a graph on
+the stated vertex set, plus a proper colouring when a chromatic-number upper bound is asked for. Every
+constraint present in the instance is checked exactly (no stored answer is compared).
+
+Upstream note: upstream verifies by fixing the edge variables in the pysms CNF and calling CaDiCaL; the
+min-chromatic-number bound of pysms_graph_builder is only a raw smsg solver parameter, not a clause, so the
+upstream verifier ignored it. It is enforced here. Edge endpoints are also type-checked (upstream accepted
+floats/bools that compare in range)."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def _has_clique(adj, n, k):
+    def rec(cand, size):
+        if size == k:
+            return True
+        while cand:
+            if size + bin(cand).count("1") < k:
+                return False
+            v = cand.bit_length() - 1
+            cand &= ~(1 << v)
+            if rec(cand & adj[v], size + 1):
+                return True
+        return False
+    return rec((1 << n) - 1, 0)
+
+
+def _connected(adj, n, removed):
+    alive = [v for v in range(n) if not removed >> v & 1]
+    if len(alive) <= 1:
+        return True
+    mask = sum(1 << v for v in alive)
+    seen = 1 << alive[0]
+    stack = [alive[0]]
+    while stack:
+        v = stack.pop()
+        new = adj[v] & mask & ~seen
+        seen |= new
+        while new:
+            u = new.bit_length() - 1
+            new &= ~(1 << u)
+            stack.append(u)
+    return seen == mask
+
+
+def _girth(adj, n):
+    best = None
+    for r in range(n):
+        dist, par, q = {r: 0}, {r: -1}, [r]
+        for v in q:
+            nb = adj[v]
+            while nb:
+                u = nb.bit_length() - 1
+                nb &= ~(1 << u)
+                if u not in dist:
+                    dist[u], par[u] = dist[v] + 1, v
+                    q.append(u)
+                elif par[v] != u:
+                    c = dist[u] + dist[v] + 1
+                    best = c if best is None else min(best, c)
+    return best  # None = acyclic
+
+
+def _colourable(adj, n, k):
+    col = [-1] * n
+    order = sorted(range(n), key=lambda v: -bin(adj[v]).count("1"))
+
+    def rec(i):
+        if i == n:
+            return True
+        v = order[i]
+        used = {col[u] for u in range(n) if adj[v] >> u & 1 and col[u] >= 0}
+        for c in range(k):
+            if c not in used:
+                col[v] = c
+                if rec(i + 1):
+                    return True
+        col[v] = -1
+        return False
+    return rec(0)
+
+
+def check(inst, ans):
+    n = inst["vertices"]
+    E = ans.get("edges")
+    if not isinstance(E, list):
+        raise Invalid("edges must be a list of [u, v] pairs")
+    edges = set()
+    for e in E:
+        u, v = int_list(e, 2, 0, n - 1, "edge")
+        if u == v:
+            raise Invalid(f"self-loop at {u}")
+        e2 = (min(u, v), max(u, v))
+        if e2 in edges:
+            raise Invalid(f"duplicate edge {e2}")
+        edges.add(e2)
+    adj = [0] * n
+    for u, v in edges:
+        adj[u] |= 1 << v
+        adj[v] |= 1 << u
+    deg = [bin(a).count("1") for a in adj]
+    for u, v in inst.get("required_edges", []):
+        if (min(u, v), max(u, v)) not in edges:
+            raise Invalid(f"required edge ({u},{v}) is missing")
+    c = inst.get
+    if c("min_degree") is not None and min(deg) < c("min_degree"):
+        raise Invalid(f"minimum degree {min(deg)} < {c('min_degree')}")
+    if c("max_degree") is not None and max(deg) > c("max_degree"):
+        raise Invalid(f"maximum degree {max(deg)} > {c('max_degree')}")
+    if c("min_edges") is not None and len(edges) < c("min_edges"):
+        raise Invalid(f"{len(edges)} edges < {c('min_edges')}")
+    if c("max_edges") is not None and len(edges) > c("max_edges"):
+        raise Invalid(f"{len(edges)} edges > {c('max_edges')}")
+    if c("max_clique") is not None and _has_clique(adj, n, c("max_clique") + 1):
+        raise Invalid(f"graph contains a clique of size {c('max_clique') + 1}")
+    if c("max_independent_set") is not None:
+        full = (1 << n) - 1
+        comp = [full & ~adj[v] & ~(1 << v) for v in range(n)]
+        if _has_clique(comp, n, c("max_independent_set") + 1):
+            raise Invalid(f"graph contains an independent set of size {c('max_independent_set') + 1}")
+    if c("min_girth") is not None:
+        g = _girth(adj, n)
+        if g is not None and g < c("min_girth"):
+            raise Invalid(f"graph has a cycle of length {g} < {c('min_girth')}")
+    if c("min_connectivity") is not None:
+        k = c("min_connectivity")
+        if n <= k:
+            raise Invalid("too few vertices for this connectivity")
+        for s in range(k):
+            for I in itertools.combinations(range(n), s):
+                if not _connected(adj, n, sum(1 << v for v in I)):
+                    raise Invalid(f"deleting vertices {list(I)} disconnects the graph")
+    if c("mtf"):
+        for u, v in itertools.combinations(range(n), 2):
+            if adj[u] >> v & 1:
+                if adj[u] & adj[v]:
+                    raise Invalid(f"triangle on edge ({u},{v})")
+            elif not adj[u] & adj[v]:
+                raise Invalid(f"non-adjacent {u}, {v} have no common neighbour (edge could be added)")
+    if c("max_chromatic_number") is not None:
+        k = c("max_chromatic_number")
+        col = int_list(ans.get("coloring"), n, 0, k - 1, "coloring")
+        for u, v in edges:
+            if col[u] == col[v]:
+                raise Invalid(f"coloring: edge ({u},{v}) is monochromatic")
+    if c("min_chromatic_number") is not None and c("min_chromatic_number") >= 2:
+        m = c("min_chromatic_number")
+        if not edges or _colourable(adj, n, m - 1):
+            raise Invalid(f"graph is {m - 1}-colourable, chromatic number must be at least {m}")
+    return None, {"num_edges": len(edges)}
diff --git a/tasks/construct-mc-pysms-min-degree-l2-s1/tests/instance.json b/tasks/construct-mc-pysms-min-degree-l2-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3f43f356871b98ae8554c9f6305455e6fdaf6d73
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-degree-l2-s1/tests/instance.json
@@ -0,0 +1 @@
+{"vertices": 10, "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-l2-s1/tests/run.py b/tasks/construct-mc-pysms-min-degree-l2-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-degree-l2-s1/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-pysms-min-degree-l2-s1/tests/test.sh b/tasks/construct-mc-pysms-min-degree-l2-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-pysms-min-degree-l2-s1/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-pysms-mtf-l1-s1/environment/Dockerfile b/tasks/construct-mc-pysms-mtf-l1-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-pysms-mtf-l1-s1/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-pysms-mtf-l1-s1/environment/instance.json b/tasks/construct-mc-pysms-mtf-l1-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e383f01632abc9b067450e694f02e7b42ce28f63
--- /dev/null
+++ b/tasks/construct-mc-pysms-mtf-l1-s1/environment/instance.json
@@ -0,0 +1 @@
+{"vertices": 6, "required_edges": [[3, 5]], "mtf": true, "family": "mc_pysms_mtf", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-mtf-l1-s1/solution/solve.sh b/tasks/construct-mc-pysms-mtf-l1-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..b591bdcb071cd75a05d4d1b37e574a4ab2844e39
--- /dev/null
+++ b/tasks/construct-mc-pysms-mtf-l1-s1/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"edges": [[0, 4], [0, 5], [1, 4], [1, 5], [2, 4], [2, 5], [3, 4], [3, 5]]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-pysms-mtf-l1-s1/tests/check.py b/tasks/construct-mc-pysms-mtf-l1-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..4ab0b12508cf59b733f007a9d1f50831aec9efdf
--- /dev/null
+++ b/tasks/construct-mc-pysms-mtf-l1-s1/tests/check.py
@@ -0,0 +1,151 @@
+"""Graph-existence constraints (MathConstraint pysms_* problem types, CC-BY-4.0). The answer is a graph on
+the stated vertex set, plus a proper colouring when a chromatic-number upper bound is asked for. Every
+constraint present in the instance is checked exactly (no stored answer is compared).
+
+Upstream note: upstream verifies by fixing the edge variables in the pysms CNF and calling CaDiCaL; the
+min-chromatic-number bound of pysms_graph_builder is only a raw smsg solver parameter, not a clause, so the
+upstream verifier ignored it. It is enforced here. Edge endpoints are also type-checked (upstream accepted
+floats/bools that compare in range)."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def _has_clique(adj, n, k):
+    def rec(cand, size):
+        if size == k:
+            return True
+        while cand:
+            if size + bin(cand).count("1") < k:
+                return False
+            v = cand.bit_length() - 1
+            cand &= ~(1 << v)
+            if rec(cand & adj[v], size + 1):
+                return True
+        return False
+    return rec((1 << n) - 1, 0)
+
+
+def _connected(adj, n, removed):
+    alive = [v for v in range(n) if not removed >> v & 1]
+    if len(alive) <= 1:
+        return True
+    mask = sum(1 << v for v in alive)
+    seen = 1 << alive[0]
+    stack = [alive[0]]
+    while stack:
+        v = stack.pop()
+        new = adj[v] & mask & ~seen
+        seen |= new
+        while new:
+            u = new.bit_length() - 1
+            new &= ~(1 << u)
+            stack.append(u)
+    return seen == mask
+
+
+def _girth(adj, n):
+    best = None
+    for r in range(n):
+        dist, par, q = {r: 0}, {r: -1}, [r]
+        for v in q:
+            nb = adj[v]
+            while nb:
+                u = nb.bit_length() - 1
+                nb &= ~(1 << u)
+                if u not in dist:
+                    dist[u], par[u] = dist[v] + 1, v
+                    q.append(u)
+                elif par[v] != u:
+                    c = dist[u] + dist[v] + 1
+                    best = c if best is None else min(best, c)
+    return best  # None = acyclic
+
+
+def _colourable(adj, n, k):
+    col = [-1] * n
+    order = sorted(range(n), key=lambda v: -bin(adj[v]).count("1"))
+
+    def rec(i):
+        if i == n:
+            return True
+        v = order[i]
+        used = {col[u] for u in range(n) if adj[v] >> u & 1 and col[u] >= 0}
+        for c in range(k):
+            if c not in used:
+                col[v] = c
+                if rec(i + 1):
+                    return True
+        col[v] = -1
+        return False
+    return rec(0)
+
+
+def check(inst, ans):
+    n = inst["vertices"]
+    E = ans.get("edges")
+    if not isinstance(E, list):
+        raise Invalid("edges must be a list of [u, v] pairs")
+    edges = set()
+    for e in E:
+        u, v = int_list(e, 2, 0, n - 1, "edge")
+        if u == v:
+            raise Invalid(f"self-loop at {u}")
+        e2 = (min(u, v), max(u, v))
+        if e2 in edges:
+            raise Invalid(f"duplicate edge {e2}")
+        edges.add(e2)
+    adj = [0] * n
+    for u, v in edges:
+        adj[u] |= 1 << v
+        adj[v] |= 1 << u
+    deg = [bin(a).count("1") for a in adj]
+    for u, v in inst.get("required_edges", []):
+        if (min(u, v), max(u, v)) not in edges:
+            raise Invalid(f"required edge ({u},{v}) is missing")
+    c = inst.get
+    if c("min_degree") is not None and min(deg) < c("min_degree"):
+        raise Invalid(f"minimum degree {min(deg)} < {c('min_degree')}")
+    if c("max_degree") is not None and max(deg) > c("max_degree"):
+        raise Invalid(f"maximum degree {max(deg)} > {c('max_degree')}")
+    if c("min_edges") is not None and len(edges) < c("min_edges"):
+        raise Invalid(f"{len(edges)} edges < {c('min_edges')}")
+    if c("max_edges") is not None and len(edges) > c("max_edges"):
+        raise Invalid(f"{len(edges)} edges > {c('max_edges')}")
+    if c("max_clique") is not None and _has_clique(adj, n, c("max_clique") + 1):
+        raise Invalid(f"graph contains a clique of size {c('max_clique') + 1}")
+    if c("max_independent_set") is not None:
+        full = (1 << n) - 1
+        comp = [full & ~adj[v] & ~(1 << v) for v in range(n)]
+        if _has_clique(comp, n, c("max_independent_set") + 1):
+            raise Invalid(f"graph contains an independent set of size {c('max_independent_set') + 1}")
+    if c("min_girth") is not None:
+        g = _girth(adj, n)
+        if g is not None and g < c("min_girth"):
+            raise Invalid(f"graph has a cycle of length {g} < {c('min_girth')}")
+    if c("min_connectivity") is not None:
+        k = c("min_connectivity")
+        if n <= k:
+            raise Invalid("too few vertices for this connectivity")
+        for s in range(k):
+            for I in itertools.combinations(range(n), s):
+                if not _connected(adj, n, sum(1 << v for v in I)):
+                    raise Invalid(f"deleting vertices {list(I)} disconnects the graph")
+    if c("mtf"):
+        for u, v in itertools.combinations(range(n), 2):
+            if adj[u] >> v & 1:
+                if adj[u] & adj[v]:
+                    raise Invalid(f"triangle on edge ({u},{v})")
+            elif not adj[u] & adj[v]:
+                raise Invalid(f"non-adjacent {u}, {v} have no common neighbour (edge could be added)")
+    if c("max_chromatic_number") is not None:
+        k = c("max_chromatic_number")
+        col = int_list(ans.get("coloring"), n, 0, k - 1, "coloring")
+        for u, v in edges:
+            if col[u] == col[v]:
+                raise Invalid(f"coloring: edge ({u},{v}) is monochromatic")
+    if c("min_chromatic_number") is not None and c("min_chromatic_number") >= 2:
+        m = c("min_chromatic_number")
+        if not edges or _colourable(adj, n, m - 1):
+            raise Invalid(f"graph is {m - 1}-colourable, chromatic number must be at least {m}")
+    return None, {"num_edges": len(edges)}
diff --git a/tasks/construct-mc-pysms-mtf-l1-s1/tests/instance.json b/tasks/construct-mc-pysms-mtf-l1-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e383f01632abc9b067450e694f02e7b42ce28f63
--- /dev/null
+++ b/tasks/construct-mc-pysms-mtf-l1-s1/tests/instance.json
@@ -0,0 +1 @@
+{"vertices": 6, "required_edges": [[3, 5]], "mtf": true, "family": "mc_pysms_mtf", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-mtf-l1-s1/tests/run.py b/tasks/construct-mc-pysms-mtf-l1-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-pysms-mtf-l1-s1/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-pysms-mtf-l1-s1/tests/test.sh b/tasks/construct-mc-pysms-mtf-l1-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-pysms-mtf-l1-s1/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-pysms-mtf-l4-s0/environment/Dockerfile b/tasks/construct-mc-pysms-mtf-l4-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-pysms-mtf-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-mtf-l4-s0/environment/instance.json b/tasks/construct-mc-pysms-mtf-l4-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..fab64344965daa2fba1dcdfc4d9a97e286a9a71e
--- /dev/null
+++ b/tasks/construct-mc-pysms-mtf-l4-s0/environment/instance.json
@@ -0,0 +1 @@
+{"vertices": 15, "required_edges": [[2, 14], [6, 13], [7, 13], [10, 13], [6, 14]], "mtf": true, "family": "mc_pysms_mtf", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-mtf-l4-s0/solution/solve.sh b/tasks/construct-mc-pysms-mtf-l4-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..9fd11f02301eb41411e1e12ae499e36f0e57d928
--- /dev/null
+++ b/tasks/construct-mc-pysms-mtf-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, 13], [0, 14], [1, 13], [1, 14], [2, 13], [2, 14], [3, 13], [3, 14], [4, 13], [4, 14], [5, 13], [5, 14], [6, 13], [6, 14], [7, 13], [7, 14], [8, 13], [8, 14], [9, 13], [9, 14], [10, 13], [10, 14], [11, 13], [11, 14], [12, 13], [12, 14]]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-pysms-mtf-l4-s0/tests/check.py b/tasks/construct-mc-pysms-mtf-l4-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..4ab0b12508cf59b733f007a9d1f50831aec9efdf
--- /dev/null
+++ b/tasks/construct-mc-pysms-mtf-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-mtf-l4-s0/tests/instance.json b/tasks/construct-mc-pysms-mtf-l4-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..fab64344965daa2fba1dcdfc4d9a97e286a9a71e
--- /dev/null
+++ b/tasks/construct-mc-pysms-mtf-l4-s0/tests/instance.json
@@ -0,0 +1 @@
+{"vertices": 15, "required_edges": [[2, 14], [6, 13], [7, 13], [10, 13], [6, 14]], "mtf": true, "family": "mc_pysms_mtf", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-pysms-mtf-l4-s0/tests/run.py b/tasks/construct-mc-pysms-mtf-l4-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-pysms-mtf-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-mtf-l4-s0/tests/test.sh b/tasks/construct-mc-pysms-mtf-l4-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-pysms-mtf-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-l2-s3/tests/check.py b/tasks/construct-mc-ramsey-l2-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..53526c0bc87af6298219d3e1eac5f95f01b57541
--- /dev/null
+++ b/tasks/construct-mc-ramsey-l2-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-l2-s3/tests/instance.json b/tasks/construct-mc-ramsey-l2-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..cbe9e07b8339380a55f80246277a1a89086a1212
--- /dev/null
+++ b/tasks/construct-mc-ramsey-l2-s3/tests/instance.json
@@ -0,0 +1 @@
+{"n": 10, "r": 3, "s": 5, "fixed": [], "family": "mc_ramsey", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-ramsey-l2-s3/tests/run.py b/tasks/construct-mc-ramsey-l2-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-ramsey-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-mc-ramsey-l2-s3/tests/test.sh b/tasks/construct-mc-ramsey-l2-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-ramsey-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-mc-ramsey-l2-s4/environment/Dockerfile b/tasks/construct-mc-ramsey-l2-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-ramsey-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-mc-ramsey-l2-s4/environment/instance.json b/tasks/construct-mc-ramsey-l2-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..86375ee09407146897731dd9d65205f5b44b7e3a
--- /dev/null
+++ b/tasks/construct-mc-ramsey-l2-s4/environment/instance.json
@@ -0,0 +1 @@
+{"n": 13, "r": 5, "s": 3, "fixed": [], "family": "mc_ramsey", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-ramsey-l2-s4/solution/solve.sh b/tasks/construct-mc-ramsey-l2-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..fe94b409c7a605ef6795e05ae8219fab070d2b34
--- /dev/null
+++ b/tasks/construct-mc-ramsey-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'
+{"colors": [0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-ramsey-l2-s4/tests/check.py b/tasks/construct-mc-ramsey-l2-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..53526c0bc87af6298219d3e1eac5f95f01b57541
--- /dev/null
+++ b/tasks/construct-mc-ramsey-l2-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-l2-s4/tests/instance.json b/tasks/construct-mc-ramsey-l2-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..86375ee09407146897731dd9d65205f5b44b7e3a
--- /dev/null
+++ b/tasks/construct-mc-ramsey-l2-s4/tests/instance.json
@@ -0,0 +1 @@
+{"n": 13, "r": 5, "s": 3, "fixed": [], "family": "mc_ramsey", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-ramsey-l2-s4/tests/run.py b/tasks/construct-mc-ramsey-l2-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-ramsey-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-mc-ramsey-l2-s4/tests/test.sh b/tasks/construct-mc-ramsey-l2-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-ramsey-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-mc-van-der-waerden-l6-s0/environment/Dockerfile b/tasks/construct-mc-van-der-waerden-l6-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-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-van-der-waerden-l6-s0/environment/instance.json b/tasks/construct-mc-van-der-waerden-l6-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f511064f5d56cf209d3d2ebd178ddf196f6919a6
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l6-s0/environment/instance.json
@@ -0,0 +1 @@
+{"k": 2, "L": 5, "n": 165, "family": "mc_van_der_waerden", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-van-der-waerden-l6-s0/solution/solve.sh b/tasks/construct-mc-van-der-waerden-l6-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..dccb14d898db7ca36861e491b6e2411ce2cf1297
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-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'
+{"colors": [0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-van-der-waerden-l6-s0/tests/check.py b/tasks/construct-mc-van-der-waerden-l6-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..9853b089b6ac1773d80eac8e9e02ceb4019ad41f
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l6-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-l6-s0/tests/instance.json b/tasks/construct-mc-van-der-waerden-l6-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f511064f5d56cf209d3d2ebd178ddf196f6919a6
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l6-s0/tests/instance.json
@@ -0,0 +1 @@
+{"k": 2, "L": 5, "n": 165, "family": "mc_van_der_waerden", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-van-der-waerden-l6-s0/tests/run.py b/tasks/construct-mc-van-der-waerden-l6-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-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-van-der-waerden-l6-s0/tests/test.sh b/tasks/construct-mc-van-der-waerden-l6-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-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-van-der-waerden-l6-s7/environment/Dockerfile b/tasks/construct-mc-van-der-waerden-l6-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l6-s7/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-mc-van-der-waerden-l6-s7/environment/instance.json b/tasks/construct-mc-van-der-waerden-l6-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..40e198c66d2c761f1fbc8a18423afe250333d521
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l6-s7/environment/instance.json
@@ -0,0 +1 @@
+{"k": 2, "L": 6, "n": 695, "family": "mc_van_der_waerden", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-van-der-waerden-l6-s7/solution/solve.sh b/tasks/construct-mc-van-der-waerden-l6-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..03e366c52a1b3f9f901efedaecc31a8afc0df4ea
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-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'
+{"colors": [0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-van-der-waerden-l6-s7/tests/check.py b/tasks/construct-mc-van-der-waerden-l6-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..9853b089b6ac1773d80eac8e9e02ceb4019ad41f
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l6-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-l6-s7/tests/instance.json b/tasks/construct-mc-van-der-waerden-l6-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..40e198c66d2c761f1fbc8a18423afe250333d521
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l6-s7/tests/instance.json
@@ -0,0 +1 @@
+{"k": 2, "L": 6, "n": 695, "family": "mc_van_der_waerden", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-van-der-waerden-l6-s7/tests/run.py b/tasks/construct-mc-van-der-waerden-l6-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l6-s7/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-mc-van-der-waerden-l6-s7/tests/test.sh b/tasks/construct-mc-van-der-waerden-l6-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l6-s7/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-mc-van-der-waerden-l6-s9/environment/Dockerfile b/tasks/construct-mc-van-der-waerden-l6-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-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-van-der-waerden-l6-s9/environment/instance.json b/tasks/construct-mc-van-der-waerden-l6-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a9e4c2f656d2386ee1821b1371959cce16bccbc8
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l6-s9/environment/instance.json
@@ -0,0 +1 @@
+{"k": 2, "L": 5, "n": 177, "family": "mc_van_der_waerden", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-van-der-waerden-l6-s9/solution/solve.sh b/tasks/construct-mc-van-der-waerden-l6-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..1abdc0de8be1370f60c9ddf12cbe9e48f1e3571e
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-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'
+{"colors": [0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-van-der-waerden-l6-s9/tests/check.py b/tasks/construct-mc-van-der-waerden-l6-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..9853b089b6ac1773d80eac8e9e02ceb4019ad41f
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l6-s9/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-l6-s9/tests/instance.json b/tasks/construct-mc-van-der-waerden-l6-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a9e4c2f656d2386ee1821b1371959cce16bccbc8
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-l6-s9/tests/instance.json
@@ -0,0 +1 @@
+{"k": 2, "L": 5, "n": 177, "family": "mc_van_der_waerden", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-van-der-waerden-l6-s9/tests/run.py b/tasks/construct-mc-van-der-waerden-l6-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-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-van-der-waerden-l6-s9/tests/test.sh b/tasks/construct-mc-van-der-waerden-l6-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-van-der-waerden-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-vertex-cover-l3-s6/environment/Dockerfile b/tasks/construct-mc-vertex-cover-l3-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-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-mc-vertex-cover-l3-s6/environment/instance.json b/tasks/construct-mc-vertex-cover-l3-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..01624c5d4bd60691b0a555e02caf0ff15d5bd38b
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-l3-s6/environment/instance.json
@@ -0,0 +1 @@
+{"n": 45, "k": 30, "edges": [[0, 2], [0, 3], [0, 4], [0, 6], [0, 17], [0, 21], [0, 30], [1, 3], [1, 4], [1, 5], [1, 15], [1, 22], [1, 42], [1, 43], [2, 5], [2, 7], [2, 10], [2, 11], [2, 18], [2, 23], [2, 26], [2, 32], [2, 33], [2, 35], [2, 39], [2, 44], [3, 25], [3, 30], [3, 31], [3, 33], [3, 35], [3, 36], [3, 38], [3, 40], [4, 18], [4, 23], [4, 24], [4, 44], [5, 15], [5, 16], [5, 21], [5, 27], [5, 33], [5, 34], [5, 41], [6, 7], [6, 11], [6, 14], [6, 19], [6, 20], [6, 23], [6, 24], [6, 26], [6, 34], [6, 35], [6, 39], [6, 40], [7, 9], [7, 12], [7, 17], [7, 23], [7, 25], [7, 27], [7, 29], [7, 33], [7, 34], [7, 43], [8, 14], [8, 16], [8, 20], [8, 21], [8, 24], [8, 26], [8, 27], [8, 28], [8, 32], [8, 35], [8, 37], [8, 44], [9, 14], [9, 21], [9, 22], [9, 24], [9, 26], [9, 33], [9, 38], [9, 39], [9, 42], [9, 43], [10, 12], [10, 16], [10, 18], [10, 21], [10, 23], [10, 32], [10, 36], [11, 13], [11, 25], [11, 32], [12, 13], [12, 18], [12, 24], [12, 37], [12, 39], [12, 41], [13, 18], [13, 20], [13, 32], [13, 38], [13, 43], [14, 20], [14, 22], [14, 23], [14, 31], [14, 32], [14, 33], [15, 16], [15, 19], [15, 23], [15, 24], [15, 25], [15, 27], [15, 29], [15, 40], [16, 21], [16, 29], [16, 37], [16, 39], [16, 42], [17, 19], [17, 21], [17, 31], [17, 34], [18, 21], [18, 25], [18, 36], [18, 39], [19, 33], [19, 34], [19, 39], [20, 32], [20, 34], [20, 37], [21, 31], [21, 33], [21, 36], [21, 39], [21, 41], [21, 43], [22, 27], [22, 28], [22, 33], [22, 35], [22, 36], [22, 39], [22, 40], [22, 44], [23, 27], [23, 30], [23, 32], [23, 38], [23, 41], [23, 43], [24, 31], [24, 37], [24, 41], [25, 28], [25, 31], [25, 32], [25, 36], [25, 43], [25, 44], [26, 34], [26, 39], [27, 28], [27, 31], [27, 36], [27, 39], [27, 41], [27, 43], [28, 30], [28, 39], [28, 43], [28, 44], [29, 32], [29, 33], [29, 40], [29, 41], [30, 33], [30, 35], [30, 40], [30, 41], [30, 42], [30, 43], [31, 42], [32, 34], [33, 34], [33, 35], [33, 37], [33, 40], [33, 41], [34, 41], [34, 44], [36, 39], [37, 38], [40, 44], [41, 42], [43, 44]], "family": "mc_vertex_cover", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-vertex-cover-l3-s6/solution/solve.sh b/tasks/construct-mc-vertex-cover-l3-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..e9a72ce5fbccd8283bcb76cc4d38cb3cf1fb02f9
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-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'
+{"cover": [2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 15, 17, 18, 21, 22, 23, 24, 25, 27, 29, 30, 32, 33, 34, 37, 39, 42, 43, 44]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-vertex-cover-l3-s6/tests/check.py b/tasks/construct-mc-vertex-cover-l3-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..e3accb3da1c773fd2126b8b5b0a97b099e29c83b
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-l3-s6/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-l3-s6/tests/instance.json b/tasks/construct-mc-vertex-cover-l3-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..01624c5d4bd60691b0a555e02caf0ff15d5bd38b
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-l3-s6/tests/instance.json
@@ -0,0 +1 @@
+{"n": 45, "k": 30, "edges": [[0, 2], [0, 3], [0, 4], [0, 6], [0, 17], [0, 21], [0, 30], [1, 3], [1, 4], [1, 5], [1, 15], [1, 22], [1, 42], [1, 43], [2, 5], [2, 7], [2, 10], [2, 11], [2, 18], [2, 23], [2, 26], [2, 32], [2, 33], [2, 35], [2, 39], [2, 44], [3, 25], [3, 30], [3, 31], [3, 33], [3, 35], [3, 36], [3, 38], [3, 40], [4, 18], [4, 23], [4, 24], [4, 44], [5, 15], [5, 16], [5, 21], [5, 27], [5, 33], [5, 34], [5, 41], [6, 7], [6, 11], [6, 14], [6, 19], [6, 20], [6, 23], [6, 24], [6, 26], [6, 34], [6, 35], [6, 39], [6, 40], [7, 9], [7, 12], [7, 17], [7, 23], [7, 25], [7, 27], [7, 29], [7, 33], [7, 34], [7, 43], [8, 14], [8, 16], [8, 20], [8, 21], [8, 24], [8, 26], [8, 27], [8, 28], [8, 32], [8, 35], [8, 37], [8, 44], [9, 14], [9, 21], [9, 22], [9, 24], [9, 26], [9, 33], [9, 38], [9, 39], [9, 42], [9, 43], [10, 12], [10, 16], [10, 18], [10, 21], [10, 23], [10, 32], [10, 36], [11, 13], [11, 25], [11, 32], [12, 13], [12, 18], [12, 24], [12, 37], [12, 39], [12, 41], [13, 18], [13, 20], [13, 32], [13, 38], [13, 43], [14, 20], [14, 22], [14, 23], [14, 31], [14, 32], [14, 33], [15, 16], [15, 19], [15, 23], [15, 24], [15, 25], [15, 27], [15, 29], [15, 40], [16, 21], [16, 29], [16, 37], [16, 39], [16, 42], [17, 19], [17, 21], [17, 31], [17, 34], [18, 21], [18, 25], [18, 36], [18, 39], [19, 33], [19, 34], [19, 39], [20, 32], [20, 34], [20, 37], [21, 31], [21, 33], [21, 36], [21, 39], [21, 41], [21, 43], [22, 27], [22, 28], [22, 33], [22, 35], [22, 36], [22, 39], [22, 40], [22, 44], [23, 27], [23, 30], [23, 32], [23, 38], [23, 41], [23, 43], [24, 31], [24, 37], [24, 41], [25, 28], [25, 31], [25, 32], [25, 36], [25, 43], [25, 44], [26, 34], [26, 39], [27, 28], [27, 31], [27, 36], [27, 39], [27, 41], [27, 43], [28, 30], [28, 39], [28, 43], [28, 44], [29, 32], [29, 33], [29, 40], [29, 41], [30, 33], [30, 35], [30, 40], [30, 41], [30, 42], [30, 43], [31, 42], [32, 34], [33, 34], [33, 35], [33, 37], [33, 40], [33, 41], [34, 41], [34, 44], [36, 39], [37, 38], [40, 44], [41, 42], [43, 44]], "family": "mc_vertex_cover", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-vertex-cover-l3-s6/tests/run.py b/tasks/construct-mc-vertex-cover-l3-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-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-mc-vertex-cover-l3-s6/tests/test.sh b/tasks/construct-mc-vertex-cover-l3-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-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-mc-vertex-cover-l6-s0/environment/Dockerfile b/tasks/construct-mc-vertex-cover-l6-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-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-vertex-cover-l6-s0/environment/instance.json b/tasks/construct-mc-vertex-cover-l6-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..689011a944ddd1b2b28244b2f5b0fc7ed5900b9c
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-l6-s0/environment/instance.json
@@ -0,0 +1 @@
+{"n": 140, "k": 107, "edges": [[0, 7], [0, 8], [0, 12], [0, 18], [0, 30], [0, 43], [0, 45], [0, 56], [0, 73], [0, 79], [0, 83], [0, 84], [0, 95], [0, 100], [0, 103], [0, 114], [0, 115], [0, 128], [1, 3], [1, 8], [1, 10], [1, 28], [1, 32], [1, 37], [1, 38], [1, 41], [1, 50], [1, 51], [1, 54], [1, 55], [1, 58], [1, 60], [1, 64], [1, 89], [1, 115], [1, 129], [2, 16], [2, 17], [2, 19], [2, 22], [2, 26], [2, 33], [2, 34], [2, 35], [2, 41], [2, 53], [2, 54], [2, 59], [2, 71], [2, 78], [2, 79], [2, 89], [2, 96], [2, 102], [2, 118], [2, 120], [3, 13], [3, 18], [3, 19], [3, 23], [3, 25], [3, 29], [3, 30], [3, 38], [3, 42], [3, 43], [3, 58], [3, 59], [3, 64], [3, 68], [3, 87], [3, 93], [3, 95], [3, 115], [3, 116], [3, 136], [4, 12], [4, 13], [4, 35], [4, 36], [4, 38], [4, 42], [4, 43], [4, 44], [4, 49], [4, 74], [4, 94], [4, 96], [4, 99], [4, 101], [4, 105], [4, 107], [4, 117], [4, 124], [4, 125], [4, 128], [4, 129], [5, 13], [5, 27], [5, 31], [5, 36], [5, 38], [5, 60], [5, 63], [5, 67], [5, 71], [5, 83], [5, 84], [5, 87], [5, 91], [5, 97], [5, 107], [5, 110], [5, 114], [5, 120], [5, 121], [5, 127], [5, 131], [6, 8], [6, 9], [6, 15], [6, 19], [6, 28], [6, 35], [6, 51], [6, 64], [6, 69], [6, 79], [6, 90], [6, 93], [6, 95], [6, 100], [6, 124], [6, 127], [6, 137], [6, 139], [7, 8], [7, 16], [7, 27], [7, 37], [7, 42], [7, 50], [7, 57], [7, 58], [7, 69], [7, 79], [7, 80], [7, 84], [7, 97], [7, 108], [7, 123], [7, 127], [7, 128], [7, 132], [8, 16], [8, 32], [8, 41], [8, 56], [8, 58], [8, 65], [8, 69], [8, 71], [8, 80], [8, 97], [8, 101], [8, 106], [8, 107], [8, 110], [8, 119], [8, 131], [8, 133], [8, 134], [8, 138], [9, 19], [9, 25], [9, 26], [9, 28], [9, 33], [9, 38], [9, 51], [9, 59], [9, 63], [9, 64], [9, 68], [9, 70], [9, 73], [9, 75], [9, 80], [9, 85], [9, 88], [9, 114], [9, 116], [9, 119], [9, 121], [9, 134], [9, 135], [9, 139], [10, 18], [10, 19], [10, 21], [10, 23], [10, 29], [10, 33], [10, 40], [10, 42], [10, 48], [10, 52], [10, 58], [10, 60], [10, 66], [10, 70], [10, 80], [10, 82], [10, 85], [10, 87], [10, 90], [10, 99], [10, 124], [10, 129], [10, 132], [10, 137], [11, 12], [11, 20], [11, 29], [11, 30], [11, 60], [11, 78], [11, 84], [11, 97], [11, 104], [11, 105], [11, 108], [11, 110], [11, 113], [11, 120], [11, 132], [12, 27], [12, 49], [12, 53], [12, 56], [12, 60], [12, 62], [12, 80], [12, 81], [12, 83], [12, 95], [12, 99], [12, 103], [12, 106], [12, 113], [12, 119], [12, 124], [12, 127], [12, 130], [12, 135], [13, 28], [13, 30], [13, 31], [13, 35], [13, 47], [13, 53], [13, 55], [13, 63], [13, 66], [13, 71], [13, 72], [13, 77], [13, 78], [13, 94], [13, 100], [13, 109], [13, 122], [13, 123], [13, 124], [13, 134], [14, 21], [14, 23], [14, 25], [14, 48], [14, 50], [14, 55], [14, 56], [14, 65], [14, 66], [14, 70], [14, 72], [14, 80], [14, 86], [14, 91], [14, 94], [14, 96], [14, 101], [14, 104], [14, 121], [14, 128], [14, 132], [14, 133], [14, 138], [15, 34], [15, 44], [15, 47], [15, 51], [15, 63], [15, 64], [15, 75], [15, 79], [15, 81], [15, 89], 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128], [113, 130], [113, 131], [114, 116], [114, 135], [115, 119], [115, 120], [115, 123], [115, 127], [115, 130], [115, 136], [116, 118], [116, 119], [116, 130], [116, 135], [116, 138], [117, 118], [117, 124], [117, 129], [117, 136], [117, 139], [118, 130], [118, 134], [119, 127], [119, 137], [120, 123], [120, 127], [120, 129], [122, 128], [122, 138], [122, 139], [123, 129], [123, 131], [123, 134], [125, 130], [125, 135], [126, 132], [127, 131], [127, 135], [128, 136], [129, 139], [130, 131], [130, 133], [132, 136], [132, 137], [132, 138]], "family": "mc_vertex_cover", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-vertex-cover-l6-s0/solution/solve.sh b/tasks/construct-mc-vertex-cover-l6-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..1a6a9fa77c4147adfefb54049ccbbb18dcadc85b
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-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'
+{"cover": [0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22, 23, 25, 26, 27, 30, 31, 33, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 51, 53, 54, 55, 56, 57, 58, 60, 63, 65, 66, 67, 68, 69, 71, 72, 73, 77, 79, 80, 81, 83, 84, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 99, 100, 101, 102, 103, 104, 105, 107, 109, 110, 111, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 125, 126, 127, 129, 130, 131, 132, 134, 135, 136, 137, 138]}
+RLMATH_EOF
diff --git a/tasks/construct-mc-vertex-cover-l6-s0/tests/check.py b/tasks/construct-mc-vertex-cover-l6-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..e3accb3da1c773fd2126b8b5b0a97b099e29c83b
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-l6-s0/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-l6-s0/tests/instance.json b/tasks/construct-mc-vertex-cover-l6-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..689011a944ddd1b2b28244b2f5b0fc7ed5900b9c
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-l6-s0/tests/instance.json
@@ -0,0 +1 @@
+{"n": 140, "k": 107, "edges": [[0, 7], [0, 8], [0, 12], [0, 18], [0, 30], [0, 43], [0, 45], [0, 56], [0, 73], [0, 79], [0, 83], [0, 84], [0, 95], [0, 100], [0, 103], [0, 114], [0, 115], [0, 128], [1, 3], [1, 8], [1, 10], [1, 28], [1, 32], [1, 37], [1, 38], [1, 41], [1, 50], [1, 51], [1, 54], [1, 55], [1, 58], [1, 60], [1, 64], [1, 89], [1, 115], [1, 129], [2, 16], [2, 17], [2, 19], [2, 22], [2, 26], [2, 33], [2, 34], [2, 35], [2, 41], [2, 53], [2, 54], [2, 59], [2, 71], [2, 78], [2, 79], [2, 89], [2, 96], [2, 102], [2, 118], [2, 120], [3, 13], [3, 18], [3, 19], [3, 23], [3, 25], [3, 29], [3, 30], [3, 38], [3, 42], [3, 43], [3, 58], [3, 59], [3, 64], [3, 68], [3, 87], [3, 93], [3, 95], [3, 115], [3, 116], [3, 136], [4, 12], [4, 13], [4, 35], [4, 36], [4, 38], [4, 42], [4, 43], [4, 44], [4, 49], [4, 74], [4, 94], [4, 96], [4, 99], [4, 101], [4, 105], [4, 107], [4, 117], [4, 124], [4, 125], [4, 128], [4, 129], [5, 13], [5, 27], [5, 31], [5, 36], [5, 38], [5, 60], [5, 63], [5, 67], [5, 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128], [113, 130], [113, 131], [114, 116], [114, 135], [115, 119], [115, 120], [115, 123], [115, 127], [115, 130], [115, 136], [116, 118], [116, 119], [116, 130], [116, 135], [116, 138], [117, 118], [117, 124], [117, 129], [117, 136], [117, 139], [118, 130], [118, 134], [119, 127], [119, 137], [120, 123], [120, 127], [120, 129], [122, 128], [122, 138], [122, 139], [123, 129], [123, 131], [123, 134], [125, 130], [125, 135], [126, 132], [127, 131], [127, 135], [128, 136], [129, 139], [130, 131], [130, 133], [132, 136], [132, 137], [132, 138]], "family": "mc_vertex_cover", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-mc-vertex-cover-l6-s0/tests/run.py b/tasks/construct-mc-vertex-cover-l6-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-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-vertex-cover-l6-s0/tests/test.sh b/tasks/construct-mc-vertex-cover-l6-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-mc-vertex-cover-l6-s0/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-nppc-3col-l3-s1/environment/Dockerfile b/tasks/construct-nppc-3col-l3-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-3col-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-nppc-3col-l3-s1/environment/instance.json b/tasks/construct-nppc-3col-l3-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d6fc4f33e602991bf181e4f8cd751583cfa339a6
--- /dev/null
+++ b/tasks/construct-nppc-3col-l3-s1/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 30, "edges": [[0, 15], [0, 18], [0, 24], [1, 14], [1, 21], [1, 23], [1, 27], [2, 7], [2, 12], [2, 25], [3, 20], [3, 22], [3, 26], [4, 10], [4, 14], [4, 19], [4, 25], [5, 25], [6, 16], [6, 24], [7, 20], [7, 21], [7, 27], [8, 16], [8, 26], [9, 10], [10, 13], [10, 26], [10, 29], [11, 12], [11, 23], [11, 28], [11, 29], [12, 13], [12, 16], [12, 25], [12, 27], [12, 28], [13, 17], [13, 20], [13, 25], [13, 26], [14, 18], [14, 21], [15, 20], [15, 23], [16, 20], [17, 28], [18, 23], [18, 28], [19, 22], [19, 28], [20, 21], [20, 24], [20, 26], [21, 22], [21, 25], [22, 25], [25, 27], [25, 29]], "family": "nppc_three_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-3col-l3-s1/solution/solve.sh b/tasks/construct-nppc-3col-l3-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..09b747f0f75189345c524399acb46e0d80cc25f3
--- /dev/null
+++ b/tasks/construct-nppc-3col-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'
+{"coloring": [2, 1, 0, 0, 0, 1, 2, 1, 2, 1, 2, 2, 1, 0, 2, 1, 0, 2, 1, 2, 2, 0, 1, 0, 0, 2, 1, 0, 0, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-3col-l3-s1/tests/check.py b/tasks/construct-nppc-3col-l3-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..72362ed9180250b6c53ce0c71867ca476ac2af20
--- /dev/null
+++ b/tasks/construct-nppc-3col-l3-s1/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-l3-s1/tests/instance.json b/tasks/construct-nppc-3col-l3-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d6fc4f33e602991bf181e4f8cd751583cfa339a6
--- /dev/null
+++ b/tasks/construct-nppc-3col-l3-s1/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 30, "edges": [[0, 15], [0, 18], [0, 24], [1, 14], [1, 21], [1, 23], [1, 27], [2, 7], [2, 12], [2, 25], [3, 20], [3, 22], [3, 26], [4, 10], [4, 14], [4, 19], [4, 25], [5, 25], [6, 16], [6, 24], [7, 20], [7, 21], [7, 27], [8, 16], [8, 26], [9, 10], [10, 13], [10, 26], [10, 29], [11, 12], [11, 23], [11, 28], [11, 29], [12, 13], [12, 16], [12, 25], [12, 27], [12, 28], [13, 17], [13, 20], [13, 25], [13, 26], [14, 18], [14, 21], [15, 20], [15, 23], [16, 20], [17, 28], [18, 23], [18, 28], [19, 22], [19, 28], [20, 21], [20, 24], [20, 26], [21, 22], [21, 25], [22, 25], [25, 27], [25, 29]], "family": "nppc_three_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-3col-l3-s1/tests/run.py b/tasks/construct-nppc-3col-l3-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-3col-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-nppc-3col-l3-s1/tests/test.sh b/tasks/construct-nppc-3col-l3-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-3col-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-nppc-3col-l3-s8/environment/Dockerfile b/tasks/construct-nppc-3col-l3-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-3col-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-nppc-3col-l3-s8/environment/instance.json b/tasks/construct-nppc-3col-l3-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a3115ee0bafb360fc96ae26632223a81cda44e3a
--- /dev/null
+++ b/tasks/construct-nppc-3col-l3-s8/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 30, "edges": [[0, 11], [0, 28], [1, 5], [1, 22], [1, 25], [1, 29], [2, 9], [2, 10], [2, 12], [2, 24], [2, 28], [2, 29], [3, 9], [3, 10], [3, 18], [4, 6], [4, 20], [4, 27], [4, 29], [5, 9], [5, 16], [5, 21], [6, 8], [6, 11], [6, 16], [6, 17], [7, 20], [8, 9], [8, 10], [8, 12], [8, 21], [8, 24], [9, 13], [9, 17], [9, 18], [9, 20], [9, 26], [9, 27], [10, 17], [10, 19], [10, 20], [11, 19], [12, 26], [13, 19], [13, 24], [14, 18], [14, 26], [15, 20], [16, 22], [17, 23], [17, 29], [18, 19], [18, 21], [19, 23], [20, 25], [22, 23], [25, 26], [27, 28], [27, 29], [28, 29]], "family": "nppc_three_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-3col-l3-s8/solution/solve.sh b/tasks/construct-nppc-3col-l3-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..f9184aa7607df83ab3a8c2069c938b17cf95a8b0
--- /dev/null
+++ b/tasks/construct-nppc-3col-l3-s8/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"coloring": [0, 1, 0, 1, 1, 0, 0, 1, 1, 2, 2, 2, 2, 0, 1, 2, 1, 1, 0, 1, 0, 2, 0, 2, 2, 2, 0, 0, 1, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-3col-l3-s8/tests/check.py b/tasks/construct-nppc-3col-l3-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..72362ed9180250b6c53ce0c71867ca476ac2af20
--- /dev/null
+++ b/tasks/construct-nppc-3col-l3-s8/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-l3-s8/tests/instance.json b/tasks/construct-nppc-3col-l3-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a3115ee0bafb360fc96ae26632223a81cda44e3a
--- /dev/null
+++ b/tasks/construct-nppc-3col-l3-s8/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 30, "edges": [[0, 11], [0, 28], [1, 5], [1, 22], [1, 25], [1, 29], [2, 9], [2, 10], [2, 12], [2, 24], [2, 28], [2, 29], [3, 9], [3, 10], [3, 18], [4, 6], [4, 20], [4, 27], [4, 29], [5, 9], [5, 16], [5, 21], [6, 8], [6, 11], [6, 16], [6, 17], [7, 20], [8, 9], [8, 10], [8, 12], [8, 21], [8, 24], [9, 13], [9, 17], [9, 18], [9, 20], [9, 26], [9, 27], [10, 17], [10, 19], [10, 20], [11, 19], [12, 26], [13, 19], [13, 24], [14, 18], [14, 26], [15, 20], [16, 22], [17, 23], [17, 29], [18, 19], [18, 21], [19, 23], [20, 25], [22, 23], [25, 26], [27, 28], [27, 29], [28, 29]], "family": "nppc_three_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-3col-l3-s8/tests/run.py b/tasks/construct-nppc-3col-l3-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-3col-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-nppc-3col-l3-s8/tests/test.sh b/tasks/construct-nppc-3col-l3-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-3col-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-nppc-bandwidth-l1-s2/environment/Dockerfile b/tasks/construct-nppc-bandwidth-l1-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-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-nppc-bandwidth-l1-s2/environment/instance.json b/tasks/construct-nppc-bandwidth-l1-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..6a1f3f5e64e7015c852ed18691a853c0a8c85897
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l1-s2/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 6, "bandwidth": 3, "edges": [[0, 1], [1, 2], [2, 4], [3, 4], [4, 5]], "family": "nppc_bandwidth", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-bandwidth-l1-s2/solution/solve.sh b/tasks/construct-nppc-bandwidth-l1-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..1da1b9fe0dd65bc12183bc0af82f0f20f3a3a943
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-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'
+{"layout": [2, 0, 1, 5, 4, 3]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-bandwidth-l1-s2/tests/check.py b/tasks/construct-nppc-bandwidth-l1-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..0f847cded257e6c404063dfafde33fa44163afa2
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l1-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-l1-s2/tests/instance.json b/tasks/construct-nppc-bandwidth-l1-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..6a1f3f5e64e7015c852ed18691a853c0a8c85897
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-l1-s2/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 6, "bandwidth": 3, "edges": [[0, 1], [1, 2], [2, 4], [3, 4], [4, 5]], "family": "nppc_bandwidth", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-bandwidth-l1-s2/tests/run.py b/tasks/construct-nppc-bandwidth-l1-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-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-nppc-bandwidth-l1-s2/tests/test.sh b/tasks/construct-nppc-bandwidth-l1-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-bandwidth-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-nppc-clique-l4-s4/environment/Dockerfile b/tasks/construct-nppc-clique-l4-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-clique-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-nppc-clique-l4-s4/environment/instance.json b/tasks/construct-nppc-clique-l4-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0352c9ee06d9a9a10dce14a51854beff7fe028a5
--- /dev/null
+++ b/tasks/construct-nppc-clique-l4-s4/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 30, "k": 15, "edges": [[0, 4], [0, 6], [0, 8], [0, 9], [0, 14], [0, 15], [0, 16], [0, 17], [0, 19], [0, 23], [0, 26], [0, 27], [0, 28], [0, 29], [1, 2], [1, 3], [1, 9], [1, 10], [1, 13], [1, 14], [1, 15], [1, 16], [1, 17], [1, 18], [1, 19], [1, 22], [1, 23], [1, 24], [1, 25], [1, 26], [1, 27], [1, 28], [1, 29], [2, 4], [2, 5], [2, 9], [2, 10], [2, 13], [2, 14], [2, 16], [2, 17], [2, 19], [2, 20], [2, 22], [2, 23], [2, 26], [2, 27], [2, 28], [3, 4], [3, 6], [3, 7], [3, 10], [3, 12], [3, 13], [3, 15], [3, 17], [3, 18], [3, 19], [3, 21], [3, 23], [3, 26], [4, 6], [4, 7], [4, 8], [4, 9], [4, 10], [4, 12], [4, 14], [4, 15], [4, 18], [4, 19], [4, 20], [4, 21], [4, 24], [4, 25], [4, 29], [5, 7], [5, 10], [5, 11], [5, 12], [5, 13], [5, 14], [5, 18], [5, 21], [5, 23], [5, 25], [5, 28], [5, 29], [6, 8], [6, 9], [6, 10], [6, 13], [6, 14], [6, 17], [6, 20], [6, 21], [6, 23], [7, 8], [7, 10], [7, 11], [7, 12], [7, 13], [7, 21], [7, 25], [7, 28], [7, 29], [8, 10], [8, 11], [8, 14], [8, 15], [8, 16], [8, 17], [8, 19], [8, 20], [8, 22], [8, 23], [8, 24], [8, 25], [8, 26], [8, 28], [8, 29], [9, 10], [9, 13], [9, 14], [9, 15], [9, 16], [9, 17], [9, 18], [9, 19], [9, 20], [9, 21], [9, 22], [9, 23], [9, 24], [9, 25], [9, 26], [9, 27], [9, 28], [9, 29], [10, 13], [10, 14], [10, 15], [10, 16], [10, 17], [10, 18], [10, 19], [10, 21], [10, 22], [10, 23], [10, 24], [10, 26], [10, 27], [10, 28], [10, 29], [11, 15], [11, 16], [11, 20], [11, 21], [11, 26], [11, 29], [12, 13], [12, 15], [12, 17], [12, 19], [12, 21], [12, 24], [12, 25], [12, 26], [12, 27], [12, 29], [13, 15], [13, 16], [13, 17], [13, 18], [13, 19], [13, 23], [13, 24], [13, 26], [13, 27], [13, 28], [13, 29], [14, 15], [14, 16], [14, 17], [14, 24], [14, 27], [14, 28], [14, 29], [15, 16], [15, 17], [15, 18], [15, 19], [15, 20], [15, 22], [15, 23], [15, 24], [15, 25], [15, 26], [15, 27], [15, 28], [15, 29], [16, 17], [16, 18], [16, 19], [16, 20], [16, 21], [16, 22], [16, 23], [16, 24], [16, 25], [16, 26], [16, 27], [16, 28], [16, 29], [17, 18], [17, 19], [17, 23], [17, 24], [17, 25], [17, 26], [17, 27], [17, 28], [17, 29], [18, 19], [18, 21], [18, 23], [18, 24], [18, 25], [18, 26], [18, 27], [18, 28], [18, 29], [19, 21], [19, 22], [19, 23], [19, 24], [19, 25], [19, 26], [19, 27], [19, 28], [19, 29], [20, 21], [20, 22], [20, 23], [20, 24], [20, 27], [20, 28], [21, 22], [21, 23], [21, 26], [21, 27], [21, 28], [22, 23], [22, 25], [22, 27], [22, 28], [22, 29], [23, 24], [23, 26], [23, 27], [23, 28], [23, 29], [24, 25], [24, 26], [24, 27], [24, 28], [24, 29], [25, 26], [25, 28], [26, 27], [26, 28], [26, 29], [27, 28], [27, 29], [28, 29]], "family": "nppc_clique", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-clique-l4-s4/solution/solve.sh b/tasks/construct-nppc-clique-l4-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..a8e7128aef47cf24b5eef7607c97df906c300555
--- /dev/null
+++ b/tasks/construct-nppc-clique-l4-s4/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"clique": [1, 9, 10, 13, 15, 16, 17, 18, 19, 23, 24, 26, 27, 28, 29]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-clique-l4-s4/tests/check.py b/tasks/construct-nppc-clique-l4-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..98eb4fc9097fcfa31988cf0a8b79280cec2b7980
--- /dev/null
+++ b/tasks/construct-nppc-clique-l4-s4/tests/check.py
@@ -0,0 +1,19 @@
+"""Clique of size >= k. Accepts any valid clique."""
+import itertools
+
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["num_nodes"], inst["k"]
+    c = int_list(ans.get("clique"), None, 0, n - 1, "clique")
+    # upstream NPPC verifier checks len(list) >= k but only tests pairs u < v, so [v, v, ..., v] passed
+    if len(set(c)) != len(c):
+        raise Invalid("clique contains duplicates")
+    if len(c) < k:
+        raise Invalid(f"clique has {len(c)} < {k} vertices")
+    es = {(min(u, v), max(u, v)) for u, v in inst["edges"]}
+    for u, v in itertools.combinations(sorted(c), 2):
+        if (u, v) not in es:
+            raise Invalid(f"no edge between {u} and {v}")
+    return None, {"size": len(c)}
diff --git a/tasks/construct-nppc-clique-l4-s4/tests/instance.json b/tasks/construct-nppc-clique-l4-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0352c9ee06d9a9a10dce14a51854beff7fe028a5
--- /dev/null
+++ b/tasks/construct-nppc-clique-l4-s4/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 30, "k": 15, "edges": [[0, 4], [0, 6], [0, 8], [0, 9], [0, 14], [0, 15], [0, 16], [0, 17], [0, 19], [0, 23], [0, 26], [0, 27], [0, 28], [0, 29], [1, 2], [1, 3], [1, 9], [1, 10], [1, 13], [1, 14], [1, 15], [1, 16], [1, 17], [1, 18], [1, 19], [1, 22], [1, 23], [1, 24], [1, 25], [1, 26], [1, 27], [1, 28], [1, 29], [2, 4], [2, 5], [2, 9], [2, 10], [2, 13], [2, 14], [2, 16], [2, 17], [2, 19], [2, 20], [2, 22], [2, 23], [2, 26], [2, 27], [2, 28], [3, 4], [3, 6], [3, 7], [3, 10], [3, 12], [3, 13], [3, 15], [3, 17], [3, 18], [3, 19], [3, 21], [3, 23], [3, 26], [4, 6], [4, 7], [4, 8], [4, 9], [4, 10], [4, 12], [4, 14], [4, 15], [4, 18], [4, 19], [4, 20], [4, 21], [4, 24], [4, 25], [4, 29], [5, 7], [5, 10], [5, 11], [5, 12], [5, 13], [5, 14], [5, 18], [5, 21], [5, 23], [5, 25], [5, 28], [5, 29], [6, 8], [6, 9], [6, 10], [6, 13], [6, 14], [6, 17], [6, 20], [6, 21], [6, 23], [7, 8], [7, 10], [7, 11], [7, 12], [7, 13], [7, 21], [7, 25], [7, 28], [7, 29], [8, 10], [8, 11], [8, 14], [8, 15], [8, 16], [8, 17], [8, 19], [8, 20], [8, 22], [8, 23], [8, 24], [8, 25], [8, 26], [8, 28], [8, 29], [9, 10], [9, 13], [9, 14], [9, 15], [9, 16], [9, 17], [9, 18], [9, 19], [9, 20], [9, 21], [9, 22], [9, 23], [9, 24], [9, 25], [9, 26], [9, 27], [9, 28], [9, 29], [10, 13], [10, 14], [10, 15], [10, 16], [10, 17], [10, 18], [10, 19], [10, 21], [10, 22], [10, 23], [10, 24], [10, 26], [10, 27], [10, 28], [10, 29], [11, 15], [11, 16], [11, 20], [11, 21], [11, 26], [11, 29], [12, 13], [12, 15], [12, 17], [12, 19], [12, 21], [12, 24], [12, 25], [12, 26], [12, 27], [12, 29], [13, 15], [13, 16], [13, 17], [13, 18], [13, 19], [13, 23], [13, 24], [13, 26], [13, 27], [13, 28], [13, 29], [14, 15], [14, 16], [14, 17], [14, 24], [14, 27], [14, 28], [14, 29], [15, 16], [15, 17], [15, 18], [15, 19], [15, 20], [15, 22], [15, 23], [15, 24], [15, 25], [15, 26], [15, 27], [15, 28], [15, 29], [16, 17], [16, 18], [16, 19], [16, 20], [16, 21], [16, 22], [16, 23], [16, 24], [16, 25], [16, 26], [16, 27], [16, 28], [16, 29], [17, 18], [17, 19], [17, 23], [17, 24], [17, 25], [17, 26], [17, 27], [17, 28], [17, 29], [18, 19], [18, 21], [18, 23], [18, 24], [18, 25], [18, 26], [18, 27], [18, 28], [18, 29], [19, 21], [19, 22], [19, 23], [19, 24], [19, 25], [19, 26], [19, 27], [19, 28], [19, 29], [20, 21], [20, 22], [20, 23], [20, 24], [20, 27], [20, 28], [21, 22], [21, 23], [21, 26], [21, 27], [21, 28], [22, 23], [22, 25], [22, 27], [22, 28], [22, 29], [23, 24], [23, 26], [23, 27], [23, 28], [23, 29], [24, 25], [24, 26], [24, 27], [24, 28], [24, 29], [25, 26], [25, 28], [26, 27], [26, 28], [26, 29], [27, 28], [27, 29], [28, 29]], "family": "nppc_clique", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-clique-l4-s4/tests/run.py b/tasks/construct-nppc-clique-l4-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-clique-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-nppc-clique-l4-s4/tests/test.sh b/tasks/construct-nppc-clique-l4-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-clique-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-nppc-dominating-set-l4-s3/environment/Dockerfile b/tasks/construct-nppc-dominating-set-l4-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-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-nppc-dominating-set-l4-s3/environment/instance.json b/tasks/construct-nppc-dominating-set-l4-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e01ec8ad51681d5565165a50c05209b51830a312
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l4-s3/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 100, "k": 20, "edges": [[0, 2], [0, 3], [0, 4], [0, 16], [0, 17], [0, 18], [0, 22], [0, 25], [0, 27], [0, 28], [0, 32], [0, 34], [0, 43], [0, 44], [0, 48], [0, 50], [0, 56], [0, 57], [0, 60], [0, 61], [0, 62], [0, 65], [0, 68], [0, 69], [0, 74], [0, 76], [0, 77], [0, 78], [0, 79], [0, 86], [0, 92], [0, 93], [0, 95], [0, 98], [1, 3], [1, 7], [1, 8], [1, 9], [1, 12], [1, 14], [1, 15], [1, 20], [1, 25], [1, 27], [1, 28], [1, 29], [1, 32], [1, 36], [1, 43], [1, 46], [1, 47], [1, 52], [1, 56], [1, 57], [1, 63], [1, 66], [1, 67], [1, 70], [1, 73], [1, 74], [1, 76], [1, 77], [1, 80], [1, 83], [1, 85], [1, 86], [1, 89], [1, 93], [1, 94], [1, 96], [1, 98], [2, 5], [2, 6], [2, 9], [2, 10], [2, 13], [2, 16], [2, 18], [2, 19], [2, 22], [2, 24], [2, 25], [2, 26], [2, 27], [2, 30], [2, 36], [2, 37], [2, 38], [2, 45], [2, 46], [2, 47], [2, 48], [2, 53], [2, 54], [2, 55], [2, 56], [2, 57], [2, 59], [2, 62], [2, 66], [2, 70], [2, 73], [2, 76], [2, 78], [2, 80], [2, 84], [2, 85], [2, 86], 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\ No newline at end of file
diff --git a/tasks/construct-nppc-dominating-set-l4-s3/solution/solve.sh b/tasks/construct-nppc-dominating-set-l4-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..2c022b0bb03598a2467a3b70b6636da1e261e845
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-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'
+{"dominating_set": [2, 6, 14, 22, 41, 51, 52, 54, 68, 74, 78, 80, 85, 86, 87, 89, 92, 96, 97, 98]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-dominating-set-l4-s3/tests/check.py b/tasks/construct-nppc-dominating-set-l4-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..0677a7bcbccb02297d022ffa4b5dd15da6371bff
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l4-s3/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-l4-s3/tests/instance.json b/tasks/construct-nppc-dominating-set-l4-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e01ec8ad51681d5565165a50c05209b51830a312
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l4-s3/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 100, "k": 20, "edges": [[0, 2], [0, 3], [0, 4], [0, 16], [0, 17], [0, 18], [0, 22], [0, 25], [0, 27], [0, 28], [0, 32], [0, 34], [0, 43], [0, 44], [0, 48], [0, 50], [0, 56], [0, 57], [0, 60], [0, 61], [0, 62], [0, 65], [0, 68], [0, 69], [0, 74], [0, 76], [0, 77], [0, 78], [0, 79], [0, 86], [0, 92], [0, 93], [0, 95], [0, 98], [1, 3], [1, 7], [1, 8], [1, 9], [1, 12], [1, 14], [1, 15], [1, 20], [1, 25], [1, 27], [1, 28], [1, 29], [1, 32], [1, 36], [1, 43], [1, 46], [1, 47], [1, 52], [1, 56], [1, 57], [1, 63], [1, 66], [1, 67], [1, 70], [1, 73], [1, 74], [1, 76], [1, 77], [1, 80], [1, 83], [1, 85], [1, 86], [1, 89], [1, 93], [1, 94], [1, 96], [1, 98], [2, 5], [2, 6], [2, 9], [2, 10], [2, 13], [2, 16], [2, 18], [2, 19], [2, 22], [2, 24], [2, 25], [2, 26], [2, 27], [2, 30], [2, 36], [2, 37], [2, 38], [2, 45], [2, 46], [2, 47], [2, 48], [2, 53], [2, 54], [2, 55], [2, 56], [2, 57], [2, 59], [2, 62], [2, 66], [2, 70], [2, 73], [2, 76], [2, 78], [2, 80], [2, 84], [2, 85], [2, 86], 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[89, 92], [89, 99], [90, 91], [90, 94], [90, 97], [91, 95], [92, 97], [92, 99], [93, 94], [94, 97], [95, 98], [95, 99], [97, 98], [97, 99], [98, 99]], "family": "nppc_dominating_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-dominating-set-l4-s3/tests/run.py b/tasks/construct-nppc-dominating-set-l4-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-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-nppc-dominating-set-l4-s3/tests/test.sh b/tasks/construct-nppc-dominating-set-l4-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-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-nppc-dominating-set-l4-s4/environment/Dockerfile b/tasks/construct-nppc-dominating-set-l4-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-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-nppc-dominating-set-l4-s4/environment/instance.json b/tasks/construct-nppc-dominating-set-l4-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e9ac531d97abeb3503b1a09366abd31865a5b781
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l4-s4/environment/instance.json
@@ -0,0 +1 @@
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\ No newline at end of file
diff --git a/tasks/construct-nppc-dominating-set-l4-s4/solution/solve.sh b/tasks/construct-nppc-dominating-set-l4-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..837df39b13e7979d5e6ba51b4d769919323bbd14
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-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'
+{"dominating_set": [5, 8, 13, 15, 19, 26, 29, 33, 34, 39, 45, 46, 49, 56, 70, 82, 84, 85, 92, 98]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-dominating-set-l4-s4/tests/check.py b/tasks/construct-nppc-dominating-set-l4-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..0677a7bcbccb02297d022ffa4b5dd15da6371bff
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l4-s4/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-l4-s4/tests/instance.json b/tasks/construct-nppc-dominating-set-l4-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e9ac531d97abeb3503b1a09366abd31865a5b781
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-l4-s4/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 100, "k": 20, "edges": [[0, 3], [0, 7], [0, 9], [0, 13], [0, 14], [0, 19], [0, 21], [0, 25], [0, 26], [0, 39], [0, 41], [0, 48], [0, 49], [0, 50], [0, 55], [0, 60], [0, 65], [0, 66], [0, 68], [0, 69], [0, 70], [0, 72], [0, 73], [0, 74], [0, 75], [0, 76], [0, 78], [0, 81], [0, 85], [0, 88], [0, 89], [0, 90], [0, 93], [0, 95], [0, 97], [1, 2], [1, 4], [1, 5], [1, 9], [1, 12], [1, 17], [1, 20], [1, 21], [1, 23], [1, 36], [1, 39], [1, 41], [1, 43], [1, 49], [1, 53], [1, 56], [1, 57], [1, 58], [1, 66], [1, 69], [1, 70], [1, 73], [1, 74], [1, 78], [1, 82], [1, 83], [1, 92], [1, 93], [1, 95], [1, 98], [2, 3], [2, 5], [2, 8], [2, 15], [2, 21], [2, 24], [2, 29], [2, 30], [2, 32], [2, 33], [2, 40], [2, 43], [2, 44], [2, 46], [2, 48], [2, 51], [2, 61], [2, 68], [2, 71], [2, 72], [2, 73], [2, 74], [2, 75], [2, 76], [2, 80], [2, 85], [2, 96], [2, 98], [3, 5], [3, 6], [3, 14], [3, 15], [3, 16], [3, 22], [3, 26], [3, 28], [3, 30], [3, 32], [3, 41], [3, 42], [3, 49], [3, 57], [3, 66], 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[71, 99], [72, 76], [72, 77], [72, 81], [72, 90], [72, 91], [72, 96], [72, 97], [73, 75], [73, 77], [73, 78], [73, 80], [73, 81], [73, 82], [73, 86], [73, 94], [73, 98], [74, 76], [74, 80], [74, 82], [74, 88], [74, 90], [74, 91], [74, 92], [74, 93], [74, 96], [75, 77], [75, 81], [75, 84], [75, 85], [75, 86], [75, 87], [75, 90], [75, 94], [75, 97], [76, 80], [76, 82], [76, 85], [76, 91], [77, 81], [77, 87], [77, 88], [77, 93], [77, 95], [77, 96], [77, 98], [78, 83], [78, 84], [78, 85], [78, 87], [78, 89], [78, 98], [79, 80], [79, 86], [79, 87], [79, 91], [79, 94], [79, 97], [80, 81], [80, 83], [80, 89], [80, 93], [80, 96], [80, 97], [81, 83], [81, 84], [81, 88], [81, 91], [81, 92], [81, 99], [82, 86], [82, 90], [82, 92], [82, 93], [83, 84], [83, 86], [83, 88], [83, 90], [83, 98], [84, 86], [84, 91], [84, 92], [84, 93], [84, 94], [84, 95], [84, 99], [85, 86], [85, 91], [85, 95], [85, 96], [85, 98], [85, 99], [86, 88], [86, 90], [86, 92], [86, 98], [86, 99], [87, 88], [87, 89], [87, 96], [88, 89], [88, 92], [89, 90], [89, 92], [89, 97], [90, 98], [92, 93], [92, 97], [93, 94], [93, 95], [93, 98], [94, 95], [94, 96], [94, 99], [96, 97], [97, 99]], "family": "nppc_dominating_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-dominating-set-l4-s4/tests/run.py b/tasks/construct-nppc-dominating-set-l4-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-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-nppc-dominating-set-l4-s4/tests/test.sh b/tasks/construct-nppc-dominating-set-l4-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-dominating-set-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-nppc-independent-set-l5-s4/environment/Dockerfile b/tasks/construct-nppc-independent-set-l5-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-independent-set-l5-s4/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-nppc-independent-set-l5-s4/environment/instance.json b/tasks/construct-nppc-independent-set-l5-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..aec79b7813993825688a47b688a82f58b1c07d5a
--- /dev/null
+++ b/tasks/construct-nppc-independent-set-l5-s4/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 48, "k": 24, "edges": [[0, 1], [0, 3], [0, 4], [0, 12], [0, 15], [0, 16], [0, 19], [0, 21], [0, 26], [0, 27], [0, 28], [0, 29], [0, 30], [0, 31], [0, 32], [0, 33], [0, 35], [0, 38], [0, 42], [0, 44], [0, 45], [0, 46], [0, 47], [1, 2], [1, 5], [1, 8], [1, 20], [1, 22], [1, 24], [1, 28], [1, 37], [1, 38], [1, 40], [1, 43], [1, 45], [2, 3], [2, 5], [2, 7], [2, 10], [2, 11], [2, 13], [2, 18], [2, 20], [2, 28], [2, 29], [2, 33], [2, 34], [2, 37], [2, 38], [2, 40], [2, 45], [2, 46], [2, 47], [3, 22], [3, 25], [3, 26], [3, 28], [3, 37], [3, 39], [3, 40], [3, 43], [3, 46], [4, 20], [4, 22], [4, 24], [4, 25], [4, 28], [4, 30], [4, 37], [4, 39], [4, 40], [4, 41], [4, 45], [5, 6], [5, 7], [5, 9], [5, 11], [5, 14], [5, 15], [5, 17], [5, 20], [5, 21], [5, 22], [5, 24], [5, 28], [5, 30], [5, 32], [5, 33], [5, 34], [5, 35], [5, 36], [5, 42], [5, 43], [5, 45], [5, 46], [5, 47], [6, 8], [6, 13], [6, 15], [6, 22], [6, 24], [6, 25], [6, 27], [6, 28], [6, 39], [6, 43], [6, 44], [6, 46], [7, 8], [7, 9], [7, 13], [7, 14], [7, 15], [7, 16], [7, 18], [7, 19], [7, 20], [7, 23], [7, 24], [7, 25], [7, 26], [7, 27], [7, 28], [7, 31], [7, 33], [7, 37], [7, 39], [7, 40], [7, 41], [7, 43], [7, 44], [7, 45], [7, 46], [8, 9], [8, 10], [8, 11], [8, 13], [8, 21], [8, 22], [8, 24], [8, 25], [8, 29], [8, 30], [8, 32], [8, 33], [8, 38], [8, 40], [8, 41], [8, 43], [8, 44], [8, 45], [9, 20], [9, 24], [9, 30], [9, 37], [9, 40], [9, 41], [9, 46], [10, 22], [10, 24], [10, 26], [10, 27], [10, 30], [10, 38], [10, 39], [10, 43], [10, 44], [10, 46], [11, 13], [11, 22], [11, 26], [11, 28], [11, 30], [11, 37], [11, 40], [11, 41], [11, 45], [11, 46], [12, 13], [12, 15], [12, 24], [12, 25], [12, 26], [12, 37], [12, 39], [12, 40], [12, 43], [12, 45], [13, 14], [13, 15], [13, 17], [13, 20], [13, 21], [13, 22], [13, 24], [13, 25], [13, 26], [13, 29], [13, 32], [13, 34], [13, 35], [13, 36], [13, 37], [13, 38], [13, 39], [13, 40], [13, 41], [13, 42], [13, 43], [13, 46], [14, 27], [14, 28], [14, 39], [14, 41], [14, 43], [14, 44], [14, 45], [14, 46], [15, 16], [15, 18], [15, 19], [15, 20], [15, 23], [15, 24], [15, 25], [15, 26], [15, 28], [15, 29], [15, 30], [15, 31], [15, 32], [15, 33], [15, 35], [15, 37], [15, 39], [15, 40], [15, 41], [15, 42], [15, 45], [16, 20], [16, 24], [16, 25], [16, 26], [16, 27], [16, 38], [16, 40], [16, 41], [16, 43], [16, 44], [16, 45], [17, 20], [17, 22], [17, 24], [17, 25], [17, 28], [17, 30], [17, 38], [17, 40], [17, 45], [18, 25], [18, 30], [18, 37], [18, 38], [18, 39], [18, 40], [18, 41], [18, 43], [18, 45], [19, 20], [19, 22], [19, 24], [19, 28], [19, 37], [19, 38], [19, 39], [19, 40], [19, 41], [19, 43], [19, 44], [20, 27], [20, 29], [20, 33], [20, 37], [20, 39], [20, 44], [21, 24], [21, 25], [21, 28], [21, 30], [21, 37], [21, 38], [21, 39], [21, 41], [21, 45], [21, 46], [22, 23], [22, 24], [22, 27], [22, 29], [22, 30], [22, 31], [22, 32], [22, 36], [22, 37], [22, 39], [22, 40], [22, 41], [22, 43], [22, 44], [22, 46], [23, 27], [23, 28], [23, 30], [23, 40], [23, 44], [23, 45], [23, 46], [24, 26], [24, 27], [24, 28], [24, 31], [24, 32], [24, 36], [24, 38], [24, 40], [24, 41], [24, 43], [25, 26], [25, 31], [25, 33], [25, 36], [25, 43], [25, 44], [25, 46], [26, 27], [26, 28], [26, 33], [26, 34], [26, 38], [26, 39], [26, 41], [26, 44], [26, 45], [26, 47], [27, 29], [27, 30], [27, 35], [27, 36], [27, 37], [27, 39], [27, 41], [27, 42], [27, 44], [27, 45], [27, 47], [28, 30], [28, 35], [28, 37], [28, 38], [28, 40], [28, 41], [28, 44], [28, 47], [29, 39], [29, 40], [29, 44], [29, 46], [30, 31], [30, 33], [30, 34], [30, 35], [30, 36], [30, 38], [30, 41], [30, 43], [30, 44], [30, 45], [30, 47], [31, 37], [31, 38], [31, 40], [31, 43], [31, 44], [31, 46], [32, 37], [32, 39], [32, 41], [32, 43], [32, 44], [33, 38], [33, 39], [33, 45], [33, 46], [34, 37], [34, 41], [34, 43], [34, 46], [35, 39], [35, 41], [35, 45], [36, 43], [36, 45], [36, 46], [37, 38], [37, 42], [37, 43], [37, 44], [38, 39], [38, 42], [38, 46], [39, 40], [39, 41], [39, 43], [39, 44], [39, 46], [40, 44], [40, 45], [40, 46], [40, 47], [41, 42], [41, 43], [41, 44], [41, 45], [42, 43], [42, 44], [42, 45], [43, 46], [44, 46], [46, 47]], "family": "nppc_independent_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-independent-set-l5-s4/solution/solve.sh b/tasks/construct-nppc-independent-set-l5-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..7d218036133f4e1eec9ab705b387f41ce89e7d43
--- /dev/null
+++ b/tasks/construct-nppc-independent-set-l5-s4/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"independent_set": [1, 3, 4, 6, 9, 10, 11, 12, 14, 16, 17, 18, 19, 21, 23, 29, 31, 32, 33, 34, 35, 36, 42, 47]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-independent-set-l5-s4/tests/check.py b/tasks/construct-nppc-independent-set-l5-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..92279a931488c7f39c08a3b11162b3f54a9c662a
--- /dev/null
+++ b/tasks/construct-nppc-independent-set-l5-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-l5-s4/tests/instance.json b/tasks/construct-nppc-independent-set-l5-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..aec79b7813993825688a47b688a82f58b1c07d5a
--- /dev/null
+++ b/tasks/construct-nppc-independent-set-l5-s4/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 48, "k": 24, "edges": [[0, 1], [0, 3], [0, 4], [0, 12], [0, 15], [0, 16], [0, 19], [0, 21], [0, 26], [0, 27], [0, 28], [0, 29], [0, 30], [0, 31], [0, 32], [0, 33], [0, 35], [0, 38], [0, 42], [0, 44], [0, 45], [0, 46], [0, 47], [1, 2], [1, 5], [1, 8], [1, 20], [1, 22], [1, 24], [1, 28], [1, 37], [1, 38], [1, 40], [1, 43], [1, 45], [2, 3], [2, 5], [2, 7], [2, 10], [2, 11], [2, 13], [2, 18], [2, 20], [2, 28], [2, 29], [2, 33], [2, 34], [2, 37], [2, 38], [2, 40], [2, 45], [2, 46], [2, 47], [3, 22], [3, 25], [3, 26], [3, 28], [3, 37], [3, 39], [3, 40], [3, 43], [3, 46], [4, 20], [4, 22], [4, 24], [4, 25], [4, 28], [4, 30], [4, 37], [4, 39], [4, 40], [4, 41], [4, 45], [5, 6], [5, 7], [5, 9], [5, 11], [5, 14], [5, 15], [5, 17], [5, 20], [5, 21], [5, 22], [5, 24], [5, 28], [5, 30], [5, 32], [5, 33], [5, 34], [5, 35], [5, 36], [5, 42], [5, 43], [5, 45], [5, 46], [5, 47], [6, 8], [6, 13], [6, 15], [6, 22], [6, 24], [6, 25], [6, 27], [6, 28], [6, 39], [6, 43], [6, 44], [6, 46], [7, 8], [7, 9], [7, 13], [7, 14], [7, 15], [7, 16], [7, 18], [7, 19], [7, 20], [7, 23], [7, 24], [7, 25], [7, 26], [7, 27], [7, 28], [7, 31], [7, 33], [7, 37], [7, 39], [7, 40], [7, 41], [7, 43], [7, 44], [7, 45], [7, 46], [8, 9], [8, 10], [8, 11], [8, 13], [8, 21], [8, 22], [8, 24], [8, 25], [8, 29], [8, 30], [8, 32], [8, 33], [8, 38], [8, 40], [8, 41], [8, 43], [8, 44], [8, 45], [9, 20], [9, 24], [9, 30], [9, 37], [9, 40], [9, 41], [9, 46], [10, 22], [10, 24], [10, 26], [10, 27], [10, 30], [10, 38], [10, 39], [10, 43], [10, 44], [10, 46], [11, 13], [11, 22], [11, 26], [11, 28], [11, 30], [11, 37], [11, 40], [11, 41], [11, 45], [11, 46], [12, 13], [12, 15], [12, 24], [12, 25], [12, 26], [12, 37], [12, 39], [12, 40], [12, 43], [12, 45], [13, 14], [13, 15], [13, 17], [13, 20], [13, 21], [13, 22], [13, 24], [13, 25], [13, 26], [13, 29], [13, 32], [13, 34], [13, 35], [13, 36], [13, 37], [13, 38], [13, 39], [13, 40], [13, 41], [13, 42], [13, 43], [13, 46], [14, 27], [14, 28], [14, 39], [14, 41], [14, 43], [14, 44], [14, 45], [14, 46], [15, 16], [15, 18], [15, 19], [15, 20], [15, 23], [15, 24], [15, 25], [15, 26], [15, 28], [15, 29], [15, 30], [15, 31], [15, 32], [15, 33], [15, 35], [15, 37], [15, 39], [15, 40], [15, 41], [15, 42], [15, 45], [16, 20], [16, 24], [16, 25], [16, 26], [16, 27], [16, 38], [16, 40], [16, 41], [16, 43], [16, 44], [16, 45], [17, 20], [17, 22], [17, 24], [17, 25], [17, 28], [17, 30], [17, 38], [17, 40], [17, 45], [18, 25], [18, 30], [18, 37], [18, 38], [18, 39], [18, 40], [18, 41], [18, 43], [18, 45], [19, 20], [19, 22], [19, 24], [19, 28], [19, 37], [19, 38], [19, 39], [19, 40], [19, 41], [19, 43], [19, 44], [20, 27], [20, 29], [20, 33], [20, 37], [20, 39], [20, 44], [21, 24], [21, 25], [21, 28], [21, 30], [21, 37], [21, 38], [21, 39], [21, 41], [21, 45], [21, 46], [22, 23], [22, 24], [22, 27], [22, 29], [22, 30], [22, 31], [22, 32], [22, 36], [22, 37], [22, 39], [22, 40], [22, 41], [22, 43], [22, 44], [22, 46], [23, 27], [23, 28], [23, 30], [23, 40], [23, 44], [23, 45], [23, 46], [24, 26], [24, 27], [24, 28], [24, 31], [24, 32], [24, 36], [24, 38], [24, 40], [24, 41], [24, 43], [25, 26], [25, 31], [25, 33], [25, 36], [25, 43], [25, 44], [25, 46], [26, 27], [26, 28], [26, 33], [26, 34], [26, 38], [26, 39], [26, 41], [26, 44], [26, 45], [26, 47], [27, 29], [27, 30], [27, 35], [27, 36], [27, 37], [27, 39], [27, 41], [27, 42], [27, 44], [27, 45], [27, 47], [28, 30], [28, 35], [28, 37], [28, 38], [28, 40], [28, 41], [28, 44], [28, 47], [29, 39], [29, 40], [29, 44], [29, 46], [30, 31], [30, 33], [30, 34], [30, 35], [30, 36], [30, 38], [30, 41], [30, 43], [30, 44], [30, 45], [30, 47], [31, 37], [31, 38], [31, 40], [31, 43], [31, 44], [31, 46], [32, 37], [32, 39], [32, 41], [32, 43], [32, 44], [33, 38], [33, 39], [33, 45], [33, 46], [34, 37], [34, 41], [34, 43], [34, 46], [35, 39], [35, 41], [35, 45], [36, 43], [36, 45], [36, 46], [37, 38], [37, 42], [37, 43], [37, 44], [38, 39], [38, 42], [38, 46], [39, 40], [39, 41], [39, 43], [39, 44], [39, 46], [40, 44], [40, 45], [40, 46], [40, 47], [41, 42], [41, 43], [41, 44], [41, 45], [42, 43], [42, 44], [42, 45], [43, 46], [44, 46], [46, 47]], "family": "nppc_independent_set", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-independent-set-l5-s4/tests/run.py b/tasks/construct-nppc-independent-set-l5-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-independent-set-l5-s4/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-nppc-independent-set-l5-s4/tests/test.sh b/tasks/construct-nppc-independent-set-l5-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-independent-set-l5-s4/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-nppc-quad-diophantine-l4-s2/environment/Dockerfile b/tasks/construct-nppc-quad-diophantine-l4-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-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-nppc-quad-diophantine-l4-s2/environment/instance.json b/tasks/construct-nppc-quad-diophantine-l4-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..ef545747c0a7504808e9d5726cbc070d7a8e9054
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-l4-s2/environment/instance.json
@@ -0,0 +1 @@
+{"a": 25872, "b": 90714, "c": 607691258881800, "family": "nppc_quad_diophantine", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-quad-diophantine-l4-s2/solution/solve.sh b/tasks/construct-nppc-quad-diophantine-l4-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..0d1e2fc2c9b679e4fc9613f4ae7b9823494731e6
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-l4-s2/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": 153258, "y": 103028}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-quad-diophantine-l4-s2/tests/check.py b/tasks/construct-nppc-quad-diophantine-l4-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..88989d2a8839b089c9b07dd2d3c26c21bfdca58c
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-l4-s2/tests/check.py
@@ -0,0 +1,11 @@
+"""a*x^2 + b*y = c in positive integers. Accepts any solution (exact integer arithmetic)."""
+from run import Invalid, as_int
+
+
+def check(inst, ans):
+    # upstream NPPC verifier only checks x > 0 and y > 0, so non-integer (float) x, y were accepted
+    x = as_int(ans.get("x"), "x", lo=1)
+    y = as_int(ans.get("y"), "y", lo=1)
+    if inst["a"] * x * x + inst["b"] * y != inst["c"]:
+        raise Invalid("a*x^2 + b*y != c")
+    return None, {}
diff --git a/tasks/construct-nppc-quad-diophantine-l4-s2/tests/instance.json b/tasks/construct-nppc-quad-diophantine-l4-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..ef545747c0a7504808e9d5726cbc070d7a8e9054
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-l4-s2/tests/instance.json
@@ -0,0 +1 @@
+{"a": 25872, "b": 90714, "c": 607691258881800, "family": "nppc_quad_diophantine", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-quad-diophantine-l4-s2/tests/run.py b/tasks/construct-nppc-quad-diophantine-l4-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-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-nppc-quad-diophantine-l4-s2/tests/test.sh b/tasks/construct-nppc-quad-diophantine-l4-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-quad-diophantine-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-nppc-quadratic-congruence-l4-s5/environment/Dockerfile b/tasks/construct-nppc-quadratic-congruence-l4-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-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-nppc-quadratic-congruence-l4-s5/environment/instance.json b/tasks/construct-nppc-quadratic-congruence-l4-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0f7c07acf083d8022ee0549498865aca0fa1aebc
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-l4-s5/environment/instance.json
@@ -0,0 +1 @@
+{"a": 1187071, "b": 1303743, "c": 3436932, "family": "nppc_quadratic_congruence", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-quadratic-congruence-l4-s5/solution/solve.sh b/tasks/construct-nppc-quadratic-congruence-l4-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..f7dfbd491fa2b9d75eb5cf033a5a932b3f825da1
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-l4-s5/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": 2029592}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-quadratic-congruence-l4-s5/tests/check.py b/tasks/construct-nppc-quadratic-congruence-l4-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..1a751c071a276ca0aae7fb05bfdcd7b6e0c6d3a6
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-l4-s5/tests/check.py
@@ -0,0 +1,11 @@
+"""Quadratic congruence: integer x with 1 <= x < c and x^2 = a (mod b). Accepts any such x."""
+from run import Invalid, as_int
+
+
+def check(inst, ans):
+    # upstream NPPC verifier did int(x) (accepting "12", 12.9 -> 12) and only checked x < c, so negative x
+    # (e.g. -r for any root r) and x = 0 passed and the bound c was meaningless
+    x = as_int(ans.get("x"), "x", lo=1, hi=inst["c"] - 1)
+    if x * x % inst["b"] != inst["a"]:
+        raise Invalid("x^2 mod b != a")
+    return None, {}
diff --git a/tasks/construct-nppc-quadratic-congruence-l4-s5/tests/instance.json b/tasks/construct-nppc-quadratic-congruence-l4-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0f7c07acf083d8022ee0549498865aca0fa1aebc
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-l4-s5/tests/instance.json
@@ -0,0 +1 @@
+{"a": 1187071, "b": 1303743, "c": 3436932, "family": "nppc_quadratic_congruence", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-quadratic-congruence-l4-s5/tests/run.py b/tasks/construct-nppc-quadratic-congruence-l4-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-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-nppc-quadratic-congruence-l4-s5/tests/test.sh b/tasks/construct-nppc-quadratic-congruence-l4-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-quadratic-congruence-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-nppc-vertex-cover-l2-s1/environment/Dockerfile b/tasks/construct-nppc-vertex-cover-l2-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-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-nppc-vertex-cover-l2-s1/environment/instance.json b/tasks/construct-nppc-vertex-cover-l2-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..aa97dc76ceaeaa7b64f661363fd3f34d136b2205
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-l2-s1/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 16, "k": 5, "edges": [[0, 1], [0, 4], [0, 6], [0, 7], [0, 8], [0, 10], [0, 11], [0, 13], [0, 14], [1, 8], [1, 10], [1, 12], [2, 8], [2, 13], [3, 8], [3, 10], [3, 13], [4, 8], [4, 12], [4, 13], [5, 10], [5, 12], [6, 8], [6, 12], [6, 13], [7, 10], [7, 12], [8, 9], [8, 10], [8, 11], [8, 12], [8, 13], [8, 15], [9, 12], [9, 13], [10, 11], [10, 12], [10, 13], [10, 14], [10, 15], [11, 12], [11, 13], [12, 13], [12, 14], [12, 15], [13, 14]], "family": "nppc_vertex_cover", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-vertex-cover-l2-s1/solution/solve.sh b/tasks/construct-nppc-vertex-cover-l2-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..106cb7122c8f56b753fde74fb1a4492a7d1c350f
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-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'
+{"cover": [0, 8, 10, 12, 13]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-vertex-cover-l2-s1/tests/check.py b/tasks/construct-nppc-vertex-cover-l2-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..d8ed740cc660c4837d941a8d70ea394c3bc9f69e
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-l2-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-l2-s1/tests/instance.json b/tasks/construct-nppc-vertex-cover-l2-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..aa97dc76ceaeaa7b64f661363fd3f34d136b2205
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-l2-s1/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 16, "k": 5, "edges": [[0, 1], [0, 4], [0, 6], [0, 7], [0, 8], [0, 10], [0, 11], [0, 13], [0, 14], [1, 8], [1, 10], [1, 12], [2, 8], [2, 13], [3, 8], [3, 10], [3, 13], [4, 8], [4, 12], [4, 13], [5, 10], [5, 12], [6, 8], [6, 12], [6, 13], [7, 10], [7, 12], [8, 9], [8, 10], [8, 11], [8, 12], [8, 13], [8, 15], [9, 12], [9, 13], [10, 11], [10, 12], [10, 13], [10, 14], [10, 15], [11, 12], [11, 13], [12, 13], [12, 14], [12, 15], [13, 14]], "family": "nppc_vertex_cover", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-vertex-cover-l2-s1/tests/run.py b/tasks/construct-nppc-vertex-cover-l2-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-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-nppc-vertex-cover-l2-s1/tests/test.sh b/tasks/construct-nppc-vertex-cover-l2-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-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-nppc-vertex-cover-l3-s8/environment/Dockerfile b/tasks/construct-nppc-vertex-cover-l3-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-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-nppc-vertex-cover-l3-s8/environment/instance.json b/tasks/construct-nppc-vertex-cover-l3-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..7e290c99a143c8b7866a1c4100be4af285ee37c7
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-l3-s8/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 24, "k": 12, "edges": [[0, 1], [0, 3], [0, 9], [0, 13], [0, 15], [0, 16], [0, 17], [0, 18], [0, 23], [1, 2], [1, 3], [1, 4], [1, 5], [1, 8], [1, 10], [1, 11], [1, 13], [1, 15], [1, 16], [1, 17], [1, 18], [1, 19], [1, 20], [1, 21], [1, 22], [1, 23], [2, 3], [2, 8], [2, 10], [2, 13], [2, 15], [2, 19], [2, 23], [3, 4], [3, 5], [3, 7], [3, 8], [3, 12], [3, 13], [3, 14], [3, 15], [3, 19], [3, 20], [3, 22], [4, 8], [4, 9], [4, 10], [4, 16], [5, 10], [5, 15], [5, 16], [5, 18], [5, 19], [5, 23], [6, 8], [6, 10], [6, 13], [6, 15], [6, 17], [6, 18], [6, 19], [6, 23], [7, 8], [7, 9], [7, 10], [7, 13], [7, 15], [7, 16], [7, 19], [7, 23], [8, 10], [8, 11], [8, 13], [8, 18], [8, 19], [8, 21], [8, 22], [8, 23], [9, 10], [9, 12], [9, 13], [9, 15], [9, 16], [9, 18], [9, 19], [9, 20], [10, 11], [10, 13], [10, 14], [10, 16], [10, 17], [10, 19], [10, 20], [10, 23], [11, 13], [11, 15], [11, 16], [11, 17], [11, 23], [12, 13], [12, 15], [12, 16], [13, 14], [13, 15], [13, 16], [13, 17], [13, 18], [13, 20], [13, 21], [13, 22], [13, 23], [14, 15], [14, 17], [14, 19], [14, 23], [15, 16], [15, 17], [15, 23], [16, 17], [16, 18], [16, 19], [16, 21], [16, 22], [16, 23], [17, 18], [17, 19], [17, 21], [17, 23], [18, 19], [18, 21], [18, 23], [19, 20], [19, 23]], "family": "nppc_vertex_cover", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-vertex-cover-l3-s8/solution/solve.sh b/tasks/construct-nppc-vertex-cover-l3-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..8688de31804054efb1e0fab771b253bb33f474d1
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-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'
+{"cover": [1, 3, 8, 9, 10, 13, 15, 16, 17, 18, 19, 23]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-vertex-cover-l3-s8/tests/check.py b/tasks/construct-nppc-vertex-cover-l3-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..d8ed740cc660c4837d941a8d70ea394c3bc9f69e
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-l3-s8/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-l3-s8/tests/instance.json b/tasks/construct-nppc-vertex-cover-l3-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..7e290c99a143c8b7866a1c4100be4af285ee37c7
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-l3-s8/tests/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 24, "k": 12, "edges": [[0, 1], [0, 3], [0, 9], [0, 13], [0, 15], [0, 16], [0, 17], [0, 18], [0, 23], [1, 2], [1, 3], [1, 4], [1, 5], [1, 8], [1, 10], [1, 11], [1, 13], [1, 15], [1, 16], [1, 17], [1, 18], [1, 19], [1, 20], [1, 21], [1, 22], [1, 23], [2, 3], [2, 8], [2, 10], [2, 13], [2, 15], [2, 19], [2, 23], [3, 4], [3, 5], [3, 7], [3, 8], [3, 12], [3, 13], [3, 14], [3, 15], [3, 19], [3, 20], [3, 22], [4, 8], [4, 9], [4, 10], [4, 16], [5, 10], [5, 15], [5, 16], [5, 18], [5, 19], [5, 23], [6, 8], [6, 10], [6, 13], [6, 15], [6, 17], [6, 18], [6, 19], [6, 23], [7, 8], [7, 9], [7, 10], [7, 13], [7, 15], [7, 16], [7, 19], [7, 23], [8, 10], [8, 11], [8, 13], [8, 18], [8, 19], [8, 21], [8, 22], [8, 23], [9, 10], [9, 12], [9, 13], [9, 15], [9, 16], [9, 18], [9, 19], [9, 20], [10, 11], [10, 13], [10, 14], [10, 16], [10, 17], [10, 19], [10, 20], [10, 23], [11, 13], [11, 15], [11, 16], [11, 17], [11, 23], [12, 13], [12, 15], [12, 16], [13, 14], [13, 15], [13, 16], [13, 17], [13, 18], [13, 20], [13, 21], [13, 22], [13, 23], [14, 15], [14, 17], [14, 19], [14, 23], [15, 16], [15, 17], [15, 23], [16, 17], [16, 18], [16, 19], [16, 21], [16, 22], [16, 23], [17, 18], [17, 19], [17, 21], [17, 23], [18, 19], [18, 21], [18, 23], [19, 20], [19, 23]], "family": "nppc_vertex_cover", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-vertex-cover-l3-s8/tests/run.py b/tasks/construct-nppc-vertex-cover-l3-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-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-nppc-vertex-cover-l3-s8/tests/test.sh b/tasks/construct-nppc-vertex-cover-l3-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-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-nppc-vertex-cover-l6-s9/environment/Dockerfile b/tasks/construct-nppc-vertex-cover-l6-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-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-nppc-vertex-cover-l6-s9/environment/instance.json b/tasks/construct-nppc-vertex-cover-l6-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d9512504f95f8f96e53798df88919afc6308b82d
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-l6-s9/environment/instance.json
@@ -0,0 +1 @@
+{"num_nodes": 160, "k": 64, "edges": [[0, 6], [0, 12], [0, 15], [0, 23], [0, 29], [0, 55], [0, 57], [0, 59], [0, 65], [0, 67], [0, 75], [0, 83], [0, 86], [0, 90], [0, 103], [0, 105], [0, 107], [0, 108], [0, 117], [0, 119], [0, 131], [0, 133], [0, 142], [0, 145], [0, 155], [0, 156], [1, 7], [1, 23], [1, 33], [1, 49], [1, 60], [1, 73], [1, 96], [1, 98], [1, 147], [2, 11], [2, 13], [2, 17], [2, 20], [2, 21], [2, 25], [2, 27], [2, 28], [2, 30], [2, 32], [2, 33], [2, 39], [2, 41], [2, 44], [2, 50], [2, 52], [2, 62], [2, 68], [2, 77], [2, 79], [2, 83], [2, 88], [2, 90], [2, 92], [2, 94], [2, 98], [2, 111], [2, 112], [2, 122], [2, 126], [2, 128], [2, 129], [2, 131], [2, 139], [2, 148], [2, 156], [3, 10], [3, 24], [3, 26], [3, 30], [3, 31], [3, 37], [3, 40], [3, 41], [3, 50], [3, 60], [3, 65], [3, 68], [3, 73], [3, 74], [3, 90], [3, 94], [3, 96], [3, 100], [3, 101], [3, 102], [3, 104], [3, 106], [3, 107], [3, 108], [3, 109], [3, 110], [3, 123], [3, 126], [3, 127], [3, 129], [3, 136], [3, 140], [3, 141], [3, 155], [4, 13], [4, 42], [4, 46], [4, 65], [4, 74], [4, 89], [4, 98], [4, 136], [4, 138], [4, 147], [4, 152], [4, 155], [5, 7], [5, 11], [5, 27], [5, 32], [5, 34], [5, 41], [5, 43], [5, 49], [5, 56], [5, 60], [5, 61], [5, 69], [5, 71], [5, 86], [5, 104], [5, 105], [5, 106], [5, 116], [5, 119], [5, 134], [5, 136], [5, 149], [5, 154], [5, 157], [6, 13], [6, 16], [6, 20], [6, 24], [6, 28], [6, 35], [6, 36], [6, 39], [6, 43], [6, 44], [6, 49], [6, 56], [6, 63], [6, 64], [6, 65], [6, 68], [6, 70], [6, 71], [6, 74], [6, 78], [6, 84], [6, 100], [6, 101], [6, 102], [6, 103], [6, 104], [6, 117], [6, 120], [6, 125], [6, 132], [6, 147], [6, 155], [7, 11], [7, 13], [7, 26], [7, 29], [7, 39], [7, 41], [7, 55], [7, 59], [7, 62], [7, 64], [7, 65], [7, 66], [7, 72], [7, 76], [7, 80], [7, 86], [7, 87], [7, 97], [7, 99], [7, 101], [7, 104], [7, 109], [7, 119], [7, 125], [7, 139], [7, 140], [7, 148], [7, 152], [7, 157], [8, 20], [8, 37], [8, 39], [8, 65], [8, 73], [8, 89], [8, 103], [8, 136], [9, 32], [9, 41], [9, 96], [9, 104], [9, 136], [9, 143], [10, 11], [10, 12], [10, 13], [10, 14], [10, 16], [10, 22], [10, 23], [10, 29], [10, 30], [10, 33], [10, 39], [10, 42], [10, 44], [10, 47], [10, 52], [10, 59], [10, 69], [10, 71], [10, 78], [10, 81], [10, 85], [10, 91], [10, 93], [10, 100], [10, 103], [10, 105], [10, 110], [10, 115], [10, 117], [10, 124], [10, 126], [10, 131], [10, 135], [10, 136], [10, 142], [10, 144], [10, 148], [10, 149], [10, 158], [11, 46], [11, 47], [11, 62], [11, 83], [11, 152], [11, 154], [12, 23], [12, 35], [12, 49], [12, 59], [12, 109], [12, 114], [12, 143], [12, 149], [12, 150], [13, 19], [13, 32], [13, 34], [13, 35], [13, 39], [13, 48], [13, 52], [13, 58], [13, 62], [13, 67], [13, 70], [13, 86], [13, 91], [13, 95], [13, 96], [13, 100], [13, 102], [13, 104], [13, 108], [13, 114], [13, 117], [13, 124], [13, 127], [13, 131], [13, 133], [13, 136], [13, 147], [13, 157], [14, 39], [14, 90], [14, 92], [14, 108], [14, 124], [14, 128], [14, 157], [15, 32], [15, 46], [15, 59], [15, 125], [15, 138], [15, 152], [16, 20], [16, 39], [16, 40], [16, 89], [16, 92], [16, 112], [16, 114], [16, 117], [16, 128], [16, 149], [17, 32], [17, 33], [17, 41], [17, 42], [17, 46], [17, 47], [17, 49], [17, 62], [17, 111], [17, 114], [17, 150], [17, 153], [17, 158], [18, 59], [18, 89], [18, 90], [18, 92], [18, 109], [18, 155], [19, 33], [19, 35], [19, 41], [19, 49], [19, 52], [19, 59], [19, 94], [19, 98], [19, 102], [19, 104], [19, 114], [19, 124], [19, 125], [19, 138], [20, 21], [20, 29], [20, 38], [20, 40], [20, 46], [20, 49], [20, 55], [20, 61], [20, 65], [20, 81], [20, 87], [20, 89], [20, 90], [20, 91], [20, 93], [20, 96], [20, 102], [20, 104], [20, 118], [20, 122], [20, 123], [20, 125], [20, 126], [20, 129], [20, 132], [20, 136], [20, 137], [20, 138], [20, 140], [20, 141], [20, 145], [21, 40], [21, 46], [21, 49], [21, 55], [21, 106], [21, 114], [21, 125], [21, 128], [21, 153], [22, 40], [22, 42], [22, 60], [22, 83], [22, 108], [22, 112], [22, 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[136, 150], [136, 152], [136, 153], [137, 157], [138, 143], [138, 145], [138, 150], [138, 151], [138, 152], [138, 153], [138, 157], [138, 158], [139, 152], [139, 153], [139, 154], [140, 143], [140, 149], [140, 154], [140, 157], [141, 153], [142, 149], [142, 153], [142, 157], [142, 159], [143, 153], [143, 154], [143, 157], [144, 155], [145, 150], [145, 155], [145, 157], [146, 147], [146, 152], [147, 153], [147, 156], [147, 158], [148, 155], [153, 157], [155, 157]], "family": "nppc_vertex_cover", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-nppc-vertex-cover-l6-s9/solution/solve.sh b/tasks/construct-nppc-vertex-cover-l6-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..42cbf2f4ba79e0e8ab5588eb933063e22fa07304
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-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'
+{"cover": [0, 2, 3, 5, 6, 7, 10, 13, 20, 23, 25, 32, 33, 35, 37, 39, 40, 41, 42, 46, 47, 49, 52, 54, 55, 59, 60, 62, 65, 73, 74, 83, 89, 90, 92, 94, 96, 98, 102, 103, 104, 106, 108, 109, 111, 112, 114, 117, 124, 125, 128, 136, 138, 143, 147, 149, 150, 152, 153, 154, 155, 157, 158, 159]}
+RLMATH_EOF
diff --git a/tasks/construct-nppc-vertex-cover-l6-s9/tests/check.py b/tasks/construct-nppc-vertex-cover-l6-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..d8ed740cc660c4837d941a8d70ea394c3bc9f69e
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-l6-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-l6-s9/tests/instance.json b/tasks/construct-nppc-vertex-cover-l6-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d9512504f95f8f96e53798df88919afc6308b82d
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-l6-s9/tests/instance.json
@@ -0,0 +1 @@
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\ No newline at end of file
diff --git a/tasks/construct-nppc-vertex-cover-l6-s9/tests/run.py b/tasks/construct-nppc-vertex-cover-l6-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-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-nppc-vertex-cover-l6-s9/tests/test.sh b/tasks/construct-nppc-vertex-cover-l6-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-nppc-vertex-cover-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-rg-graph-color-l1-s3/environment/Dockerfile b/tasks/construct-rg-graph-color-l1-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l1-s3/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rg-graph-color-l1-s3/environment/instance.json b/tasks/construct-rg-graph-color-l1-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a3a66add9ac442f7d5f1e02110c9e12c1b03b153
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l1-s3/environment/instance.json
@@ -0,0 +1 @@
+{"num_vertices": 6, "num_colors": 4, "edges": [[0, 3], [0, 4], [1, 4], [1, 5], [4, 5]], "family": "rg_graph_color", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-graph-color-l1-s3/solution/solve.sh b/tasks/construct-rg-graph-color-l1-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..c6c0e5f292d10d5c43f15ac3eb48381b61d3cb19
--- /dev/null
+++ b/tasks/construct-rg-graph-color-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'
+{"coloring": [1, 1, 1, 2, 2, 3]}
+RLMATH_EOF
diff --git a/tasks/construct-rg-graph-color-l1-s3/tests/check.py b/tasks/construct-rg-graph-color-l1-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..e9328912405d1c894970c90415be62bf9556ed36
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l1-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-l1-s3/tests/instance.json b/tasks/construct-rg-graph-color-l1-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a3a66add9ac442f7d5f1e02110c9e12c1b03b153
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l1-s3/tests/instance.json
@@ -0,0 +1 @@
+{"num_vertices": 6, "num_colors": 4, "edges": [[0, 3], [0, 4], [1, 4], [1, 5], [4, 5]], "family": "rg_graph_color", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-graph-color-l1-s3/tests/run.py b/tasks/construct-rg-graph-color-l1-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l1-s3/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rg-graph-color-l1-s3/tests/test.sh b/tasks/construct-rg-graph-color-l1-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l1-s3/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rg-graph-color-l1-s4/environment/Dockerfile b/tasks/construct-rg-graph-color-l1-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rg-graph-color-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-rg-graph-color-l1-s4/environment/instance.json b/tasks/construct-rg-graph-color-l1-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..225d54eb2b6c15f30d48472df63e104e434949c7
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l1-s4/environment/instance.json
@@ -0,0 +1 @@
+{"num_vertices": 6, "num_colors": 4, "edges": [[0, 1], [0, 2], [0, 3], [0, 4], [1, 4], [2, 4], [2, 5], [3, 5]], "family": "rg_graph_color", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-graph-color-l1-s4/solution/solve.sh b/tasks/construct-rg-graph-color-l1-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..8d1fd615bee27aaf16cd8e173b252be32380586c
--- /dev/null
+++ b/tasks/construct-rg-graph-color-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'
+{"coloring": [1, 2, 2, 2, 3, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-rg-graph-color-l1-s4/tests/check.py b/tasks/construct-rg-graph-color-l1-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..e9328912405d1c894970c90415be62bf9556ed36
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l1-s4/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-l1-s4/tests/instance.json b/tasks/construct-rg-graph-color-l1-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..225d54eb2b6c15f30d48472df63e104e434949c7
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l1-s4/tests/instance.json
@@ -0,0 +1 @@
+{"num_vertices": 6, "num_colors": 4, "edges": [[0, 1], [0, 2], [0, 3], [0, 4], [1, 4], [2, 4], [2, 5], [3, 5]], "family": "rg_graph_color", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-graph-color-l1-s4/tests/run.py b/tasks/construct-rg-graph-color-l1-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rg-graph-color-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-rg-graph-color-l1-s4/tests/test.sh b/tasks/construct-rg-graph-color-l1-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rg-graph-color-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-rg-graph-color-l5-s2/environment/Dockerfile b/tasks/construct-rg-graph-color-l5-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rg-graph-color-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-rg-graph-color-l5-s2/environment/instance.json b/tasks/construct-rg-graph-color-l5-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..48eeaf21335f73d9fce8db482a40c5b52a7af449
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l5-s2/environment/instance.json
@@ -0,0 +1 @@
+{"num_vertices": 25, "num_colors": 3, "edges": [[0, 2], [0, 4], [0, 13], [0, 14], [0, 15], [1, 8], [1, 9], [1, 14], [1, 15], [2, 5], [2, 11], [2, 19], [2, 24], [3, 5], [3, 10], [3, 14], [3, 16], [3, 21], [4, 7], [4, 10], [4, 13], [5, 15], [5, 18], [6, 8], [6, 10], [6, 22], [6, 23], [7, 8], [7, 15], [7, 20], [8, 12], [8, 22], [8, 23], [9, 11], [9, 16], [9, 18], [9, 23], [10, 18], [11, 12], [11, 17], [11, 22], [12, 17], [13, 21], [14, 16], [14, 22], [16, 17], [17, 24], [18, 19], [20, 22], [21, 24]], "family": "rg_graph_color", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-graph-color-l5-s2/solution/solve.sh b/tasks/construct-rg-graph-color-l5-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..7708bbfd92a6de7811a7e6059e8544102c64f0a4
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l5-s2/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"coloring": [1, 1, 2, 1, 2, 3, 1, 1, 2, 2, 3, 1, 3, 3, 2, 2, 3, 2, 1, 3, 2, 2, 3, 3, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-rg-graph-color-l5-s2/tests/check.py b/tasks/construct-rg-graph-color-l5-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..e9328912405d1c894970c90415be62bf9556ed36
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l5-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-l5-s2/tests/instance.json b/tasks/construct-rg-graph-color-l5-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..48eeaf21335f73d9fce8db482a40c5b52a7af449
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l5-s2/tests/instance.json
@@ -0,0 +1 @@
+{"num_vertices": 25, "num_colors": 3, "edges": [[0, 2], [0, 4], [0, 13], [0, 14], [0, 15], [1, 8], [1, 9], [1, 14], [1, 15], [2, 5], [2, 11], [2, 19], [2, 24], [3, 5], [3, 10], [3, 14], [3, 16], [3, 21], [4, 7], [4, 10], [4, 13], [5, 15], [5, 18], [6, 8], [6, 10], [6, 22], [6, 23], [7, 8], [7, 15], [7, 20], [8, 12], [8, 22], [8, 23], [9, 11], [9, 16], [9, 18], [9, 23], [10, 18], [11, 12], [11, 17], [11, 22], [12, 17], [13, 21], [14, 16], [14, 22], [16, 17], [17, 24], [18, 19], [20, 22], [21, 24]], "family": "rg_graph_color", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-graph-color-l5-s2/tests/run.py b/tasks/construct-rg-graph-color-l5-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rg-graph-color-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-rg-graph-color-l5-s2/tests/test.sh b/tasks/construct-rg-graph-color-l5-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rg-graph-color-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-rg-graph-color-l5-s5/environment/Dockerfile b/tasks/construct-rg-graph-color-l5-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rg-graph-color-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-rg-graph-color-l5-s5/environment/instance.json b/tasks/construct-rg-graph-color-l5-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b903fb88d811acf183cdc7d5fea3f95246b31f53
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l5-s5/environment/instance.json
@@ -0,0 +1 @@
+{"num_vertices": 25, "num_colors": 3, "edges": [[0, 1], [0, 7], [0, 21], [1, 21], [2, 7], [2, 8], [2, 20], [3, 6], [3, 18], [4, 6], [4, 7], [4, 12], [5, 7], [5, 10], [5, 15], [5, 20], [5, 21], [6, 10], [6, 14], [6, 20], [7, 14], [7, 16], [7, 18], [8, 16], [8, 20], [9, 14], [9, 15], [9, 16], [10, 17], [12, 16], [13, 16], [13, 17], [13, 23], [14, 17], [16, 17], [17, 20], [17, 21], [17, 22], [18, 19], [19, 21], [20, 23], [21, 23]], "family": "rg_graph_color", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-graph-color-l5-s5/solution/solve.sh b/tasks/construct-rg-graph-color-l5-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..b4f34b50d83afd670994d9c4379db45f32483a72
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l5-s5/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"coloring": [1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 3, 1, 2, 1, 3, 2, 3, 2, 3, 1, 3, 3, 1, 2, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-rg-graph-color-l5-s5/tests/check.py b/tasks/construct-rg-graph-color-l5-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..e9328912405d1c894970c90415be62bf9556ed36
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l5-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-l5-s5/tests/instance.json b/tasks/construct-rg-graph-color-l5-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b903fb88d811acf183cdc7d5fea3f95246b31f53
--- /dev/null
+++ b/tasks/construct-rg-graph-color-l5-s5/tests/instance.json
@@ -0,0 +1 @@
+{"num_vertices": 25, "num_colors": 3, "edges": [[0, 1], [0, 7], [0, 21], [1, 21], [2, 7], [2, 8], [2, 20], [3, 6], [3, 18], [4, 6], [4, 7], [4, 12], [5, 7], [5, 10], [5, 15], [5, 20], [5, 21], [6, 10], [6, 14], [6, 20], [7, 14], [7, 16], [7, 18], [8, 16], [8, 20], [9, 14], [9, 15], [9, 16], [10, 17], [12, 16], [13, 16], [13, 17], [13, 23], [14, 17], [16, 17], [17, 20], [17, 21], [17, 22], [18, 19], [19, 21], [20, 23], [21, 23]], "family": "rg_graph_color", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-graph-color-l5-s5/tests/run.py b/tasks/construct-rg-graph-color-l5-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rg-graph-color-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-rg-graph-color-l5-s5/tests/test.sh b/tasks/construct-rg-graph-color-l5-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rg-graph-color-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-rg-intermediate-integration-l3-s6/environment/Dockerfile b/tasks/construct-rg-intermediate-integration-l3-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-l3-s6/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rg-intermediate-integration-l3-s6/environment/instance.json b/tasks/construct-rg-intermediate-integration-l3-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..6e5644058c1ea1917c2450352d3a0823cb79dd88
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-l3-s6/environment/instance.json
@@ -0,0 +1 @@
+{"variable": "x", "integrand": "-8*sin(8*x + 2)*cos(8*x + 2)**3", "problem_type": "trigonometric", "family": "rg_intermediate_integration", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-intermediate-integration-l3-s6/solution/solve.sh b/tasks/construct-rg-intermediate-integration-l3-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..a87febe6c61e912469a29c7e58b44ec147d9da02
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-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'
+{"antiderivative": "cos(8*x + 2)**4/4 + C"}
+RLMATH_EOF
diff --git a/tasks/construct-rg-intermediate-integration-l3-s6/tests/check.py b/tasks/construct-rg-intermediate-integration-l3-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..a9e364a8ede6c8215a16a04ee6b0d3529adc647d
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-l3-s6/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-l3-s6/tests/instance.json b/tasks/construct-rg-intermediate-integration-l3-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..6e5644058c1ea1917c2450352d3a0823cb79dd88
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-l3-s6/tests/instance.json
@@ -0,0 +1 @@
+{"variable": "x", "integrand": "-8*sin(8*x + 2)*cos(8*x + 2)**3", "problem_type": "trigonometric", "family": "rg_intermediate_integration", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-intermediate-integration-l3-s6/tests/run.py b/tasks/construct-rg-intermediate-integration-l3-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-l3-s6/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rg-intermediate-integration-l3-s6/tests/test.sh b/tasks/construct-rg-intermediate-integration-l3-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rg-intermediate-integration-l3-s6/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rg-simple-integration-l1-s5/environment/Dockerfile b/tasks/construct-rg-simple-integration-l1-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l1-s5/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rg-simple-integration-l1-s5/environment/instance.json b/tasks/construct-rg-simple-integration-l1-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..28fbb257a83db7461b91fbe209e3c7ba20905305
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l1-s5/environment/instance.json
@@ -0,0 +1 @@
+{"variable": "x", "integrand": "-27*x**2", "family": "rg_simple_integration", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-simple-integration-l1-s5/solution/solve.sh b/tasks/construct-rg-simple-integration-l1-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..b795c63d702f444f715401bf70803cb2211ae585
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l1-s5/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"antiderivative": "-9*x**3 + C"}
+RLMATH_EOF
diff --git a/tasks/construct-rg-simple-integration-l1-s5/tests/check.py b/tasks/construct-rg-simple-integration-l1-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..a9e364a8ede6c8215a16a04ee6b0d3529adc647d
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l1-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-l1-s5/tests/instance.json b/tasks/construct-rg-simple-integration-l1-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..28fbb257a83db7461b91fbe209e3c7ba20905305
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l1-s5/tests/instance.json
@@ -0,0 +1 @@
+{"variable": "x", "integrand": "-27*x**2", "family": "rg_simple_integration", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-simple-integration-l1-s5/tests/run.py b/tasks/construct-rg-simple-integration-l1-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l1-s5/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rg-simple-integration-l1-s5/tests/test.sh b/tasks/construct-rg-simple-integration-l1-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l1-s5/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rg-simple-integration-l5-s3/environment/Dockerfile b/tasks/construct-rg-simple-integration-l5-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l5-s3/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rg-simple-integration-l5-s3/environment/instance.json b/tasks/construct-rg-simple-integration-l5-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0604390e2c655bbdfdc3e3b8875e7ca8ce16e530
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l5-s3/environment/instance.json
@@ -0,0 +1 @@
+{"variable": "X", "integrand": "-8*X**7/63 + 15*X**5/2 - 10*X**4/3 - 3/X**4 + 20/X**6", "family": "rg_simple_integration", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-simple-integration-l5-s3/solution/solve.sh b/tasks/construct-rg-simple-integration-l5-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..b5c5d6d30b4b7f51ec6aae8e16b997429157f162
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l5-s3/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"antiderivative": "-X**8/63 + 5*X**6/4 - 2*X**5/3 + X**(-3) - 4/X**5 + C"}
+RLMATH_EOF
diff --git a/tasks/construct-rg-simple-integration-l5-s3/tests/check.py b/tasks/construct-rg-simple-integration-l5-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..a9e364a8ede6c8215a16a04ee6b0d3529adc647d
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l5-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-l5-s3/tests/instance.json b/tasks/construct-rg-simple-integration-l5-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0604390e2c655bbdfdc3e3b8875e7ca8ce16e530
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l5-s3/tests/instance.json
@@ -0,0 +1 @@
+{"variable": "X", "integrand": "-8*X**7/63 + 15*X**5/2 - 10*X**4/3 - 3/X**4 + 20/X**6", "family": "rg_simple_integration", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-simple-integration-l5-s3/tests/run.py b/tasks/construct-rg-simple-integration-l5-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l5-s3/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rg-simple-integration-l5-s3/tests/test.sh b/tasks/construct-rg-simple-integration-l5-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l5-s3/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rg-simple-integration-l5-s4/environment/Dockerfile b/tasks/construct-rg-simple-integration-l5-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l5-s4/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rg-simple-integration-l5-s4/environment/instance.json b/tasks/construct-rg-simple-integration-l5-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..65b30db3efcf1b53137130ea543a32216e10c511
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l5-s4/environment/instance.json
@@ -0,0 +1 @@
+{"variable": "X", "integrand": "20*X**9 + 14*X**6 + 4*X**3/3 + 25/(2*X**6)", "family": "rg_simple_integration", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-simple-integration-l5-s4/solution/solve.sh b/tasks/construct-rg-simple-integration-l5-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..a534dafe5b94999fac4f5ce3fb41137e1fb170ca
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l5-s4/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"antiderivative": "2*X**10 + 2*X**7 + X**4/3 + 5/4 - 5/(2*X**5) + C"}
+RLMATH_EOF
diff --git a/tasks/construct-rg-simple-integration-l5-s4/tests/check.py b/tasks/construct-rg-simple-integration-l5-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..a9e364a8ede6c8215a16a04ee6b0d3529adc647d
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l5-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-l5-s4/tests/instance.json b/tasks/construct-rg-simple-integration-l5-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..65b30db3efcf1b53137130ea543a32216e10c511
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l5-s4/tests/instance.json
@@ -0,0 +1 @@
+{"variable": "X", "integrand": "20*X**9 + 14*X**6 + 4*X**3/3 + 25/(2*X**6)", "family": "rg_simple_integration", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rg-simple-integration-l5-s4/tests/run.py b/tasks/construct-rg-simple-integration-l5-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l5-s4/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rg-simple-integration-l5-s4/tests/test.sh b/tasks/construct-rg-simple-integration-l5-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rg-simple-integration-l5-s4/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-antiderivative-l4-s3/environment/Dockerfile b/tasks/construct-rlve-antiderivative-l4-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-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-antiderivative-l4-s3/environment/instance.json b/tasks/construct-rlve-antiderivative-l4-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e2781a71c3b4ac58230fc12358d2151be3983a41
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l4-s3/environment/instance.json
@@ -0,0 +1 @@
+{"f_prime": "(x**2*cos(x)/sin(x) + 2*x*log(sin(x)))*sin(x)**(x**2)", "max_ops": 32, "max_len": 680, "family": "rlve_antiderivative", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-antiderivative-l4-s3/solution/solve.sh b/tasks/construct-rlve-antiderivative-l4-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..e49585d2fec3033210f8ebbec98336baabc691ba
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-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'
+{"F": "sin(x)**(x**2)"}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-antiderivative-l4-s3/tests/check.py b/tasks/construct-rlve-antiderivative-l4-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..c6b30b93bbbb7795b54d973e43f25b23c82cb7fc
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l4-s3/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-l4-s3/tests/instance.json b/tasks/construct-rlve-antiderivative-l4-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e2781a71c3b4ac58230fc12358d2151be3983a41
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l4-s3/tests/instance.json
@@ -0,0 +1 @@
+{"f_prime": "(x**2*cos(x)/sin(x) + 2*x*log(sin(x)))*sin(x)**(x**2)", "max_ops": 32, "max_len": 680, "family": "rlve_antiderivative", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-antiderivative-l4-s3/tests/run.py b/tasks/construct-rlve-antiderivative-l4-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-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-antiderivative-l4-s3/tests/test.sh b/tasks/construct-rlve-antiderivative-l4-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-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-antiderivative-l4-s4/environment/instance.json b/tasks/construct-rlve-antiderivative-l4-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4768a7934d28ff6565bef8de3fc4c1430fde827c
--- /dev/null
+++ b/tasks/construct-rlve-antiderivative-l4-s4/environment/instance.json
@@ -0,0 +1 @@
+{"f_prime": "-12*sqrt(x) + 3*x*(-16 - 2/sqrt(x)) - 48*x", "max_ops": 52, "max_len": 860, "family": "rlve_antiderivative", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l1-s1/environment/Dockerfile b/tasks/construct-rlve-bezout-identity-l1-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-bezout-identity-l1-s1/environment/instance.json b/tasks/construct-rlve-bezout-identity-l1-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..de3a9de06f823442ec4acf255faa58b70fcd3276
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l1-s1/environment/instance.json
@@ -0,0 +1 @@
+{"A": [2, -28], "target": 2, "family": "rlve_bezout_identity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l1-s1/solution/solve.sh b/tasks/construct-rlve-bezout-identity-l1-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..c6b25a4830fb3b1d7b72d79b8431fb28419b4f1c
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l1-s1/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"X": [1, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-bezout-identity-l1-s1/tests/check.py b/tasks/construct-rlve-bezout-identity-l1-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..b4cb860ed36974a12b71adc9a0fa3f1ad85c814a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l1-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-l1-s1/tests/instance.json b/tasks/construct-rlve-bezout-identity-l1-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..de3a9de06f823442ec4acf255faa58b70fcd3276
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l1-s1/tests/instance.json
@@ -0,0 +1 @@
+{"A": [2, -28], "target": 2, "family": "rlve_bezout_identity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l1-s1/tests/run.py b/tasks/construct-rlve-bezout-identity-l1-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-bezout-identity-l1-s1/tests/test.sh b/tasks/construct-rlve-bezout-identity-l1-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-bezout-identity-l1-s6/environment/Dockerfile b/tasks/construct-rlve-bezout-identity-l1-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-bezout-identity-l1-s6/environment/instance.json b/tasks/construct-rlve-bezout-identity-l1-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..031a49bfd347bad54896f3645bc5e126cb6abd44
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l1-s6/environment/instance.json
@@ -0,0 +1 @@
+{"A": [-31, 31], "target": 31, "family": "rlve_bezout_identity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l1-s6/solution/solve.sh b/tasks/construct-rlve-bezout-identity-l1-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..53e6e3b84866036b2097d25dc9ec4119c0348d4a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l1-s6/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"X": [0, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-bezout-identity-l1-s6/tests/check.py b/tasks/construct-rlve-bezout-identity-l1-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..b4cb860ed36974a12b71adc9a0fa3f1ad85c814a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l1-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-l1-s6/tests/instance.json b/tasks/construct-rlve-bezout-identity-l1-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..031a49bfd347bad54896f3645bc5e126cb6abd44
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l1-s6/tests/instance.json
@@ -0,0 +1 @@
+{"A": [-31, 31], "target": 31, "family": "rlve_bezout_identity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l1-s6/tests/run.py b/tasks/construct-rlve-bezout-identity-l1-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-bezout-identity-l1-s6/tests/test.sh b/tasks/construct-rlve-bezout-identity-l1-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-bezout-identity-l1-s8/environment/Dockerfile b/tasks/construct-rlve-bezout-identity-l1-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-bezout-identity-l1-s8/environment/instance.json b/tasks/construct-rlve-bezout-identity-l1-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c818a6af96456ea1d322997e9007ab2dc7144422
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l1-s8/environment/instance.json
@@ -0,0 +1 @@
+{"A": [-22, -22], "target": 22, "family": "rlve_bezout_identity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l1-s8/solution/solve.sh b/tasks/construct-rlve-bezout-identity-l1-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..ac9af345c90a4a9109b47c2fae0aa39fc703e603
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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'
+{"X": [0, -1]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-bezout-identity-l1-s8/tests/check.py b/tasks/construct-rlve-bezout-identity-l1-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..b4cb860ed36974a12b71adc9a0fa3f1ad85c814a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l1-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-l1-s8/tests/instance.json b/tasks/construct-rlve-bezout-identity-l1-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c818a6af96456ea1d322997e9007ab2dc7144422
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l1-s8/tests/instance.json
@@ -0,0 +1 @@
+{"A": [-22, -22], "target": 22, "family": "rlve_bezout_identity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l1-s8/tests/run.py b/tasks/construct-rlve-bezout-identity-l1-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-bezout-identity-l1-s8/tests/test.sh b/tasks/construct-rlve-bezout-identity-l1-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-bezout-identity-l5-s0/environment/Dockerfile b/tasks/construct-rlve-bezout-identity-l5-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-bezout-identity-l5-s0/environment/instance.json b/tasks/construct-rlve-bezout-identity-l5-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c69d627d3b3ce70bed825ac4b91cb00d875529d2
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l5-s0/environment/instance.json
@@ -0,0 +1 @@
+{"A": [7820, 2512, -7314, -3611, 7958, -2990, -4186, 5014, 6486, -2070], "target": 1, "family": "rlve_bezout_identity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l5-s0/solution/solve.sh b/tasks/construct-rlve-bezout-identity-l5-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..0053953c7521be88a239cefb79e2a7018f6ad46a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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": [379447100, -1181235320, 1805, -1, 0, 0, 0, 0, 0, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-bezout-identity-l5-s0/tests/check.py b/tasks/construct-rlve-bezout-identity-l5-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..b4cb860ed36974a12b71adc9a0fa3f1ad85c814a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l5-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-l5-s0/tests/instance.json b/tasks/construct-rlve-bezout-identity-l5-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c69d627d3b3ce70bed825ac4b91cb00d875529d2
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l5-s0/tests/instance.json
@@ -0,0 +1 @@
+{"A": [7820, 2512, -7314, -3611, 7958, -2990, -4186, 5014, 6486, -2070], "target": 1, "family": "rlve_bezout_identity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l5-s0/tests/run.py b/tasks/construct-rlve-bezout-identity-l5-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-bezout-identity-l5-s0/tests/test.sh b/tasks/construct-rlve-bezout-identity-l5-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-bezout-identity-l5-s7/environment/Dockerfile b/tasks/construct-rlve-bezout-identity-l5-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-bezout-identity-l5-s7/environment/instance.json b/tasks/construct-rlve-bezout-identity-l5-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..291ffc760277e283c3a6041143e1aa58fbdb3c0b
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l5-s7/environment/instance.json
@@ -0,0 +1 @@
+{"A": [-8177, 680, -7208, -3145, 85, 3519, 7157, 1734, -4369, -3689], "target": 17, "family": "rlve_bezout_identity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l5-s7/solution/solve.sh b/tasks/construct-rlve-bezout-identity-l5-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..317ec934d3b83988c00686b827b173b7d5ec2817
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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'
+{"X": [-1, -12, 0, 0, 0, 0, 0, 0, 0, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-bezout-identity-l5-s7/tests/check.py b/tasks/construct-rlve-bezout-identity-l5-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..b4cb860ed36974a12b71adc9a0fa3f1ad85c814a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l5-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-l5-s7/tests/instance.json b/tasks/construct-rlve-bezout-identity-l5-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..291ffc760277e283c3a6041143e1aa58fbdb3c0b
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l5-s7/tests/instance.json
@@ -0,0 +1 @@
+{"A": [-8177, 680, -7208, -3145, 85, 3519, 7157, 1734, -4369, -3689], "target": 17, "family": "rlve_bezout_identity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l5-s7/tests/run.py b/tasks/construct-rlve-bezout-identity-l5-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-bezout-identity-l5-s7/tests/test.sh b/tasks/construct-rlve-bezout-identity-l5-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-bezout-identity-l5-s9/environment/Dockerfile b/tasks/construct-rlve-bezout-identity-l5-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-bezout-identity-l5-s9/environment/instance.json b/tasks/construct-rlve-bezout-identity-l5-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..57867292a78cea4a5bd4b65de1b21def8e52fda5
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l5-s9/environment/instance.json
@@ -0,0 +1 @@
+{"A": [-8108, -5536, 5622, -412, -4698, -6226, -3468, -6662, 4436, -1202], "target": 2, "family": "rlve_bezout_identity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l5-s9/solution/solve.sh b/tasks/construct-rlve-bezout-identity-l5-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..861b657976971d223a4cd17eddf2beadd92b4e23
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l5-s9/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"X": [813495, -1191440, 1, 0, 0, 0, 0, 0, 0, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-bezout-identity-l5-s9/tests/check.py b/tasks/construct-rlve-bezout-identity-l5-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..b4cb860ed36974a12b71adc9a0fa3f1ad85c814a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l5-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-l5-s9/tests/instance.json b/tasks/construct-rlve-bezout-identity-l5-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..57867292a78cea4a5bd4b65de1b21def8e52fda5
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-l5-s9/tests/instance.json
@@ -0,0 +1 @@
+{"A": [-8108, -5536, 5622, -412, -4698, -6226, -3468, -6662, 4436, -1202], "target": 2, "family": "rlve_bezout_identity", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-bezout-identity-l5-s9/tests/run.py b/tasks/construct-rlve-bezout-identity-l5-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-bezout-identity-l5-s9/tests/test.sh b/tasks/construct-rlve-bezout-identity-l5-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-bezout-identity-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-construct-hack-interval-l2-s5/environment/Dockerfile b/tasks/construct-rlve-construct-hack-interval-l2-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-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-construct-hack-interval-l2-s5/environment/instance.json b/tasks/construct-rlve-construct-hack-interval-l2-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..141e6cd9aae0b5e74926ea45e74c96224c998607
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l2-s5/environment/instance.json
@@ -0,0 +1 @@
+{"mod": 3252, "family": "rlve_construct_hack_interval", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-construct-hack-interval-l2-s5/solution/solve.sh b/tasks/construct-rlve-construct-hack-interval-l2-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..b3d3a536a2e00538206d2d618a911f58ecb01576
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-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'
+{"L": 3, "R": 329}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-construct-hack-interval-l2-s5/tests/check.py b/tasks/construct-rlve-construct-hack-interval-l2-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..34b46e9758e3a50ed16c9c2a29cfc0abe663926b
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l2-s5/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-l2-s5/tests/instance.json b/tasks/construct-rlve-construct-hack-interval-l2-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..141e6cd9aae0b5e74926ea45e74c96224c998607
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-l2-s5/tests/instance.json
@@ -0,0 +1 @@
+{"mod": 3252, "family": "rlve_construct_hack_interval", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-construct-hack-interval-l2-s5/tests/run.py b/tasks/construct-rlve-construct-hack-interval-l2-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-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-construct-hack-interval-l2-s5/tests/test.sh b/tasks/construct-rlve-construct-hack-interval-l2-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-construct-hack-interval-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-crt-l2-s3/environment/Dockerfile b/tasks/construct-rlve-crt-l2-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-crt-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-crt-l2-s3/environment/instance.json b/tasks/construct-rlve-crt-l2-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a0f841e03d78f56fcfda7ec9b3f7d3ff91f786b2
--- /dev/null
+++ b/tasks/construct-rlve-crt-l2-s3/environment/instance.json
@@ -0,0 +1 @@
+{"moduli": [76, 72, 132], "residues": [73, 5, 17], "family": "rlve_crt", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-crt-l2-s3/solution/solve.sh b/tasks/construct-rlve-crt-l2-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..ea86580df467f07314621d231c26e2b8c53a65c9
--- /dev/null
+++ b/tasks/construct-rlve-crt-l2-s3/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": 149}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-crt-l2-s3/tests/check.py b/tasks/construct-rlve-crt-l2-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..886d95606ef325930b24794c0959c2ba7b254ebd
--- /dev/null
+++ b/tasks/construct-rlve-crt-l2-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-l2-s3/tests/instance.json b/tasks/construct-rlve-crt-l2-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a0f841e03d78f56fcfda7ec9b3f7d3ff91f786b2
--- /dev/null
+++ b/tasks/construct-rlve-crt-l2-s3/tests/instance.json
@@ -0,0 +1 @@
+{"moduli": [76, 72, 132], "residues": [73, 5, 17], "family": "rlve_crt", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-crt-l2-s3/tests/run.py b/tasks/construct-rlve-crt-l2-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-crt-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-crt-l2-s3/tests/test.sh b/tasks/construct-rlve-crt-l2-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-crt-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-crt-l2-s4/environment/Dockerfile b/tasks/construct-rlve-crt-l2-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-crt-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-crt-l2-s4/environment/instance.json b/tasks/construct-rlve-crt-l2-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..6a18ab2b5f41bde9a923991a4163389e799e1205
--- /dev/null
+++ b/tasks/construct-rlve-crt-l2-s4/environment/instance.json
@@ -0,0 +1 @@
+{"moduli": [16, 51, 78], "residues": [15, 25, 49], "family": "rlve_crt", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-crt-l2-s4/solution/solve.sh b/tasks/construct-rlve-crt-l2-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..d2bdc3cb9a53a5a37433016598bf7890c8ddf315
--- /dev/null
+++ b/tasks/construct-rlve-crt-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'
+{"x": 127}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-crt-l2-s4/tests/check.py b/tasks/construct-rlve-crt-l2-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..886d95606ef325930b24794c0959c2ba7b254ebd
--- /dev/null
+++ b/tasks/construct-rlve-crt-l2-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-l2-s4/tests/instance.json b/tasks/construct-rlve-crt-l2-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..6a18ab2b5f41bde9a923991a4163389e799e1205
--- /dev/null
+++ b/tasks/construct-rlve-crt-l2-s4/tests/instance.json
@@ -0,0 +1 @@
+{"moduli": [16, 51, 78], "residues": [15, 25, 49], "family": "rlve_crt", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-crt-l2-s4/tests/run.py b/tasks/construct-rlve-crt-l2-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-crt-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-crt-l2-s4/tests/test.sh b/tasks/construct-rlve-crt-l2-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-crt-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-crt-l6-s2/environment/Dockerfile b/tasks/construct-rlve-crt-l6-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-crt-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-crt-l6-s2/environment/instance.json b/tasks/construct-rlve-crt-l6-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c43b92f7d86e18ae2390f16edb9195c4a26e6a35
--- /dev/null
+++ b/tasks/construct-rlve-crt-l6-s2/environment/instance.json
@@ -0,0 +1 @@
+{"moduli": [486808087331710, 210298158358916, 53210879947514, 83528608205695, 11910582829170, 628853003240228, 447710323222005, 361360120193456], "residues": [182711631042351, 38625243297313, 30989159003893, 1290852728501, 2527079940541, 40666715133833, 221809395152056, 308159598180605], "family": "rlve_crt", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-crt-l6-s2/solution/solve.sh b/tasks/construct-rlve-crt-l6-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..f780dcb988202a36a18c12a576843d75ffbf9097
--- /dev/null
+++ b/tasks/construct-rlve-crt-l6-s2/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": 669519718374061}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-crt-l6-s2/tests/check.py b/tasks/construct-rlve-crt-l6-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..886d95606ef325930b24794c0959c2ba7b254ebd
--- /dev/null
+++ b/tasks/construct-rlve-crt-l6-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-l6-s2/tests/instance.json b/tasks/construct-rlve-crt-l6-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c43b92f7d86e18ae2390f16edb9195c4a26e6a35
--- /dev/null
+++ b/tasks/construct-rlve-crt-l6-s2/tests/instance.json
@@ -0,0 +1 @@
+{"moduli": [486808087331710, 210298158358916, 53210879947514, 83528608205695, 11910582829170, 628853003240228, 447710323222005, 361360120193456], "residues": [182711631042351, 38625243297313, 30989159003893, 1290852728501, 2527079940541, 40666715133833, 221809395152056, 308159598180605], "family": "rlve_crt", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-crt-l6-s2/tests/run.py b/tasks/construct-rlve-crt-l6-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-crt-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-crt-l6-s2/tests/test.sh b/tasks/construct-rlve-crt-l6-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-crt-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-crt-l6-s5/environment/Dockerfile b/tasks/construct-rlve-crt-l6-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-crt-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-crt-l6-s5/environment/instance.json b/tasks/construct-rlve-crt-l6-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a26cb8cdc8d1ac1a83e48bb5de7f71b139ee0a3b
--- /dev/null
+++ b/tasks/construct-rlve-crt-l6-s5/environment/instance.json
@@ -0,0 +1 @@
+{"moduli": [856749491349613, 21330735087207, 44454512816847, 107307491893967, 772378457491186, 528753489930346, 665853847519480, 196894342895235], "residues": [83837508535991, 2034656048496, 7042230731817, 82127064733868, 168208542394418, 411833509955258, 274733152366124, 153009628304664], "family": "rlve_crt", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-crt-l6-s5/solution/solve.sh b/tasks/construct-rlve-crt-l6-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..a34ba0d0f3e5383d0bef2226a8cf8d606050ac23
--- /dev/null
+++ b/tasks/construct-rlve-crt-l6-s5/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": 940586999885604}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-crt-l6-s5/tests/check.py b/tasks/construct-rlve-crt-l6-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..886d95606ef325930b24794c0959c2ba7b254ebd
--- /dev/null
+++ b/tasks/construct-rlve-crt-l6-s5/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-l6-s5/tests/instance.json b/tasks/construct-rlve-crt-l6-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a26cb8cdc8d1ac1a83e48bb5de7f71b139ee0a3b
--- /dev/null
+++ b/tasks/construct-rlve-crt-l6-s5/tests/instance.json
@@ -0,0 +1 @@
+{"moduli": [856749491349613, 21330735087207, 44454512816847, 107307491893967, 772378457491186, 528753489930346, 665853847519480, 196894342895235], "residues": [83837508535991, 2034656048496, 7042230731817, 82127064733868, 168208542394418, 411833509955258, 274733152366124, 153009628304664], "family": "rlve_crt", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-crt-l6-s5/tests/run.py b/tasks/construct-rlve-crt-l6-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-crt-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-crt-l6-s5/tests/test.sh b/tasks/construct-rlve-crt-l6-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-crt-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-distinct-subset-sum-permutation-l3-s3/environment/Dockerfile b/tasks/construct-rlve-distinct-subset-sum-permutation-l3-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-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-distinct-subset-sum-permutation-l3-s3/environment/instance.json b/tasks/construct-rlve-distinct-subset-sum-permutation-l3-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..501e5aa3243f834badf2a2d9fc4fd7aa1e09f17f
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l3-s3/environment/instance.json
@@ -0,0 +1 @@
+{"A": [9, 13, 12, 16, 3, 18, 11, 10, 15, 0], "family": "rlve_distinct_subset_sum_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l3-s3/solution/solve.sh b/tasks/construct-rlve-distinct-subset-sum-permutation-l3-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..92285ad4889eddccb6a8e81119b14da17c35fbe7
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-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'
+{"B": [10, 15, 13, 18, 9, 0, 12, 11, 16, 3]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l3-s3/tests/check.py b/tasks/construct-rlve-distinct-subset-sum-permutation-l3-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..06bbbcf872c2fca05785f40fdfdcf85a5e5a0e7c
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l3-s3/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-l3-s3/tests/instance.json b/tasks/construct-rlve-distinct-subset-sum-permutation-l3-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..501e5aa3243f834badf2a2d9fc4fd7aa1e09f17f
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l3-s3/tests/instance.json
@@ -0,0 +1 @@
+{"A": [9, 13, 12, 16, 3, 18, 11, 10, 15, 0], "family": "rlve_distinct_subset_sum_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l3-s3/tests/run.py b/tasks/construct-rlve-distinct-subset-sum-permutation-l3-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-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-distinct-subset-sum-permutation-l3-s3/tests/test.sh b/tasks/construct-rlve-distinct-subset-sum-permutation-l3-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-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-distinct-subset-sum-permutation-l3-s4/environment/Dockerfile b/tasks/construct-rlve-distinct-subset-sum-permutation-l3-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-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-distinct-subset-sum-permutation-l3-s4/environment/instance.json b/tasks/construct-rlve-distinct-subset-sum-permutation-l3-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..38d2c91fa1f89848564a74e46995549023a70d45
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l3-s4/environment/instance.json
@@ -0,0 +1 @@
+{"A": [17, 14, 3, 11, 0, 15, 9, 6, 1, 7], "family": "rlve_distinct_subset_sum_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l3-s4/solution/solve.sh b/tasks/construct-rlve-distinct-subset-sum-permutation-l3-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..d941e827010e3f63ebcf727a6dc410999a066cf3
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-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'
+{"B": [0, 15, 6, 14, 1, 17, 11, 7, 3, 9]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l3-s4/tests/check.py b/tasks/construct-rlve-distinct-subset-sum-permutation-l3-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..06bbbcf872c2fca05785f40fdfdcf85a5e5a0e7c
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l3-s4/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-l3-s4/tests/instance.json b/tasks/construct-rlve-distinct-subset-sum-permutation-l3-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..38d2c91fa1f89848564a74e46995549023a70d45
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-sum-permutation-l3-s4/tests/instance.json
@@ -0,0 +1 @@
+{"A": [17, 14, 3, 11, 0, 15, 9, 6, 1, 7], "family": "rlve_distinct_subset_sum_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-distinct-subset-sum-permutation-l3-s4/tests/run.py b/tasks/construct-rlve-distinct-subset-sum-permutation-l3-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-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-distinct-subset-sum-permutation-l3-s4/tests/test.sh b/tasks/construct-rlve-distinct-subset-sum-permutation-l3-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-distinct-subset-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-even-degree-partition-l2-s4/environment/Dockerfile b/tasks/construct-rlve-even-degree-partition-l2-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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-even-degree-partition-l2-s4/environment/instance.json b/tasks/construct-rlve-even-degree-partition-l2-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..db23d5c4d5649c8dc3f9cc6995224d2b8966a46a
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l2-s4/environment/instance.json
@@ -0,0 +1 @@
+{"n": 8, "edges": [[0, 1], [0, 4], [0, 5], [1, 3], [1, 7], [2, 4], [2, 5], [3, 7]], "family": "rlve_even_degree_partition", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-even-degree-partition-l2-s4/solution/solve.sh b/tasks/construct-rlve-even-degree-partition-l2-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..bc7f1927129b174e515eb343daae2c1478e390e7
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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'
+{"groups": [1, 2, 1, 2, 1, 1, 2, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-even-degree-partition-l2-s4/tests/check.py b/tasks/construct-rlve-even-degree-partition-l2-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..9c7f0f71ea6d10264ececcc933b97a4a0e6eaf8e
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l2-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-l2-s4/tests/instance.json b/tasks/construct-rlve-even-degree-partition-l2-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..db23d5c4d5649c8dc3f9cc6995224d2b8966a46a
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l2-s4/tests/instance.json
@@ -0,0 +1 @@
+{"n": 8, "edges": [[0, 1], [0, 4], [0, 5], [1, 3], [1, 7], [2, 4], [2, 5], [3, 7]], "family": "rlve_even_degree_partition", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-even-degree-partition-l2-s4/tests/run.py b/tasks/construct-rlve-even-degree-partition-l2-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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-even-degree-partition-l2-s4/tests/test.sh b/tasks/construct-rlve-even-degree-partition-l2-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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-even-degree-partition-l6-s2/environment/Dockerfile b/tasks/construct-rlve-even-degree-partition-l6-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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-even-degree-partition-l6-s2/environment/instance.json b/tasks/construct-rlve-even-degree-partition-l6-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1e9503db72722e4d43d7bf06f82d20be301329e0
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l6-s2/environment/instance.json
@@ -0,0 +1 @@
+{"n": 40, "edges": [[0, 5], [0, 8], [0, 10], [0, 14], [0, 18], [0, 19], [0, 21], [0, 23], [0, 24], [0, 25], [0, 26], [0, 27], [0, 28], [0, 31], [0, 33], [0, 34], [0, 35], [0, 36], [0, 37], [0, 38], [0, 39], [1, 4], [1, 5], [1, 6], [1, 7], [1, 9], [1, 11], [1, 14], [1, 15], [1, 16], [1, 17], [1, 18], [1, 19], [1, 20], [1, 21], [1, 22], [1, 23], [1, 24], [1, 25], [1, 26], [1, 28], [1, 29], [1, 31], [1, 34], [1, 38], [1, 39], [2, 3], [2, 5], [2, 7], [2, 10], [2, 12], [2, 13], [2, 14], [2, 15], [2, 16], [2, 17], [2, 18], [2, 19], [2, 22], [2, 25], [2, 26], [2, 27], [2, 28], [2, 29], [2, 31], [2, 32], [2, 38], [2, 39], [3, 5], [3, 9], [3, 10], [3, 11], [3, 14], [3, 15], [3, 16], [3, 17], [3, 18], [3, 19], [3, 20], [3, 21], [3, 22], [3, 24], [3, 27], [3, 30], [3, 31], [3, 33], [3, 35], [3, 39], [4, 7], [4, 9], [4, 13], [4, 14], [4, 17], [4, 19], [4, 20], [4, 21], [4, 22], [4, 23], [4, 24], [4, 26], [4, 27], [4, 30], [4, 33], [5, 6], [5, 7], [5, 8], [5, 10], [5, 12], [5, 13], [5, 14], [5, 15], [5, 17], [5, 18], [5, 19], [5, 21], [5, 22], [5, 23], [5, 24], [5, 25], [5, 26], [5, 27], [5, 28], [5, 29], [5, 30], [5, 33], [5, 34], [5, 35], [5, 37], [5, 38], [5, 39], [6, 13], [6, 14], [6, 18], [6, 20], [6, 24], [6, 25], [6, 27], [6, 28], [6, 29], [6, 30], [6, 31], [6, 34], [6, 36], [6, 37], [6, 38], [6, 39], [7, 8], [7, 10], [7, 11], [7, 13], [7, 14], [7, 16], [7, 18], [7, 19], [7, 21], [7, 22], [7, 23], [7, 26], [7, 27], [7, 28], [7, 29], [7, 30], [7, 34], [7, 35], [7, 36], [7, 38], [7, 39], [8, 9], [8, 11], [8, 12], [8, 15], [8, 16], [8, 17], [8, 18], [8, 20], [8, 22], [8, 23], [8, 24], [8, 26], [8, 27], [8, 28], [8, 29], [8, 33], [8, 34], [8, 35], [8, 36], [8, 37], [8, 38], [8, 39], [9, 12], [9, 13], [9, 15], [9, 19], [9, 20], [9, 23], [9, 25], [9, 26], [9, 27], [9, 28], [9, 29], [9, 31], [9, 32], [9, 34], [9, 36], [9, 37], [9, 38], [9, 39], [10, 11], [10, 13], [10, 14], [10, 16], [10, 18], [10, 19], [10, 24], [10, 25], [10, 26], [10, 27], [10, 28], [10, 29], [10, 31], [10, 33], [10, 34], [10, 35], [10, 36], [10, 37], [11, 12], [11, 13], [11, 14], [11, 17], [11, 18], [11, 20], [11, 21], [11, 23], [11, 24], [11, 25], [11, 27], [11, 29], [11, 33], [11, 35], [11, 36], [11, 39], [12, 13], [12, 15], [12, 16], [12, 20], [12, 22], [12, 23], [12, 24], [12, 27], [12, 28], [12, 30], [12, 33], [12, 35], [12, 36], [12, 37], [12, 39], [13, 14], [13, 15], [13, 16], [13, 17], [13, 18], [13, 21], [13, 22], [13, 26], [13, 27], [13, 30], [13, 31], [13, 32], [13, 33], [13, 35], [13, 37], [13, 38], [14, 15], [14, 17], [14, 22], [14, 24], [14, 26], [14, 27], [14, 29], [14, 34], [14, 36], [15, 16], [15, 18], [15, 20], [15, 22], [15, 24], [15, 25], [15, 26], [15, 27], [15, 34], [15, 36], [15, 38], [16, 17], [16, 18], [16, 21], [16, 22], [16, 26], [16, 27], [16, 29], [16, 30], [16, 33], [16, 36], [16, 37], [16, 38], [16, 39], [17, 18], [17, 21], [17, 22], [17, 23], [17, 27], [17, 28], [17, 31], [17, 34], [17, 37], [17, 38], [17, 39], [18, 19], [18, 20], [18, 23], [18, 25], [18, 26], [18, 27], [18, 28], [18, 30], [18, 33], [18, 34], [18, 35], [18, 38], [18, 39], [19, 20], [19, 21], [19, 22], [19, 23], [19, 24], [19, 25], [19, 26], [19, 27], [19, 28], [19, 29], [19, 31], [19, 32], [19, 36], [19, 38], [19, 39], [20, 21], [20, 24], [20, 29], [20, 30], [20, 32], [20, 35], [20, 37], [20, 38], [21, 22], [21, 23], [21, 24], [21, 25], [21, 26], [21, 27], [21, 28], [21, 29], [21, 30], [21, 31], [21, 33], [21, 34], [21, 37], [22, 23], [22, 24], [22, 26], [22, 27], [22, 28], [22, 30], [22, 31], [22, 32], [22, 34], [22, 36], [22, 39], [23, 27], [23, 30], [23, 33], [23, 34], [23, 35], [23, 37], [23, 39], [24, 26], [24, 27], [24, 28], [24, 29], [24, 33], [24, 34], [24, 35], [24, 36], [24, 38], [25, 29], [25, 31], [25, 34], [25, 35], [25, 36], [25, 38], [26, 27], [26, 28], [26, 29], [26, 31], [26, 32], [26, 33], [26, 34], [26, 36], [26, 37], [26, 39], [27, 28], [27, 29], [27, 30], [27, 31], [27, 32], [27, 33], [27, 36], [27, 37], [27, 39], [28, 32], [28, 34], [28, 36], [28, 38], [28, 39], [29, 30], [29, 33], [29, 35], [29, 36], [29, 37], [30, 37], [31, 34], [31, 38], [32, 33], [32, 35], [32, 37], [33, 38], [34, 35], [34, 36], [34, 37], [34, 39], [35, 36], [35, 38], [36, 37], [36, 38], [36, 39], [37, 38], [38, 39]], "family": "rlve_even_degree_partition", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-even-degree-partition-l6-s2/solution/solve.sh b/tasks/construct-rlve-even-degree-partition-l6-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..0823b89ae2a438ff011aec743bba37c14ebf2375
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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'
+{"groups": [2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-even-degree-partition-l6-s2/tests/check.py b/tasks/construct-rlve-even-degree-partition-l6-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..9c7f0f71ea6d10264ececcc933b97a4a0e6eaf8e
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l6-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-l6-s2/tests/instance.json b/tasks/construct-rlve-even-degree-partition-l6-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1e9503db72722e4d43d7bf06f82d20be301329e0
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l6-s2/tests/instance.json
@@ -0,0 +1 @@
+{"n": 40, "edges": [[0, 5], [0, 8], [0, 10], [0, 14], [0, 18], [0, 19], [0, 21], [0, 23], [0, 24], [0, 25], [0, 26], [0, 27], [0, 28], [0, 31], [0, 33], [0, 34], [0, 35], [0, 36], [0, 37], [0, 38], [0, 39], [1, 4], [1, 5], [1, 6], [1, 7], [1, 9], [1, 11], [1, 14], [1, 15], [1, 16], [1, 17], [1, 18], [1, 19], [1, 20], [1, 21], [1, 22], [1, 23], [1, 24], [1, 25], [1, 26], [1, 28], [1, 29], [1, 31], [1, 34], [1, 38], [1, 39], [2, 3], [2, 5], [2, 7], [2, 10], [2, 12], [2, 13], [2, 14], [2, 15], [2, 16], [2, 17], [2, 18], [2, 19], [2, 22], [2, 25], [2, 26], [2, 27], [2, 28], [2, 29], [2, 31], [2, 32], [2, 38], [2, 39], [3, 5], [3, 9], [3, 10], [3, 11], [3, 14], [3, 15], [3, 16], [3, 17], [3, 18], [3, 19], [3, 20], [3, 21], [3, 22], [3, 24], [3, 27], [3, 30], [3, 31], [3, 33], [3, 35], [3, 39], [4, 7], [4, 9], [4, 13], [4, 14], [4, 17], [4, 19], [4, 20], [4, 21], [4, 22], [4, 23], [4, 24], [4, 26], [4, 27], [4, 30], [4, 33], [5, 6], [5, 7], [5, 8], [5, 10], [5, 12], [5, 13], [5, 14], [5, 15], [5, 17], [5, 18], [5, 19], [5, 21], [5, 22], [5, 23], [5, 24], [5, 25], [5, 26], [5, 27], [5, 28], [5, 29], [5, 30], [5, 33], [5, 34], [5, 35], [5, 37], [5, 38], [5, 39], [6, 13], [6, 14], [6, 18], [6, 20], [6, 24], [6, 25], [6, 27], [6, 28], [6, 29], [6, 30], [6, 31], [6, 34], [6, 36], [6, 37], [6, 38], [6, 39], [7, 8], [7, 10], [7, 11], [7, 13], [7, 14], [7, 16], [7, 18], [7, 19], [7, 21], [7, 22], [7, 23], [7, 26], [7, 27], [7, 28], [7, 29], [7, 30], [7, 34], [7, 35], [7, 36], [7, 38], [7, 39], [8, 9], [8, 11], [8, 12], [8, 15], [8, 16], [8, 17], [8, 18], [8, 20], [8, 22], [8, 23], [8, 24], [8, 26], [8, 27], [8, 28], [8, 29], [8, 33], [8, 34], [8, 35], [8, 36], [8, 37], [8, 38], [8, 39], [9, 12], [9, 13], [9, 15], [9, 19], [9, 20], [9, 23], [9, 25], [9, 26], [9, 27], [9, 28], [9, 29], [9, 31], [9, 32], [9, 34], [9, 36], [9, 37], [9, 38], [9, 39], [10, 11], [10, 13], [10, 14], [10, 16], [10, 18], [10, 19], [10, 24], [10, 25], [10, 26], [10, 27], [10, 28], [10, 29], [10, 31], [10, 33], [10, 34], [10, 35], [10, 36], [10, 37], [11, 12], [11, 13], [11, 14], [11, 17], [11, 18], [11, 20], [11, 21], [11, 23], [11, 24], [11, 25], [11, 27], [11, 29], [11, 33], [11, 35], [11, 36], [11, 39], [12, 13], [12, 15], [12, 16], [12, 20], [12, 22], [12, 23], [12, 24], [12, 27], [12, 28], [12, 30], [12, 33], [12, 35], [12, 36], [12, 37], [12, 39], [13, 14], [13, 15], [13, 16], [13, 17], [13, 18], [13, 21], [13, 22], [13, 26], [13, 27], [13, 30], [13, 31], [13, 32], [13, 33], [13, 35], [13, 37], [13, 38], [14, 15], [14, 17], [14, 22], [14, 24], [14, 26], [14, 27], [14, 29], [14, 34], [14, 36], [15, 16], [15, 18], [15, 20], [15, 22], [15, 24], [15, 25], [15, 26], [15, 27], [15, 34], [15, 36], [15, 38], [16, 17], [16, 18], [16, 21], [16, 22], [16, 26], [16, 27], [16, 29], [16, 30], [16, 33], [16, 36], [16, 37], [16, 38], [16, 39], [17, 18], [17, 21], [17, 22], [17, 23], [17, 27], [17, 28], [17, 31], [17, 34], [17, 37], [17, 38], [17, 39], [18, 19], [18, 20], [18, 23], [18, 25], [18, 26], [18, 27], [18, 28], [18, 30], [18, 33], [18, 34], [18, 35], [18, 38], [18, 39], [19, 20], [19, 21], [19, 22], [19, 23], [19, 24], [19, 25], [19, 26], [19, 27], [19, 28], [19, 29], [19, 31], [19, 32], [19, 36], [19, 38], [19, 39], [20, 21], [20, 24], [20, 29], [20, 30], [20, 32], [20, 35], [20, 37], [20, 38], [21, 22], [21, 23], [21, 24], [21, 25], [21, 26], [21, 27], [21, 28], [21, 29], [21, 30], [21, 31], [21, 33], [21, 34], [21, 37], [22, 23], [22, 24], [22, 26], [22, 27], [22, 28], [22, 30], [22, 31], [22, 32], [22, 34], [22, 36], [22, 39], [23, 27], [23, 30], [23, 33], [23, 34], [23, 35], [23, 37], [23, 39], [24, 26], [24, 27], [24, 28], [24, 29], [24, 33], [24, 34], [24, 35], [24, 36], [24, 38], [25, 29], [25, 31], [25, 34], [25, 35], [25, 36], [25, 38], [26, 27], [26, 28], [26, 29], [26, 31], [26, 32], [26, 33], [26, 34], [26, 36], [26, 37], [26, 39], [27, 28], [27, 29], [27, 30], [27, 31], [27, 32], [27, 33], [27, 36], [27, 37], [27, 39], [28, 32], [28, 34], [28, 36], [28, 38], [28, 39], [29, 30], [29, 33], [29, 35], [29, 36], [29, 37], [30, 37], [31, 34], [31, 38], [32, 33], [32, 35], [32, 37], [33, 38], [34, 35], [34, 36], [34, 37], [34, 39], [35, 36], [35, 38], [36, 37], [36, 38], [36, 39], [37, 38], [38, 39]], "family": "rlve_even_degree_partition", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-even-degree-partition-l6-s2/tests/run.py b/tasks/construct-rlve-even-degree-partition-l6-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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-even-degree-partition-l6-s2/tests/test.sh b/tasks/construct-rlve-even-degree-partition-l6-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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-even-degree-partition-l6-s5/environment/Dockerfile b/tasks/construct-rlve-even-degree-partition-l6-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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-even-degree-partition-l6-s5/environment/instance.json b/tasks/construct-rlve-even-degree-partition-l6-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..31014e41602ece8622e2951053c723ad4ea1c51b
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l6-s5/environment/instance.json
@@ -0,0 +1 @@
+{"n": 40, "edges": [[0, 2], [0, 5], [0, 6], [0, 8], [0, 10], [0, 12], [0, 13], [0, 15], [0, 16], [0, 24], [0, 28], [0, 30], [0, 31], [0, 32], [0, 33], [0, 35], [0, 36], [0, 37], [0, 39], [1, 2], [1, 5], [1, 6], [1, 9], [1, 11], [1, 13], [1, 14], [1, 15], [1, 19], [1, 20], [1, 21], [1, 22], [1, 25], [1, 27], [1, 30], [1, 31], [1, 32], [1, 33], [1, 37], [1, 38], [1, 39], [2, 3], [2, 4], [2, 5], [2, 6], [2, 7], [2, 8], [2, 10], [2, 13], [2, 14], [2, 15], [2, 16], [2, 19], [2, 20], [2, 22], [2, 24], [2, 25], [2, 26], [2, 29], [2, 38], [2, 39], [3, 4], [3, 8], [3, 11], [3, 13], [3, 14], [3, 15], [3, 18], [3, 19], [3, 22], [3, 24], [3, 25], [3, 28], [3, 30], [3, 31], [3, 33], [3, 35], [3, 39], [4, 5], [4, 6], [4, 10], [4, 11], [4, 12], [4, 13], [4, 15], [4, 16], [4, 18], [4, 19], [4, 21], [4, 24], [4, 25], [4, 32], [4, 33], [4, 37], [4, 38], [4, 39], [5, 6], [5, 7], [5, 9], [5, 11], [5, 16], [5, 17], [5, 20], [5, 21], [5, 23], [5, 24], [5, 25], [5, 26], [5, 27], [5, 29], [5, 31], [5, 32], [5, 33], [5, 34], [5, 36], [5, 37], [5, 39], [6, 7], [6, 9], [6, 10], [6, 13], [6, 14], [6, 15], [6, 20], [6, 21], [6, 23], [6, 24], [6, 25], [6, 26], [6, 27], [6, 28], [6, 30], [6, 31], [6, 34], [6, 36], [6, 39], [7, 8], [7, 9], [7, 11], [7, 13], [7, 15], [7, 16], [7, 18], [7, 19], [7, 20], [7, 24], [7, 26], [7, 28], [7, 29], [7, 31], [7, 34], [7, 35], [8, 11], [8, 14], [8, 15], [8, 16], [8, 19], [8, 22], [8, 23], [8, 24], [8, 25], [8, 26], [8, 27], [8, 29], [8, 31], [8, 32], [8, 33], [8, 34], [8, 36], [8, 38], [8, 39], [9, 18], [9, 21], [9, 22], [9, 24], [9, 25], [9, 28], [9, 29], [9, 30], [9, 34], [9, 36], [9, 38], [9, 39], [10, 11], [10, 12], [10, 14], [10, 21], [10, 27], [10, 28], [10, 29], [10, 30], [10, 31], [10, 33], [10, 38], [10, 39], [11, 13], [11, 14], [11, 15], [11, 17], [11, 18], [11, 19], [11, 20], [11, 21], [11, 25], [11, 27], [11, 30], [11, 31], [11, 32], [11, 34], [11, 35], [11, 37], [11, 38], [12, 13], [12, 14], [12, 15], [12, 16], [12, 17], [12, 19], [12, 22], [12, 24], [12, 25], [12, 26], [12, 28], [12, 35], [12, 36], [13, 15], [13, 16], [13, 19], [13, 21], [13, 24], [13, 25], [13, 27], [13, 29], [13, 30], [13, 31], [13, 32], [13, 33], [13, 34], [13, 35], [13, 38], [13, 39], [14, 15], [14, 16], [14, 20], [14, 21], [14, 22], [14, 23], [14, 24], [14, 25], [14, 26], [14, 27], [14, 28], [14, 29], [14, 32], [14, 35], [14, 36], [14, 38], [14, 39], [15, 16], [15, 19], [15, 20], [15, 21], [15, 22], [15, 23], [15, 24], [15, 25], [15, 26], [15, 27], [15, 28], [15, 29], [15, 30], [15, 31], [15, 36], [16, 19], [16, 20], [16, 22], [16, 23], [16, 28], [16, 29], [16, 32], [16, 37], [16, 38], [16, 39], [17, 19], [17, 25], [17, 27], [17, 29], [17, 30], [18, 23], [18, 27], [18, 32], [18, 34], [18, 38], [19, 20], [19, 23], [19, 25], [19, 29], [19, 30], [19, 34], [19, 36], [20, 22], [20, 24], [20, 25], [20, 26], [20, 28], [20, 29], [20, 30], [20, 32], [20, 34], [20, 36], [20, 37], [20, 38], [20, 39], [21, 23], [21, 24], [21, 25], [21, 26], [21, 27], [21, 28], [21, 29], [21, 30], [21, 31], [21, 33], [21, 35], [21, 37], [21, 39], [22, 24], [22, 25], [22, 26], [22, 27], [22, 29], [22, 30], [22, 32], [22, 33], [22, 34], [22, 35], [22, 36], [22, 37], [22, 39], [23, 24], [23, 25], [23, 26], [23, 28], [23, 31], [23, 32], [23, 33], [23, 36], [23, 37], [23, 38], [23, 39], [24, 25], [24, 29], [24, 30], [24, 32], [24, 33], [24, 35], [24, 36], [24, 39], [25, 26], [25, 27], [25, 28], [25, 29], [25, 30], [25, 34], [25, 37], [25, 38], [25, 39], [26, 27], [26, 28], [26, 29], [26, 30], [26, 31], [26, 32], [26, 34], [26, 35], [26, 36], [26, 37], [26, 38], [27, 28], [27, 31], [27, 32], [27, 33], [27, 35], [27, 37], [27, 38], [27, 39], [28, 34], [28, 37], [28, 38], [28, 39], [29, 31], [29, 33], [29, 35], [29, 37], [30, 32], [30, 33], [30, 36], [30, 39], [31, 32], [31, 34], [31, 35], [31, 38], [31, 39], [32, 33], [32, 35], [32, 37], [32, 39], [33, 34], [33, 35], [33, 38], [34, 35], [34, 36], [34, 37], [34, 38], [34, 39], [35, 37], [35, 38], [35, 39], [36, 39], [37, 39], [38, 39]], "family": "rlve_even_degree_partition", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-even-degree-partition-l6-s5/solution/solve.sh b/tasks/construct-rlve-even-degree-partition-l6-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..088834ca70c405a1a23521733a7bd2baf76a55d8
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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'
+{"groups": [1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-even-degree-partition-l6-s5/tests/check.py b/tasks/construct-rlve-even-degree-partition-l6-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..9c7f0f71ea6d10264ececcc933b97a4a0e6eaf8e
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l6-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-l6-s5/tests/instance.json b/tasks/construct-rlve-even-degree-partition-l6-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..31014e41602ece8622e2951053c723ad4ea1c51b
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-l6-s5/tests/instance.json
@@ -0,0 +1 @@
+{"n": 40, "edges": [[0, 2], [0, 5], [0, 6], [0, 8], [0, 10], [0, 12], [0, 13], [0, 15], [0, 16], [0, 24], [0, 28], [0, 30], [0, 31], [0, 32], [0, 33], [0, 35], [0, 36], [0, 37], [0, 39], [1, 2], [1, 5], [1, 6], [1, 9], [1, 11], [1, 13], [1, 14], [1, 15], [1, 19], [1, 20], [1, 21], [1, 22], [1, 25], [1, 27], [1, 30], [1, 31], [1, 32], [1, 33], [1, 37], [1, 38], [1, 39], [2, 3], [2, 4], [2, 5], [2, 6], [2, 7], [2, 8], [2, 10], [2, 13], [2, 14], [2, 15], [2, 16], [2, 19], [2, 20], [2, 22], [2, 24], [2, 25], [2, 26], [2, 29], [2, 38], [2, 39], [3, 4], [3, 8], [3, 11], [3, 13], [3, 14], [3, 15], [3, 18], [3, 19], [3, 22], [3, 24], [3, 25], [3, 28], [3, 30], [3, 31], [3, 33], [3, 35], [3, 39], [4, 5], [4, 6], [4, 10], [4, 11], [4, 12], [4, 13], [4, 15], [4, 16], [4, 18], [4, 19], [4, 21], [4, 24], [4, 25], [4, 32], [4, 33], [4, 37], [4, 38], [4, 39], [5, 6], [5, 7], [5, 9], [5, 11], [5, 16], [5, 17], [5, 20], [5, 21], [5, 23], [5, 24], [5, 25], [5, 26], [5, 27], [5, 29], [5, 31], [5, 32], [5, 33], [5, 34], [5, 36], [5, 37], [5, 39], [6, 7], [6, 9], [6, 10], [6, 13], [6, 14], [6, 15], [6, 20], [6, 21], [6, 23], [6, 24], [6, 25], [6, 26], [6, 27], [6, 28], [6, 30], [6, 31], [6, 34], [6, 36], [6, 39], [7, 8], [7, 9], [7, 11], [7, 13], [7, 15], [7, 16], [7, 18], [7, 19], [7, 20], [7, 24], [7, 26], [7, 28], [7, 29], [7, 31], [7, 34], [7, 35], [8, 11], [8, 14], [8, 15], [8, 16], [8, 19], [8, 22], [8, 23], [8, 24], [8, 25], [8, 26], [8, 27], [8, 29], [8, 31], [8, 32], [8, 33], [8, 34], [8, 36], [8, 38], [8, 39], [9, 18], [9, 21], [9, 22], [9, 24], [9, 25], [9, 28], [9, 29], [9, 30], [9, 34], [9, 36], [9, 38], [9, 39], [10, 11], [10, 12], [10, 14], [10, 21], [10, 27], [10, 28], [10, 29], [10, 30], [10, 31], [10, 33], [10, 38], [10, 39], [11, 13], [11, 14], [11, 15], [11, 17], [11, 18], [11, 19], [11, 20], [11, 21], [11, 25], [11, 27], [11, 30], [11, 31], [11, 32], [11, 34], [11, 35], [11, 37], [11, 38], [12, 13], [12, 14], [12, 15], [12, 16], [12, 17], [12, 19], [12, 22], [12, 24], [12, 25], [12, 26], [12, 28], [12, 35], [12, 36], [13, 15], [13, 16], [13, 19], [13, 21], [13, 24], [13, 25], [13, 27], [13, 29], [13, 30], [13, 31], [13, 32], [13, 33], [13, 34], [13, 35], [13, 38], [13, 39], [14, 15], [14, 16], [14, 20], [14, 21], [14, 22], [14, 23], [14, 24], [14, 25], [14, 26], [14, 27], [14, 28], [14, 29], [14, 32], [14, 35], [14, 36], [14, 38], [14, 39], [15, 16], [15, 19], [15, 20], [15, 21], [15, 22], [15, 23], [15, 24], [15, 25], [15, 26], [15, 27], [15, 28], [15, 29], [15, 30], [15, 31], [15, 36], [16, 19], [16, 20], [16, 22], [16, 23], [16, 28], [16, 29], [16, 32], [16, 37], [16, 38], [16, 39], [17, 19], [17, 25], [17, 27], [17, 29], [17, 30], [18, 23], [18, 27], [18, 32], [18, 34], [18, 38], [19, 20], [19, 23], [19, 25], [19, 29], [19, 30], [19, 34], [19, 36], [20, 22], [20, 24], [20, 25], [20, 26], [20, 28], [20, 29], [20, 30], [20, 32], [20, 34], [20, 36], [20, 37], [20, 38], [20, 39], [21, 23], [21, 24], [21, 25], [21, 26], [21, 27], [21, 28], [21, 29], [21, 30], [21, 31], [21, 33], [21, 35], [21, 37], [21, 39], [22, 24], [22, 25], [22, 26], [22, 27], [22, 29], [22, 30], [22, 32], [22, 33], [22, 34], [22, 35], [22, 36], [22, 37], [22, 39], [23, 24], [23, 25], [23, 26], [23, 28], [23, 31], [23, 32], [23, 33], [23, 36], [23, 37], [23, 38], [23, 39], [24, 25], [24, 29], [24, 30], [24, 32], [24, 33], [24, 35], [24, 36], [24, 39], [25, 26], [25, 27], [25, 28], [25, 29], [25, 30], [25, 34], [25, 37], [25, 38], [25, 39], [26, 27], [26, 28], [26, 29], [26, 30], [26, 31], [26, 32], [26, 34], [26, 35], [26, 36], [26, 37], [26, 38], [27, 28], [27, 31], [27, 32], [27, 33], [27, 35], [27, 37], [27, 38], [27, 39], [28, 34], [28, 37], [28, 38], [28, 39], [29, 31], [29, 33], [29, 35], [29, 37], [30, 32], [30, 33], [30, 36], [30, 39], [31, 32], [31, 34], [31, 35], [31, 38], [31, 39], [32, 33], [32, 35], [32, 37], [32, 39], [33, 34], [33, 35], [33, 38], [34, 35], [34, 36], [34, 37], [34, 38], [34, 39], [35, 37], [35, 38], [35, 39], [36, 39], [37, 39], [38, 39]], "family": "rlve_even_degree_partition", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-even-degree-partition-l6-s5/tests/run.py b/tasks/construct-rlve-even-degree-partition-l6-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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-even-degree-partition-l6-s5/tests/test.sh b/tasks/construct-rlve-even-degree-partition-l6-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-even-degree-partition-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-graph-isomorphism-l4-s0/environment/Dockerfile b/tasks/construct-rlve-graph-isomorphism-l4-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-graph-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-graph-isomorphism-l4-s0/environment/instance.json b/tasks/construct-rlve-graph-isomorphism-l4-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..640f0b76df07eb37bf9c35598646474298eb6a9c
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l4-s0/environment/instance.json
@@ -0,0 +1 @@
+{"n": 18, "edges1": [[14, 17], [12, 17], [5, 8], [1, 11], [12, 13], [10, 14], [16, 17], [3, 12], [8, 16], [15, 17], [6, 13], [3, 17], [9, 14], [3, 13], [9, 16], [7, 12], [9, 12], [4, 6], [5, 6], [3, 9], [1, 6], [8, 12], [7, 16], [0, 9], [9, 17], [13, 14], [3, 6], [11, 16], [0, 12], [13, 16], [3, 5], [11, 15], [0, 3], [6, 8], [0, 2], [7, 13], [11, 14], [1, 7], [6, 9], [7, 14], [9, 10], [3, 11], [5, 16], [3, 14], [8, 15], [0, 5], [7, 15], [5, 14], [3, 16], [7, 17], [4, 13], [0, 6], [1, 2], [0, 14], [4, 7], [7, 8], [2, 13], [2, 5], [5, 15], [1, 13], [10, 13], [0, 8], [4, 16], [3, 4], [2, 9], [0, 11], [4, 10], [6, 15], [5, 11], [6, 7], [9, 15], [6, 10], [9, 13], [10, 15], [1, 8], [8, 14], [2, 10], [6, 16], [1, 16], [1, 9], [4, 9], [1, 4], [11, 13], [12, 14]], "edges2": [[10, 12], [3, 7], [3, 17], [3, 10], [4, 7], [8, 16], [0, 12], [4, 5], [16, 17], [2, 12], [8, 17], [0, 14], [6, 17], [1, 2], [5, 9], [11, 16], [10, 16], [6, 15], [2, 17], [6, 9], [8, 9], [9, 16], [4, 6], [0, 8], [12, 13], [7, 14], [11, 17], [7, 12], [12, 17], [11, 13], [4, 13], [2, 11], [7, 8], [3, 9], [1, 9], [1, 16], [8, 10], [5, 8], [0, 11], [1, 6], [3, 6], [5, 6], [4, 11], [8, 11], [5, 13], [1, 13], [6, 8], [2, 9], [4, 17], [1, 3], [14, 16], [8, 14], [4, 10], [0, 10], [0, 16], [0, 17], [5, 12], [1, 11], [14, 15], [9, 17], [10, 17], [3, 5], [9, 12], [1, 12], [10, 15], [12, 16], [7, 11], [1, 4], [8, 12], [4, 8], [1, 14], [13, 16], [15, 16], [3, 16], [6, 7], [1, 5], [5, 15], [2, 7], [10, 11], [9, 13], [12, 14], [0, 9], [2, 13], [12, 15]], "family": "rlve_graph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-graph-isomorphism-l4-s0/solution/solve.sh b/tasks/construct-rlve-graph-isomorphism-l4-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..9afe556befb840cb3863b545459670c0dc900be0
--- /dev/null
+++ b/tasks/construct-rlve-graph-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": [5, 10, 15, 9, 0, 6, 8, 11, 4, 12, 14, 3, 13, 16, 1, 7, 17, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-graph-isomorphism-l4-s0/tests/check.py b/tasks/construct-rlve-graph-isomorphism-l4-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..64b755f5ab12e7270811ccfc2f02d104715517b7
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l4-s0/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-l4-s0/tests/instance.json b/tasks/construct-rlve-graph-isomorphism-l4-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..640f0b76df07eb37bf9c35598646474298eb6a9c
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l4-s0/tests/instance.json
@@ -0,0 +1 @@
+{"n": 18, "edges1": [[14, 17], [12, 17], [5, 8], [1, 11], [12, 13], [10, 14], [16, 17], [3, 12], [8, 16], [15, 17], [6, 13], [3, 17], [9, 14], [3, 13], [9, 16], [7, 12], [9, 12], [4, 6], [5, 6], [3, 9], [1, 6], [8, 12], [7, 16], [0, 9], [9, 17], [13, 14], [3, 6], [11, 16], [0, 12], [13, 16], [3, 5], [11, 15], [0, 3], [6, 8], [0, 2], [7, 13], [11, 14], [1, 7], [6, 9], [7, 14], [9, 10], [3, 11], [5, 16], [3, 14], [8, 15], [0, 5], [7, 15], [5, 14], [3, 16], [7, 17], [4, 13], [0, 6], [1, 2], [0, 14], [4, 7], [7, 8], [2, 13], [2, 5], [5, 15], [1, 13], [10, 13], [0, 8], [4, 16], [3, 4], [2, 9], [0, 11], [4, 10], [6, 15], [5, 11], [6, 7], [9, 15], [6, 10], [9, 13], [10, 15], [1, 8], [8, 14], [2, 10], [6, 16], [1, 16], [1, 9], [4, 9], [1, 4], [11, 13], [12, 14]], "edges2": [[10, 12], [3, 7], [3, 17], [3, 10], [4, 7], [8, 16], [0, 12], [4, 5], [16, 17], [2, 12], [8, 17], [0, 14], [6, 17], [1, 2], [5, 9], [11, 16], [10, 16], [6, 15], [2, 17], [6, 9], [8, 9], [9, 16], [4, 6], [0, 8], [12, 13], [7, 14], [11, 17], [7, 12], [12, 17], [11, 13], [4, 13], [2, 11], [7, 8], [3, 9], [1, 9], [1, 16], [8, 10], [5, 8], [0, 11], [1, 6], [3, 6], [5, 6], [4, 11], [8, 11], [5, 13], [1, 13], [6, 8], [2, 9], [4, 17], [1, 3], [14, 16], [8, 14], [4, 10], [0, 10], [0, 16], [0, 17], [5, 12], [1, 11], [14, 15], [9, 17], [10, 17], [3, 5], [9, 12], [1, 12], [10, 15], [12, 16], [7, 11], [1, 4], [8, 12], [4, 8], [1, 14], [13, 16], [15, 16], [3, 16], [6, 7], [1, 5], [5, 15], [2, 7], [10, 11], [9, 13], [12, 14], [0, 9], [2, 13], [12, 15]], "family": "rlve_graph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-graph-isomorphism-l4-s0/tests/run.py b/tasks/construct-rlve-graph-isomorphism-l4-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-graph-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-graph-isomorphism-l4-s0/tests/test.sh b/tasks/construct-rlve-graph-isomorphism-l4-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-graph-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-graph-isomorphism-l4-s7/environment/Dockerfile b/tasks/construct-rlve-graph-isomorphism-l4-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-graph-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-graph-isomorphism-l4-s7/environment/instance.json b/tasks/construct-rlve-graph-isomorphism-l4-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..ad1302f7657e2a3fe73cc96db2dba388bc3a65ea
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l4-s7/environment/instance.json
@@ -0,0 +1 @@
+{"n": 18, "edges1": [[1, 16], [4, 15], [4, 17], [1, 12], [3, 6], [13, 16], [2, 3], [7, 16], [8, 11], [8, 9], [4, 5], [2, 13], [11, 13], [3, 9], [7, 8], [7, 15], [3, 12], [8, 13], [1, 6], [0, 8], [5, 13], [7, 12], [9, 16], [4, 9], [5, 9], [13, 14], [7, 11], [0, 2], [4, 12], [11, 15], [14, 15], [5, 12], [7, 13], [3, 8], [10, 14], [11, 17], [2, 11], [10, 17], [9, 14], [10, 12], [8, 14], [1, 7], [0, 5], [6, 10], [3, 5], [6, 17], [0, 1], [12, 13], [4, 11], [9, 15], [4, 6], [6, 7], [0, 10]], "edges2": [[4, 8], [1, 2], [13, 16], [3, 15], [10, 13], [5, 17], [0, 3], [0, 10], [0, 1], [4, 6], [1, 12], [9, 17], [13, 14], [3, 5], [8, 14], [3, 4], [14, 17], [1, 6], [6, 16], [5, 9], [7, 16], [7, 10], [2, 11], [3, 11], [2, 17], [13, 17], [2, 13], [0, 8], [15, 16], [7, 8], [5, 7], [3, 12], [6, 14], [5, 10], [0, 16], [4, 15], [10, 11], [3, 14], [0, 5], [0, 2], [2, 12], [4, 17], [4, 9], [1, 10], [2, 8], [5, 12], [6, 9], [14, 15], [12, 17], [8, 13], [11, 15], [3, 10], [10, 14]], "family": "rlve_graph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-graph-isomorphism-l4-s7/solution/solve.sh b/tasks/construct-rlve-graph-isomorphism-l4-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..7ca9e90c94655a15745e213af746e4d59a5d9eee
--- /dev/null
+++ b/tasks/construct-rlve-graph-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": [16, 15, 7, 8, 17, 13, 4, 3, 0, 2, 6, 5, 14, 10, 1, 12, 11, 9]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-graph-isomorphism-l4-s7/tests/check.py b/tasks/construct-rlve-graph-isomorphism-l4-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..64b755f5ab12e7270811ccfc2f02d104715517b7
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l4-s7/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-l4-s7/tests/instance.json b/tasks/construct-rlve-graph-isomorphism-l4-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..ad1302f7657e2a3fe73cc96db2dba388bc3a65ea
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l4-s7/tests/instance.json
@@ -0,0 +1 @@
+{"n": 18, "edges1": [[1, 16], [4, 15], [4, 17], [1, 12], [3, 6], [13, 16], [2, 3], [7, 16], [8, 11], [8, 9], [4, 5], [2, 13], [11, 13], [3, 9], [7, 8], [7, 15], [3, 12], [8, 13], [1, 6], [0, 8], [5, 13], [7, 12], [9, 16], [4, 9], [5, 9], [13, 14], [7, 11], [0, 2], [4, 12], [11, 15], [14, 15], [5, 12], [7, 13], [3, 8], [10, 14], [11, 17], [2, 11], [10, 17], [9, 14], [10, 12], [8, 14], [1, 7], [0, 5], [6, 10], [3, 5], [6, 17], [0, 1], [12, 13], [4, 11], [9, 15], [4, 6], [6, 7], [0, 10]], "edges2": [[4, 8], [1, 2], [13, 16], [3, 15], [10, 13], [5, 17], [0, 3], [0, 10], [0, 1], [4, 6], [1, 12], [9, 17], [13, 14], [3, 5], [8, 14], [3, 4], [14, 17], [1, 6], [6, 16], [5, 9], [7, 16], [7, 10], [2, 11], [3, 11], [2, 17], [13, 17], [2, 13], [0, 8], [15, 16], [7, 8], [5, 7], [3, 12], [6, 14], [5, 10], [0, 16], [4, 15], [10, 11], [3, 14], [0, 5], [0, 2], [2, 12], [4, 17], [4, 9], [1, 10], [2, 8], [5, 12], [6, 9], [14, 15], [12, 17], [8, 13], [11, 15], [3, 10], [10, 14]], "family": "rlve_graph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-graph-isomorphism-l4-s7/tests/run.py b/tasks/construct-rlve-graph-isomorphism-l4-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-graph-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-graph-isomorphism-l4-s7/tests/test.sh b/tasks/construct-rlve-graph-isomorphism-l4-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-graph-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/construct-rlve-graph-isomorphism-l4-s9/environment/Dockerfile b/tasks/construct-rlve-graph-isomorphism-l4-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-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-graph-isomorphism-l4-s9/environment/instance.json b/tasks/construct-rlve-graph-isomorphism-l4-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..10afabbdc787c12315febd23b0d07cf7015623ef
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l4-s9/environment/instance.json
@@ -0,0 +1 @@
+{"n": 18, "edges1": [[3, 16], [4, 17], [4, 6], [11, 15], [2, 11], [14, 16], [11, 14], [6, 17], [13, 17], [10, 14], [11, 17], [2, 6], [5, 6], [2, 15], [1, 14], [7, 15], [2, 16], [10, 13], [4, 16], [0, 10], [1, 17], [3, 11], [4, 9], [8, 13], [1, 7], [1, 9], [7, 8], [5, 12], [1, 2], [10, 12], [5, 17], [3, 8], [7, 14], [9, 14], [0, 13], [1, 12], [0, 7], [4, 10], [7, 11], [6, 12], [2, 9], [2, 12], [1, 11], [4, 12], [8, 11], [0, 12], [1, 8], [4, 13], [8, 12], [5, 10], [2, 3], [1, 16], [9, 15], [9, 12], [6, 14], [2, 13], [6, 16], [8, 16], [3, 4], [5, 7], [2, 10], [10, 16], [0, 1], [8, 15], [11, 13], [3, 7], [6, 13], [6, 8], [16, 17], [3, 13], [7, 9], [2, 8], [2, 14], [4, 11], [4, 8], [13, 16], [6, 11], [11, 12], [0, 3], [13, 14], [0, 14], [6, 9], [0, 15], [12, 17]], "edges2": [[4, 14], [3, 10], [0, 4], [7, 9], [0, 9], [15, 16], [3, 12], [9, 12], [5, 9], [7, 10], [15, 17], [0, 10], [0, 8], [1, 4], [8, 13], [1, 5], [10, 14], [10, 13], [10, 15], [0, 5], [2, 12], [6, 11], [10, 16], [13, 17], [6, 15], [8, 16], [0, 15], [0, 13], [5, 17], [13, 15], [9, 11], [0, 17], [4, 5], [1, 2], [7, 14], [5, 6], [1, 8], [6, 17], [8, 10], [2, 6], [1, 7], [2, 14], [11, 17], [0, 6], [9, 10], [4, 15], [5, 16], [0, 3], [4, 6], [3, 11], [1, 3], [9, 16], [4, 9], [4, 12], [2, 11], [3, 6], [3, 17], [0, 7], [14, 17], [3, 14], [6, 13], [5, 7], [7, 12], [1, 14], [11, 15], [9, 14], [1, 11], [3, 15], [2, 16], [4, 13], [5, 14], [9, 17], [13, 14], [1, 12], [4, 11], [2, 10], [7, 15], [6, 9], [2, 5], [16, 17], [4, 10], [6, 16], [3, 4], [3, 9]], "family": "rlve_graph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-graph-isomorphism-l4-s9/solution/solve.sh b/tasks/construct-rlve-graph-isomorphism-l4-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..47580dbda5fca30ec171d75b6d47165c6c93e202
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-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'
+{"p": [2, 14, 9, 11, 15, 8, 0, 1, 3, 7, 16, 4, 10, 6, 5, 12, 17, 13]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-graph-isomorphism-l4-s9/tests/check.py b/tasks/construct-rlve-graph-isomorphism-l4-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..64b755f5ab12e7270811ccfc2f02d104715517b7
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l4-s9/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-l4-s9/tests/instance.json b/tasks/construct-rlve-graph-isomorphism-l4-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..10afabbdc787c12315febd23b0d07cf7015623ef
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-l4-s9/tests/instance.json
@@ -0,0 +1 @@
+{"n": 18, "edges1": [[3, 16], [4, 17], [4, 6], [11, 15], [2, 11], [14, 16], [11, 14], [6, 17], [13, 17], [10, 14], [11, 17], [2, 6], [5, 6], [2, 15], [1, 14], [7, 15], [2, 16], [10, 13], [4, 16], [0, 10], [1, 17], [3, 11], [4, 9], [8, 13], [1, 7], [1, 9], [7, 8], [5, 12], [1, 2], [10, 12], [5, 17], [3, 8], [7, 14], [9, 14], [0, 13], [1, 12], [0, 7], [4, 10], [7, 11], [6, 12], [2, 9], [2, 12], [1, 11], [4, 12], [8, 11], [0, 12], [1, 8], [4, 13], [8, 12], [5, 10], [2, 3], [1, 16], [9, 15], [9, 12], [6, 14], [2, 13], [6, 16], [8, 16], [3, 4], [5, 7], [2, 10], [10, 16], [0, 1], [8, 15], [11, 13], [3, 7], [6, 13], [6, 8], [16, 17], [3, 13], [7, 9], [2, 8], [2, 14], [4, 11], [4, 8], [13, 16], [6, 11], [11, 12], [0, 3], [13, 14], [0, 14], [6, 9], [0, 15], [12, 17]], "edges2": [[4, 14], [3, 10], [0, 4], [7, 9], [0, 9], [15, 16], [3, 12], [9, 12], [5, 9], [7, 10], [15, 17], [0, 10], [0, 8], [1, 4], [8, 13], [1, 5], [10, 14], [10, 13], [10, 15], [0, 5], [2, 12], [6, 11], [10, 16], [13, 17], [6, 15], [8, 16], [0, 15], [0, 13], [5, 17], [13, 15], [9, 11], [0, 17], [4, 5], [1, 2], [7, 14], [5, 6], [1, 8], [6, 17], [8, 10], [2, 6], [1, 7], [2, 14], [11, 17], [0, 6], [9, 10], [4, 15], [5, 16], [0, 3], [4, 6], [3, 11], [1, 3], [9, 16], [4, 9], [4, 12], [2, 11], [3, 6], [3, 17], [0, 7], [14, 17], [3, 14], [6, 13], [5, 7], [7, 12], [1, 14], [11, 15], [9, 14], [1, 11], [3, 15], [2, 16], [4, 13], [5, 14], [9, 17], [13, 14], [1, 12], [4, 11], [2, 10], [7, 15], [6, 9], [2, 5], [16, 17], [4, 10], [6, 16], [3, 4], [3, 9]], "family": "rlve_graph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-graph-isomorphism-l4-s9/tests/run.py b/tasks/construct-rlve-graph-isomorphism-l4-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-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-graph-isomorphism-l4-s9/tests/test.sh b/tasks/construct-rlve-graph-isomorphism-l4-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-graph-isomorphism-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-grid-parity-construction-l4-s0/environment/Dockerfile b/tasks/construct-rlve-grid-parity-construction-l4-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-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-grid-parity-construction-l4-s0/environment/instance.json b/tasks/construct-rlve-grid-parity-construction-l4-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c5ffff57177b9c73be3c5fb9d63349b74d930079
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l4-s0/environment/instance.json
@@ -0,0 +1 @@
+{"parity": ["111", "111", "111", "100", "100", "100", "001", "110", "110", "011", "110"], "family": "rlve_grid_parity_construction", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-grid-parity-construction-l4-s0/solution/solve.sh b/tasks/construct-rlve-grid-parity-construction-l4-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..30bf1035ec1bb0a7e1472d87f55b4298e7120d7d
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-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'
+{"grid": ["111", "101", "101", "111", "011", "111", "101", "011", "111", "111", "110"]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-grid-parity-construction-l4-s0/tests/check.py b/tasks/construct-rlve-grid-parity-construction-l4-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..89d5d286593348d0000ac10db74e1c6242be6c97
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l4-s0/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-l4-s0/tests/instance.json b/tasks/construct-rlve-grid-parity-construction-l4-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c5ffff57177b9c73be3c5fb9d63349b74d930079
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l4-s0/tests/instance.json
@@ -0,0 +1 @@
+{"parity": ["111", "111", "111", "100", "100", "100", "001", "110", "110", "011", "110"], "family": "rlve_grid_parity_construction", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-grid-parity-construction-l4-s0/tests/run.py b/tasks/construct-rlve-grid-parity-construction-l4-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-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-grid-parity-construction-l4-s0/tests/test.sh b/tasks/construct-rlve-grid-parity-construction-l4-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-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-grid-parity-construction-l4-s7/environment/Dockerfile b/tasks/construct-rlve-grid-parity-construction-l4-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-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-grid-parity-construction-l4-s7/environment/instance.json b/tasks/construct-rlve-grid-parity-construction-l4-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..572ee6a139bc1d57ab1fb25f79af092453b8a034
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l4-s7/environment/instance.json
@@ -0,0 +1 @@
+{"parity": ["11000", "10000", "10000", "11010", "10010", "01111", "00001", "01110"], "family": "rlve_grid_parity_construction", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-grid-parity-construction-l4-s7/solution/solve.sh b/tasks/construct-rlve-grid-parity-construction-l4-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..6edcebc94002f40c26366adde1c2440eee9607eb
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-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'
+{"grid": ["10000", "00000", "00000", "10000", "00010", "00101", "00000", "00100"]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-grid-parity-construction-l4-s7/tests/check.py b/tasks/construct-rlve-grid-parity-construction-l4-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..89d5d286593348d0000ac10db74e1c6242be6c97
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l4-s7/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-l4-s7/tests/instance.json b/tasks/construct-rlve-grid-parity-construction-l4-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..572ee6a139bc1d57ab1fb25f79af092453b8a034
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-l4-s7/tests/instance.json
@@ -0,0 +1 @@
+{"parity": ["11000", "10000", "10000", "11010", "10010", "01111", "00001", "01110"], "family": "rlve_grid_parity_construction", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-grid-parity-construction-l4-s7/tests/run.py b/tasks/construct-rlve-grid-parity-construction-l4-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-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-grid-parity-construction-l4-s7/tests/test.sh b/tasks/construct-rlve-grid-parity-construction-l4-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-grid-parity-construction-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-imp-party-l3-s7/environment/Dockerfile b/tasks/construct-rlve-imp-party-l3-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-l3-s7/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-imp-party-l3-s7/environment/instance.json b/tasks/construct-rlve-imp-party-l3-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..9758223c6827a413db66f954757a32871e6ddcee
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-l3-s7/environment/instance.json
@@ -0,0 +1 @@
+{"n": 8, "edges": [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [0, 6], [0, 7], [0, 8], [0, 11], [0, 12], [0, 13], [0, 14], [0, 17], [0, 18], [0, 19], [0, 20], [0, 22], [0, 23], [1, 2], [1, 3], [1, 4], [1, 5], [1, 6], [1, 8], [1, 9], [1, 11], [1, 12], [1, 13], [1, 14], [1, 16], [1, 17], [1, 18], [1, 19], [1, 20], [1, 22], [1, 23], [2, 3], [2, 4], [2, 5], [2, 6], [2, 7], [2, 8], [2, 9], [2, 11], [2, 12], [2, 13], [2, 14], [2, 16], [2, 18], [2, 19], [2, 20], [2, 21], [2, 22], [2, 23], [3, 5], [3, 13], [3, 14], [3, 18], [3, 19], [3, 20], [3, 23], [4, 5], [4, 6], [4, 8], [4, 9], [4, 11], [4, 12], [4, 13], [4, 14], [4, 15], [4, 16], [4, 18], [4, 19], [4, 20], [4, 21], [4, 22], [4, 23], [5, 6], [5, 7], [5, 8], [5, 11], [5, 12], [5, 13], [5, 14], [5, 15], [5, 18], [5, 19], [5, 20], [5, 22], [5, 23], [6, 7], [6, 8], [6, 11], [6, 12], [6, 13], [6, 14], [6, 15], [6, 16], [6, 17], [6, 18], [6, 19], [6, 20], [6, 21], [6, 22], [6, 23], [7, 8], [7, 11], [7, 12], [7, 14], [7, 18], [7, 19], [8, 9], [8, 10], [8, 11], [8, 12], [8, 13], [8, 14], [8, 16], [8, 18], [8, 19], [8, 20], [8, 22], [8, 23], [9, 11], [10, 11], [10, 14], [11, 12], [11, 13], [11, 14], [11, 15], [11, 18], [11, 19], [11, 20], [11, 21], [11, 22], [11, 23], [12, 13], [12, 14], [12, 17], [12, 18], [12, 19], [12, 20], [12, 21], [12, 22], [12, 23], [13, 14], [13, 18], [13, 19], [13, 20], [13, 22], [13, 23], [14, 16], [14, 17], [14, 18], [14, 19], [14, 20], [14, 21], [14, 22], [14, 23], [15, 18], [15, 19], [15, 22], [15, 23], [16, 19], [16, 20], [16, 23], [17, 22], [17, 23], [18, 19], [18, 20], [18, 22], [18, 23], [19, 20], [19, 21], [19, 22], [19, 23], [20, 21], [20, 22], [20, 23], [22, 23]], "family": "rlve_imp_party", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-imp-party-l3-s7/solution/solve.sh b/tasks/construct-rlve-imp-party-l3-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..89124c90de0d8e6ece2abe032085337807304d08
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-l3-s7/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"clique": [1, 4, 5, 6, 8, 12, 13, 18]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-imp-party-l3-s7/tests/check.py b/tasks/construct-rlve-imp-party-l3-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..97b5a037a2ba8bf4495cf87cc074d1c60b11311c
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-l3-s7/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-l3-s7/tests/instance.json b/tasks/construct-rlve-imp-party-l3-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..9758223c6827a413db66f954757a32871e6ddcee
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-l3-s7/tests/instance.json
@@ -0,0 +1 @@
+{"n": 8, "edges": [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [0, 6], [0, 7], [0, 8], [0, 11], [0, 12], [0, 13], [0, 14], [0, 17], [0, 18], [0, 19], [0, 20], [0, 22], [0, 23], [1, 2], [1, 3], [1, 4], [1, 5], [1, 6], [1, 8], [1, 9], [1, 11], [1, 12], [1, 13], [1, 14], [1, 16], [1, 17], [1, 18], [1, 19], [1, 20], [1, 22], [1, 23], [2, 3], [2, 4], [2, 5], [2, 6], [2, 7], [2, 8], [2, 9], [2, 11], [2, 12], [2, 13], [2, 14], [2, 16], [2, 18], [2, 19], [2, 20], [2, 21], [2, 22], [2, 23], [3, 5], [3, 13], [3, 14], [3, 18], [3, 19], [3, 20], [3, 23], [4, 5], [4, 6], [4, 8], [4, 9], [4, 11], [4, 12], [4, 13], [4, 14], [4, 15], [4, 16], [4, 18], [4, 19], [4, 20], [4, 21], [4, 22], [4, 23], [5, 6], [5, 7], [5, 8], [5, 11], [5, 12], [5, 13], [5, 14], [5, 15], [5, 18], [5, 19], [5, 20], [5, 22], [5, 23], [6, 7], [6, 8], [6, 11], [6, 12], [6, 13], [6, 14], [6, 15], [6, 16], [6, 17], [6, 18], [6, 19], [6, 20], [6, 21], [6, 22], [6, 23], [7, 8], [7, 11], [7, 12], [7, 14], [7, 18], [7, 19], [8, 9], [8, 10], [8, 11], [8, 12], [8, 13], [8, 14], [8, 16], [8, 18], [8, 19], [8, 20], [8, 22], [8, 23], [9, 11], [10, 11], [10, 14], [11, 12], [11, 13], [11, 14], [11, 15], [11, 18], [11, 19], [11, 20], [11, 21], [11, 22], [11, 23], [12, 13], [12, 14], [12, 17], [12, 18], [12, 19], [12, 20], [12, 21], [12, 22], [12, 23], [13, 14], [13, 18], [13, 19], [13, 20], [13, 22], [13, 23], [14, 16], [14, 17], [14, 18], [14, 19], [14, 20], [14, 21], [14, 22], [14, 23], [15, 18], [15, 19], [15, 22], [15, 23], [16, 19], [16, 20], [16, 23], [17, 22], [17, 23], [18, 19], [18, 20], [18, 22], [18, 23], [19, 20], [19, 21], [19, 22], [19, 23], [20, 21], [20, 22], [20, 23], [22, 23]], "family": "rlve_imp_party", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-imp-party-l3-s7/tests/run.py b/tasks/construct-rlve-imp-party-l3-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-l3-s7/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-imp-party-l3-s7/tests/test.sh b/tasks/construct-rlve-imp-party-l3-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-l3-s7/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-imp-party-l6-s1/environment/Dockerfile b/tasks/construct-rlve-imp-party-l6-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-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-imp-party-l6-s1/environment/instance.json b/tasks/construct-rlve-imp-party-l6-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4f96444b5091c3395beb0ad8a37bbae64cf1118d
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-l6-s1/environment/instance.json
@@ -0,0 +1 @@
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70], [65, 71], [65, 72], [65, 73], [65, 74], [66, 68], [66, 70], [66, 71], [66, 72], [66, 74], [67, 68], [67, 69], [67, 70], [67, 71], [67, 72], [67, 73], [67, 74], [68, 69], [68, 70], [68, 71], [68, 72], [68, 73], [68, 74], [69, 70], [69, 71], [69, 72], [69, 73], [69, 74], [70, 71], [70, 72], [70, 73], [70, 74], [71, 72], [71, 73], [71, 74], [72, 73], [72, 74], [73, 74]], "family": "rlve_imp_party", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-imp-party-l6-s1/solution/solve.sh b/tasks/construct-rlve-imp-party-l6-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..74486daa90c663ae9d26fa88a6fb98ea9ea9fcdb
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-l6-s1/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"clique": [1, 3, 15, 19, 20, 21, 22, 23, 26, 28, 30, 31, 33, 42, 43, 47, 51, 55, 57, 59, 63, 65, 69, 72, 74]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-imp-party-l6-s1/tests/check.py b/tasks/construct-rlve-imp-party-l6-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..97b5a037a2ba8bf4495cf87cc074d1c60b11311c
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-l6-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-l6-s1/tests/instance.json b/tasks/construct-rlve-imp-party-l6-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4f96444b5091c3395beb0ad8a37bbae64cf1118d
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-l6-s1/tests/instance.json
@@ -0,0 +1 @@
+{"n": 25, "edges": [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [0, 6], [0, 7], [0, 8], [0, 9], [0, 12], [0, 13], [0, 14], [0, 15], [0, 16], [0, 17], [0, 18], [0, 19], [0, 20], [0, 21], [0, 22], [0, 23], [0, 24], [0, 25], [0, 26], [0, 27], [0, 28], [0, 29], [0, 30], [0, 31], [0, 32], [0, 33], [0, 34], [0, 35], [0, 36], [0, 37], [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, 52], [0, 53], [0, 55], [0, 57], [0, 58], [0, 59], [0, 60], [0, 61], [0, 62], [0, 63], [0, 64], [0, 65], [0, 68], [0, 69], [0, 70], [0, 71], [0, 72], [0, 73], [0, 74], [1, 2], [1, 3], [1, 4], [1, 5], [1, 7], [1, 8], [1, 9], [1, 10], [1, 11], [1, 12], [1, 13], [1, 14], [1, 15], [1, 16], [1, 17], [1, 19], [1, 20], [1, 21], [1, 22], [1, 23], [1, 24], [1, 25], [1, 26], [1, 27], [1, 28], [1, 30], [1, 31], [1, 32], [1, 33], [1, 34], [1, 35], [1, 36], [1, 37], [1, 38], [1, 39], [1, 40], [1, 41], [1, 42], [1, 43], [1, 44], [1, 45], [1, 47], [1, 48], [1, 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70], [65, 71], [65, 72], [65, 73], [65, 74], [66, 68], [66, 70], [66, 71], [66, 72], [66, 74], [67, 68], [67, 69], [67, 70], [67, 71], [67, 72], [67, 73], [67, 74], [68, 69], [68, 70], [68, 71], [68, 72], [68, 73], [68, 74], [69, 70], [69, 71], [69, 72], [69, 73], [69, 74], [70, 71], [70, 72], [70, 73], [70, 74], [71, 72], [71, 73], [71, 74], [72, 73], [72, 74], [73, 74]], "family": "rlve_imp_party", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-imp-party-l6-s1/tests/run.py b/tasks/construct-rlve-imp-party-l6-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-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-imp-party-l6-s1/tests/test.sh b/tasks/construct-rlve-imp-party-l6-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-imp-party-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-integer-linear-system-l1-s1/environment/Dockerfile b/tasks/construct-rlve-integer-linear-system-l1-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-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-integer-linear-system-l1-s1/environment/instance.json b/tasks/construct-rlve-integer-linear-system-l1-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..5eae145904b95034667b5e0ea2ee512a3b0773f1
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-l1-s1/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [[0, 0, 1], [0, 1, 1], [0, 1, 0], [1, 1, 1], [1, 0, 0], [1, 1, 1], [0, 1, 1]], "rhs": [1, 2, 1, 4, 2, 4, 2], "family": "rlve_integer_linear_system", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-integer-linear-system-l1-s1/solution/solve.sh b/tasks/construct-rlve-integer-linear-system-l1-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..f6b30568ba05b24ec2c17acfa018526c65019a04
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-l1-s1/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": [2, 1, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-integer-linear-system-l1-s1/tests/check.py b/tasks/construct-rlve-integer-linear-system-l1-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..ff19002b3fddc9597f84ee31ef5ffbee5d3fac32
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-l1-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-l1-s1/tests/instance.json b/tasks/construct-rlve-integer-linear-system-l1-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..5eae145904b95034667b5e0ea2ee512a3b0773f1
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-l1-s1/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [[0, 0, 1], [0, 1, 1], [0, 1, 0], [1, 1, 1], [1, 0, 0], [1, 1, 1], [0, 1, 1]], "rhs": [1, 2, 1, 4, 2, 4, 2], "family": "rlve_integer_linear_system", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-integer-linear-system-l1-s1/tests/run.py b/tasks/construct-rlve-integer-linear-system-l1-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-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-integer-linear-system-l1-s1/tests/test.sh b/tasks/construct-rlve-integer-linear-system-l1-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-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-integer-linear-system-l1-s6/environment/Dockerfile b/tasks/construct-rlve-integer-linear-system-l1-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-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-integer-linear-system-l1-s6/environment/instance.json b/tasks/construct-rlve-integer-linear-system-l1-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..704ed3c0f589fcab1db5829d6165c6a7ce06bb80
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-l1-s6/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [[1, 0, 1], [0, 0, 1], [1, 1, 1], [0, 0, 1], [1, 0, 0], [1, 0, 1], [0, 0, 1]], "rhs": [2, 1, 5, 1, 1, 2, 1], "family": "rlve_integer_linear_system", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-integer-linear-system-l1-s6/solution/solve.sh b/tasks/construct-rlve-integer-linear-system-l1-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..1455aece84d7e45241f5617b0022773181ac12c4
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-l1-s6/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": [1, 3, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-integer-linear-system-l1-s6/tests/check.py b/tasks/construct-rlve-integer-linear-system-l1-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..ff19002b3fddc9597f84ee31ef5ffbee5d3fac32
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-l1-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-l1-s6/tests/instance.json b/tasks/construct-rlve-integer-linear-system-l1-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..704ed3c0f589fcab1db5829d6165c6a7ce06bb80
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-l1-s6/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [[1, 0, 1], [0, 0, 1], [1, 1, 1], [0, 0, 1], [1, 0, 0], [1, 0, 1], [0, 0, 1]], "rhs": [2, 1, 5, 1, 1, 2, 1], "family": "rlve_integer_linear_system", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-integer-linear-system-l1-s6/tests/run.py b/tasks/construct-rlve-integer-linear-system-l1-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-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-integer-linear-system-l1-s6/tests/test.sh b/tasks/construct-rlve-integer-linear-system-l1-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-integer-linear-system-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-l2-s1/environment/Dockerfile b/tasks/construct-rlve-largest-convex-polygon-l2-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l2-s1/environment/instance.json b/tasks/construct-rlve-largest-convex-polygon-l2-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3b57296b790d427a0d773312c93e2cfd4edd8854
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l2-s1/environment/instance.json
@@ -0,0 +1 @@
+{"points": [[0, 1], [0, 7], [3, 7], [6, 8], [1, 8], [6, 4], [3, 2], [4, 1], [5, 2]], "k": 7, "family": "rlve_largest_convex_polygon", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-convex-polygon-l2-s1/solution/solve.sh b/tasks/construct-rlve-largest-convex-polygon-l2-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..4eaa06306b778b1fb695958e3e35cbc1ebf2f38a
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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'
+{"indices": [0, 1, 4, 2, 5, 8, 7]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-largest-convex-polygon-l2-s1/tests/check.py b/tasks/construct-rlve-largest-convex-polygon-l2-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3a09cc858f41ce5745e4e491188cd42360c65dae
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l2-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-l2-s1/tests/instance.json b/tasks/construct-rlve-largest-convex-polygon-l2-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3b57296b790d427a0d773312c93e2cfd4edd8854
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l2-s1/tests/instance.json
@@ -0,0 +1 @@
+{"points": [[0, 1], [0, 7], [3, 7], [6, 8], [1, 8], [6, 4], [3, 2], [4, 1], [5, 2]], "k": 7, "family": "rlve_largest_convex_polygon", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-convex-polygon-l2-s1/tests/run.py b/tasks/construct-rlve-largest-convex-polygon-l2-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l2-s1/tests/test.sh b/tasks/construct-rlve-largest-convex-polygon-l2-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l2-s8/environment/Dockerfile b/tasks/construct-rlve-largest-convex-polygon-l2-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l2-s8/environment/instance.json b/tasks/construct-rlve-largest-convex-polygon-l2-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..61cd3fa7867b2f6c360a83bfcedec112cd061098
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l2-s8/environment/instance.json
@@ -0,0 +1 @@
+{"points": [[0, 7], [1, 2], [2, 1], [5, 8], [6, 8], [2, 0], [0, 6], [3, 3], [6, 6]], "k": 7, "family": "rlve_largest_convex_polygon", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-convex-polygon-l2-s8/solution/solve.sh b/tasks/construct-rlve-largest-convex-polygon-l2-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..4918818b1c00b8cd7b3bce0b9e5b9fb074c1f7ad
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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'
+{"indices": [6, 0, 3, 4, 8, 7, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-largest-convex-polygon-l2-s8/tests/check.py b/tasks/construct-rlve-largest-convex-polygon-l2-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3a09cc858f41ce5745e4e491188cd42360c65dae
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l2-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-l2-s8/tests/instance.json b/tasks/construct-rlve-largest-convex-polygon-l2-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..61cd3fa7867b2f6c360a83bfcedec112cd061098
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l2-s8/tests/instance.json
@@ -0,0 +1 @@
+{"points": [[0, 7], [1, 2], [2, 1], [5, 8], [6, 8], [2, 0], [0, 6], [3, 3], [6, 6]], "k": 7, "family": "rlve_largest_convex_polygon", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-convex-polygon-l2-s8/tests/run.py b/tasks/construct-rlve-largest-convex-polygon-l2-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l2-s8/tests/test.sh b/tasks/construct-rlve-largest-convex-polygon-l2-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l6-s0/environment/Dockerfile b/tasks/construct-rlve-largest-convex-polygon-l6-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l6-s0/environment/instance.json b/tasks/construct-rlve-largest-convex-polygon-l6-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..726a4cdd4e7029adbf98be4d74b136ddc1f35478
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l6-s0/environment/instance.json
@@ -0,0 +1 @@
+{"points": [[6, 18], [4, 0], [20, 14], [2, 30], [4, 9], [21, 31], [0, 2], [14, 25], [2, 11], [16, 1], [27, 31], [18, 7], [20, 2], [24, 32], [3, 9], [34, 12], [27, 6], [5, 6], [9, 1], [28, 26], [28, 35], [31, 0], [21, 8], [23, 5], [13, 34], [34, 8], [14, 20], [5, 20], [8, 19], [3, 32], [24, 18], [33, 30], [1, 25], [16, 11], [7, 14]], "k": 14, "family": "rlve_largest_convex_polygon", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-convex-polygon-l6-s0/solution/solve.sh b/tasks/construct-rlve-largest-convex-polygon-l6-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..7e7e3035ba8a7cfc23a8e12b56c7dd64c2651ea4
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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'
+{"indices": [32, 3, 29, 24, 13, 10, 19, 23, 12, 9, 18, 17, 14, 8]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-largest-convex-polygon-l6-s0/tests/check.py b/tasks/construct-rlve-largest-convex-polygon-l6-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3a09cc858f41ce5745e4e491188cd42360c65dae
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l6-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-l6-s0/tests/instance.json b/tasks/construct-rlve-largest-convex-polygon-l6-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..726a4cdd4e7029adbf98be4d74b136ddc1f35478
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l6-s0/tests/instance.json
@@ -0,0 +1 @@
+{"points": [[6, 18], [4, 0], [20, 14], [2, 30], [4, 9], [21, 31], [0, 2], [14, 25], [2, 11], [16, 1], [27, 31], [18, 7], [20, 2], [24, 32], [3, 9], [34, 12], [27, 6], [5, 6], [9, 1], [28, 26], [28, 35], [31, 0], [21, 8], [23, 5], [13, 34], [34, 8], [14, 20], [5, 20], [8, 19], [3, 32], [24, 18], [33, 30], [1, 25], [16, 11], [7, 14]], "k": 14, "family": "rlve_largest_convex_polygon", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-convex-polygon-l6-s0/tests/run.py b/tasks/construct-rlve-largest-convex-polygon-l6-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l6-s0/tests/test.sh b/tasks/construct-rlve-largest-convex-polygon-l6-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l6-s7/environment/Dockerfile b/tasks/construct-rlve-largest-convex-polygon-l6-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l6-s7/environment/instance.json b/tasks/construct-rlve-largest-convex-polygon-l6-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2829bf8479c2dcb0945898d702b4e45321b023ed
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l6-s7/environment/instance.json
@@ -0,0 +1 @@
+{"points": [[12, 10], [15, 27], [21, 31], [28, 6], [10, 0], [5, 34], [35, 20], [24, 20], [20, 1], [34, 0], [25, 34], [3, 21], [9, 19], [0, 25], [0, 31], [35, 28], [26, 28], [22, 9], [3, 8], [23, 23], [19, 4], [8, 4], [5, 14], [21, 32], [23, 29], [34, 35], [9, 15], [32, 13], [19, 13], [18, 2], [8, 25], [16, 14], [24, 21], [7, 8], [2, 30]], "k": 13, "family": "rlve_largest_convex_polygon", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-convex-polygon-l6-s7/solution/solve.sh b/tasks/construct-rlve-largest-convex-polygon-l6-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..08049a90824aa69fc08947839e90b02d41d05013
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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'
+{"indices": [11, 30, 23, 10, 25, 15, 6, 27, 3, 8, 29, 33, 22]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-largest-convex-polygon-l6-s7/tests/check.py b/tasks/construct-rlve-largest-convex-polygon-l6-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3a09cc858f41ce5745e4e491188cd42360c65dae
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l6-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-l6-s7/tests/instance.json b/tasks/construct-rlve-largest-convex-polygon-l6-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2829bf8479c2dcb0945898d702b4e45321b023ed
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l6-s7/tests/instance.json
@@ -0,0 +1 @@
+{"points": [[12, 10], [15, 27], [21, 31], [28, 6], [10, 0], [5, 34], [35, 20], [24, 20], [20, 1], [34, 0], [25, 34], [3, 21], [9, 19], [0, 25], [0, 31], [35, 28], [26, 28], [22, 9], [3, 8], [23, 23], [19, 4], [8, 4], [5, 14], [21, 32], [23, 29], [34, 35], [9, 15], [32, 13], [19, 13], [18, 2], [8, 25], [16, 14], [24, 21], [7, 8], [2, 30]], "k": 13, "family": "rlve_largest_convex_polygon", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-convex-polygon-l6-s7/tests/run.py b/tasks/construct-rlve-largest-convex-polygon-l6-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l6-s7/tests/test.sh b/tasks/construct-rlve-largest-convex-polygon-l6-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l6-s9/environment/Dockerfile b/tasks/construct-rlve-largest-convex-polygon-l6-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l6-s9/environment/instance.json b/tasks/construct-rlve-largest-convex-polygon-l6-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3dcfe6fbd5ef76fd6fdf5c1581ba206c08bdee80
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l6-s9/environment/instance.json
@@ -0,0 +1 @@
+{"points": [[2, 27], [7, 23], [12, 10], [28, 0], [34, 16], [16, 32], [6, 33], [22, 23], [23, 25], [15, 11], [25, 13], [21, 24], [8, 2], [6, 35], [22, 28], [5, 18], [35, 1], [24, 7], [29, 6], [7, 9], [20, 6], [16, 15], [31, 21], [28, 7], [21, 32], [14, 26], [13, 0], [33, 9], [9, 15], [30, 22], [8, 28], [2, 21], [20, 2], [19, 34], [13, 24]], "k": 14, "family": "rlve_largest_convex_polygon", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-convex-polygon-l6-s9/solution/solve.sh b/tasks/construct-rlve-largest-convex-polygon-l6-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..16aa89be34ef60cb82c805d6ebe31691e0f86e0f
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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'
+{"indices": [31, 0, 6, 5, 14, 29, 22, 4, 27, 23, 17, 9, 28, 15]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-largest-convex-polygon-l6-s9/tests/check.py b/tasks/construct-rlve-largest-convex-polygon-l6-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3a09cc858f41ce5745e4e491188cd42360c65dae
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l6-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-l6-s9/tests/instance.json b/tasks/construct-rlve-largest-convex-polygon-l6-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3dcfe6fbd5ef76fd6fdf5c1581ba206c08bdee80
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-l6-s9/tests/instance.json
@@ -0,0 +1 @@
+{"points": [[2, 27], [7, 23], [12, 10], [28, 0], [34, 16], [16, 32], [6, 33], [22, 23], [23, 25], [15, 11], [25, 13], [21, 24], [8, 2], [6, 35], [22, 28], [5, 18], [35, 1], [24, 7], [29, 6], [7, 9], [20, 6], [16, 15], [31, 21], [28, 7], [21, 32], [14, 26], [13, 0], [33, 9], [9, 15], [30, 22], [8, 28], [2, 21], [20, 2], [19, 34], [13, 24]], "k": 14, "family": "rlve_largest_convex_polygon", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-convex-polygon-l6-s9/tests/run.py b/tasks/construct-rlve-largest-convex-polygon-l6-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-largest-convex-polygon-l6-s9/tests/test.sh b/tasks/construct-rlve-largest-convex-polygon-l6-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-largest-convex-polygon-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-rectangle-l4-s0/environment/Dockerfile b/tasks/construct-rlve-largest-rectangle-l4-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-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-largest-rectangle-l4-s0/environment/instance.json b/tasks/construct-rlve-largest-rectangle-l4-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..5006ae652f2ba08853a5af6506f56c6c5c63a6d2
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l4-s0/environment/instance.json
@@ -0,0 +1 @@
+{"points": [[-23, -21], [-18, 15], [-21, -2], [-8, -6], [-20, -6], [0, -7], [-1, -12], [25, 4], [-2, -21], [-10, -14], [-19, 8], [16, 4], [9, -15], [24, 23], [4, -13], [20, 5], [6, -18], [-14, -11], [16, 6], [17, -2], [25, 18], [-16, 3], [17, -9], [5, -8], [7, 4]], "area": 26, "family": "rlve_largest_rectangle", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-rectangle-l4-s0/solution/solve.sh b/tasks/construct-rlve-largest-rectangle-l4-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..216831594e8dfb517ea85a07b92174c287f7ba73
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-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": [5, 6, 14, 23]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-largest-rectangle-l4-s0/tests/check.py b/tasks/construct-rlve-largest-rectangle-l4-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2251b2af23a481e598fa1e48d0a8249f70b6cc44
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l4-s0/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-l4-s0/tests/instance.json b/tasks/construct-rlve-largest-rectangle-l4-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..5006ae652f2ba08853a5af6506f56c6c5c63a6d2
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l4-s0/tests/instance.json
@@ -0,0 +1 @@
+{"points": [[-23, -21], [-18, 15], [-21, -2], [-8, -6], [-20, -6], [0, -7], [-1, -12], [25, 4], [-2, -21], [-10, -14], [-19, 8], [16, 4], [9, -15], [24, 23], [4, -13], [20, 5], [6, -18], [-14, -11], [16, 6], [17, -2], [25, 18], [-16, 3], [17, -9], [5, -8], [7, 4]], "area": 26, "family": "rlve_largest_rectangle", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-rectangle-l4-s0/tests/run.py b/tasks/construct-rlve-largest-rectangle-l4-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-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-largest-rectangle-l4-s0/tests/test.sh b/tasks/construct-rlve-largest-rectangle-l4-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-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-largest-rectangle-l4-s7/environment/Dockerfile b/tasks/construct-rlve-largest-rectangle-l4-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-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-largest-rectangle-l4-s7/environment/instance.json b/tasks/construct-rlve-largest-rectangle-l4-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..31fd1c0dc72c96300266b67d6f61389fb7ce54bf
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l4-s7/environment/instance.json
@@ -0,0 +1 @@
+{"points": [[8, 19], [-7, -2], [5, -18], [24, -19], [-1, -12], [-16, 1], [2, -20], [3, -9], [-20, -2], [-16, 22], [-21, -22], [-21, 11], [-17, -17], [-24, -20], [-2, -19], [-2, -14], [-1, -18], [5, -12], [-11, -21], [13, -8], [23, 19], [22, -4], [-6, 24], [-23, -12], [-2, -23]], "area": 36, "family": "rlve_largest_rectangle", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-rectangle-l4-s7/solution/solve.sh b/tasks/construct-rlve-largest-rectangle-l4-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..d4ee12a5b37285e41e4e5289228559fc6c12c810
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-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": [2, 16, 4, 17]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-largest-rectangle-l4-s7/tests/check.py b/tasks/construct-rlve-largest-rectangle-l4-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2251b2af23a481e598fa1e48d0a8249f70b6cc44
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l4-s7/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-l4-s7/tests/instance.json b/tasks/construct-rlve-largest-rectangle-l4-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..31fd1c0dc72c96300266b67d6f61389fb7ce54bf
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l4-s7/tests/instance.json
@@ -0,0 +1 @@
+{"points": [[8, 19], [-7, -2], [5, -18], [24, -19], [-1, -12], [-16, 1], [2, -20], [3, -9], [-20, -2], [-16, 22], [-21, -22], [-21, 11], [-17, -17], [-24, -20], [-2, -19], [-2, -14], [-1, -18], [5, -12], [-11, -21], [13, -8], [23, 19], [22, -4], [-6, 24], [-23, -12], [-2, -23]], "area": 36, "family": "rlve_largest_rectangle", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-rectangle-l4-s7/tests/run.py b/tasks/construct-rlve-largest-rectangle-l4-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-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-largest-rectangle-l4-s7/tests/test.sh b/tasks/construct-rlve-largest-rectangle-l4-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-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-largest-rectangle-l4-s9/environment/Dockerfile b/tasks/construct-rlve-largest-rectangle-l4-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-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-largest-rectangle-l4-s9/environment/instance.json b/tasks/construct-rlve-largest-rectangle-l4-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..473a457e0e7612097bd2275ec9ac65cf2e18baa2
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l4-s9/environment/instance.json
@@ -0,0 +1 @@
+{"points": [[-10, -1], [16, -19], [7, -17], [-21, -16], [-19, -19], [-5, -18], [1, -1], [10, -7], [19, -18], [-2, -5], [5, 16], [-12, -5], [12, -6], [-11, 12], [12, -25], [-4, -12], [20, 7], [-23, -10], [12, 6], [-12, -12], [-17, 8], [1, 23], [-23, 5], [14, -20], [9, 14]], "area": 146, "family": "rlve_largest_rectangle", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-rectangle-l4-s9/solution/solve.sh b/tasks/construct-rlve-largest-rectangle-l4-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..5be960a201c3bd0a90f35bf44b49bba58515aa83
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-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": [2, 12, 6, 15]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-largest-rectangle-l4-s9/tests/check.py b/tasks/construct-rlve-largest-rectangle-l4-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2251b2af23a481e598fa1e48d0a8249f70b6cc44
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l4-s9/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-l4-s9/tests/instance.json b/tasks/construct-rlve-largest-rectangle-l4-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..473a457e0e7612097bd2275ec9ac65cf2e18baa2
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-l4-s9/tests/instance.json
@@ -0,0 +1 @@
+{"points": [[-10, -1], [16, -19], [7, -17], [-21, -16], [-19, -19], [-5, -18], [1, -1], [10, -7], [19, -18], [-2, -5], [5, 16], [-12, -5], [12, -6], [-11, 12], [12, -25], [-4, -12], [20, 7], [-23, -10], [12, 6], [-12, -12], [-17, 8], [1, 23], [-23, 5], [14, -20], [9, 14]], "area": 146, "family": "rlve_largest_rectangle", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-largest-rectangle-l4-s9/tests/run.py b/tasks/construct-rlve-largest-rectangle-l4-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-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-largest-rectangle-l4-s9/tests/test.sh b/tasks/construct-rlve-largest-rectangle-l4-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-largest-rectangle-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-max-achromatic-coloring-l2-s0/environment/Dockerfile b/tasks/construct-rlve-max-achromatic-coloring-l2-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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-max-achromatic-coloring-l2-s0/environment/instance.json b/tasks/construct-rlve-max-achromatic-coloring-l2-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..976e10a57d015ed40992128cc54e90d150d4c6c2
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l2-s0/environment/instance.json
@@ -0,0 +1 @@
+{"n": 8, "k": 4, "edges": [[0, 1], [0, 2], [0, 7], [1, 7], [3, 7], [4, 5], [5, 7], [6, 7]], "family": "rlve_max_achromatic_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l2-s0/solution/solve.sh b/tasks/construct-rlve-max-achromatic-coloring-l2-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..0c8ebe4ded05a35b6eba0b2cfeed7255ef89d8f3
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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'
+{"colors": [0, 1, 2, 0, 1, 2, 0, 3]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l2-s0/tests/check.py b/tasks/construct-rlve-max-achromatic-coloring-l2-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..bb676a518264e5db4fb0964e27ccffccf6980807
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l2-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-l2-s0/tests/instance.json b/tasks/construct-rlve-max-achromatic-coloring-l2-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..976e10a57d015ed40992128cc54e90d150d4c6c2
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l2-s0/tests/instance.json
@@ -0,0 +1 @@
+{"n": 8, "k": 4, "edges": [[0, 1], [0, 2], [0, 7], [1, 7], [3, 7], [4, 5], [5, 7], [6, 7]], "family": "rlve_max_achromatic_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l2-s0/tests/run.py b/tasks/construct-rlve-max-achromatic-coloring-l2-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l2-s0/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l2-s0/tests/test.sh b/tasks/construct-rlve-max-achromatic-coloring-l2-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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-max-achromatic-coloring-l2-s9/environment/Dockerfile b/tasks/construct-rlve-max-achromatic-coloring-l2-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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-max-achromatic-coloring-l2-s9/environment/instance.json b/tasks/construct-rlve-max-achromatic-coloring-l2-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..13101c41c46acb194c4a7743587f91939b152c87
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l2-s9/environment/instance.json
@@ -0,0 +1 @@
+{"n": 8, "k": 4, "edges": [[0, 1], [1, 2], [1, 5], [1, 6], [2, 3], [2, 7], [3, 6], [5, 6]], "family": "rlve_max_achromatic_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l2-s9/solution/solve.sh b/tasks/construct-rlve-max-achromatic-coloring-l2-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..55fd54b26730cb395b97c21b112fd8b225db486c
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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'
+{"colors": [0, 1, 0, 2, 0, 2, 3, 3]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l2-s9/tests/check.py b/tasks/construct-rlve-max-achromatic-coloring-l2-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..bb676a518264e5db4fb0964e27ccffccf6980807
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l2-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-l2-s9/tests/instance.json b/tasks/construct-rlve-max-achromatic-coloring-l2-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..13101c41c46acb194c4a7743587f91939b152c87
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l2-s9/tests/instance.json
@@ -0,0 +1 @@
+{"n": 8, "k": 4, "edges": [[0, 1], [1, 2], [1, 5], [1, 6], [2, 3], [2, 7], [3, 6], [5, 6]], "family": "rlve_max_achromatic_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l2-s9/tests/run.py b/tasks/construct-rlve-max-achromatic-coloring-l2-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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-max-achromatic-coloring-l2-s9/tests/test.sh b/tasks/construct-rlve-max-achromatic-coloring-l2-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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-max-achromatic-coloring-l6-s1/environment/Dockerfile b/tasks/construct-rlve-max-achromatic-coloring-l6-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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-max-achromatic-coloring-l6-s1/environment/instance.json b/tasks/construct-rlve-max-achromatic-coloring-l6-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..6b2cce3531e3be4f35b91f6e888e51ed066bf93f
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l6-s1/environment/instance.json
@@ -0,0 +1 @@
+{"n": 14, "k": 9, "edges": [[0, 2], [0, 3], [0, 4], [0, 5], [0, 6], [0, 8], [0, 9], [0, 10], [0, 11], [0, 13], [1, 6], [1, 7], [1, 8], [1, 9], [1, 10], [1, 12], [1, 13], [2, 3], [2, 4], [2, 7], [2, 8], [2, 9], [2, 10], [2, 11], [2, 12], [2, 13], [3, 4], [3, 8], [3, 9], [3, 10], [3, 11], [3, 13], [4, 10], [4, 11], [4, 13], [5, 9], [5, 10], [5, 11], [5, 12], [6, 7], [6, 8], [6, 9], [6, 10], [6, 11], [6, 12], [6, 13], [7, 10], [7, 13], [8, 9], [8, 10], [8, 11], [8, 12], [9, 10], [10, 11], [10, 12], [10, 13], [11, 12], [11, 13], [12, 13]], "family": "rlve_max_achromatic_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l6-s1/solution/solve.sh b/tasks/construct-rlve-max-achromatic-coloring-l6-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..11d3665ea2085dc8f23da62a9202571d8993bda4
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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'
+{"colors": [0, 0, 1, 2, 3, 1, 3, 4, 4, 5, 6, 7, 5, 8]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l6-s1/tests/check.py b/tasks/construct-rlve-max-achromatic-coloring-l6-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..bb676a518264e5db4fb0964e27ccffccf6980807
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l6-s1/tests/check.py
@@ -0,0 +1,16 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    c = int_list(ans.get("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-l6-s1/tests/instance.json b/tasks/construct-rlve-max-achromatic-coloring-l6-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..6b2cce3531e3be4f35b91f6e888e51ed066bf93f
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l6-s1/tests/instance.json
@@ -0,0 +1 @@
+{"n": 14, "k": 9, "edges": [[0, 2], [0, 3], [0, 4], [0, 5], [0, 6], [0, 8], [0, 9], [0, 10], [0, 11], [0, 13], [1, 6], [1, 7], [1, 8], [1, 9], [1, 10], [1, 12], [1, 13], [2, 3], [2, 4], [2, 7], [2, 8], [2, 9], [2, 10], [2, 11], [2, 12], [2, 13], [3, 4], [3, 8], [3, 9], [3, 10], [3, 11], [3, 13], [4, 10], [4, 11], [4, 13], [5, 9], [5, 10], [5, 11], [5, 12], [6, 7], [6, 8], [6, 9], [6, 10], [6, 11], [6, 12], [6, 13], [7, 10], [7, 13], [8, 9], [8, 10], [8, 11], [8, 12], [9, 10], [10, 11], [10, 12], [10, 13], [11, 12], [11, 13], [12, 13]], "family": "rlve_max_achromatic_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l6-s1/tests/run.py b/tasks/construct-rlve-max-achromatic-coloring-l6-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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-max-achromatic-coloring-l6-s1/tests/test.sh b/tasks/construct-rlve-max-achromatic-coloring-l6-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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-max-achromatic-coloring-l6-s6/environment/Dockerfile b/tasks/construct-rlve-max-achromatic-coloring-l6-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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-max-achromatic-coloring-l6-s6/environment/instance.json b/tasks/construct-rlve-max-achromatic-coloring-l6-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..459f87578301ff99b4ad3de1bd7791a7b3157f12
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l6-s6/environment/instance.json
@@ -0,0 +1 @@
+{"n": 14, "k": 9, "edges": [[0, 1], [0, 2], [0, 3], [0, 5], [0, 6], [0, 7], [0, 8], [0, 9], [0, 10], [1, 2], [1, 3], [1, 4], [1, 9], [1, 11], [2, 3], [2, 6], [2, 7], [2, 8], [2, 9], [2, 11], [2, 12], [2, 13], [3, 4], [3, 5], [3, 6], [3, 7], [3, 12], [3, 13], [4, 9], [4, 10], [5, 6], [5, 7], [5, 9], [5, 10], [5, 12], [6, 8], [6, 9], [6, 11], [6, 12], [7, 8], [7, 9], [7, 10], [7, 13], [8, 9], [8, 10], [8, 13], [9, 10], [9, 11], [10, 13], [11, 12]], "family": "rlve_max_achromatic_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l6-s6/solution/solve.sh b/tasks/construct-rlve-max-achromatic-coloring-l6-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..4d266c138c2451343bee643b9bd89c8c07cd6c01
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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'
+{"colors": [0, 1, 2, 3, 4, 1, 5, 6, 4, 7, 8, 3, 8, 5]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l6-s6/tests/check.py b/tasks/construct-rlve-max-achromatic-coloring-l6-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..bb676a518264e5db4fb0964e27ccffccf6980807
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l6-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-l6-s6/tests/instance.json b/tasks/construct-rlve-max-achromatic-coloring-l6-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..459f87578301ff99b4ad3de1bd7791a7b3157f12
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l6-s6/tests/instance.json
@@ -0,0 +1 @@
+{"n": 14, "k": 9, "edges": [[0, 1], [0, 2], [0, 3], [0, 5], [0, 6], [0, 7], [0, 8], [0, 9], [0, 10], [1, 2], [1, 3], [1, 4], [1, 9], [1, 11], [2, 3], [2, 6], [2, 7], [2, 8], [2, 9], [2, 11], [2, 12], [2, 13], [3, 4], [3, 5], [3, 6], [3, 7], [3, 12], [3, 13], [4, 9], [4, 10], [5, 6], [5, 7], [5, 9], [5, 10], [5, 12], [6, 8], [6, 9], [6, 11], [6, 12], [7, 8], [7, 9], [7, 10], [7, 13], [8, 9], [8, 10], [8, 13], [9, 10], [9, 11], [10, 13], [11, 12]], "family": "rlve_max_achromatic_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l6-s6/tests/run.py b/tasks/construct-rlve-max-achromatic-coloring-l6-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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-max-achromatic-coloring-l6-s6/tests/test.sh b/tasks/construct-rlve-max-achromatic-coloring-l6-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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-max-achromatic-coloring-l6-s8/environment/Dockerfile b/tasks/construct-rlve-max-achromatic-coloring-l6-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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-max-achromatic-coloring-l6-s8/environment/instance.json b/tasks/construct-rlve-max-achromatic-coloring-l6-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2c2645d9f072d693d848a64059619c0e4bb92882
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l6-s8/environment/instance.json
@@ -0,0 +1 @@
+{"n": 14, "k": 8, "edges": [[0, 2], [0, 6], [0, 7], [0, 8], [1, 2], [1, 3], [1, 4], [1, 6], [1, 7], [1, 8], [1, 9], [2, 7], [2, 11], [2, 13], [3, 5], [3, 6], [3, 7], [3, 8], [3, 10], [3, 12], [4, 5], [4, 9], [4, 10], [5, 7], [5, 10], [5, 12], [5, 13], [6, 7], [6, 8], [6, 9], [6, 12], [7, 9], [7, 10], [7, 12], [9, 12], [9, 13], [10, 12], [10, 13], [11, 13], [12, 13]], "family": "rlve_max_achromatic_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l6-s8/solution/solve.sh b/tasks/construct-rlve-max-achromatic-coloring-l6-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..6f7419b73dfea7c78533c5e6fd17235f694fe626
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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'
+{"colors": [0, 1, 2, 2, 3, 4, 5, 6, 4, 7, 7, 1, 3, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l6-s8/tests/check.py b/tasks/construct-rlve-max-achromatic-coloring-l6-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..bb676a518264e5db4fb0964e27ccffccf6980807
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l6-s8/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-l6-s8/tests/instance.json b/tasks/construct-rlve-max-achromatic-coloring-l6-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2c2645d9f072d693d848a64059619c0e4bb92882
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-l6-s8/tests/instance.json
@@ -0,0 +1 @@
+{"n": 14, "k": 8, "edges": [[0, 2], [0, 6], [0, 7], [0, 8], [1, 2], [1, 3], [1, 4], [1, 6], [1, 7], [1, 8], [1, 9], [2, 7], [2, 11], [2, 13], [3, 5], [3, 6], [3, 7], [3, 8], [3, 10], [3, 12], [4, 5], [4, 9], [4, 10], [5, 7], [5, 10], [5, 12], [5, 13], [6, 7], [6, 8], [6, 9], [6, 12], [7, 9], [7, 10], [7, 12], [9, 12], [9, 13], [10, 12], [10, 13], [11, 13], [12, 13]], "family": "rlve_max_achromatic_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-max-achromatic-coloring-l6-s8/tests/run.py b/tasks/construct-rlve-max-achromatic-coloring-l6-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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-max-achromatic-coloring-l6-s8/tests/test.sh b/tasks/construct-rlve-max-achromatic-coloring-l6-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-max-achromatic-coloring-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-maximum-clique-l4-s5/environment/Dockerfile b/tasks/construct-rlve-maximum-clique-l4-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-maximum-clique-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-maximum-clique-l4-s5/environment/instance.json b/tasks/construct-rlve-maximum-clique-l4-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..99d5bf86443cc1c3d3f25b38eb3f43879097e946
--- /dev/null
+++ b/tasks/construct-rlve-maximum-clique-l4-s5/environment/instance.json
@@ -0,0 +1 @@
+{"n": 32, "k": 8, "edges": [[0, 1], [0, 2], [0, 3], [0, 6], [0, 8], [0, 9], [0, 11], [0, 12], [0, 14], [0, 15], [0, 16], [0, 20], [0, 22], [0, 23], [0, 24], [0, 26], [0, 27], [0, 29], [0, 31], [1, 2], [1, 4], [1, 5], [1, 6], [1, 7], [1, 8], [1, 9], [1, 15], [1, 18], [1, 21], [1, 22], [1, 24], [1, 25], [1, 26], [1, 27], [1, 28], [1, 30], [2, 3], [2, 4], [2, 6], [2, 8], [2, 9], [2, 10], [2, 11], [2, 12], [2, 13], [2, 15], [2, 16], [2, 17], [2, 18], [2, 19], [2, 20], [2, 21], [2, 23], [2, 25], [2, 26], [2, 27], [2, 28], [2, 29], [2, 31], [3, 5], [3, 7], [3, 8], [3, 9], [3, 11], [3, 12], [3, 13], [3, 14], [3, 18], [3, 22], [3, 23], [3, 25], [3, 28], [3, 29], [3, 30], [4, 5], [4, 6], [4, 7], [4, 10], [4, 15], [4, 18], [4, 20], [4, 24], [4, 25], [4, 27], [5, 6], [5, 7], [5, 8], [5, 9], [5, 13], [5, 14], [5, 16], [5, 19], [5, 22], [5, 23], [5, 24], [5, 27], [5, 28], [5, 29], [5, 31], [6, 7], [6, 8], [6, 10], [6, 12], [6, 14], [6, 15], [6, 16], [6, 18], [6, 20], [6, 21], [6, 22], [6, 23], [6, 24], [6, 25], [6, 27], [6, 28], [6, 29], [6, 30], [6, 31], [7, 8], [7, 9], [7, 10], [7, 12], [7, 13], [7, 14], [7, 15], [7, 17], [7, 19], [7, 21], [7, 22], [7, 23], [7, 25], [7, 27], [7, 28], [7, 29], [7, 30], [7, 31], [8, 9], [8, 12], [8, 13], [8, 14], [8, 15], [8, 16], [8, 17], [8, 18], [8, 19], [8, 21], [8, 22], [8, 23], [8, 25], [8, 26], [8, 27], [8, 28], [8, 30], [9, 10], [9, 14], [9, 15], [9, 16], [9, 17], [9, 18], [9, 20], [9, 21], [9, 22], [9, 23], [9, 24], [9, 25], [9, 27], [9, 29], [9, 30], [9, 31], [10, 11], [10, 12], [10, 13], [10, 16], [10, 17], [10, 18], [10, 21], [10, 25], [10, 27], [10, 28], [11, 12], [11, 13], [11, 15], [11, 17], [11, 18], [11, 20], [11, 21], [11, 22], [11, 23], [11, 24], [11, 25], [11, 26], [11, 31], [12, 13], [12, 14], [12, 15], [12, 16], [12, 17], [12, 18], [12, 19], [12, 20], [12, 21], [12, 22], [12, 23], [12, 24], [12, 29], [12, 30], [12, 31], [13, 14], [13, 15], [13, 16], [13, 17], [13, 18], [13, 22], [13, 23], [13, 24], [13, 29], [13, 30], [14, 17], [14, 18], [14, 19], [14, 20], [14, 21], [14, 23], [14, 24], [14, 25], [14, 26], [14, 27], [14, 29], [14, 30], [14, 31], [15, 16], [15, 19], [15, 20], [15, 24], [15, 25], [15, 26], [15, 27], [15, 28], [15, 29], [15, 30], [15, 31], [16, 17], [16, 18], [16, 19], [16, 20], [16, 23], [16, 24], [16, 25], [16, 27], [16, 28], [17, 18], [17, 20], [17, 22], [17, 25], [17, 26], [17, 27], [17, 29], [17, 31], [18, 19], [18, 20], [18, 21], [18, 23], [18, 25], [18, 26], [18, 27], [18, 28], [18, 29], [18, 30], [18, 31], [19, 20], [19, 21], [19, 23], [19, 24], [19, 25], [19, 27], [19, 28], [19, 31], [20, 22], [20, 23], [20, 24], [20, 26], [20, 28], [20, 29], [21, 22], [21, 23], [21, 25], [21, 28], [21, 29], [21, 30], [21, 31], [22, 23], [22, 25], [22, 26], [22, 27], [22, 29], [22, 30], [22, 31], [23, 24], [23, 25], [23, 26], [23, 27], [23, 29], [23, 30], [23, 31], [24, 25], [24, 26], [24, 28], [24, 29], [24, 31], [25, 26], [25, 27], [25, 30], [25, 31], [26, 28], [26, 29], [26, 31], [27, 28], [27, 29], [27, 31], [28, 29], [28, 30], [28, 31]], "family": "rlve_maximum_clique", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-maximum-clique-l4-s5/solution/solve.sh b/tasks/construct-rlve-maximum-clique-l4-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..5e335c8685cef095b12a1f59a4e7b01b7a53b558
--- /dev/null
+++ b/tasks/construct-rlve-maximum-clique-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'
+{"clique": [0, 6, 12, 14, 20, 23, 24, 29]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-maximum-clique-l4-s5/tests/check.py b/tasks/construct-rlve-maximum-clique-l4-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..95e00c78f7c986ba3aecbdc8af3cf8ed221abd05
--- /dev/null
+++ b/tasks/construct-rlve-maximum-clique-l4-s5/tests/check.py
@@ -0,0 +1,16 @@
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n, k = inst["n"], inst["k"]
+    c = int_list(ans.get("clique"), None, 0, n - 1, "clique")
+    if len(set(c)) != len(c):
+        raise Invalid("vertices must be distinct")
+    if len(c) < k:
+        raise Invalid(f"clique has {len(c)} vertices, need {k}")
+    es = {(min(u, v), max(u, v)) for u, v in inst["edges"]}
+    for i in range(len(c)):
+        for j in range(i + 1, len(c)):
+            if (min(c[i], c[j]), max(c[i], c[j])) not in es:
+                raise Invalid(f"{c[i]} and {c[j]} are not adjacent")
+    return None, {}
diff --git a/tasks/construct-rlve-maximum-clique-l4-s5/tests/instance.json b/tasks/construct-rlve-maximum-clique-l4-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..99d5bf86443cc1c3d3f25b38eb3f43879097e946
--- /dev/null
+++ b/tasks/construct-rlve-maximum-clique-l4-s5/tests/instance.json
@@ -0,0 +1 @@
+{"n": 32, "k": 8, "edges": [[0, 1], [0, 2], [0, 3], [0, 6], [0, 8], [0, 9], [0, 11], [0, 12], [0, 14], [0, 15], [0, 16], [0, 20], [0, 22], [0, 23], [0, 24], [0, 26], [0, 27], [0, 29], [0, 31], [1, 2], [1, 4], [1, 5], [1, 6], [1, 7], [1, 8], [1, 9], [1, 15], [1, 18], [1, 21], [1, 22], [1, 24], [1, 25], [1, 26], [1, 27], [1, 28], [1, 30], [2, 3], [2, 4], [2, 6], [2, 8], [2, 9], [2, 10], [2, 11], [2, 12], [2, 13], [2, 15], [2, 16], [2, 17], [2, 18], [2, 19], [2, 20], [2, 21], [2, 23], [2, 25], [2, 26], [2, 27], [2, 28], [2, 29], [2, 31], [3, 5], [3, 7], [3, 8], [3, 9], [3, 11], [3, 12], [3, 13], [3, 14], [3, 18], [3, 22], [3, 23], [3, 25], [3, 28], [3, 29], [3, 30], [4, 5], [4, 6], [4, 7], [4, 10], [4, 15], [4, 18], [4, 20], [4, 24], [4, 25], [4, 27], [5, 6], [5, 7], [5, 8], [5, 9], [5, 13], [5, 14], [5, 16], [5, 19], [5, 22], [5, 23], [5, 24], [5, 27], [5, 28], [5, 29], [5, 31], [6, 7], [6, 8], [6, 10], [6, 12], [6, 14], [6, 15], [6, 16], [6, 18], [6, 20], [6, 21], [6, 22], [6, 23], [6, 24], [6, 25], [6, 27], [6, 28], [6, 29], [6, 30], [6, 31], [7, 8], [7, 9], [7, 10], [7, 12], [7, 13], [7, 14], [7, 15], [7, 17], [7, 19], [7, 21], [7, 22], [7, 23], [7, 25], [7, 27], [7, 28], [7, 29], [7, 30], [7, 31], [8, 9], [8, 12], [8, 13], [8, 14], [8, 15], [8, 16], [8, 17], [8, 18], [8, 19], [8, 21], [8, 22], [8, 23], [8, 25], [8, 26], [8, 27], [8, 28], [8, 30], [9, 10], [9, 14], [9, 15], [9, 16], [9, 17], [9, 18], [9, 20], [9, 21], [9, 22], [9, 23], [9, 24], [9, 25], [9, 27], [9, 29], [9, 30], [9, 31], [10, 11], [10, 12], [10, 13], [10, 16], [10, 17], [10, 18], [10, 21], [10, 25], [10, 27], [10, 28], [11, 12], [11, 13], [11, 15], [11, 17], [11, 18], [11, 20], [11, 21], [11, 22], [11, 23], [11, 24], [11, 25], [11, 26], [11, 31], [12, 13], [12, 14], [12, 15], [12, 16], [12, 17], [12, 18], [12, 19], [12, 20], [12, 21], [12, 22], [12, 23], [12, 24], [12, 29], [12, 30], [12, 31], [13, 14], [13, 15], [13, 16], [13, 17], [13, 18], [13, 22], [13, 23], [13, 24], [13, 29], [13, 30], [14, 17], [14, 18], [14, 19], [14, 20], [14, 21], [14, 23], [14, 24], [14, 25], [14, 26], [14, 27], [14, 29], [14, 30], [14, 31], [15, 16], [15, 19], [15, 20], [15, 24], [15, 25], [15, 26], [15, 27], [15, 28], [15, 29], [15, 30], [15, 31], [16, 17], [16, 18], [16, 19], [16, 20], [16, 23], [16, 24], [16, 25], [16, 27], [16, 28], [17, 18], [17, 20], [17, 22], [17, 25], [17, 26], [17, 27], [17, 29], [17, 31], [18, 19], [18, 20], [18, 21], [18, 23], [18, 25], [18, 26], [18, 27], [18, 28], [18, 29], [18, 30], [18, 31], [19, 20], [19, 21], [19, 23], [19, 24], [19, 25], [19, 27], [19, 28], [19, 31], [20, 22], [20, 23], [20, 24], [20, 26], [20, 28], [20, 29], [21, 22], [21, 23], [21, 25], [21, 28], [21, 29], [21, 30], [21, 31], [22, 23], [22, 25], [22, 26], [22, 27], [22, 29], [22, 30], [22, 31], [23, 24], [23, 25], [23, 26], [23, 27], [23, 29], [23, 30], [23, 31], [24, 25], [24, 26], [24, 28], [24, 29], [24, 31], [25, 26], [25, 27], [25, 30], [25, 31], [26, 28], [26, 29], [26, 31], [27, 28], [27, 29], [27, 31], [28, 29], [28, 30], [28, 31]], "family": "rlve_maximum_clique", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-maximum-clique-l4-s5/tests/run.py b/tasks/construct-rlve-maximum-clique-l4-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-maximum-clique-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-maximum-clique-l4-s5/tests/test.sh b/tasks/construct-rlve-maximum-clique-l4-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-maximum-clique-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-min-chromatic-number-l3-s0/environment/Dockerfile b/tasks/construct-rlve-min-chromatic-number-l3-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l3-s0/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-min-chromatic-number-l3-s0/environment/instance.json b/tasks/construct-rlve-min-chromatic-number-l3-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..87538b9f5d663cf8aa7b61d7fae69bdd1720a0a8
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l3-s0/environment/instance.json
@@ -0,0 +1 @@
+{"n": 14, "k": 6, "edges": [[0, 1], [0, 4], [0, 5], [0, 6], [0, 7], [0, 8], [0, 9], [0, 10], [0, 11], [0, 12], [0, 13], [1, 4], [1, 5], [1, 6], [1, 7], [1, 8], [1, 10], [1, 12], [2, 3], [2, 5], [2, 6], [2, 7], [2, 8], [2, 10], [2, 13], [3, 4], [3, 5], [3, 6], [3, 7], [3, 8], [3, 9], [3, 11], [3, 12], [3, 13], [4, 5], [4, 6], [4, 7], [4, 8], [4, 9], [4, 10], [4, 11], [4, 12], [5, 6], [5, 7], [5, 9], [5, 10], [5, 12], [5, 13], [6, 7], [6, 8], [6, 9], [6, 13], [7, 8], [7, 9], [7, 11], [7, 13], [8, 10], [8, 12], [8, 13], [9, 10], [9, 11], [9, 12], [9, 13], [10, 11], [10, 13], [11, 12], [11, 13], [12, 13]], "family": "rlve_min_chromatic_number", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-chromatic-number-l3-s0/solution/solve.sh b/tasks/construct-rlve-min-chromatic-number-l3-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..b7b7672cdd537e3f1ac722371dcc99093a7a7429
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l3-s0/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"colors": [0, 1, 1, 0, 2, 3, 4, 5, 3, 1, 4, 3, 4, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-min-chromatic-number-l3-s0/tests/check.py b/tasks/construct-rlve-min-chromatic-number-l3-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..d2b77793fb15e0d97174b3c28e69a86e2a26996f
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l3-s0/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-l3-s0/tests/instance.json b/tasks/construct-rlve-min-chromatic-number-l3-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..87538b9f5d663cf8aa7b61d7fae69bdd1720a0a8
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l3-s0/tests/instance.json
@@ -0,0 +1 @@
+{"n": 14, "k": 6, "edges": [[0, 1], [0, 4], [0, 5], [0, 6], [0, 7], [0, 8], [0, 9], [0, 10], [0, 11], [0, 12], [0, 13], [1, 4], [1, 5], [1, 6], [1, 7], [1, 8], [1, 10], [1, 12], [2, 3], [2, 5], [2, 6], [2, 7], [2, 8], [2, 10], [2, 13], [3, 4], [3, 5], [3, 6], [3, 7], [3, 8], [3, 9], [3, 11], [3, 12], [3, 13], [4, 5], [4, 6], [4, 7], [4, 8], [4, 9], [4, 10], [4, 11], [4, 12], [5, 6], [5, 7], [5, 9], [5, 10], [5, 12], [5, 13], [6, 7], [6, 8], [6, 9], [6, 13], [7, 8], [7, 9], [7, 11], [7, 13], [8, 10], [8, 12], [8, 13], [9, 10], [9, 11], [9, 12], [9, 13], [10, 11], [10, 13], [11, 12], [11, 13], [12, 13]], "family": "rlve_min_chromatic_number", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-chromatic-number-l3-s0/tests/run.py b/tasks/construct-rlve-min-chromatic-number-l3-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l3-s0/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-min-chromatic-number-l3-s0/tests/test.sh b/tasks/construct-rlve-min-chromatic-number-l3-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l3-s0/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-min-chromatic-number-l3-s7/environment/Dockerfile b/tasks/construct-rlve-min-chromatic-number-l3-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l3-s7/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-min-chromatic-number-l3-s7/environment/instance.json b/tasks/construct-rlve-min-chromatic-number-l3-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..959479caae7c563a2e1438ae116e6e50fb8ac7a2
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l3-s7/environment/instance.json
@@ -0,0 +1 @@
+{"n": 14, "k": 7, "edges": [[0, 1], [0, 2], [0, 4], [0, 5], [0, 8], [0, 10], [0, 11], [0, 12], [0, 13], [1, 2], [1, 3], [1, 5], [1, 6], [1, 7], [1, 8], [1, 9], [1, 10], [1, 13], [2, 3], [2, 4], [2, 5], [2, 6], [2, 8], [2, 9], [2, 11], [2, 12], [2, 13], [3, 4], [3, 5], [3, 6], [3, 7], [3, 8], [3, 10], [3, 12], [3, 13], [4, 5], [4, 6], [4, 8], [4, 10], [4, 11], [4, 12], [4, 13], [5, 6], [5, 8], [5, 9], [5, 13], [6, 7], [6, 8], [6, 10], [6, 11], [6, 12], [6, 13], [7, 9], [7, 10], [7, 11], [7, 12], [7, 13], [8, 9], [8, 10], [8, 11], [8, 13], [9, 10], [9, 12], [9, 13], [10, 11], [10, 12], [10, 13], [11, 12]], "family": "rlve_min_chromatic_number", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-chromatic-number-l3-s7/solution/solve.sh b/tasks/construct-rlve-min-chromatic-number-l3-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..67aec7c7a649676dc5ad818004990b5c6ced2f44
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l3-s7/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"colors": [0, 1, 2, 0, 1, 3, 4, 2, 5, 0, 3, 6, 5, 6]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-min-chromatic-number-l3-s7/tests/check.py b/tasks/construct-rlve-min-chromatic-number-l3-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..d2b77793fb15e0d97174b3c28e69a86e2a26996f
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l3-s7/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-l3-s7/tests/instance.json b/tasks/construct-rlve-min-chromatic-number-l3-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..959479caae7c563a2e1438ae116e6e50fb8ac7a2
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l3-s7/tests/instance.json
@@ -0,0 +1 @@
+{"n": 14, "k": 7, "edges": [[0, 1], [0, 2], [0, 4], [0, 5], [0, 8], [0, 10], [0, 11], [0, 12], [0, 13], [1, 2], [1, 3], [1, 5], [1, 6], [1, 7], [1, 8], [1, 9], [1, 10], [1, 13], [2, 3], [2, 4], [2, 5], [2, 6], [2, 8], [2, 9], [2, 11], [2, 12], [2, 13], [3, 4], [3, 5], [3, 6], [3, 7], [3, 8], [3, 10], [3, 12], [3, 13], [4, 5], [4, 6], [4, 8], [4, 10], [4, 11], [4, 12], [4, 13], [5, 6], [5, 8], [5, 9], [5, 13], [6, 7], [6, 8], [6, 10], [6, 11], [6, 12], [6, 13], [7, 9], [7, 10], [7, 11], [7, 12], [7, 13], [8, 9], [8, 10], [8, 11], [8, 13], [9, 10], [9, 12], [9, 13], [10, 11], [10, 12], [10, 13], [11, 12]], "family": "rlve_min_chromatic_number", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-chromatic-number-l3-s7/tests/run.py b/tasks/construct-rlve-min-chromatic-number-l3-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l3-s7/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-min-chromatic-number-l3-s7/tests/test.sh b/tasks/construct-rlve-min-chromatic-number-l3-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l3-s7/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-min-chromatic-number-l3-s9/environment/Dockerfile b/tasks/construct-rlve-min-chromatic-number-l3-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l3-s9/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-min-chromatic-number-l3-s9/environment/instance.json b/tasks/construct-rlve-min-chromatic-number-l3-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..7df4515e828921be53b018e7b6e316a4a9f4c760
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l3-s9/environment/instance.json
@@ -0,0 +1 @@
+{"n": 14, "k": 4, "edges": [[0, 1], [0, 4], [0, 6], [0, 7], [1, 3], [1, 7], [1, 9], [1, 12], [1, 13], [2, 3], [2, 5], [2, 7], [2, 8], [2, 9], [2, 11], [3, 4], [3, 7], [3, 10], [4, 5], [4, 6], [4, 9], [5, 6], [5, 7], [5, 10], [5, 11], [6, 7], [6, 12], [7, 8], [7, 10], [7, 11], [7, 13], [8, 10], [8, 11], [8, 13], [9, 11], [9, 13], [10, 11], [10, 12], [11, 12], [11, 13]], "family": "rlve_min_chromatic_number", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-chromatic-number-l3-s9/solution/solve.sh b/tasks/construct-rlve-min-chromatic-number-l3-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..02cd853d5102c7197ac6510d4206d0600af50140
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l3-s9/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"colors": [0, 1, 0, 2, 3, 2, 1, 3, 2, 2, 0, 1, 2, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-min-chromatic-number-l3-s9/tests/check.py b/tasks/construct-rlve-min-chromatic-number-l3-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..d2b77793fb15e0d97174b3c28e69a86e2a26996f
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l3-s9/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-l3-s9/tests/instance.json b/tasks/construct-rlve-min-chromatic-number-l3-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..7df4515e828921be53b018e7b6e316a4a9f4c760
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l3-s9/tests/instance.json
@@ -0,0 +1 @@
+{"n": 14, "k": 4, "edges": [[0, 1], [0, 4], [0, 6], [0, 7], [1, 3], [1, 7], [1, 9], [1, 12], [1, 13], [2, 3], [2, 5], [2, 7], [2, 8], [2, 9], [2, 11], [3, 4], [3, 7], [3, 10], [4, 5], [4, 6], [4, 9], [5, 6], [5, 7], [5, 10], [5, 11], [6, 7], [6, 12], [7, 8], [7, 10], [7, 11], [7, 13], [8, 10], [8, 11], [8, 13], [9, 11], [9, 13], [10, 11], [10, 12], [11, 12], [11, 13]], "family": "rlve_min_chromatic_number", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-chromatic-number-l3-s9/tests/run.py b/tasks/construct-rlve-min-chromatic-number-l3-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l3-s9/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-min-chromatic-number-l3-s9/tests/test.sh b/tasks/construct-rlve-min-chromatic-number-l3-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-min-chromatic-number-l3-s9/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l3-s3/environment/Dockerfile b/tasks/construct-rlve-min-harmonious-coloring-l3-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-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-min-harmonious-coloring-l3-s3/environment/instance.json b/tasks/construct-rlve-min-harmonious-coloring-l3-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..da2941277952c7eecb535ba1bce60ee7a4ad8740
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l3-s3/environment/instance.json
@@ -0,0 +1 @@
+{"n": 10, "k": 4, "edges": [[0, 3], [0, 8], [1, 8], [7, 9]], "family": "rlve_min_harmonious_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l3-s3/solution/solve.sh b/tasks/construct-rlve-min-harmonious-coloring-l3-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..73d016e429284bbce8a41c1921a93a453d1e90cc
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-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'
+{"colors": [0, 1, 0, 1, 0, 0, 0, 0, 2, 3]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l3-s3/tests/check.py b/tasks/construct-rlve-min-harmonious-coloring-l3-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..91282a31c765b6103e6575180adb47abddb6a506
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l3-s3/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-l3-s3/tests/instance.json b/tasks/construct-rlve-min-harmonious-coloring-l3-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..da2941277952c7eecb535ba1bce60ee7a4ad8740
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l3-s3/tests/instance.json
@@ -0,0 +1 @@
+{"n": 10, "k": 4, "edges": [[0, 3], [0, 8], [1, 8], [7, 9]], "family": "rlve_min_harmonious_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l3-s3/tests/run.py b/tasks/construct-rlve-min-harmonious-coloring-l3-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-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-min-harmonious-coloring-l3-s3/tests/test.sh b/tasks/construct-rlve-min-harmonious-coloring-l3-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-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-min-harmonious-coloring-l3-s4/environment/Dockerfile b/tasks/construct-rlve-min-harmonious-coloring-l3-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-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-min-harmonious-coloring-l3-s4/environment/instance.json b/tasks/construct-rlve-min-harmonious-coloring-l3-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a837b366ace0d7f669824385d69b6591282de7fb
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l3-s4/environment/instance.json
@@ -0,0 +1 @@
+{"n": 10, "k": 6, "edges": [[0, 9], [1, 5], [1, 7], [1, 9], [2, 7], [3, 5], [3, 6], [3, 7], [4, 5], [4, 8], [7, 9]], "family": "rlve_min_harmonious_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l3-s4/solution/solve.sh b/tasks/construct-rlve-min-harmonious-coloring-l3-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..2378a4bc51560c10e544c64ea25c954370a98e0a
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-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'
+{"colors": [0, 1, 0, 2, 0, 3, 0, 4, 1, 5]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l3-s4/tests/check.py b/tasks/construct-rlve-min-harmonious-coloring-l3-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..91282a31c765b6103e6575180adb47abddb6a506
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l3-s4/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-l3-s4/tests/instance.json b/tasks/construct-rlve-min-harmonious-coloring-l3-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a837b366ace0d7f669824385d69b6591282de7fb
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-l3-s4/tests/instance.json
@@ -0,0 +1 @@
+{"n": 10, "k": 6, "edges": [[0, 9], [1, 5], [1, 7], [1, 9], [2, 7], [3, 5], [3, 6], [3, 7], [4, 5], [4, 8], [7, 9]], "family": "rlve_min_harmonious_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-min-harmonious-coloring-l3-s4/tests/run.py b/tasks/construct-rlve-min-harmonious-coloring-l3-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-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-min-harmonious-coloring-l3-s4/tests/test.sh b/tasks/construct-rlve-min-harmonious-coloring-l3-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-min-harmonious-coloring-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-pairwise-sums-recovery-l1-s3/environment/Dockerfile b/tasks/construct-rlve-pairwise-sums-recovery-l1-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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-pairwise-sums-recovery-l1-s3/environment/instance.json b/tasks/construct-rlve-pairwise-sums-recovery-l1-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c0f1934bdf8575bf933a32071abc262a547e70c4
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l1-s3/environment/instance.json
@@ -0,0 +1 @@
+{"n": 3, "sums": [9, 10, 11], "family": "rlve_pairwise_sums_recovery", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l1-s3/solution/solve.sh b/tasks/construct-rlve-pairwise-sums-recovery-l1-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..180965935c946f6416bc02f45f0b7a9d44022dec
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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'
+{"numbers": [4, 5, 6]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l1-s3/tests/check.py b/tasks/construct-rlve-pairwise-sums-recovery-l1-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..01b5b475e927688c0209fb93478a95f6dd1d3407
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l1-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-l1-s3/tests/instance.json b/tasks/construct-rlve-pairwise-sums-recovery-l1-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c0f1934bdf8575bf933a32071abc262a547e70c4
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l1-s3/tests/instance.json
@@ -0,0 +1 @@
+{"n": 3, "sums": [9, 10, 11], "family": "rlve_pairwise_sums_recovery", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l1-s3/tests/run.py b/tasks/construct-rlve-pairwise-sums-recovery-l1-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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-pairwise-sums-recovery-l1-s3/tests/test.sh b/tasks/construct-rlve-pairwise-sums-recovery-l1-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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-pairwise-sums-recovery-l1-s4/environment/Dockerfile b/tasks/construct-rlve-pairwise-sums-recovery-l1-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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-pairwise-sums-recovery-l1-s4/environment/instance.json b/tasks/construct-rlve-pairwise-sums-recovery-l1-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4927399c9842f55ec5371d2c2a47afb791206c9a
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l1-s4/environment/instance.json
@@ -0,0 +1 @@
+{"n": 3, "sums": [9, 5, 8], "family": "rlve_pairwise_sums_recovery", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l1-s4/solution/solve.sh b/tasks/construct-rlve-pairwise-sums-recovery-l1-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..507a5fc8d263b75c630d4cbe0920cf724b4b6d37
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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'
+{"numbers": [2, 3, 6]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l1-s4/tests/check.py b/tasks/construct-rlve-pairwise-sums-recovery-l1-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..01b5b475e927688c0209fb93478a95f6dd1d3407
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l1-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-l1-s4/tests/instance.json b/tasks/construct-rlve-pairwise-sums-recovery-l1-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4927399c9842f55ec5371d2c2a47afb791206c9a
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l1-s4/tests/instance.json
@@ -0,0 +1 @@
+{"n": 3, "sums": [9, 5, 8], "family": "rlve_pairwise_sums_recovery", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l1-s4/tests/run.py b/tasks/construct-rlve-pairwise-sums-recovery-l1-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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-pairwise-sums-recovery-l1-s4/tests/test.sh b/tasks/construct-rlve-pairwise-sums-recovery-l1-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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-pairwise-sums-recovery-l5-s2/environment/Dockerfile b/tasks/construct-rlve-pairwise-sums-recovery-l5-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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-pairwise-sums-recovery-l5-s2/environment/instance.json b/tasks/construct-rlve-pairwise-sums-recovery-l5-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3f6bc8bfd0e361d5c5ebc5cd01f8136b3126bd19
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l5-s2/environment/instance.json
@@ -0,0 +1 @@
+{"n": 18, "sums": [35, 32, 59, 28, 37, 46, 30, 57, 36, 23, 23, 34, 11, 25, 29, 46, 68, 60, 64, 30, 34, 65, 28, 38, 20, 13, 14, 7, 18, 11, 34, 42, 33, 50, 9, 14, 43, 16, 33, 31, 42, 3, 62, 49, 43, 40, 32, 8, 51, 56, 7, 24, 40, 43, 29, 38, 37, 9, 10, 18, 27, 53, 29, 45, 8, 50, 53, 30, 37, 55, 52, 35, 22, 47, 39, 14, 54, 30, 58, 34, 22, 63, 17, 43, 48, 39, 44, 51, 38, 55, 13, 34, 57, 15, 31, 12, 38, 45, 46, 54, 11, 29, 19, 35, 33, 49, 30, 58, 12, 42, 37, 57, 15, 31, 36, 31, 21, 54, 19, 31, 17, 16, 37, 34, 30, 41, 26, 32, 23, 52, 40, 59, 36, 10, 63, 27, 61, 39, 15, 35, 41, 26, 36, 15, 41, 26, 35, 6, 32, 35, 27, 31, 38], "family": "rlve_pairwise_sums_recovery", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l5-s2/solution/solve.sh b/tasks/construct-rlve-pairwise-sums-recovery-l5-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..f015380e299555e7c48d0a2470f0bd3ed41f2259
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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'
+{"numbers": [1, 2, 5, 6, 7, 8, 9, 10, 13, 21, 22, 24, 25, 28, 29, 30, 33, 35]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l5-s2/tests/check.py b/tasks/construct-rlve-pairwise-sums-recovery-l5-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..01b5b475e927688c0209fb93478a95f6dd1d3407
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l5-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-l5-s2/tests/instance.json b/tasks/construct-rlve-pairwise-sums-recovery-l5-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3f6bc8bfd0e361d5c5ebc5cd01f8136b3126bd19
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l5-s2/tests/instance.json
@@ -0,0 +1 @@
+{"n": 18, "sums": [35, 32, 59, 28, 37, 46, 30, 57, 36, 23, 23, 34, 11, 25, 29, 46, 68, 60, 64, 30, 34, 65, 28, 38, 20, 13, 14, 7, 18, 11, 34, 42, 33, 50, 9, 14, 43, 16, 33, 31, 42, 3, 62, 49, 43, 40, 32, 8, 51, 56, 7, 24, 40, 43, 29, 38, 37, 9, 10, 18, 27, 53, 29, 45, 8, 50, 53, 30, 37, 55, 52, 35, 22, 47, 39, 14, 54, 30, 58, 34, 22, 63, 17, 43, 48, 39, 44, 51, 38, 55, 13, 34, 57, 15, 31, 12, 38, 45, 46, 54, 11, 29, 19, 35, 33, 49, 30, 58, 12, 42, 37, 57, 15, 31, 36, 31, 21, 54, 19, 31, 17, 16, 37, 34, 30, 41, 26, 32, 23, 52, 40, 59, 36, 10, 63, 27, 61, 39, 15, 35, 41, 26, 36, 15, 41, 26, 35, 6, 32, 35, 27, 31, 38], "family": "rlve_pairwise_sums_recovery", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l5-s2/tests/run.py b/tasks/construct-rlve-pairwise-sums-recovery-l5-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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-pairwise-sums-recovery-l5-s2/tests/test.sh b/tasks/construct-rlve-pairwise-sums-recovery-l5-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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-pairwise-sums-recovery-l5-s5/environment/Dockerfile b/tasks/construct-rlve-pairwise-sums-recovery-l5-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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-pairwise-sums-recovery-l5-s5/solution/solve.sh b/tasks/construct-rlve-pairwise-sums-recovery-l5-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..a962f26852f79b68f02cfab9ebfb5c43616132ef
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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'
+{"numbers": [1, 2, 3, 6, 11, 12, 13, 14, 15, 18, 20, 21, 25, 29, 31, 32, 35, 36]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l5-s5/tests/check.py b/tasks/construct-rlve-pairwise-sums-recovery-l5-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..01b5b475e927688c0209fb93478a95f6dd1d3407
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l5-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-l5-s5/tests/instance.json b/tasks/construct-rlve-pairwise-sums-recovery-l5-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f190cb9cf01aeaa6e8cda2be28287e100e940d86
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-l5-s5/tests/instance.json
@@ -0,0 +1 @@
+{"n": 18, "sums": [43, 37, 50, 33, 23, 33, 50, 51, 32, 13, 16, 41, 54, 39, 32, 30, 35, 18, 19, 14, 27, 47, 20, 44, 7, 21, 42, 47, 45, 24, 63, 49, 61, 17, 36, 61, 22, 31, 36, 38, 42, 34, 53, 35, 32, 57, 47, 15, 12, 46, 41, 53, 41, 17, 21, 22, 31, 67, 56, 48, 37, 46, 26, 38, 49, 48, 51, 35, 32, 4, 38, 24, 29, 49, 26, 28, 43, 46, 55, 9, 49, 24, 17, 60, 67, 37, 33, 64, 66, 34, 44, 36, 43, 71, 57, 45, 13, 68, 43, 35, 46, 50, 15, 52, 34, 60, 52, 29, 37, 20, 31, 42, 34, 50, 25, 27, 19, 39, 45, 25, 33, 40, 26, 39, 28, 5, 38, 38, 18, 32, 44, 15, 33, 16, 40, 16, 56, 47, 23, 23, 31, 14, 26, 3, 56, 8, 54, 21, 14, 65, 27, 27, 30], "family": "rlve_pairwise_sums_recovery", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pairwise-sums-recovery-l5-s5/tests/run.py b/tasks/construct-rlve-pairwise-sums-recovery-l5-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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-pairwise-sums-recovery-l5-s5/tests/test.sh b/tasks/construct-rlve-pairwise-sums-recovery-l5-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-pairwise-sums-recovery-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-pan-solar-panels-l4-s0/environment/Dockerfile b/tasks/construct-rlve-pan-solar-panels-l4-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-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-pan-solar-panels-l4-s0/environment/instance.json b/tasks/construct-rlve-pan-solar-panels-l4-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b8b4138b25fbee2090261f6cfc4afb87ce3dba11
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l4-s0/environment/instance.json
@@ -0,0 +1 @@
+{"A": 2131042, "B": 5755140, "C": 6961510, "D": 7520703, "target": 3760351, "family": "rlve_pan_solar_panels", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pan-solar-panels-l4-s0/solution/solve.sh b/tasks/construct-rlve-pan-solar-panels-l4-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..39c058082ec7ce66857ef5df5d9c1f0e0e201d20
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-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'
+{"X": 3760351, "Y": 7520702}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-pan-solar-panels-l4-s0/tests/check.py b/tasks/construct-rlve-pan-solar-panels-l4-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..8bb2e7cc0b0280ab877755cc5d97b510680f7aa3
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l4-s0/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-l4-s0/tests/instance.json b/tasks/construct-rlve-pan-solar-panels-l4-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..b8b4138b25fbee2090261f6cfc4afb87ce3dba11
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l4-s0/tests/instance.json
@@ -0,0 +1 @@
+{"A": 2131042, "B": 5755140, "C": 6961510, "D": 7520703, "target": 3760351, "family": "rlve_pan_solar_panels", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pan-solar-panels-l4-s0/tests/run.py b/tasks/construct-rlve-pan-solar-panels-l4-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-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-pan-solar-panels-l4-s0/tests/test.sh b/tasks/construct-rlve-pan-solar-panels-l4-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-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-pan-solar-panels-l4-s7/environment/Dockerfile b/tasks/construct-rlve-pan-solar-panels-l4-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-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-pan-solar-panels-l4-s7/environment/instance.json b/tasks/construct-rlve-pan-solar-panels-l4-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..70105c5a6ed1a184ef7c883e0547c01af1af8998
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l4-s7/environment/instance.json
@@ -0,0 +1 @@
+{"A": 8308601, "B": 9820980, "C": 5572394, "D": 5585589, "target": 2792794, "family": "rlve_pan_solar_panels", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pan-solar-panels-l4-s7/solution/solve.sh b/tasks/construct-rlve-pan-solar-panels-l4-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..caf322b268587f32a34f203b50171a9ebad8a25e
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l4-s7/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"X": 8378382, "Y": 5585588}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-pan-solar-panels-l4-s7/tests/check.py b/tasks/construct-rlve-pan-solar-panels-l4-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..8bb2e7cc0b0280ab877755cc5d97b510680f7aa3
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l4-s7/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-l4-s7/tests/instance.json b/tasks/construct-rlve-pan-solar-panels-l4-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..70105c5a6ed1a184ef7c883e0547c01af1af8998
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l4-s7/tests/instance.json
@@ -0,0 +1 @@
+{"A": 8308601, "B": 9820980, "C": 5572394, "D": 5585589, "target": 2792794, "family": "rlve_pan_solar_panels", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pan-solar-panels-l4-s7/tests/run.py b/tasks/construct-rlve-pan-solar-panels-l4-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-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-pan-solar-panels-l4-s7/tests/test.sh b/tasks/construct-rlve-pan-solar-panels-l4-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-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-pan-solar-panels-l4-s9/environment/Dockerfile b/tasks/construct-rlve-pan-solar-panels-l4-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-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-pan-solar-panels-l4-s9/environment/instance.json b/tasks/construct-rlve-pan-solar-panels-l4-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8a1adbb912a5cf8b2399919658b824c0ae36c067
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l4-s9/environment/instance.json
@@ -0,0 +1 @@
+{"A": 6973908, "B": 8464741, "C": 389181, "D": 4252966, "target": 4232370, "family": "rlve_pan_solar_panels", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pan-solar-panels-l4-s9/solution/solve.sh b/tasks/construct-rlve-pan-solar-panels-l4-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..7ad596a649c0729edd634f72af01193d4838713d
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l4-s9/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"X": 8464740, "Y": 4232370}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-pan-solar-panels-l4-s9/tests/check.py b/tasks/construct-rlve-pan-solar-panels-l4-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..8bb2e7cc0b0280ab877755cc5d97b510680f7aa3
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l4-s9/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-l4-s9/tests/instance.json b/tasks/construct-rlve-pan-solar-panels-l4-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..8a1adbb912a5cf8b2399919658b824c0ae36c067
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-l4-s9/tests/instance.json
@@ -0,0 +1 @@
+{"A": 6973908, "B": 8464741, "C": 389181, "D": 4252966, "target": 4232370, "family": "rlve_pan_solar_panels", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-pan-solar-panels-l4-s9/tests/run.py b/tasks/construct-rlve-pan-solar-panels-l4-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-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-pan-solar-panels-l4-s9/tests/test.sh b/tasks/construct-rlve-pan-solar-panels-l4-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-pan-solar-panels-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-polynomial-global-minimum-l1-s2/environment/Dockerfile b/tasks/construct-rlve-polynomial-global-minimum-l1-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l1-s2/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l1-s2/environment/instance.json b/tasks/construct-rlve-polynomial-global-minimum-l1-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3ea1def13ca1971c64cb1fc8ef3c041aac5afd92
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l1-s2/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [0, 0, 1], "fmin": "0", "family": "rlve_polynomial_global_minimum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l1-s2/solution/solve.sh b/tasks/construct-rlve-polynomial-global-minimum-l1-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..48ffae347b6a0545978b9cdd566b602d31a7dd61
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-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'
+{"x0": 0.0}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l1-s2/tests/check.py b/tasks/construct-rlve-polynomial-global-minimum-l1-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3be068f4391ad0838c29a4f7d521b8072cf39991
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l1-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-l1-s2/tests/instance.json b/tasks/construct-rlve-polynomial-global-minimum-l1-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3ea1def13ca1971c64cb1fc8ef3c041aac5afd92
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l1-s2/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [0, 0, 1], "fmin": "0", "family": "rlve_polynomial_global_minimum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l1-s2/tests/run.py b/tasks/construct-rlve-polynomial-global-minimum-l1-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l1-s2/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l1-s2/tests/test.sh b/tasks/construct-rlve-polynomial-global-minimum-l1-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l1-s2/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l1-s5/environment/Dockerfile b/tasks/construct-rlve-polynomial-global-minimum-l1-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l1-s5/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l1-s5/environment/instance.json b/tasks/construct-rlve-polynomial-global-minimum-l1-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..95217a1e278c4ff84720d1b8910304bcefc27c11
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l1-s5/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [8, -8, 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-l1-s5/solution/solve.sh b/tasks/construct-rlve-polynomial-global-minimum-l1-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..d7f36b28cb2085c08c88b0949c334857d56020c5
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l1-s5/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x0": 2.0}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l1-s5/tests/check.py b/tasks/construct-rlve-polynomial-global-minimum-l1-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3be068f4391ad0838c29a4f7d521b8072cf39991
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l1-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-l1-s5/tests/instance.json b/tasks/construct-rlve-polynomial-global-minimum-l1-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..95217a1e278c4ff84720d1b8910304bcefc27c11
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l1-s5/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [8, -8, 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-l1-s5/tests/run.py b/tasks/construct-rlve-polynomial-global-minimum-l1-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l1-s5/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l1-s5/tests/test.sh b/tasks/construct-rlve-polynomial-global-minimum-l1-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l1-s5/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l5-s3/environment/Dockerfile b/tasks/construct-rlve-polynomial-global-minimum-l5-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l5-s3/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l5-s3/environment/instance.json b/tasks/construct-rlve-polynomial-global-minimum-l5-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f139a2a1125437ce0ecee7d5484dc698f861be96
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l5-s3/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [2115, 10048, 23321, 30512, 27012, 16060, 6749, 1912, 361, 40, 2], "fmin": "385.307096773224604010444346575792992643649679", "family": "rlve_polynomial_global_minimum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l5-s3/solution/solve.sh b/tasks/construct-rlve-polynomial-global-minimum-l5-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..515d9ee3dc1739e9198670230fad02093a1b625b
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l5-s3/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x0": -0.5072723875200363}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l5-s3/tests/check.py b/tasks/construct-rlve-polynomial-global-minimum-l5-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3be068f4391ad0838c29a4f7d521b8072cf39991
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l5-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-l5-s3/tests/instance.json b/tasks/construct-rlve-polynomial-global-minimum-l5-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f139a2a1125437ce0ecee7d5484dc698f861be96
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l5-s3/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [2115, 10048, 23321, 30512, 27012, 16060, 6749, 1912, 361, 40, 2], "fmin": "385.307096773224604010444346575792992643649679", "family": "rlve_polynomial_global_minimum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l5-s3/tests/run.py b/tasks/construct-rlve-polynomial-global-minimum-l5-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l5-s3/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l5-s3/tests/test.sh b/tasks/construct-rlve-polynomial-global-minimum-l5-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l5-s3/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l5-s4/environment/Dockerfile b/tasks/construct-rlve-polynomial-global-minimum-l5-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l5-s4/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l5-s4/environment/instance.json b/tasks/construct-rlve-polynomial-global-minimum-l5-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a9fde872436e514be4a896c0325f58ad3972439e
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l5-s4/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [9, -4, 162, -248, 522, -548, 450, -248, 91, -20, 2], "fmin": "8.97483437977983751981436996655876362984717222", "family": "rlve_polynomial_global_minimum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l5-s4/solution/solve.sh b/tasks/construct-rlve-polynomial-global-minimum-l5-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..ce43bb71b5c41abda2ebe119eadebb2134613ced
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l5-s4/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x0": 0.012703244588695565}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l5-s4/tests/check.py b/tasks/construct-rlve-polynomial-global-minimum-l5-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3be068f4391ad0838c29a4f7d521b8072cf39991
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l5-s4/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-l5-s4/tests/instance.json b/tasks/construct-rlve-polynomial-global-minimum-l5-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a9fde872436e514be4a896c0325f58ad3972439e
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l5-s4/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [9, -4, 162, -248, 522, -548, 450, -248, 91, -20, 2], "fmin": "8.97483437977983751981436996655876362984717222", "family": "rlve_polynomial_global_minimum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l5-s4/tests/run.py b/tasks/construct-rlve-polynomial-global-minimum-l5-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l5-s4/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-polynomial-global-minimum-l5-s4/tests/test.sh b/tasks/construct-rlve-polynomial-global-minimum-l5-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-global-minimum-l5-s4/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l2-s6/environment/Dockerfile b/tasks/construct-rlve-polynomial-integer-roots-l2-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l2-s6/environment/instance.json b/tasks/construct-rlve-polynomial-integer-roots-l2-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3d812cf8ace43d9226aebcf66a8c4c33cd793932
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l2-s6/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [0, -3, -4, 2, 4, 1], "family": "rlve_polynomial_integer_roots", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l2-s6/solution/solve.sh b/tasks/construct-rlve-polynomial-integer-roots-l2-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..8f1b971cfe3c57ccb38c863a03cecfd2eb321dd8
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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'
+{"roots": [-3, -1, -1, 0, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l2-s6/tests/check.py b/tasks/construct-rlve-polynomial-integer-roots-l2-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2d30553cb884c5e03c8afe73ef652efc483ab791
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l2-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-l2-s6/tests/instance.json b/tasks/construct-rlve-polynomial-integer-roots-l2-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3d812cf8ace43d9226aebcf66a8c4c33cd793932
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l2-s6/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [0, -3, -4, 2, 4, 1], "family": "rlve_polynomial_integer_roots", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l2-s6/tests/run.py b/tasks/construct-rlve-polynomial-integer-roots-l2-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l2-s6/tests/test.sh b/tasks/construct-rlve-polynomial-integer-roots-l2-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l2-s8/environment/Dockerfile b/tasks/construct-rlve-polynomial-integer-roots-l2-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l2-s8/environment/instance.json b/tasks/construct-rlve-polynomial-integer-roots-l2-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e7fb6e3b2152a17d608809500c84fa492430f008
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l2-s8/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [0, -64, 32, 12, -8, 1], "family": "rlve_polynomial_integer_roots", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l2-s8/solution/solve.sh b/tasks/construct-rlve-polynomial-integer-roots-l2-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..3f0edb40c92b5c1a088b1392b6c9310bee00bbaa
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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'
+{"roots": [-2, 0, 2, 4, 4]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l2-s8/tests/check.py b/tasks/construct-rlve-polynomial-integer-roots-l2-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2d30553cb884c5e03c8afe73ef652efc483ab791
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l2-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-l2-s8/tests/instance.json b/tasks/construct-rlve-polynomial-integer-roots-l2-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e7fb6e3b2152a17d608809500c84fa492430f008
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l2-s8/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [0, -64, 32, 12, -8, 1], "family": "rlve_polynomial_integer_roots", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l2-s8/tests/run.py b/tasks/construct-rlve-polynomial-integer-roots-l2-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l2-s8/tests/test.sh b/tasks/construct-rlve-polynomial-integer-roots-l2-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l6-s0/environment/Dockerfile b/tasks/construct-rlve-polynomial-integer-roots-l6-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l6-s0/environment/instance.json b/tasks/construct-rlve-polynomial-integer-roots-l6-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..9d8408c81a0498d468964cddc4b26ea76409fc17
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l6-s0/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [39046947400581120, 18305450790617088, -70709098325188608, -21290079982491648, 39715032787637760, 2321955476341632, -8762688795400128, 764149280711904, 737673598403856, -106065431697240, -28486741314564, 4684860931266, 629771958317, -95987700892, -9804238155, 997485548, 108257258, -4190664, -698546, -7070, 1689, 76, 1], "family": "rlve_polynomial_integer_roots", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l6-s0/solution/solve.sh b/tasks/construct-rlve-polynomial-integer-roots-l6-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..4cbe46bcbdf3ad94e4de391a35a6d7d0c53ef510
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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'
+{"roots": [-22, -21, -16, -16, -15, -13, -12, -6, -4, -2, -1, -1, 1, 3, 3, 3, 6, 6, 6, 6, 7, 12]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l6-s0/tests/check.py b/tasks/construct-rlve-polynomial-integer-roots-l6-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2d30553cb884c5e03c8afe73ef652efc483ab791
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l6-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-l6-s0/tests/instance.json b/tasks/construct-rlve-polynomial-integer-roots-l6-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..9d8408c81a0498d468964cddc4b26ea76409fc17
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l6-s0/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [39046947400581120, 18305450790617088, -70709098325188608, -21290079982491648, 39715032787637760, 2321955476341632, -8762688795400128, 764149280711904, 737673598403856, -106065431697240, -28486741314564, 4684860931266, 629771958317, -95987700892, -9804238155, 997485548, 108257258, -4190664, -698546, -7070, 1689, 76, 1], "family": "rlve_polynomial_integer_roots", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l6-s0/tests/run.py b/tasks/construct-rlve-polynomial-integer-roots-l6-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l6-s0/tests/test.sh b/tasks/construct-rlve-polynomial-integer-roots-l6-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l6-s7/environment/Dockerfile b/tasks/construct-rlve-polynomial-integer-roots-l6-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l6-s7/environment/instance.json b/tasks/construct-rlve-polynomial-integer-roots-l6-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0208e7b642fb74765bb773daf86f6fa0487ca9b4
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l6-s7/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [0, -44749048878755020800, 15949006336864972800, 52352242939548744192, -20742496031367652224, -6910071544332905312, 4847041226049601128, -715428355440016756, -52532728949551270, 22529163551062419, -1042085612162327, -223997664018170, 23458575611316, 670249276961, -176169609769, 2868316320, 609639514, -25503611, -849033, 64814, -136, -57, 1], "family": "rlve_polynomial_integer_roots", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l6-s7/solution/solve.sh b/tasks/construct-rlve-polynomial-integer-roots-l6-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..adc4b7fc03758c75f03880bceb4c1ebe5afb612c
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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'
+{"roots": [-19, -17, -17, -13, -11, -11, -2, -1, 0, 1, 4, 5, 6, 8, 11, 12, 13, 13, 16, 18, 20, 21]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l6-s7/tests/check.py b/tasks/construct-rlve-polynomial-integer-roots-l6-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2d30553cb884c5e03c8afe73ef652efc483ab791
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l6-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-l6-s7/tests/instance.json b/tasks/construct-rlve-polynomial-integer-roots-l6-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0208e7b642fb74765bb773daf86f6fa0487ca9b4
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l6-s7/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [0, -44749048878755020800, 15949006336864972800, 52352242939548744192, -20742496031367652224, -6910071544332905312, 4847041226049601128, -715428355440016756, -52532728949551270, 22529163551062419, -1042085612162327, -223997664018170, 23458575611316, 670249276961, -176169609769, 2868316320, 609639514, -25503611, -849033, 64814, -136, -57, 1], "family": "rlve_polynomial_integer_roots", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l6-s7/tests/run.py b/tasks/construct-rlve-polynomial-integer-roots-l6-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l6-s7/tests/test.sh b/tasks/construct-rlve-polynomial-integer-roots-l6-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l6-s9/environment/Dockerfile b/tasks/construct-rlve-polynomial-integer-roots-l6-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l6-s9/environment/instance.json b/tasks/construct-rlve-polynomial-integer-roots-l6-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e7e1e17f3940b3331bbc72f39b1fe8c8c35fe9f6
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l6-s9/environment/instance.json
@@ -0,0 +1 @@
+{"coeffs": [1852877437931520000, 1870256519239680000, -2396717772750144000, -2392063778290348800, 630235687185930528, 565387120895442064, -91604824366524608, -45284714102086664, 5348888846648050, 1741609881434413, -141264852793811, -37224452533118, 1859812536496, 470380099915, -11829167657, -3567493676, 21865642, 15843571, 130075, -37742, -716, 37, 1], "family": "rlve_polynomial_integer_roots", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l6-s9/solution/solve.sh b/tasks/construct-rlve-polynomial-integer-roots-l6-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..bdcdd227cfe53ea0e647d9d3d9601d2ddb61b89a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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'
+{"roots": [-22, -18, -17, -15, -13, -11, -10, -8, -5, -2, -1, -1, 1, 3, 4, 5, 8, 9, 12, 13, 14, 17]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l6-s9/tests/check.py b/tasks/construct-rlve-polynomial-integer-roots-l6-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..2d30553cb884c5e03c8afe73ef652efc483ab791
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l6-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-l6-s9/tests/instance.json b/tasks/construct-rlve-polynomial-integer-roots-l6-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e7e1e17f3940b3331bbc72f39b1fe8c8c35fe9f6
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-l6-s9/tests/instance.json
@@ -0,0 +1 @@
+{"coeffs": [1852877437931520000, 1870256519239680000, -2396717772750144000, -2392063778290348800, 630235687185930528, 565387120895442064, -91604824366524608, -45284714102086664, 5348888846648050, 1741609881434413, -141264852793811, -37224452533118, 1859812536496, 470380099915, -11829167657, -3567493676, 21865642, 15843571, 130075, -37742, -716, 37, 1], "family": "rlve_polynomial_integer_roots", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-integer-roots-l6-s9/tests/run.py b/tasks/construct-rlve-polynomial-integer-roots-l6-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-integer-roots-l6-s9/tests/test.sh b/tasks/construct-rlve-polynomial-integer-roots-l6-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-integer-roots-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-polynomial-interpolation-l1-s0/environment/Dockerfile b/tasks/construct-rlve-polynomial-interpolation-l1-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l1-s0/environment/instance.json b/tasks/construct-rlve-polynomial-interpolation-l1-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2535a494b7a374b851b2648fa5f8ef654160ce95
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l1-s0/environment/instance.json
@@ -0,0 +1 @@
+{"X": [-1, 2, -2], "Y": [3, -3, 13], "family": "rlve_polynomial_interpolation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-interpolation-l1-s0/solution/solve.sh b/tasks/construct-rlve-polynomial-interpolation-l1-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..a59625d5994486067e129126dab8bfcca07e9b62
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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'
+{"coeffs": [-3, -4, 2]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-interpolation-l1-s0/tests/check.py b/tasks/construct-rlve-polynomial-interpolation-l1-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..f05264e3fd4584db7cb3768800e4a2497564b4f5
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l1-s0/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-l1-s0/tests/instance.json b/tasks/construct-rlve-polynomial-interpolation-l1-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..2535a494b7a374b851b2648fa5f8ef654160ce95
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l1-s0/tests/instance.json
@@ -0,0 +1 @@
+{"X": [-1, 2, -2], "Y": [3, -3, 13], "family": "rlve_polynomial_interpolation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-interpolation-l1-s0/tests/run.py b/tasks/construct-rlve-polynomial-interpolation-l1-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l1-s0/tests/test.sh b/tasks/construct-rlve-polynomial-interpolation-l1-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l1-s7/environment/Dockerfile b/tasks/construct-rlve-polynomial-interpolation-l1-s7/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l1-s7/environment/instance.json b/tasks/construct-rlve-polynomial-interpolation-l1-s7/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..5b958a2eb7acb7e312634b6ad7cf48e7d9c59551
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l1-s7/environment/instance.json
@@ -0,0 +1 @@
+{"X": [2, -2, -1], "Y": [31, 11, 1], "family": "rlve_polynomial_interpolation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-interpolation-l1-s7/solution/solve.sh b/tasks/construct-rlve-polynomial-interpolation-l1-s7/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..58e28d23d96533dab36c9869755ed116bbaedc87
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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'
+{"coeffs": [1, 5, 5]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-interpolation-l1-s7/tests/check.py b/tasks/construct-rlve-polynomial-interpolation-l1-s7/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..f05264e3fd4584db7cb3768800e4a2497564b4f5
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l1-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-l1-s7/tests/instance.json b/tasks/construct-rlve-polynomial-interpolation-l1-s7/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..5b958a2eb7acb7e312634b6ad7cf48e7d9c59551
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l1-s7/tests/instance.json
@@ -0,0 +1 @@
+{"X": [2, -2, -1], "Y": [31, 11, 1], "family": "rlve_polynomial_interpolation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-interpolation-l1-s7/tests/run.py b/tasks/construct-rlve-polynomial-interpolation-l1-s7/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l1-s7/tests/test.sh b/tasks/construct-rlve-polynomial-interpolation-l1-s7/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l1-s9/environment/Dockerfile b/tasks/construct-rlve-polynomial-interpolation-l1-s9/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l1-s9/environment/instance.json b/tasks/construct-rlve-polynomial-interpolation-l1-s9/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..88e2226839db030c93886ecfa85d7a5ef7e9819c
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l1-s9/environment/instance.json
@@ -0,0 +1 @@
+{"X": [-2, -1, 0], "Y": [0, -2, -2], "family": "rlve_polynomial_interpolation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-interpolation-l1-s9/solution/solve.sh b/tasks/construct-rlve-polynomial-interpolation-l1-s9/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..2837bde3cf3a022ad36bb2b35222ab747b266e28
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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'
+{"coeffs": [-2, 1, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-interpolation-l1-s9/tests/check.py b/tasks/construct-rlve-polynomial-interpolation-l1-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..f05264e3fd4584db7cb3768800e4a2497564b4f5
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l1-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-l1-s9/tests/instance.json b/tasks/construct-rlve-polynomial-interpolation-l1-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..88e2226839db030c93886ecfa85d7a5ef7e9819c
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l1-s9/tests/instance.json
@@ -0,0 +1 @@
+{"X": [-2, -1, 0], "Y": [0, -2, -2], "family": "rlve_polynomial_interpolation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-interpolation-l1-s9/tests/run.py b/tasks/construct-rlve-polynomial-interpolation-l1-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l1-s9/tests/test.sh b/tasks/construct-rlve-polynomial-interpolation-l1-s9/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l5-s1/environment/Dockerfile b/tasks/construct-rlve-polynomial-interpolation-l5-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l5-s1/environment/instance.json b/tasks/construct-rlve-polynomial-interpolation-l5-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..cc4f41f1101fcdc88413da051b006c3f8203f372
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l5-s1/environment/instance.json
@@ -0,0 +1 @@
+{"X": [0, 6, 4, -3, -4, -1, 5, 10, 8, -12, 12, -9, 11, -11], "Y": [3, 20423572461, 117571991, 640827, 5767311, -9, 2010860283, 13654740012113, 794124358731, -69436597451769, 140143647547167, -1337523794889, 46102028143911, -21255261499909], "family": "rlve_polynomial_interpolation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-interpolation-l5-s1/solution/solve.sh b/tasks/construct-rlve-polynomial-interpolation-l5-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..308da45bdb6f7c391e19519293a2667dd31156c3
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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'
+{"coeffs": [3, 1, 1, 2, 1, 0, 0, 4, -3, 5, -5, -3, 4, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-interpolation-l5-s1/tests/check.py b/tasks/construct-rlve-polynomial-interpolation-l5-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..f05264e3fd4584db7cb3768800e4a2497564b4f5
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l5-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-l5-s1/tests/instance.json b/tasks/construct-rlve-polynomial-interpolation-l5-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..cc4f41f1101fcdc88413da051b006c3f8203f372
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l5-s1/tests/instance.json
@@ -0,0 +1 @@
+{"X": [0, 6, 4, -3, -4, -1, 5, 10, 8, -12, 12, -9, 11, -11], "Y": [3, 20423572461, 117571991, 640827, 5767311, -9, 2010860283, 13654740012113, 794124358731, -69436597451769, 140143647547167, -1337523794889, 46102028143911, -21255261499909], "family": "rlve_polynomial_interpolation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-interpolation-l5-s1/tests/run.py b/tasks/construct-rlve-polynomial-interpolation-l5-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l5-s1/tests/test.sh b/tasks/construct-rlve-polynomial-interpolation-l5-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l5-s6/environment/Dockerfile b/tasks/construct-rlve-polynomial-interpolation-l5-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l5-s6/environment/instance.json b/tasks/construct-rlve-polynomial-interpolation-l5-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d9e3a6b854b43060db621ef80033f5de5c0741ce
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l5-s6/environment/instance.json
@@ -0,0 +1 @@
+{"X": [-6, 4, -9, 2, 7, 8, -12, 12, -2, -5, 6, 5, 1, -8], "Y": [-31640648116, 277998204, -6595360798270, 43404, 351249186354, 1947264050172, -287646577881988, 358729425883004, -13908, -2842982754, 48819662060, 4758573996, 0, -1401736635396], "family": "rlve_polynomial_interpolation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-interpolation-l5-s6/solution/solve.sh b/tasks/construct-rlve-polynomial-interpolation-l5-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..6cabc37b7f7d4c1f45f50840f2d4297065cd3057
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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'
+{"coeffs": [-4, 0, 0, 2, -2, -1, -1, 0, 2, -4, -2, 3, 4, 3]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-interpolation-l5-s6/tests/check.py b/tasks/construct-rlve-polynomial-interpolation-l5-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..f05264e3fd4584db7cb3768800e4a2497564b4f5
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l5-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-l5-s6/tests/instance.json b/tasks/construct-rlve-polynomial-interpolation-l5-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d9e3a6b854b43060db621ef80033f5de5c0741ce
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l5-s6/tests/instance.json
@@ -0,0 +1 @@
+{"X": [-6, 4, -9, 2, 7, 8, -12, 12, -2, -5, 6, 5, 1, -8], "Y": [-31640648116, 277998204, -6595360798270, 43404, 351249186354, 1947264050172, -287646577881988, 358729425883004, -13908, -2842982754, 48819662060, 4758573996, 0, -1401736635396], "family": "rlve_polynomial_interpolation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-interpolation-l5-s6/tests/run.py b/tasks/construct-rlve-polynomial-interpolation-l5-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l5-s6/tests/test.sh b/tasks/construct-rlve-polynomial-interpolation-l5-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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-polynomial-interpolation-l5-s8/environment/Dockerfile b/tasks/construct-rlve-polynomial-interpolation-l5-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l5-s8/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-polynomial-interpolation-l5-s8/environment/instance.json b/tasks/construct-rlve-polynomial-interpolation-l5-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..063ffb8e8bec3b4e729923543e96e9412e9a3456
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l5-s8/environment/instance.json
@@ -0,0 +1 @@
+{"X": [4, 1, -1, 10, 0, -5, -8, -7, 11, 7, -13, 13, -2, -12], "Y": [252410183, -1, -7, 45014920470953, 3, -7295439547, -3089578439269, -552891987337, 156961329485909, 415597707521, -1630538153725699, 1398107114488763, -57187, -579397392985737], "family": "rlve_polynomial_interpolation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-interpolation-l5-s8/solution/solve.sh b/tasks/construct-rlve-polynomial-interpolation-l5-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..6a7a9a3f4205b5caeeb84480e9d78827cc45aa79
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-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'
+{"coeffs": [3, 5, -1, 1, -3, -5, 1, 2, -1, -5, 2, 0, -5, 5]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-polynomial-interpolation-l5-s8/tests/check.py b/tasks/construct-rlve-polynomial-interpolation-l5-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..f05264e3fd4584db7cb3768800e4a2497564b4f5
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l5-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-l5-s8/tests/instance.json b/tasks/construct-rlve-polynomial-interpolation-l5-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..063ffb8e8bec3b4e729923543e96e9412e9a3456
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l5-s8/tests/instance.json
@@ -0,0 +1 @@
+{"X": [4, 1, -1, 10, 0, -5, -8, -7, 11, 7, -13, 13, -2, -12], "Y": [252410183, -1, -7, 45014920470953, 3, -7295439547, -3089578439269, -552891987337, 156961329485909, 415597707521, -1630538153725699, 1398107114488763, -57187, -579397392985737], "family": "rlve_polynomial_interpolation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-polynomial-interpolation-l5-s8/tests/run.py b/tasks/construct-rlve-polynomial-interpolation-l5-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l5-s8/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-polynomial-interpolation-l5-s8/tests/test.sh b/tasks/construct-rlve-polynomial-interpolation-l5-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-polynomial-interpolation-l5-s8/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-prefix-product-permutation-l1-s2/environment/Dockerfile b/tasks/construct-rlve-prefix-product-permutation-l1-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l1-s2/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-prefix-product-permutation-l1-s2/environment/instance.json b/tasks/construct-rlve-prefix-product-permutation-l1-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e9006d38f3d386dd49a497f08925a69f34578550
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l1-s2/environment/instance.json
@@ -0,0 +1 @@
+{"n": 11, "family": "rlve_prefix_product_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-product-permutation-l1-s2/solution/solve.sh b/tasks/construct-rlve-prefix-product-permutation-l1-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..54fa6973c41492e61e97995c66ac22479cc809b2
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-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'
+{"perm": [1, 2, 7, 5, 4, 10, 3, 9, 8, 6, 11]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-prefix-product-permutation-l1-s2/tests/check.py b/tasks/construct-rlve-prefix-product-permutation-l1-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3d93850acd79cc0f8fff05cdbb0c63aabc484815
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l1-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-l1-s2/tests/instance.json b/tasks/construct-rlve-prefix-product-permutation-l1-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..e9006d38f3d386dd49a497f08925a69f34578550
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l1-s2/tests/instance.json
@@ -0,0 +1 @@
+{"n": 11, "family": "rlve_prefix_product_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-product-permutation-l1-s2/tests/run.py b/tasks/construct-rlve-prefix-product-permutation-l1-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l1-s2/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-prefix-product-permutation-l1-s2/tests/test.sh b/tasks/construct-rlve-prefix-product-permutation-l1-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l1-s2/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-prefix-product-permutation-l5-s3/environment/Dockerfile b/tasks/construct-rlve-prefix-product-permutation-l5-s3/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l5-s3/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-prefix-product-permutation-l5-s3/environment/instance.json b/tasks/construct-rlve-prefix-product-permutation-l5-s3/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c74d4ece8dec15d554bd1480ff4f60f33c180dd6
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l5-s3/environment/instance.json
@@ -0,0 +1 @@
+{"n": 2237, "family": "rlve_prefix_product_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-product-permutation-l5-s3/solution/solve.sh b/tasks/construct-rlve-prefix-product-permutation-l5-s3/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..d1322e2e6266e239e6f44e7c81cc0f063fa39ab9
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l5-s3/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"perm": [1, 2, 1120, 747, 1679, 896, 374, 1599, 840, 1741, 1567, 1628, 1306, 2066, 800, 1045, 1539, 659, 871, 472, 784, 2025, 1933, 1849, 1772, 180, 2152, 581, 1519, 541, 523, 434, 770, 1289, 330, 768, 436, 787, 1355, 1435, 1511, 874, 1013, 2186, 967, 349, 925, 239, 2005, 1188, 1209, 966, 2195, 1014, 291, 2116, 760, 158, 271, 456, 262, 2128, 1336, 2167, 1504, 414, 645, 602, 1284, 617, 1503, 2175, 1337, 1809, 394, 1552, 678, 553, 718, 1700, 756, 940, 1556, 567, 507, 580, 2212, 181, 484, 931, 175, 296, 463, 891, 120, 990, 1003, 370, 1713, 1176, 605, 1640, 1602, 696, 1098, 1748, 1626, 1569, 146, 1335, 2177, 263, 1499, 1169, 1198, 818, 136, 479, 1347, 95, 1250, 759, 2183, 292, 1787, 1379, 1084, 1234, 1871, 1475, 1326, 223, 323, 1027, 1420, 117, 1761, 1144, 309, 516, 752, 826, 1088, 2019, 669, 109, 905, 1888, 1316, 1112, 1895, 401, 1458, 1814, 277, 535, 1478, 58, 1969, 1830, 1497, 265, 1589, 399, 1897, 706, 284, 577, 254, 504, 1409, 799, 2225, 1307, 91, 1944, 1361, 1644, 466, 26, 88, 1756, 1267, 710, 232, 1053, 446, 1281, 1179, 723, 1614, 774, 502, 256, 1304, 1630, 857, 1375, 1707, 327, 303, 947, 1939, 1676, 1920, 623, 1467, 952, 1668, 1264, 1993, 1093, 1932, 2217, 785, 438, 1192, 702, 668, 2095, 1089, 912, 132, 1917, 750, 518, 585, 1696, 1718, 1183, 1528, 185, 1187, 2190, 240, 496, 674, 1313, 48, 235, 1744, 1876, 380, 314, 1092, 2028, 1265, 1758, 894, 1681, 690, 1730, 1661, 169, 1736, 194, 936, 1620, 738, 1072, 1782, 61, 112, 741, 162, 651, 514, 311, 1829, 2080, 59, 1784, 881, 845, 1691, 424, 155, 1705, 1377, 1789, 1495, 1832, 1532, 167, 1663, 1822, 1010, 1404, 335, 1224, 55, 124, 453, 1360, 2063, 92, 1777, 1884, 1675, 2036, 948, 1911, 201, 1293, 1848, 2216, 2026, 1094, 139, 1724, 268, 411, 1858, 244, 1148, 1329, 985, 622, 2034, 1677, 749, 2015, 133, 1213, 795, 531, 200, 1937, 949, 35, 1472, 812, 1261, 834, 289, 1016, 1246, 1853, 1371, 1136, 705, 2074, 400, 2088, 1113, 1019, 654, 1271, 46, 1315, 2091, 906, 681, 1674, 1941, 1778, 1352, 778, 1132, 1079, 1163, 379, 1997, 1745, 634, 810, 1474, 2110, 1235, 98, 527, 1424, 1342, 7, 641, 1206, 590, 1075, 362, 243, 1926, 412, 1506, 1578, 1370, 1901, 1247, 1238, 1771, 2215, 1934, 1294, 429, 75, 688, 1683, 854, 649, 164, 343, 152, 782, 474, 1583, 970, 1531, 1957, 1496, 2079, 1970, 312, 382, 734, 66, 1595, 1009, 1953, 1664, 1751, 1277, 997, 628, 547, 276, 2085, 1459, 1109, 944, 393, 2165, 1338, 1174, 1715, 32, 1470, 37, 1453, 216, 1048, 1652, 545, 630, 1575, 1526, 1185, 187, 959, 1117, 1494, 1959, 1378, 2114, 293, 880, 1967, 60, 1978, 1073, 592, 1351, 1883, 1942, 93, 1349, 594, 179, 2214, 1850, 1239, 726, 1367, 20, 1456, 403, 657, 1541, 1143, 2102, 118, 893, 1991, 1266, 2057, 89, 1309, 1417, 1276, 1820, 1665, 1625, 2133, 1099, 633, 1875, 1998, 236, 1566, 2229, 841, 1449, 1464, 193, 1984, 170, 831, 1671, 85, 1660, 1987, 691, 1216, 1218, 1587, 267, 1929, 140, 1488, 226, 1655, 1182, 2010, 1697, 31, 1806, 1175, 2140, 371, 816, 1200, 1443, 326, 2040, 1376, 1961, 156, 762, 915, 755, 2159, 719, 30, 1717, 2011, 586, 441, 1610, 423, 1964, 846, 983, 1331, 1560, 78, 1522, 853, 1843, 689, 1989, 895, 2234, 748, 1919, 2035, 1940, 1885, 682, 84, 1733, 832, 1263, 2030, 953, 1624, 1750, 1821, 1954, 168, 1986, 1731, 86, 28, 721, 1181, 1720, 227, 544, 1799, 1049, 1032, 377, 1165, 459, 889, 465, 2061, 1362, 838, 1601, 2137, 606, 1593, 68, 956, 1062, 101, 600, 647, 856, 2043, 1305, 2227, 1568, 2132, 1749, 1666, 954, 70, 737, 1981, 937, 1253, 318, 206, 773, 2048, 724, 1241, 422, 1693, 442, 665, 1139, 493, 365, 1430, 695, 2136, 1641, 839, 2231, 375, 1034, 1008, 1824, 67, 1638, 607, 1384, 398, 2076, 266, 1726, 1219, 347, 969, 1835, 475, 699, 18, 1369, 1855, 1507, 1525, 1796, 631, 1101, 1536, 219, 145, 2131, 1627, 2228, 1742, 237, 927, 919, 686, 77, 1687, 1332, 353, 566, 2156, 941, 1319, 677, 2163, 395, 557, 251, 510, 1024, 1446, 808, 636, 104, 1142, 1763, 658, 2222, 1046, 218, 1572, 1102, 341, 166, 1956, 1833, 971, 184, 2008, 1184, 1795, 1576, 1508, 852, 1685, 79, 540, 2210, 582, 1059, 190, 626, 999, 470, 873, 2198, 1436, 851, 1524, 1577, 1856, 413, 2174, 2168, 618, 259, 1168, 2126, 264, 2078, 1831, 1958, 1790, 1118, 4, 561, 321, 225, 1722, 141, 1414, 1323, 538, 81, 1300, 988, 122, 57, 2082, 536, 1325, 2109, 1872, 811, 1908, 36, 1804, 33, 951, 2032, 624, 192, 1738, 1450, 886, 358, 1108, 1813, 2086, 402, 1766, 21, 215, 1802, 38, 885, 1463, 1739, 842, 807, 1546, 1025, 325, 1709, 1201, 829, 172, 15, 1195, 850, 1510, 2199, 1356, 1396, 792, 694, 1604, 366, 766, 332, 979, 1341, 1867, 528, 1040, 116, 2104, 1028, 1275, 1753, 1310, 1322, 1486, 142, 1152, 1037, 798, 2068, 505, 569, 977, 334, 1951, 1011, 876, 597, 639, 9, 499, 791, 1433, 1357, 274, 549, 1257, 1068, 1043, 802, 730, 716, 555, 397, 1591, 608, 197, 865, 1083, 2113, 1788, 1960, 1706, 2041, 858, 1157, 1135, 1900, 1854, 1579, 19, 1768, 727, 42, 1227, 837, 1643, 2062, 1945, 454, 273, 1395, 1434, 2200, 788, 777, 1882, 1779, 593, 1775, 94, 2120, 480, 249, 559, 6, 1866, 1425, 980, 1173, 1808, 2166, 2176, 2129, 147, 352, 1559, 1688, 984, 1923, 1149, 222, 2108, 1476, 537, 1485, 1415, 1311, 676, 1554, 942, 1111, 2090, 1889, 47, 2001, 675, 1321, 1416, 1754, 90, 2065, 2226, 1629, 2044, 257, 620, 987, 1482, 82, 684, 921, 1129, 428, 1847, 1935, 202, 301, 329, 2205, 771, 208, 572, 616, 2170, 603, 1178, 2051, 447, 1123, 996, 1819, 1752, 1418, 1029, 52, 45, 1891, 655, 405, 709, 2056, 1757, 1992, 2029, 1669, 833, 1906, 813, 899, 1067, 1392, 550, 909, 317, 1618, 938, 758, 2118, 96, 1237, 1852, 1902, 1017, 1115, 961, 421, 1612, 725, 1770, 1851, 1248, 97, 1870, 2111, 1085, 1204, 643, 416, 287, 836, 1364, 43, 54, 1949, 336, 994, 1125, 346, 1586, 1727, 1217, 1728, 692, 794, 1915, 134, 820, 965, 2188, 1189, 589, 1863, 642, 1232, 1086, 828, 1442, 1710, 817, 2124, 1170, 849, 1438, 16, 701, 2022, 439, 588, 1208, 2189, 2006, 186, 1794, 1527, 2009, 1719, 1656, 722, 2050, 1282, 604, 2139, 1714, 1807, 1339, 981, 848, 1197, 2125, 1500, 260, 458, 1648, 378, 1878, 1080, 1106, 360, 1077, 1134, 1373, 859, 127, 1006, 1036, 1412, 143, 221, 1328, 1924, 245, 212, 308, 2101, 1762, 1542, 105, 492, 1607, 666, 704, 1899, 1372, 1158, 1078, 1880, 779, 427, 1296, 922, 150, 345, 1221, 995, 1279, 448, 746, 2236, 3, 1493, 1791, 960, 1244, 1018, 1894, 2089, 1317, 943, 1812, 1460, 359, 1161, 1081, 867, 340, 1535, 1573, 632, 1747, 2134, 697, 477, 138, 1931, 2027, 1994, 315, 911, 2018, 2096, 827, 1203, 1233, 2112, 1380, 866, 1105, 1162, 1879, 1133, 1159, 361, 1861, 591, 1781, 1979, 739, 114, 1042, 1391, 1258, 900, 432, 525, 100, 1635, 957, 189, 1517, 583, 520, 230, 712, 445, 2053, 233, 50, 1031, 1651, 1800, 217, 1538, 2223, 801, 1390, 1069, 115, 1422, 529, 797, 1411, 1153, 1007, 1597, 376, 1650, 1050, 51, 1274, 1419, 2105, 324, 1445, 1547, 511, 1022, 512, 653, 1893, 1114, 1245, 1903, 290, 2185, 2196, 875, 1403, 1952, 1823, 1596, 1035, 1154, 128, 369, 2142, 991, 388, 469, 1514, 627, 1818, 1278, 1124, 1222, 337, 387, 1002, 2143, 121, 1481, 1301, 621, 1922, 1330, 1689, 847, 1172, 1340, 1426, 333, 1406, 570, 210, 247, 482, 183, 1530, 1834, 1584, 348, 2194, 2187, 1210, 821, 487, 420, 1243, 1116, 1792, 188, 1061, 1636, 69, 1623, 1667, 2031, 1468, 34, 1910, 1938, 2037, 304, 392, 1811, 1110, 1318, 1555, 2157, 757, 1252, 1619, 1982, 195, 610, 13, 174, 2149, 485, 823, 918, 1564, 238, 2192, 350, 149, 1128, 1297, 685, 1563, 928, 824, 754, 1702, 763, 131, 2017, 1090, 316, 1255, 551, 680, 1887, 2092, 110, 63, 73, 431, 1066, 1259, 814, 373, 2233, 1680, 1990, 1759, 119, 2145, 464, 1646, 460, 357, 1462, 1451, 39, 805, 844, 1966, 1785, 294, 177, 596, 1402, 1012, 2197, 1512, 471, 2220, 660, 385, 339, 1104, 1082, 1381, 198, 533, 279, 451, 126, 1156, 1374, 2042, 1631, 648, 1842, 1684, 1523, 1509, 1437, 1196, 1171, 982, 1690, 1965, 882, 806, 1448, 1740, 2230, 1600, 1642, 1363, 1228, 288, 1905, 1262, 1670, 1734, 171, 1441, 1202, 1087, 2097, 753, 917, 929, 486, 964, 1211, 135, 2123, 1199, 1711, 372, 898, 1260, 1907, 1473, 1873, 635, 1545, 1447, 843, 883, 40, 729, 1389, 1044, 2224, 2067, 1410, 1038, 530, 1914, 1214, 693, 1432, 1397, 500, 776, 1354, 2201, 437, 2024, 2218, 473, 1837, 153, 426, 1131, 1881, 1353, 789, 501, 2047, 1615, 207, 1288, 2206, 435, 2203, 331, 1428, 367, 130, 914, 1703, 157, 2182, 2117, 1251, 939, 2158, 1701, 916, 825, 2098, 517, 2014, 1918, 1678, 2235, 1121, 449, 281, 408, 161, 1975, 113, 1071, 1980, 1621, 71, 65, 1826, 383, 662, 715, 1388, 803, 41, 1366, 1769, 1240, 1613, 2049, 1180, 1657, 29, 1699, 2160, 554, 1387, 731, 663, 444, 1055, 231, 2055, 1268, 406, 283, 2073, 1898, 1137, 667, 2021, 1193, 17, 1581, 476, 1097, 2135, 1603, 1431, 793, 1215, 1729, 1988, 1682, 1844, 76, 1562, 920, 1298, 83, 1673, 1886, 907, 552, 2162, 1553, 1320, 1312, 2002, 497, 11, 612, 108, 2094, 2020, 703, 1138, 1608, 443, 714, 732, 384, 870, 2221, 1540, 1764, 404, 1270, 1892, 1020, 513, 1973, 163, 1841, 855, 1632, 601, 2172, 415, 1231, 1205, 1864, 8, 1400, 598, 103, 1544, 809, 1874, 1746, 1100, 1574, 1797, 546, 1817, 998, 1515, 191, 1466, 2033, 1921, 986, 1302, 258, 1502, 2169, 1285, 573, 490, 107, 671, 12, 934, 196, 1383, 1592, 1639, 2138, 1177, 1283, 2171, 646, 1633, 102, 638, 1401, 877, 178, 1774, 1350, 1780, 1074, 1862, 1207, 1190, 440, 1695, 2012, 519, 1058, 1518, 2211, 2153, 508, 253, 2071, 285, 418, 489, 615, 1286, 209, 976, 1407, 506, 2155, 1557, 354, 299, 204, 320, 1491, 5, 1344, 250, 1550, 396, 1386, 717, 2161, 679, 908, 1256, 1393, 275, 1816, 629, 1798, 1653, 228, 522, 2209, 1520, 80, 1484, 1324, 1477, 2083, 278, 863, 199, 1913, 796, 1039, 1423, 1868, 99, 1064, 433, 2208, 542, 229, 1057, 584, 2013, 751, 2099, 310, 1972, 652, 1021, 1023, 1548, 252, 579, 2154, 568, 1408, 2069, 255, 2046, 775, 790, 1398, 10, 673, 2003, 241, 364, 1606, 1140, 106, 614, 574, 419, 963, 822, 930, 2150, 182, 973, 248, 1346, 2121, 137, 1096, 698, 1582, 1836, 783, 2219, 872, 1513, 1000, 389, 25, 2060, 1645, 890, 2146, 297, 356, 888, 1647, 1166, 261, 2179, 272, 1359, 1946, 125, 861, 280, 745, 1122, 1280, 2052, 1054, 713, 664, 1609, 1694, 587, 1191, 2023, 786, 2202, 769, 2207, 524, 1065, 901, 74, 1846, 1295, 1130, 780, 154, 1963, 1692, 1611, 1242, 962, 488, 575, 286, 1230, 644, 2173, 1505, 1857, 1927, 269, 160, 743, 282, 708, 1269, 656, 1765, 1457, 2087, 1896, 2075, 1590, 1385, 556, 1551, 2164, 1810, 945, 305, 24, 468, 1001, 992, 338, 869, 661, 733, 1827, 313, 1996, 1877, 1164, 1649, 1033, 1598, 2232, 897, 815, 1712, 2141, 1004, 129, 765, 1429, 1605, 494, 242, 1860, 1076, 1160, 1107, 1461, 887, 461, 298, 565, 1558, 1333, 148, 924, 2193, 968, 1585, 1220, 1126, 151, 1839, 165, 1534, 1103, 868, 386, 993, 1223, 1950, 1405, 978, 1427, 767, 2204, 1290, 302, 2039, 1708, 1444, 1026, 2106, 224, 1490, 562, 205, 1617, 1254, 910, 1091, 1995, 381, 1828, 1971, 515, 2100, 1145, 213, 23, 391, 946, 2038, 328, 1291, 203, 564, 355, 462, 2147, 176, 879, 1786, 2115, 2184, 1015, 1904, 835, 1229, 417, 576, 2072, 707, 407, 744, 450, 862, 534, 2084, 1815, 548, 1394, 1358, 455, 2180, 159, 410, 1928, 1725, 1588, 2077, 1498, 2127, 2178, 457, 1167, 1501, 619, 1303, 2045, 503, 2070, 578, 509, 1549, 558, 1345, 481, 974, 211, 1147, 1925, 1859, 363, 495, 2004, 2191, 926, 1565, 1743, 1999, 49, 1052, 2054, 711, 1056, 521, 543, 1654, 1721, 1489, 322, 2107, 1327, 1150, 144, 1571, 1537, 1047, 1801, 1454, 22, 307, 1146, 246, 975, 571, 1287, 772, 1616, 319, 563, 300, 1292, 1936, 1912, 532, 864, 1382, 609, 935, 1983, 1737, 1465, 625, 1516, 1060, 958, 1793, 1186, 2007, 1529, 972, 483, 2151, 2213, 1773, 595, 878, 295, 2148, 932, 14, 1440, 830, 1735, 1985, 1662, 1955, 1533, 342, 1840, 650, 1974, 742, 409, 270, 2181, 761, 1704, 1962, 425, 781, 1838, 344, 1127, 923, 351, 1334, 2130, 1570, 220, 1151, 1413, 1487, 1723, 1930, 1095, 478, 2122, 819, 1212, 1916, 2016, 913, 764, 368, 1005, 1155, 860, 452, 1947, 56, 1480, 989, 2144, 892, 1760, 2103, 1421, 1041, 1070, 740, 1976, 62, 904, 2093, 670, 613, 491, 1141, 1543, 637, 599, 1634, 1063, 526, 1869, 1236, 1249, 2119, 1348, 1776, 1943, 2064, 1308, 1755, 2058, 27, 1659, 1732, 1672, 683, 1299, 1483, 539, 1521, 1686, 1561, 687, 1845, 430, 902, 64, 736, 1622, 955, 1637, 1594, 1825, 735, 72, 903, 111, 1977, 1783, 1968, 2081, 1479, 123, 1948, 1225, 44, 1273, 1030, 1051, 234, 2000, 1314, 1890, 1272, 53, 1226, 1365, 728, 804, 884, 1452, 1803, 1471, 1909, 950, 1469, 1805, 1716, 1698, 720, 1658, 87, 2059, 467, 390, 306, 214, 1455, 1767, 1368, 1580, 700, 1194, 1439, 173, 933, 611, 672, 498, 1399, 640, 1865, 1343, 560, 1492, 1119, 2237]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-prefix-product-permutation-l5-s3/tests/check.py b/tasks/construct-rlve-prefix-product-permutation-l5-s3/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3d93850acd79cc0f8fff05cdbb0c63aabc484815
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l5-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-l5-s3/tests/instance.json b/tasks/construct-rlve-prefix-product-permutation-l5-s3/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..c74d4ece8dec15d554bd1480ff4f60f33c180dd6
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l5-s3/tests/instance.json
@@ -0,0 +1 @@
+{"n": 2237, "family": "rlve_prefix_product_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-product-permutation-l5-s3/tests/run.py b/tasks/construct-rlve-prefix-product-permutation-l5-s3/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l5-s3/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-prefix-product-permutation-l5-s3/tests/test.sh b/tasks/construct-rlve-prefix-product-permutation-l5-s3/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l5-s3/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-prefix-product-permutation-l5-s4/environment/Dockerfile b/tasks/construct-rlve-prefix-product-permutation-l5-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l5-s4/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-prefix-product-permutation-l5-s4/environment/instance.json b/tasks/construct-rlve-prefix-product-permutation-l5-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..52f20345984ccb0391d9d91f1202b67759a2a3ce
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l5-s4/environment/instance.json
@@ -0,0 +1 @@
+{"n": 5009, "family": "rlve_prefix_product_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-product-permutation-l5-s4/solution/solve.sh b/tasks/construct-rlve-prefix-product-permutation-l5-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..553f023a34b49d62f52ca8bbc520def22919a2fd
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l5-s4/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"perm": [1, 2, 2506, 1671, 3758, 1003, 836, 3579, 4384, 3897, 502, 3644, 2923, 1157, 1790, 335, 4697, 885, 1949, 2901, 2756, 4533, 4327, 3050, 1462, 2205, 579, 4639, 3400, 1901, 168, 1940, 2349, 2885, 443, 3722, 975, 3927, 1451, 2056, 3883, 4277, 2267, 234, 2164, 1782, 4030, 4264, 3236, 4090, 1103, 3635, 290, 4821, 2320, 3735, 4205, 4307, 951, 850, 2589, 2136, 3475, 4851, 1175, 1234, 1443, 1571, 222, 2687, 4366, 1412, 488, 4118, 1964, 4075, 726, 1237, 3533, 2347, 1942, 1547, 2139, 1208, 1134, 4185, 2622, 2304, 3587, 1802, 3396, 3744, 4520, 2317, 4637, 581, 4123, 3151, 4550, 2632, 552, 3423, 1818, 925, 2650, 2911, 2411, 4261, 3665, 1012, 1868, 4649, 2103, 2572, 2154, 4618, 476, 686, 2930, 2274, 1295, 2609, 3573, 3096, 1738, 4449, 2426, 1342, 588, 3418, 3122, 1454, 722, 2562, 786, 3934, 2616, 586, 1344, 4001, 4688, 1422, 3211, 3749, 2749, 381, 4564, 1364, 3487, 1144, 2038, 963, 2868, 2882, 619, 4396, 1767, 671, 1174, 4947, 3476, 4730, 774, 1937, 1070, 4585, 3109, 31, 3072, 3943, 2093, 3106, 3816, 1883, 3657, 1747, 1794, 284, 3406, 1680, 4203, 3737, 4377, 4052, 4765, 1788, 1159, 4635, 2319, 4957, 291, 3961, 2062, 1117, 1576, 412, 4780, 1730, 3821, 3978, 2781, 2194, 1712, 988, 3414, 1858, 463, 2566, 3830, 720, 1456, 4464, 1206, 2141, 2131, 3053, 1833, 278, 3011, 3043, 3439, 69, 2325, 4718, 1052, 1359, 3791, 2671, 3582, 1729, 4814, 413, 2743, 44, 2848, 3859, 3970, 783, 3642, 504, 648, 1061, 1305, 2186, 1787, 4826, 4053, 1380, 3374, 2073, 2225, 2735, 3718, 2555, 3176, 4735, 2799, 4016, 4214, 1993, 4066, 4108, 3232, 4591, 2866, 965, 3786, 2271, 2898, 4470, 4472, 3901, 3813, 4588, 2798, 4755, 3177, 218, 2001, 773, 4849, 3477, 3216, 178, 1606, 1864, 1875, 2043, 1375, 53, 191, 1051, 4787, 2326, 3187, 4178, 1744, 878, 3077, 3318, 3524, 2181, 482, 4481, 3939, 428, 3946, 4700, 310, 309, 4703, 3947, 884, 4994, 336, 971, 3092, 2466, 2474, 4272, 4243, 1421, 4870, 4002, 2892, 3253, 969, 338, 3040, 610, 2293, 2619, 1555, 3220, 16, 315, 4041, 328, 1972, 858, 1047, 1088, 4058, 4163, 4413, 1540, 942, 1560, 1829, 1752, 874, 1899, 3402, 3066, 2647, 1285, 4208, 4098, 3345, 2359, 2102, 4899, 1869, 2540, 2189, 2828, 4531, 2758, 2383, 4372, 3399, 4983, 580, 4916, 2318, 4823, 1160, 2459, 2479, 2261, 146, 448, 1981, 2839, 3536, 1250, 559, 1140, 3293, 1984, 2711, 475, 4895, 2155, 3370, 1472, 1911, 2209, 4494, 4194, 1391, 4298, 3602, 2462, 3361, 1312, 2999, 1724, 4212, 4018, 3434, 196, 232, 2269, 3788, 2246, 4420, 3004, 2865, 4747, 3233, 2797, 4737, 3814, 3108, 4845, 1071, 306, 1066, 2920, 1527, 3197, 917, 510, 2644, 3467, 1506, 1216, 1522, 3869, 1720, 2125, 35, 1290, 1163, 1363, 4864, 382, 3031, 2076, 680, 4488, 1896, 3661, 1336, 3755, 4296, 1393, 865, 2631, 4912, 3152, 207, 2446, 1372, 1466, 2527, 1449, 3929, 2372, 1930, 1894, 4490, 2299, 392, 117, 4326, 4989, 2757, 4644, 2829, 303, 531, 3247, 653, 631, 3598, 73, 894, 2316, 4918, 3745, 2027, 1586, 3195, 1529, 4192, 4496, 1037, 1055, 1113, 11, 1368, 240, 4364, 2689, 1278, 1315, 4093, 434, 2368, 4167, 1400, 1036, 4513, 4193, 4612, 2210, 997, 2298, 4538, 1895, 4559, 681, 4121, 583, 2296, 999, 3938, 4707, 483, 1765, 4398, 4183, 1136, 853, 3954, 3900, 4740, 4471, 4741, 2899, 1951, 2916, 1907, 1205, 4799, 1457, 3904, 3021, 2378, 101, 1589, 3914, 2614, 3936, 1001, 3760, 387, 3872, 2425, 4885, 1739, 3542, 4113, 1517, 2594, 1653, 3308, 3892, 3437, 3045, 938, 1321, 1022, 2928, 688, 2805, 27, 373, 96, 889, 526, 2716, 2394, 138, 3668, 129, 1594, 3003, 4594, 2247, 3377, 2307, 2944, 3461, 1539, 4666, 4164, 2794, 4267, 2493, 1091, 732, 2746, 3528, 2241, 1998, 1970, 330, 2719, 4182, 4478, 1766, 4855, 620, 2660, 2128, 155, 616, 2352, 856, 1974, 2405, 2947, 3896, 5002, 3580, 2673, 4357, 486, 1414, 4051, 4828, 3738, 1271, 3742, 3398, 4641, 2384, 2122, 3158, 711, 1411, 4940, 2688, 4506, 241, 3951, 2355, 1627, 1763, 485, 4381, 2674, 3297, 4025, 1015, 2810, 1658, 1147, 1168, 1310, 3363, 778, 1515, 4115, 2697, 2513, 3243, 158, 3838, 2021, 3514, 2669, 3793, 3491, 3767, 2934, 451, 524, 891, 3049, 4988, 4534, 118, 2082, 577, 2207, 1913, 3275, 3350, 2976, 1808, 3285, 801, 915, 3199, 3381, 1685, 2942, 2309, 950, 4953, 4206, 1287, 4038, 1921, 1324, 1852, 643, 3601, 4609, 1392, 4554, 3756, 1673, 3508, 1180, 210, 3556, 133, 2450, 2024, 935, 77, 3775, 3570, 1095, 2399, 3919, 604, 2266, 4969, 3884, 3279, 1192, 4242, 4691, 2475, 1700, 3800, 2492, 4410, 2795, 3235, 4963, 4031, 3664, 4903, 2412, 3576, 3085, 2582, 1230, 2197, 1240, 1331, 1131, 2522, 2578, 2285, 2729, 861, 991, 1579, 1420, 4690, 4273, 1193, 3130, 3009, 280, 163, 3075, 880, 1647, 665, 3497, 3126, 1356, 1040, 238, 1370, 2448, 135, 1078, 2776, 4190, 1531, 3241, 2515, 956, 3633, 1105, 1992, 4752, 4017, 4602, 1725, 696, 4097, 4654, 1286, 4306, 4954, 3736, 4830, 1681, 3685, 3161, 2558, 1500, 1301, 3367, 1390, 4611, 4495, 4514, 1530, 4222, 2777, 2603, 1553, 2621, 4925, 1135, 4477, 4399, 2720, 3628, 1743, 4715, 3188, 4007, 7, 1433, 1928, 2374, 3207, 1617, 3521, 1399, 4499, 2369, 2793, 4412, 4667, 4059, 60, 2423, 3874, 536, 1058, 2658, 622, 3038, 340, 3965, 2281, 764, 4021, 1599, 3617, 459, 2381, 2760, 1688, 3827, 1125, 1734, 1663, 3258, 351, 3113, 1352, 3266, 298, 1935, 776, 3365, 1303, 1063, 313, 18, 3063, 3150, 4914, 582, 4486, 682, 1963, 4937, 489, 2696, 4344, 1516, 4446, 3543, 2486, 2845, 3231, 4749, 4067, 3453, 1750, 1831, 3055, 3173, 3164, 1878, 3344, 4653, 4209, 697, 1813, 433, 4502, 1316, 1102, 4961, 3237, 4081, 3192, 104, 2362, 3728, 3443, 3191, 4088, 3238, 1798, 1264, 2986, 725, 4935, 1965, 573, 3691, 3335, 3470, 347, 3452, 4107, 4750, 1994, 1150, 1043, 2701, 703, 59, 4162, 4668, 1089, 2495, 793, 1379, 4764, 4827, 4378, 1415, 2972, 152, 2144, 266, 1226, 1624, 1757, 327, 4674, 316, 1920, 4304, 1288, 37, 1085, 2952, 1334, 3663, 4263, 4964, 1783, 1873, 1866, 1014, 4354, 3298, 204, 1598, 4149, 765, 3433, 4601, 4213, 4753, 2800, 519, 2714, 528, 1074, 557, 1252, 6, 4176, 3189, 3445, 2625, 2891, 4687, 4871, 1345, 110, 3144, 985, 658, 2202, 2047, 3626, 2722, 3088, 3689, 575, 2084, 2033, 3205, 2376, 3023, 2108, 2257, 3058, 2097, 2780, 4811, 3822, 3610, 514, 499, 3957, 3654, 782, 4774, 3860, 948, 2311, 2280, 4152, 341, 3924, 2061, 4819, 292, 225, 3653, 3973, 500, 3899, 4474, 854, 2354, 4362, 242, 3707, 883, 4699, 4704, 429, 2092, 4841, 3073, 165, 427, 4706, 4482, 1000, 4455, 2615, 4875, 787, 2236, 2791, 2371, 4542, 1450, 4973, 976, 2060, 3963, 342, 954, 2517, 603, 4280, 2400, 1440, 729, 2613, 4457, 1590, 1511, 2959, 3694, 920, 51, 1377, 795, 3020, 4462, 1458, 3812, 4739, 4473, 3955, 501, 5001, 4385, 2948, 194, 3436, 4441, 3309, 1213, 2682, 2443, 1183, 870, 3278, 4276, 4970, 2057, 3679, 759, 2490, 3802, 85, 827, 535, 4159, 2424, 4451, 388, 1719, 4571, 1523, 150, 2974, 3352, 661, 3844, 3016, 947, 3969, 4775, 2849, 1187, 1403, 2087, 14, 3222, 187, 377, 2553, 3720, 445, 3649, 2768, 3015, 3863, 662, 3702, 2152, 2574, 2020, 4339, 159, 65, 3778, 3302, 1299, 1502, 719, 4802, 2567, 1124, 4142, 1689, 2161, 1154, 3609, 3977, 4812, 1731, 737, 770, 1882, 4838, 3107, 4587, 4738, 3902, 1459, 2134, 2591, 1247, 3103, 546, 213, 2871, 84, 3878, 2491, 4269, 1701, 1121, 2330, 3504, 438, 3490, 4335, 2670, 4784, 1360, 2245, 4596, 2270, 4744, 966, 3388, 1340, 2428, 755, 2815, 3301, 3835, 66, 3569, 4284, 78, 1479, 2813, 757, 3681, 2895, 2933, 4333, 3492, 2419, 1203, 1909, 1474, 386, 4453, 1002, 5006, 1672, 4295, 4555, 1337, 1583, 2179, 3526, 2748, 4867, 3212, 933, 2026, 4519, 4919, 3397, 4374, 1272, 636, 1270, 4376, 4829, 4204, 4955, 2321, 507, 3697, 2014, 3566, 3442, 4084, 2363, 356, 804, 706, 974, 4975, 444, 3849, 2554, 4758, 2736, 121, 2403, 1976, 1708, 1178, 3510, 814, 1645, 882, 3949, 243, 2826, 2191, 2151, 3842, 663, 1649, 406, 2013, 3732, 508, 919, 3910, 2960, 3334, 4072, 574, 3990, 3089, 709, 3160, 4201, 1682, 3251, 2894, 3770, 758, 3881, 2058, 978, 1349, 455, 1257, 3429, 1622, 1228, 2584, 128, 4424, 139, 1011, 4902, 4262, 4032, 1335, 4557, 1897, 876, 1746, 4836, 1884, 781, 3972, 3958, 226, 1221, 2767, 3847, 446, 148, 1525, 2922, 4999, 503, 4772, 784, 2564, 465, 3546, 2967, 289, 4959, 1104, 4217, 957, 248, 1638, 1742, 4180, 2721, 3993, 2048, 3409, 2854, 1643, 816, 401, 714, 458, 4147, 1600, 2159, 1691, 1489, 843, 513, 3976, 3823, 1155, 2925, 2980, 1698, 2477, 2461, 4608, 4299, 644, 72, 4524, 632, 961, 2040, 3167, 1620, 3431, 767, 322, 142, 1801, 4922, 2305, 3379, 3201, 1728, 4782, 2672, 4383, 5003, 837, 3084, 4259, 2413, 3095, 4888, 2610, 1094, 4283, 3776, 67, 3441, 3730, 2015, 3517, 2483, 468, 1083, 39, 2956, 1888, 132, 4290, 211, 548, 1108, 1200, 2878, 1960, 25, 2807, 2966, 3638, 466, 2485, 4112, 4447, 1740, 1640, 396, 3101, 1249, 4626, 2840, 2346, 4932, 1238, 2199, 3355, 2240, 4405, 2747, 3751, 2180, 4710, 3319, 1398, 4169, 1618, 3169, 2482, 3564, 2016, 2668, 4337, 2022, 2452, 813, 3711, 1179, 4293, 1674, 1543, 437, 3796, 2331, 3120, 3420, 1099, 2053, 3125, 4232, 666, 897, 566, 2418, 3766, 4334, 3794, 439, 1143, 4862, 1365, 2090, 431, 1815, 496, 820, 790, 1771, 3215, 4729, 4850, 4948, 2137, 1549, 597, 346, 4070, 3336, 1505, 4575, 2645, 3068, 82, 2873, 1538, 4415, 2945, 2407, 824, 2391, 333, 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1979, 450, 4332, 3768, 2896, 2273, 4892, 687, 4435, 1023, 2979, 3607, 1156, 4998, 3645, 1526, 4581, 1067, 171, 1906, 4467, 1952, 1031, 2232, 2410, 4905, 2651, 361, 113, 2440, 1827, 1562, 1987, 1028, 2755, 4991, 1950, 4469, 4742, 2272, 2932, 3769, 3682, 3252, 4686, 4003, 2626, 641, 1854, 3290, 442, 4977, 2350, 618, 4857, 2869, 215, 1959, 3551, 1201, 2421, 62, 1537, 3463, 83, 3804, 214, 2881, 4858, 964, 4746, 4592, 3005, 2509, 2228, 2819, 1308, 1170, 2438, 115, 394, 1642, 3623, 3410, 1396, 3321, 1186, 3858, 4776, 45, 3230, 4110, 2487, 2435, 2993, 2345, 3535, 4627, 1982, 3295, 2676, 1696, 2982, 3427, 1259, 1486, 302, 4530, 4645, 2190, 3705, 244, 3225, 2470, 365, 2184, 1307, 2861, 2229, 202, 3300, 3780, 756, 3772, 1480, 1657, 4352, 1016, 2965, 3548, 26, 4433, 689, 3099, 398, 518, 4015, 4754, 4736, 4589, 3234, 4266, 4411, 4165, 2370, 3931, 2237, 1706, 1978, 2937, 251, 2277, 2501, 2149, 2193, 4810, 3979, 2098, 2602, 4189, 4223, 1079, 2221, 3306, 1655, 1482, 607, 3014, 3846, 3650, 1222, 416, 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3583, 3202, 695, 4211, 4603, 3000, 3156, 2124, 4570, 3870, 389, 3028, 2174, 2336, 656, 987, 4808, 2195, 1232, 1177, 3713, 1977, 2789, 2238, 3357, 569, 1120, 3799, 4270, 2476, 3605, 2981, 2835, 2677, 1933, 300, 1488, 3614, 2160, 3826, 4143, 2761, 2941, 4311, 3382, 3250, 3684, 4202, 4831, 3407, 2050, 1319, 940, 1542, 3507, 4294, 3757, 5007, 2507, 3007, 3132, 912, 359, 2653, 3257, 4139, 1735, 692, 2036, 1146, 4351, 2811, 1481, 2772, 3307, 4443, 2595, 2548, 405, 3700, 664, 4234, 881, 3709, 815, 3622, 2855, 395, 3540, 1741, 3630, 249, 2939, 2763, 593, 2705, 1426, 1811, 699, 3327, 1762, 4360, 2356, 1756, 4044, 1227, 3672, 3430, 3593, 3168, 3520, 4170, 3208, 91, 1268, 638, 371, 29, 3111, 353, 1946, 1863, 4726, 179, 3332, 2962, 1387, 2158, 3616, 4148, 4022, 205, 3154, 3002, 4422, 130, 1890, 1510, 3913, 4458, 102, 3194, 4517, 2028, 2178, 3753, 1338, 3390, 1419, 4245, 992, 411, 4816, 1118, 571, 1967, 221, 4943, 1444, 1282, 928, 1821, 1006, 2387, 2709, 1986, 2905, 1828, 4663, 943, 905, 3262, 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2290, 831, 3269, 3373, 4763, 4054, 794, 3907, 52, 4722, 2044, 1465, 4546, 2447, 4227, 239, 4508, 12, 2089, 3486, 4863, 4565, 1164, 2244, 3790, 4785, 1053, 1039, 4230, 3127, 175, 3265, 4135, 3114, 454, 3676, 979, 749, 109, 4000, 4872, 587, 4883, 2427, 3783, 3389, 1582, 3754, 4556, 3662, 4033, 2953, 1130, 4253, 1241, 2117, 1760, 3329, 810, 1851, 4302, 1922, 1021, 4437, 939, 1677, 2051, 1101, 4092, 4503, 1279, 2998, 4605, 3362, 4348, 1169, 2860, 2820, 2185, 4768, 1062, 4129, 3366, 4197, 1501, 3833, 3303, 3035, 2608, 4890, 2275, 253, 2457, 1162, 4567, 36, 4037, 4305, 4207, 4655, 2648, 927, 1569, 1445, 2997, 1314, 4504, 2690, 56, 807, 182, 635, 3741, 4375, 3739, 637, 1614, 92, 492, 2985, 4078, 1799, 144, 2263, 1485, 2832, 3428, 3674, 456, 716, 3339, 5, 4009, 558, 4625, 3537, 3102, 3808, 2592, 1519, 678, 2078, 2116, 1330, 4254, 2198, 3532, 4933, 727, 1442, 4945, 1176, 1710, 2196, 4256, 2583, 3671, 1623, 4045, 267, 2741, 415, 2766, 3651, 227, 2341, 676, 1521, 4573, 1507, 2681, 3890, 3310, 742, 2520, 1133, 4927, 2140, 4798, 4465, 1908, 3764, 2420, 2877, 3552, 1109, 273, 424, 1904, 173, 3129, 4241, 4274, 3280, 199, 1034, 1402, 3857, 2850, 3322, 869, 3887, 2444, 209, 4292, 3509, 3712, 1709, 1233, 4946, 4852, 672, 2991, 2437, 2859, 1309, 4349, 1148, 1996, 2243, 1362, 4566, 1291, 2458, 4634, 4824, 1789, 4997, 2924, 3608, 3824, 2162, 236, 1042, 4064, 1995, 1167, 4350, 1659, 2037, 4861, 3488, 440, 3292, 4623, 560, 2587, 852, 4476, 4184, 4926, 1209, 2521, 4252, 1332, 2954, 41, 735, 1733, 4141, 3828, 2568, 2329, 3798, 1702, 570, 1575, 4817, 2063, 626, 10, 4510, 1056, 538, 272, 1199, 3553, 549, 1991, 4216, 3634, 4960, 4091, 1317, 2052, 3500, 3421, 554, 2398, 4282, 3571, 2611, 731, 4408, 2494, 4057, 4669, 1048, 2951, 4035, 38, 3561, 469, 2640, 2220, 2775, 4224, 136, 2396, 556, 4011, 529, 305, 4584, 4846, 1938, 170, 2919, 4582, 307, 312, 4128, 1304, 4769, 649, 2657, 4157, 537, 1112, 4511, 1038, 1358, 4786, 4719, 192, 2950, 1087, 4670, 859, 2731, 2700, 4063, 1151, 237, 4229, 1357, 1054, 4512, 4497, 1401, 1189, 200, 2231, 2914, 1953, 2754, 2903, 1988, 2635, 1806, 2978, 2927, 4436, 1322, 1923, 2289, 1385, 2964, 2809, 4353, 4026, 1867, 4901, 3666, 140, 324, 2120, 2386, 1566, 1822, 835, 5005, 3759, 4454, 3937, 4483, 2297, 4492, 2211, 258, 798, 410, 1578, 4246, 862, 3413, 4807, 1713, 657, 3997, 3145, 3138, 3228, 47, 748, 1348, 3677, 2059, 3926, 4974, 3723, 707, 3091, 4695, 337, 4684, 3254, 3387, 3785, 4745, 2867, 4859, 2039, 3596, 633, 184, 247, 3632, 4218, 2516, 3922, 343, 849, 4952, 4308, 2310, 3968, 3861, 3017, 261, 904, 1559, 4664, 1541, 1676, 1320, 4438, 3046, 76, 4286, 2025, 3747, 3213, 1773, 923, 1820, 1568, 1283, 2649, 4907, 1819, 930, 1774, 50, 3909, 3695, 509, 4578, 3198, 4314, 802, 358, 1667, 3133, 1847, 1838, 1956, 3180, 3261, 1558, 944, 262, 1780, 2166, 2525, 1468, 565, 3495, 667, 2315, 4522, 74, 3048, 4329, 525, 4429, 97, 1861, 1948, 4993, 4698, 3948, 3708, 1646, 4235, 3076, 4713, 1745, 3659, 1898, 4660, 1753, 3348, 3277, 3886, 1184, 3323, 2251, 2630, 4552, 1394, 3412, 990, 4247, 2730, 1046, 4671, 1973, 4389, 2353, 3953, 4475, 1137, 2588, 4951, 952, 344, 599, 2218, 2642, 512, 3612, 1490, 3394, 1804, 2637, 3083, 3578, 5004, 1004, 1823, 296, 3268, 1383, 2291, 612, 534, 3876, 86, 2390, 3458, 2408, 2234, 789, 3481, 497, 516, 400, 3621, 1644, 3710, 3511, 2453, 1850, 1326, 3330, 181, 1275, 57, 705, 3725, 357, 914, 4315, 3286, 409, 994, 259, 3019, 3906, 1378, 4055, 2496, 1770, 3480, 821, 2235, 3933, 4876, 2563, 3641, 4773, 3971, 3655, 1885, 1514, 4346, 3364, 4131, 1936, 4848, 4731, 2002, 1881, 3818, 738, 321, 3591, 3432, 4020, 4150, 2282, 2726, 2433, 2489, 3880, 3680, 3771, 2814, 3781, 2429, 1926, 1435, 2599, 108, 1347, 980, 48, 1776, 2216, 601, 2519, 1211, 3311, 2536, 320, 769, 3819, 1732, 1127, 42, 2745, 4407, 1092, 2612, 3916, 1441, 1236, 4934, 4076, 2987, 2561, 4878, 1455, 4801, 3831, 1503, 3338, 1255, 457, 3619, 402, 1410, 4368, 3159, 3687, 3090, 973, 3724, 805, 58, 4061, 2702, 2069, 3326, 1630, 1812, 4096, 4210, 1726, 3203, 2035, 1661, 1736, 3098, 2804, 4434, 2929, 4893, 477, 23, 1962, 4120, 4487, 4560, 2077, 1244, 1520, 1218, 2342, 1497, 2990, 1173, 4853, 1768, 2498, 2314, 896, 3496, 4233, 1648, 3701, 3843, 3864, 3353, 2201, 3996, 986, 1714, 2337, 630, 4526, 3248, 3384, 2656, 1060, 4770, 505, 2323, 71, 3600, 4300, 1853, 2889, 2627, 370, 1613, 1269, 3740, 1273, 183, 960, 3597, 4525, 654, 2338, 1431, 9, 1115, 2064, 2606, 3037, 4155, 2659, 4395, 4856, 2883, 2351, 4391, 156, 3245, 533, 829, 2292, 4681, 3041, 3013, 2770, 1483, 2265, 4279, 3920, 2518, 744, 2217, 847, 345, 3472, 1550, 2067, 2704, 1634, 2764, 417, 2008, 3417, 4882, 1343, 4873, 2617, 2295, 4485, 4122, 4915, 4638, 4984, 2206, 4323, 2083, 3989, 3690, 4073, 1966, 1574, 1119, 1703, 3358, 2417, 3494, 898, 1469, 3272, 126, 2586, 1139, 4624, 1251, 4010, 1075, 2397, 1097, 3422, 4910, 2633, 1990, 1107, 3554, 212, 3806, 3104, 2095, 3060, 2112, 270, 540, 271, 1111, 1057, 4158, 3875, 828, 613, 3246, 4528, 304, 1073, 4012, 2715, 4428, 890, 4330, 452, 3116, 473, 2713, 4014, 2801, 399, 818, 498, 3975, 3611, 844, 2643, 4577, 918, 3696, 3733, 2322, 647, 4771, 3643, 5000, 3898, 3956, 3974, 515, 819, 3482, 1816, 3425, 2984, 1266, 93, 2695, 4117, 4938, 1413, 4380, 4358, 1764, 4480, 4708, 2182, 367, 2254, 22, 685, 4894, 4619, 2712, 521, 3117, 3081, 2639, 1082, 3562, 2484, 3545, 3639, 2565, 4804, 1859, 99, 2380, 4146, 3618, 715, 1256, 3675, 1350, 3115, 523, 4331, 2935, 1980, 4629, 147, 3648, 3848, 3721, 4976, 2886, 3291, 1142, 3489, 3795, 3505, 1544, 2367, 4501, 4094, 1814, 3484, 2091, 3945, 4705, 3940, 166, 1903, 1197, 274, 2214, 1778, 264, 2146, 2007, 591, 2765, 1223, 2742, 4779, 4815, 1577, 993, 799, 3287, 2012, 3699, 1650, 2549, 1409, 713, 3620, 817, 517, 2802, 3100, 3539, 1641, 2856, 116, 4536, 2300, 3027, 1718, 3871, 4452, 3761, 1475, 2172, 3030, 4563, 4865, 2750, 2532, 2552, 3851, 188, 2693, 95, 4431, 28, 1612, 639, 2628, 2253, 480, 2183, 2822, 2471, 3142, 112, 2909, 2652, 1666, 913, 803, 3726, 2364, 1945, 1609, 3112, 4137, 3259, 3182, 3451, 4069, 3471, 598, 848, 953, 3923, 3964, 4153, 3039, 4683, 970, 4696, 4995, 1791, 3456, 2392, 2718, 4401, 1971, 4673, 4042, 1758, 2119, 1009, 141, 3590, 768, 739, 2537, 2545, 1919, 4040, 4675, 17, 4127, 1064, 308, 4702, 4701, 311, 1065, 4583, 1072, 530, 4529, 2830, 1487, 1693, 1934, 4133, 3267, 833, 1824, 2685, 224, 3960, 4820, 4958, 3636, 2968, 3136, 3147, 3405, 4833, 1795, 1534, 162, 4238, 3010, 4793, 1834, 256, 2213, 423, 1198, 1110, 539, 541, 2113, 2740, 1225, 4046, 2145, 420, 1779, 903, 945, 3018, 797, 995, 2212, 276, 1835, 2456, 1293, 2276, 2786, 2938, 1637, 3631, 958, 185, 3224, 2825, 3706, 3950, 4363, 4507, 1369, 4228, 1041, 1152, 2163, 4967, 2268, 4598, 197, 3282, 1429, 2340, 1220, 3652, 3959, 293, 2686, 4942, 1572, 1968, 2000, 4733, 3178, 1958, 2880, 2870, 3805, 547, 3555, 4291, 1181, 2445, 4548, 3153, 1597, 4023, 3299, 2817, 2230, 1033, 1190, 3281, 231, 4599, 3435, 3894, 2949, 1050, 4720, 54, 2692, 376, 3852, 3223, 246, 959, 634, 1274, 808, 3331, 1605, 4727, 3217, 3264, 1354, 3128, 1195, 1905, 2918, 1068, 1939, 4980, 1902, 426, 3941, 3074, 4237, 281, 1535, 64, 3837, 4340, 3244, 615, 4392, 2129, 2143, 4048, 2973, 3867, 1524, 3647, 447, 4630, 2262, 1262, 1800, 3589, 323, 1010, 3667, 4425, 2395, 1077, 4225, 2449, 4289, 3557, 1889, 1593, 4423, 3669, 2585, 562, 3273, 1915, 1438, 2402, 3716, 2737, 2081, 4325, 4535, 393, 2857, 2439, 2908, 362, 3143, 3999, 1346, 750, 2600, 2100, 2361, 4086, 3193, 1588, 4459, 2379, 461, 1860, 888, 4430, 374, 2694, 491, 1267, 1615, 3209, 1424, 2707, 2389, 826, 3877, 3803, 2872, 3464, 3069, 2664, 1478, 3774, 4285, 936, 3047, 893, 4523, 3599, 645, 2324, 4789, 3440, 3568, 3777, 3836, 160, 1536, 2875, 2422, 4161, 4060, 704, 806, 1276, 2691, 190, 4721, 1376, 3908, 921, 1775, 747, 981, 3229, 2847, 4777, 2744, 734, 1128, 2955, 3560, 1084, 4036, 1289, 4568, 2126, 2662, 3071, 4843, 3110, 1611, 372, 4432, 2806, 3549, 1961, 684, 478, 2255, 2110, 3062, 4126, 314, 4676, 3221, 3854, 2088, 1367, 4509, 1114, 627, 1432, 4175, 4008, 1253, 3340, 2505, 5009]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-prefix-product-permutation-l5-s4/tests/check.py b/tasks/construct-rlve-prefix-product-permutation-l5-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..3d93850acd79cc0f8fff05cdbb0c63aabc484815
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l5-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-l5-s4/tests/instance.json b/tasks/construct-rlve-prefix-product-permutation-l5-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..52f20345984ccb0391d9d91f1202b67759a2a3ce
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l5-s4/tests/instance.json
@@ -0,0 +1 @@
+{"n": 5009, "family": "rlve_prefix_product_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-product-permutation-l5-s4/tests/run.py b/tasks/construct-rlve-prefix-product-permutation-l5-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l5-s4/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-prefix-product-permutation-l5-s4/tests/test.sh b/tasks/construct-rlve-prefix-product-permutation-l5-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-prefix-product-permutation-l5-s4/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l5-s5/environment/Dockerfile b/tasks/construct-rlve-prefix-sum-permutation-l5-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-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-prefix-sum-permutation-l5-s5/environment/instance.json b/tasks/construct-rlve-prefix-sum-permutation-l5-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..847b43c413e1bf01c7d21c51c3752ba9147d50e5
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l5-s5/environment/instance.json
@@ -0,0 +1 @@
+{"n": 8154, "family": "rlve_prefix_sum_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l5-s5/solution/solve.sh b/tasks/construct-rlve-prefix-sum-permutation-l5-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..0a550594b3e051a2ec2904280692b94a6d579fb3
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-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'
+{"perm": [8154, 1, 8152, 3, 8150, 5, 8148, 7, 8146, 9, 8144, 11, 8142, 13, 8140, 15, 8138, 17, 8136, 19, 8134, 21, 8132, 23, 8130, 25, 8128, 27, 8126, 29, 8124, 31, 8122, 33, 8120, 35, 8118, 37, 8116, 39, 8114, 41, 8112, 43, 8110, 45, 8108, 47, 8106, 49, 8104, 51, 8102, 53, 8100, 55, 8098, 57, 8096, 59, 8094, 61, 8092, 63, 8090, 65, 8088, 67, 8086, 69, 8084, 71, 8082, 73, 8080, 75, 8078, 77, 8076, 79, 8074, 81, 8072, 83, 8070, 85, 8068, 87, 8066, 89, 8064, 91, 8062, 93, 8060, 95, 8058, 97, 8056, 99, 8054, 101, 8052, 103, 8050, 105, 8048, 107, 8046, 109, 8044, 111, 8042, 113, 8040, 115, 8038, 117, 8036, 119, 8034, 121, 8032, 123, 8030, 125, 8028, 127, 8026, 129, 8024, 131, 8022, 133, 8020, 135, 8018, 137, 8016, 139, 8014, 141, 8012, 143, 8010, 145, 8008, 147, 8006, 149, 8004, 151, 8002, 153, 8000, 155, 7998, 157, 7996, 159, 7994, 161, 7992, 163, 7990, 165, 7988, 167, 7986, 169, 7984, 171, 7982, 173, 7980, 175, 7978, 177, 7976, 179, 7974, 181, 7972, 183, 7970, 185, 7968, 187, 7966, 189, 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7423, 730, 7425, 728, 7427, 726, 7429, 724, 7431, 722, 7433, 720, 7435, 718, 7437, 716, 7439, 714, 7441, 712, 7443, 710, 7445, 708, 7447, 706, 7449, 704, 7451, 702, 7453, 700, 7455, 698, 7457, 696, 7459, 694, 7461, 692, 7463, 690, 7465, 688, 7467, 686, 7469, 684, 7471, 682, 7473, 680, 7475, 678, 7477, 676, 7479, 674, 7481, 672, 7483, 670, 7485, 668, 7487, 666, 7489, 664, 7491, 662, 7493, 660, 7495, 658, 7497, 656, 7499, 654, 7501, 652, 7503, 650, 7505, 648, 7507, 646, 7509, 644, 7511, 642, 7513, 640, 7515, 638, 7517, 636, 7519, 634, 7521, 632, 7523, 630, 7525, 628, 7527, 626, 7529, 624, 7531, 622, 7533, 620, 7535, 618, 7537, 616, 7539, 614, 7541, 612, 7543, 610, 7545, 608, 7547, 606, 7549, 604, 7551, 602, 7553, 600, 7555, 598, 7557, 596, 7559, 594, 7561, 592, 7563, 590, 7565, 588, 7567, 586, 7569, 584, 7571, 582, 7573, 580, 7575, 578, 7577, 576, 7579, 574, 7581, 572, 7583, 570, 7585, 568, 7587, 566, 7589, 564, 7591, 562, 7593, 560, 7595, 558, 7597, 556, 7599, 554, 7601, 552, 7603, 550, 7605, 548, 7607, 546, 7609, 544, 7611, 542, 7613, 540, 7615, 538, 7617, 536, 7619, 534, 7621, 532, 7623, 530, 7625, 528, 7627, 526, 7629, 524, 7631, 522, 7633, 520, 7635, 518, 7637, 516, 7639, 514, 7641, 512, 7643, 510, 7645, 508, 7647, 506, 7649, 504, 7651, 502, 7653, 500, 7655, 498, 7657, 496, 7659, 494, 7661, 492, 7663, 490, 7665, 488, 7667, 486, 7669, 484, 7671, 482, 7673, 480, 7675, 478, 7677, 476, 7679, 474, 7681, 472, 7683, 470, 7685, 468, 7687, 466, 7689, 464, 7691, 462, 7693, 460, 7695, 458, 7697, 456, 7699, 454, 7701, 452, 7703, 450, 7705, 448, 7707, 446, 7709, 444, 7711, 442, 7713, 440, 7715, 438, 7717, 436, 7719, 434, 7721, 432, 7723, 430, 7725, 428, 7727, 426, 7729, 424, 7731, 422, 7733, 420, 7735, 418, 7737, 416, 7739, 414, 7741, 412, 7743, 410, 7745, 408, 7747, 406, 7749, 404, 7751, 402, 7753, 400, 7755, 398, 7757, 396, 7759, 394, 7761, 392, 7763, 390, 7765, 388, 7767, 386, 7769, 384, 7771, 382, 7773, 380, 7775, 378, 7777, 376, 7779, 374, 7781, 372, 7783, 370, 7785, 368, 7787, 366, 7789, 364, 7791, 362, 7793, 360, 7795, 358, 7797, 356, 7799, 354, 7801, 352, 7803, 350, 7805, 348, 7807, 346, 7809, 344, 7811, 342, 7813, 340, 7815, 338, 7817, 336, 7819, 334, 7821, 332, 7823, 330, 7825, 328, 7827, 326, 7829, 324, 7831, 322, 7833, 320, 7835, 318, 7837, 316, 7839, 314, 7841, 312, 7843, 310, 7845, 308, 7847, 306, 7849, 304, 7851, 302, 7853, 300, 7855, 298, 7857, 296, 7859, 294, 7861, 292, 7863, 290, 7865, 288, 7867, 286, 7869, 284, 7871, 282, 7873, 280, 7875, 278, 7877, 276, 7879, 274, 7881, 272, 7883, 270, 7885, 268, 7887, 266, 7889, 264, 7891, 262, 7893, 260, 7895, 258, 7897, 256, 7899, 254, 7901, 252, 7903, 250, 7905, 248, 7907, 246, 7909, 244, 7911, 242, 7913, 240, 7915, 238, 7917, 236, 7919, 234, 7921, 232, 7923, 230, 7925, 228, 7927, 226, 7929, 224, 7931, 222, 7933, 220, 7935, 218, 7937, 216, 7939, 214, 7941, 212, 7943, 210, 7945, 208, 7947, 206, 7949, 204, 7951, 202, 7953, 200, 7955, 198, 7957, 196, 7959, 194, 7961, 192, 7963, 190, 7965, 188, 7967, 186, 7969, 184, 7971, 182, 7973, 180, 7975, 178, 7977, 176, 7979, 174, 7981, 172, 7983, 170, 7985, 168, 7987, 166, 7989, 164, 7991, 162, 7993, 160, 7995, 158, 7997, 156, 7999, 154, 8001, 152, 8003, 150, 8005, 148, 8007, 146, 8009, 144, 8011, 142, 8013, 140, 8015, 138, 8017, 136, 8019, 134, 8021, 132, 8023, 130, 8025, 128, 8027, 126, 8029, 124, 8031, 122, 8033, 120, 8035, 118, 8037, 116, 8039, 114, 8041, 112, 8043, 110, 8045, 108, 8047, 106, 8049, 104, 8051, 102, 8053, 100, 8055, 98, 8057, 96, 8059, 94, 8061, 92, 8063, 90, 8065, 88, 8067, 86, 8069, 84, 8071, 82, 8073, 80, 8075, 78, 8077, 76, 8079, 74, 8081, 72, 8083, 70, 8085, 68, 8087, 66, 8089, 64, 8091, 62, 8093, 60, 8095, 58, 8097, 56, 8099, 54, 8101, 52, 8103, 50, 8105, 48, 8107, 46, 8109, 44, 8111, 42, 8113, 40, 8115, 38, 8117, 36, 8119, 34, 8121, 32, 8123, 30, 8125, 28, 8127, 26, 8129, 24, 8131, 22, 8133, 20, 8135, 18, 8137, 16, 8139, 14, 8141, 12, 8143, 10, 8145, 8, 8147, 6, 8149, 4, 8151, 2, 8153]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l5-s5/tests/check.py b/tasks/construct-rlve-prefix-sum-permutation-l5-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..60c3e0996cdfa8673fd7f34ad70cccb789b17654
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l5-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(), 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-l5-s5/tests/instance.json b/tasks/construct-rlve-prefix-sum-permutation-l5-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..847b43c413e1bf01c7d21c51c3752ba9147d50e5
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-l5-s5/tests/instance.json
@@ -0,0 +1 @@
+{"n": 8154, "family": "rlve_prefix_sum_permutation", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prefix-sum-permutation-l5-s5/tests/run.py b/tasks/construct-rlve-prefix-sum-permutation-l5-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-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-prefix-sum-permutation-l5-s5/tests/test.sh b/tasks/construct-rlve-prefix-sum-permutation-l5-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-prefix-sum-permutation-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-prime-graph-coloring-l1-s5/environment/Dockerfile b/tasks/construct-rlve-prime-graph-coloring-l1-s5/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-l1-s5/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/construct-rlve-prime-graph-coloring-l1-s5/environment/instance.json b/tasks/construct-rlve-prime-graph-coloring-l1-s5/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d1fb3d738d3f8de6d6f4f84d0d05e2a55c20345e
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-l1-s5/environment/instance.json
@@ -0,0 +1 @@
+{"n": 12, "k": 4, "family": "rlve_prime_graph_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prime-graph-coloring-l1-s5/solution/solve.sh b/tasks/construct-rlve-prime-graph-coloring-l1-s5/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..ce7b0874c436cba5fdc38bd700f889f77dcaeea2
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-l1-s5/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"colors": [1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-prime-graph-coloring-l1-s5/tests/check.py b/tasks/construct-rlve-prime-graph-coloring-l1-s5/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..b3736f1375f1862241999142c975ced361b8fcb5
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-l1-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-l1-s5/tests/instance.json b/tasks/construct-rlve-prime-graph-coloring-l1-s5/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..d1fb3d738d3f8de6d6f4f84d0d05e2a55c20345e
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-l1-s5/tests/instance.json
@@ -0,0 +1 @@
+{"n": 12, "k": 4, "family": "rlve_prime_graph_coloring", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-prime-graph-coloring-l1-s5/tests/run.py b/tasks/construct-rlve-prime-graph-coloring-l1-s5/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-l1-s5/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-prime-graph-coloring-l1-s5/tests/test.sh b/tasks/construct-rlve-prime-graph-coloring-l1-s5/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-prime-graph-coloring-l1-s5/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l3-s1/environment/Dockerfile b/tasks/construct-rlve-recursive-sequence-sum-l3-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-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-recursive-sequence-sum-l3-s1/environment/instance.json b/tasks/construct-rlve-recursive-sequence-sum-l3-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a2456b7dd2c10ff69a8b4c876f837121bb778e44
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l3-s1/environment/instance.json
@@ -0,0 +1 @@
+{"F0": 80, "A": 16, "B": 3847, "S": 6289138856459747, "family": "rlve_recursive_sequence_sum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l3-s1/solution/solve.sh b/tasks/construct-rlve-recursive-sequence-sum-l3-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..dcbf42d81268f8f9a0b07cb0bec637ae2f2d749a
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-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, 3, 6, 10, 11]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l3-s1/tests/check.py b/tasks/construct-rlve-recursive-sequence-sum-l3-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..feda076c3f95e5e12198151dee30e1b8223924bc
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l3-s1/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-l3-s1/tests/instance.json b/tasks/construct-rlve-recursive-sequence-sum-l3-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a2456b7dd2c10ff69a8b4c876f837121bb778e44
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l3-s1/tests/instance.json
@@ -0,0 +1 @@
+{"F0": 80, "A": 16, "B": 3847, "S": 6289138856459747, "family": "rlve_recursive_sequence_sum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l3-s1/tests/run.py b/tasks/construct-rlve-recursive-sequence-sum-l3-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-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-recursive-sequence-sum-l3-s1/tests/test.sh b/tasks/construct-rlve-recursive-sequence-sum-l3-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-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-recursive-sequence-sum-l3-s6/environment/Dockerfile b/tasks/construct-rlve-recursive-sequence-sum-l3-s6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-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-recursive-sequence-sum-l3-s6/environment/instance.json b/tasks/construct-rlve-recursive-sequence-sum-l3-s6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..ac75911f2fc48d355b4d488dea6a6e94e4395101
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l3-s6/environment/instance.json
@@ -0,0 +1 @@
+{"F0": 24, "A": 12, "B": 5463, "S": 32236431435261, "family": "rlve_recursive_sequence_sum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l3-s6/solution/solve.sh b/tasks/construct-rlve-recursive-sequence-sum-l3-s6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..b6e92c63b57cf44816bee63a601a02fa1778f628
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-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": [1, 2, 10]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l3-s6/tests/check.py b/tasks/construct-rlve-recursive-sequence-sum-l3-s6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..feda076c3f95e5e12198151dee30e1b8223924bc
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l3-s6/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-l3-s6/tests/instance.json b/tasks/construct-rlve-recursive-sequence-sum-l3-s6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..ac75911f2fc48d355b4d488dea6a6e94e4395101
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l3-s6/tests/instance.json
@@ -0,0 +1 @@
+{"F0": 24, "A": 12, "B": 5463, "S": 32236431435261, "family": "rlve_recursive_sequence_sum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l3-s6/tests/run.py b/tasks/construct-rlve-recursive-sequence-sum-l3-s6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-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-recursive-sequence-sum-l3-s6/tests/test.sh b/tasks/construct-rlve-recursive-sequence-sum-l3-s6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-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-recursive-sequence-sum-l3-s8/environment/Dockerfile b/tasks/construct-rlve-recursive-sequence-sum-l3-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-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-recursive-sequence-sum-l3-s8/environment/instance.json b/tasks/construct-rlve-recursive-sequence-sum-l3-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f99c967fcc1d3bc0a134d785fc9912f99f0ac759
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l3-s8/environment/instance.json
@@ -0,0 +1 @@
+{"F0": 7, "A": 15, "B": 1944, "S": 1346111077753815, "family": "rlve_recursive_sequence_sum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l3-s8/solution/solve.sh b/tasks/construct-rlve-recursive-sequence-sum-l3-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..b4f732e10ab7709a8e3c04984b55fbf07a268069
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-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": [1, 4, 8, 10, 11]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l3-s8/tests/check.py b/tasks/construct-rlve-recursive-sequence-sum-l3-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..feda076c3f95e5e12198151dee30e1b8223924bc
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l3-s8/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-l3-s8/tests/instance.json b/tasks/construct-rlve-recursive-sequence-sum-l3-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..f99c967fcc1d3bc0a134d785fc9912f99f0ac759
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-l3-s8/tests/instance.json
@@ -0,0 +1 @@
+{"F0": 7, "A": 15, "B": 1944, "S": 1346111077753815, "family": "rlve_recursive_sequence_sum", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-recursive-sequence-sum-l3-s8/tests/run.py b/tasks/construct-rlve-recursive-sequence-sum-l3-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-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-recursive-sequence-sum-l3-s8/tests/test.sh b/tasks/construct-rlve-recursive-sequence-sum-l3-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-recursive-sequence-sum-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-round-robin-scores-l3-s2/environment/Dockerfile b/tasks/construct-rlve-round-robin-scores-l3-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-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-round-robin-scores-l3-s2/environment/instance.json b/tasks/construct-rlve-round-robin-scores-l3-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..de93e6699e9791980b9a5098b90d8d00451a5d06
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-l3-s2/environment/instance.json
@@ -0,0 +1 @@
+{"W": [10, 8, 8, 10, 10, 9, 7, 9, 6], "family": "rlve_round_robin_scores", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-round-robin-scores-l3-s2/solution/solve.sh b/tasks/construct-rlve-round-robin-scores-l3-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..7e5c442e3f41b0c5987c8bd63faa04994c6ed9e2
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-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'
+{"A": ["011111211", "101111111", "110111111", "111012111", "111101112", "111010121", "011111011", "111110102", "111101100"]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-round-robin-scores-l3-s2/tests/check.py b/tasks/construct-rlve-round-robin-scores-l3-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..214446acaa90ce170daad3989b42317353060f9d
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-l3-s2/tests/check.py
@@ -0,0 +1,24 @@
+from run import Invalid
+
+
+def check(inst, ans):
+    W = inst["W"]
+    n = len(W)
+    A = ans.get("A")
+    if not isinstance(A, list) or len(A) != n:
+        raise Invalid(f"A must be a list of {n} strings")
+    for r in A:
+        if not isinstance(r, str) or len(r) != n or set(r) - {"0", "1", "2"}:
+            raise Invalid(f"each row must be a string of {n} digits 0/1/2")
+    for i in range(n):
+        if A[i][i] != "0":
+            raise Invalid("diagonal must be 0")
+        w = 0
+        for j in range(n):
+            if i != j:
+                if int(A[i][j]) + int(A[j][i]) != 2:
+                    raise Invalid(f"A[{i}][{j}] + A[{j}][{i}] != 2")
+                w += 3 * (A[i][j] == "2") + (A[i][j] == "1")
+        if w != W[i]:
+            raise Invalid(f"W[{i}] = {w}, need {W[i]}")
+    return None, {}
diff --git a/tasks/construct-rlve-round-robin-scores-l3-s2/tests/instance.json b/tasks/construct-rlve-round-robin-scores-l3-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..de93e6699e9791980b9a5098b90d8d00451a5d06
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-l3-s2/tests/instance.json
@@ -0,0 +1 @@
+{"W": [10, 8, 8, 10, 10, 9, 7, 9, 6], "family": "rlve_round_robin_scores", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-round-robin-scores-l3-s2/tests/run.py b/tasks/construct-rlve-round-robin-scores-l3-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-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-round-robin-scores-l3-s2/tests/test.sh b/tasks/construct-rlve-round-robin-scores-l3-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-round-robin-scores-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-shortest-path-count-l4-s0/environment/Dockerfile b/tasks/construct-rlve-shortest-path-count-l4-s0/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-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-shortest-path-count-l4-s0/environment/instance.json b/tasks/construct-rlve-shortest-path-count-l4-s0/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..24cf34d14e529e7404a759a557042f1909f2dd55
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-l4-s0/environment/instance.json
@@ -0,0 +1 @@
+{"K": 80742759, "n_max": 165, "family": "rlve_shortest_path_count", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-shortest-path-count-l4-s0/solution/solve.sh b/tasks/construct-rlve-shortest-path-count-l4-s0/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..5febadc0f2bddcc84760524ed4d1561525e95d60
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-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'
+{"n": 81, "adj": ["NNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNY", "YNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNN", "YNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNN", "NNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNN", "NNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNY", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNY", "YNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNN", "NNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNNNNNNNNNN", "NNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYNN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNYN", "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNY", "NYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNYN"]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-shortest-path-count-l4-s0/tests/check.py b/tasks/construct-rlve-shortest-path-count-l4-s0/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..c59d4c4f4bb4e824a2019380643a5b3ade0ca974
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-l4-s0/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-l4-s0/tests/instance.json b/tasks/construct-rlve-shortest-path-count-l4-s0/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..24cf34d14e529e7404a759a557042f1909f2dd55
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-l4-s0/tests/instance.json
@@ -0,0 +1 @@
+{"K": 80742759, "n_max": 165, "family": "rlve_shortest_path_count", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-shortest-path-count-l4-s0/tests/run.py b/tasks/construct-rlve-shortest-path-count-l4-s0/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-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-shortest-path-count-l4-s0/tests/test.sh b/tasks/construct-rlve-shortest-path-count-l4-s0/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-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-shortest-path-count-l5-s9/tests/check.py b/tasks/construct-rlve-shortest-path-count-l5-s9/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..c59d4c4f4bb4e824a2019380643a5b3ade0ca974
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-l5-s9/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-l5-s9/tests/instance.json b/tasks/construct-rlve-shortest-path-count-l5-s9/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..856c84378b222f37d34c1231d6096a87cb4f6aee
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-l5-s9/tests/instance.json
@@ -0,0 +1 @@
+{"K": 3025324468125, "n_max": 255, "family": "rlve_shortest_path_count", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-shortest-path-count-l5-s9/tests/run.py b/tasks/construct-rlve-shortest-path-count-l5-s9/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-shortest-path-count-l5-s9/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l3-s4/environment/Dockerfile b/tasks/construct-rlve-smallest-enclosing-circle-l3-s4/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-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-smallest-enclosing-circle-l3-s4/environment/instance.json b/tasks/construct-rlve-smallest-enclosing-circle-l3-s4/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..6375bb55c4cad95c90d796e23dbeae7a1ce99983
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l3-s4/environment/instance.json
@@ -0,0 +1 @@
+{"points": [[0, 6], [2, 3], [5, 30], [6, 22], [7, 6], [7, 23], [10, 5], [18, 3], [19, 15], [21, 4], [22, 20], [25, 7], [25, 17], [27, 5], [28, 28]], "r_opt": 18.034688796871432, "r2_num": 1301, "r2_den": 4, "family": "rlve_smallest_enclosing_circle", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l3-s4/solution/solve.sh b/tasks/construct-rlve-smallest-enclosing-circle-l3-s4/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..edf5716fa6b7f4c307d6d6506e4953689ea9cca9
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l3-s4/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": 15.0, "y": 15.5, "r": 18.034688797871432}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l3-s4/tests/check.py b/tasks/construct-rlve-smallest-enclosing-circle-l3-s4/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..0eabf3219588f98db42bf5fd305dc24ce21ca495
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l3-s4/tests/check.py
@@ -0,0 +1,17 @@
+import math
+
+from run import Invalid, num
+
+TOL = 1e-6
+
+
+def check(inst, ans):
+    x, y, r = num(ans.get("x"), "x"), num(ans.get("y"), "y"), num(ans.get("r"), "r")
+    if r <= 0:
+        raise Invalid("r must be positive")
+    if r > inst["r_opt"] + TOL:
+        raise Invalid(f"r = {r} exceeds the minimum radius {inst['r_opt']} by more than {TOL}")
+    for px, py in inst["points"]:
+        if math.hypot(px - x, py - y) > r + TOL:
+            raise Invalid(f"point ({px}, {py}) is outside the circle")
+    return None, {"r": r}
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l3-s4/tests/instance.json b/tasks/construct-rlve-smallest-enclosing-circle-l3-s4/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..6375bb55c4cad95c90d796e23dbeae7a1ce99983
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l3-s4/tests/instance.json
@@ -0,0 +1 @@
+{"points": [[0, 6], [2, 3], [5, 30], [6, 22], [7, 6], [7, 23], [10, 5], [18, 3], [19, 15], [21, 4], [22, 20], [25, 7], [25, 17], [27, 5], [28, 28]], "r_opt": 18.034688796871432, "r2_num": 1301, "r2_den": 4, "family": "rlve_smallest_enclosing_circle", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l3-s4/tests/run.py b/tasks/construct-rlve-smallest-enclosing-circle-l3-s4/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-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-smallest-enclosing-circle-l3-s4/tests/test.sh b/tasks/construct-rlve-smallest-enclosing-circle-l3-s4/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-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-smallest-enclosing-circle-l6-s2/environment/Dockerfile b/tasks/construct-rlve-smallest-enclosing-circle-l6-s2/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-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-smallest-enclosing-circle-l6-s2/environment/instance.json b/tasks/construct-rlve-smallest-enclosing-circle-l6-s2/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1b82114cf7fd5fc7f30d8764d8d36a844b5f46cb
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l6-s2/environment/instance.json
@@ -0,0 +1 @@
+{"points": [[1, 49], [3, 59], [3, 67], [4, 0], [7, 35], [15, 113], [15, 209], [18, 23], [19, 170], [27, 201], [28, 186], [28, 233], [29, 4], [29, 89], [31, 182], [33, 140], [33, 213], [42, 89], [42, 227], [44, 137], [45, 63], [47, 105], [48, 38], [51, 24], [52, 67], [52, 96], [56, 164], [58, 134], [62, 155], [71, 124], [75, 57], [80, 71], [82, 118], [83, 236], [84, 49], [86, 40], [86, 225], [87, 107], [88, 129], [88, 208], [92, 103], [93, 202], [96, 52], [97, 143], [100, 114], [109, 5], [110, 174], [111, 139], [111, 239], [116, 194], [119, 196], [120, 78], [124, 135], [124, 203], [127, 14], [129, 188], [136, 222], [137, 182], [139, 60], [140, 3], [142, 79], [144, 122], [146, 44], [149, 212], [152, 123], [153, 21], [154, 38], [154, 186], [157, 205], [157, 210], [161, 155], [163, 232], [165, 221], [165, 235], [166, 166], [166, 238], [167, 26], [168, 152], [169, 220], [170, 152], [170, 224], [180, 227], [185, 53], [185, 214], [186, 36], [186, 79], [187, 141], [187, 238], [190, 8], [193, 114], [194, 192], [194, 203], [196, 219], [198, 84], [198, 124], [200, 11], [201, 198], [201, 210], [205, 223], [206, 28], [209, 180], [211, 52], [213, 35], [213, 208], [214, 31], [214, 212], [217, 5], [218, 140], [220, 223], [222, 120], [223, 84], [224, 121], [226, 222], [227, 43], [227, 240], [234, 204], [235, 176], [238, 55], [238, 201], [239, 174]], "r_opt": 163.80552493734757, "r2_num": 107329, "r2_den": 4, "family": "rlve_smallest_enclosing_circle", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l6-s2/solution/solve.sh b/tasks/construct-rlve-smallest-enclosing-circle-l6-s2/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..1a00edf25b9a2ac17bb6cea51453e76e2c62cbd3
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l6-s2/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"x": 115.5, "y": 120.0, "r": 163.80552493834756}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l6-s2/tests/check.py b/tasks/construct-rlve-smallest-enclosing-circle-l6-s2/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..0eabf3219588f98db42bf5fd305dc24ce21ca495
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l6-s2/tests/check.py
@@ -0,0 +1,17 @@
+import math
+
+from run import Invalid, num
+
+TOL = 1e-6
+
+
+def check(inst, ans):
+    x, y, r = num(ans.get("x"), "x"), num(ans.get("y"), "y"), num(ans.get("r"), "r")
+    if r <= 0:
+        raise Invalid("r must be positive")
+    if r > inst["r_opt"] + TOL:
+        raise Invalid(f"r = {r} exceeds the minimum radius {inst['r_opt']} by more than {TOL}")
+    for px, py in inst["points"]:
+        if math.hypot(px - x, py - y) > r + TOL:
+            raise Invalid(f"point ({px}, {py}) is outside the circle")
+    return None, {"r": r}
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l6-s2/tests/instance.json b/tasks/construct-rlve-smallest-enclosing-circle-l6-s2/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..1b82114cf7fd5fc7f30d8764d8d36a844b5f46cb
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-l6-s2/tests/instance.json
@@ -0,0 +1 @@
+{"points": [[1, 49], [3, 59], [3, 67], [4, 0], [7, 35], [15, 113], [15, 209], [18, 23], [19, 170], [27, 201], [28, 186], [28, 233], [29, 4], [29, 89], [31, 182], [33, 140], [33, 213], [42, 89], [42, 227], [44, 137], [45, 63], [47, 105], [48, 38], [51, 24], [52, 67], [52, 96], [56, 164], [58, 134], [62, 155], [71, 124], [75, 57], [80, 71], [82, 118], [83, 236], [84, 49], [86, 40], [86, 225], [87, 107], [88, 129], [88, 208], [92, 103], [93, 202], [96, 52], [97, 143], [100, 114], [109, 5], [110, 174], [111, 139], [111, 239], [116, 194], [119, 196], [120, 78], [124, 135], [124, 203], [127, 14], [129, 188], [136, 222], [137, 182], [139, 60], [140, 3], [142, 79], [144, 122], [146, 44], [149, 212], [152, 123], [153, 21], [154, 38], [154, 186], [157, 205], [157, 210], [161, 155], [163, 232], [165, 221], [165, 235], [166, 166], [166, 238], [167, 26], [168, 152], [169, 220], [170, 152], [170, 224], [180, 227], [185, 53], [185, 214], [186, 36], [186, 79], [187, 141], [187, 238], [190, 8], [193, 114], [194, 192], [194, 203], [196, 219], [198, 84], [198, 124], [200, 11], [201, 198], [201, 210], [205, 223], [206, 28], [209, 180], [211, 52], [213, 35], [213, 208], [214, 31], [214, 212], [217, 5], [218, 140], [220, 223], [222, 120], [223, 84], [224, 121], [226, 222], [227, 43], [227, 240], [234, 204], [235, 176], [238, 55], [238, 201], [239, 174]], "r_opt": 163.80552493734757, "r2_num": 107329, "r2_den": 4, "family": "rlve_smallest_enclosing_circle", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-smallest-enclosing-circle-l6-s2/tests/run.py b/tasks/construct-rlve-smallest-enclosing-circle-l6-s2/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-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-smallest-enclosing-circle-l6-s2/tests/test.sh b/tasks/construct-rlve-smallest-enclosing-circle-l6-s2/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-smallest-enclosing-circle-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-subgraph-isomorphism-l3-s1/environment/Dockerfile b/tasks/construct-rlve-subgraph-isomorphism-l3-s1/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-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-subgraph-isomorphism-l3-s1/environment/instance.json b/tasks/construct-rlve-subgraph-isomorphism-l3-s1/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3c671c0e6d53ae8e935f33b296aa1d2299ed3a0b
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l3-s1/environment/instance.json
@@ -0,0 +1 @@
+{"n1": 8, "n2": 12, "edges1": [[4, 6], [2, 4], [0, 4], [6, 7], [1, 4], [1, 5], [0, 6], [2, 3], [4, 5], [3, 7], [1, 7], [0, 1], [0, 3], [4, 7], [3, 5], [2, 7]], "edges2": [[1, 6], [5, 9], [0, 10], [3, 4], [3, 10], [0, 3], [1, 8], [6, 10], [4, 6], [0, 8], [7, 8], [0, 6], [2, 8], [1, 11], [4, 5], [1, 5], [8, 11], [5, 10], [4, 9], [0, 2], [7, 10], [9, 10], [0, 9], [3, 7], [4, 8], [0, 1], [4, 10], [2, 7], [10, 11], [0, 7], [1, 3], [1, 2], [2, 3], [1, 7], [5, 8], [6, 11]], "family": "rlve_subgraph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l3-s1/solution/solve.sh b/tasks/construct-rlve-subgraph-isomorphism-l3-s1/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..59d55b438050a66bb2ed83845c66f29276ebdbe4
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-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": [4, 6, 7, 8, 10, 11, 9, 0]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l3-s1/tests/check.py b/tasks/construct-rlve-subgraph-isomorphism-l3-s1/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..b8467dd00c27c8902c06206c376ff7a11ac78ae0
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l3-s1/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-l3-s1/tests/instance.json b/tasks/construct-rlve-subgraph-isomorphism-l3-s1/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3c671c0e6d53ae8e935f33b296aa1d2299ed3a0b
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l3-s1/tests/instance.json
@@ -0,0 +1 @@
+{"n1": 8, "n2": 12, "edges1": [[4, 6], [2, 4], [0, 4], [6, 7], [1, 4], [1, 5], [0, 6], [2, 3], [4, 5], [3, 7], [1, 7], [0, 1], [0, 3], [4, 7], [3, 5], [2, 7]], "edges2": [[1, 6], [5, 9], [0, 10], [3, 4], [3, 10], [0, 3], [1, 8], [6, 10], [4, 6], [0, 8], [7, 8], [0, 6], [2, 8], [1, 11], [4, 5], [1, 5], [8, 11], [5, 10], [4, 9], [0, 2], [7, 10], [9, 10], [0, 9], [3, 7], [4, 8], [0, 1], [4, 10], [2, 7], [10, 11], [0, 7], [1, 3], [1, 2], [2, 3], [1, 7], [5, 8], [6, 11]], "family": "rlve_subgraph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l3-s1/tests/run.py b/tasks/construct-rlve-subgraph-isomorphism-l3-s1/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-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-subgraph-isomorphism-l3-s1/tests/test.sh b/tasks/construct-rlve-subgraph-isomorphism-l3-s1/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-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-subgraph-isomorphism-l3-s8/environment/Dockerfile b/tasks/construct-rlve-subgraph-isomorphism-l3-s8/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-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-subgraph-isomorphism-l3-s8/environment/instance.json b/tasks/construct-rlve-subgraph-isomorphism-l3-s8/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a5ea1edb54f494df8e009de59b6d186c9013ba26
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l3-s8/environment/instance.json
@@ -0,0 +1 @@
+{"n1": 6, "n2": 12, "edges1": [[2, 5], [3, 4], [1, 5], [2, 3], [2, 4], [0, 3], [1, 2]], "edges2": [[3, 5], [6, 11], [0, 10], [2, 5], [7, 8], [6, 9], [9, 10], [0, 6], [4, 6], [0, 11], [5, 10], [7, 11], [3, 8], [0, 8], [6, 8], [2, 9], [10, 11], [0, 2], [5, 6], [1, 8], [1, 11], [4, 8], [0, 3], [4, 11], [0, 1], [1, 7], [0, 9], [1, 4], [9, 11], [8, 11], [2, 7], [5, 8], [3, 6], [6, 10], [5, 11], [2, 4]], "family": "rlve_subgraph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l3-s8/solution/solve.sh b/tasks/construct-rlve-subgraph-isomorphism-l3-s8/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..fa2f4cb7d0da45aeb1a1bc02be2b29a593cb7ede
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-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": [9, 7, 8, 6, 5, 1]}
+RLMATH_EOF
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l3-s8/tests/check.py b/tasks/construct-rlve-subgraph-isomorphism-l3-s8/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..b8467dd00c27c8902c06206c376ff7a11ac78ae0
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l3-s8/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-l3-s8/tests/instance.json b/tasks/construct-rlve-subgraph-isomorphism-l3-s8/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..a5ea1edb54f494df8e009de59b6d186c9013ba26
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-isomorphism-l3-s8/tests/instance.json
@@ -0,0 +1 @@
+{"n1": 6, "n2": 12, "edges1": [[2, 5], [3, 4], [1, 5], [2, 3], [2, 4], [0, 3], [1, 2]], "edges2": [[3, 5], [6, 11], [0, 10], [2, 5], [7, 8], [6, 9], [9, 10], [0, 6], [4, 6], [0, 11], [5, 10], [7, 11], [3, 8], [0, 8], [6, 8], [2, 9], [10, 11], [0, 2], [5, 6], [1, 8], [1, 11], [4, 8], [0, 3], [4, 11], [0, 1], [1, 7], [0, 9], [1, 4], [9, 11], [8, 11], [2, 7], [5, 8], [3, 6], [6, 10], [5, 11], [2, 4]], "family": "rlve_subgraph_isomorphism", "subset": "construct"}
\ No newline at end of file
diff --git a/tasks/construct-rlve-subgraph-isomorphism-l3-s8/tests/run.py b/tasks/construct-rlve-subgraph-isomorphism-l3-s8/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-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-subgraph-isomorphism-l3-s8/tests/test.sh b/tasks/construct-rlve-subgraph-isomorphism-l3-s8/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/construct-rlve-subgraph-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/optimize-finch-4subset-3dist-n16/environment/Dockerfile b/tasks/optimize-finch-4subset-3dist-n16/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/optimize-finch-4subset-3dist-n16/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==5.1.0.0 PuLP==3.3.2 ortools==9.15.6755
+WORKDIR /workdir
+COPY instance.json /workdir/instance.json
diff --git a/tasks/optimize-finch-4subset-3dist-n16/environment/instance.json b/tasks/optimize-finch-4subset-3dist-n16/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..234d6ae23bbbe4c8bb577d34c84511fb0739bba0
--- /dev/null
+++ b/tasks/optimize-finch-4subset-3dist-n16/environment/instance.json
@@ -0,0 +1 @@
+{"n": 16, "direction": "minimize", "family": "finch_4subset_3dist_16", "subset": "optimize"}
\ No newline at end of file
diff --git a/tasks/optimize-finch-4subset-3dist-n16/solution/solve.sh b/tasks/optimize-finch-4subset-3dist-n16/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..53de67aa9a9f80c7c50033d8964342a23944d0e9
--- /dev/null
+++ b/tasks/optimize-finch-4subset-3dist-n16/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"points": [[6, 5], [3, 3], [4, 5], [5, 3], [4, 6], [4, 4], [2, 5], [5, 6], [6, 4], [6, 2], [3, 6], [2, 4], [2, 2], [2, 7], [6, 7], [4, 3]]}
+RLMATH_EOF
diff --git a/tasks/optimize-finch-4subset-3dist-n16/tests/check.py b/tasks/optimize-finch-4subset-3dist-n16/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..bc93325cc8cb26a11bde0b73b15ad88bec05f735
--- /dev/null
+++ b/tasks/optimize-finch-4subset-3dist-n16/tests/check.py
@@ -0,0 +1,50 @@
+"""Erdos #659, n = 16: every 4 points determine at least 3 distinct distances; minimize the total
+number of distinct distances (Finch/EFT skydiscover erdos_659_4subset_3dist_construction, Apache-2.0).
+
+Upstream bug fixed: distances were grouped by a chained relative tolerance 1e-9 over the sorted list, so
+many slightly different distances could be merged into one class. Here coordinates are exact rationals
+and squared distances are compared exactly."""
+import itertools
+import math
+import re
+from fractions import Fraction
+
+from run import Invalid
+
+RAT = re.compile(r"^[+-]?\d{1,30}(\.\d{1,30})?(/[1-9]\d{0,29})?$")
+
+
+def rat(x):
+    if isinstance(x, bool):
+        raise Invalid("coordinates must be numbers or rational strings")
+    if isinstance(x, int):
+        return Fraction(x)
+    if isinstance(x, float):
+        if not math.isfinite(x):
+            raise Invalid("non-finite number")
+        return Fraction(x)
+    if isinstance(x, str) and RAT.fullmatch(x.strip()):
+        return Fraction(x.strip())
+    raise Invalid("coordinates must be numbers or rational strings such as '1/3'")
+
+
+def check(inst, ans):
+    n = inst["n"]
+    pts = ans.get("points")
+    if not isinstance(pts, list) or len(pts) != n:
+        raise Invalid(f"points must be a list of exactly {n} pairs")
+    P = []
+    for q in pts:
+        if not isinstance(q, list) or len(q) != 2:
+            raise Invalid("each point is [x, y]")
+        P.append((rat(q[0]), rat(q[1])))
+    if len(set(P)) != n:
+        raise Invalid("points must be distinct")
+    D = {}
+    for i in range(n):
+        for j in range(i + 1, n):
+            D[i, j] = (P[i][0] - P[j][0]) ** 2 + (P[i][1] - P[j][1]) ** 2
+    for q in itertools.combinations(range(n), 4):
+        if len({D[a, b] for a, b in itertools.combinations(q, 2)}) < 3:
+            raise Invalid(f"points {list(q)} determine fewer than 3 distinct distances")
+    return float(len(set(D.values()))), {}
diff --git a/tasks/optimize-finch-4subset-3dist-n16/tests/instance.json b/tasks/optimize-finch-4subset-3dist-n16/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..0bf60c32cb19ebc50e4a7fce9e4db42fcb98b811
--- /dev/null
+++ b/tasks/optimize-finch-4subset-3dist-n16/tests/instance.json
@@ -0,0 +1 @@
+{"n": 16, "direction": "minimize", "best_known": 13.0, "baseline": 15.0, "family": "finch_4subset_3dist_16", "subset": "optimize"}
\ No newline at end of file
diff --git a/tasks/optimize-finch-4subset-3dist-n16/tests/run.py b/tasks/optimize-finch-4subset-3dist-n16/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/optimize-finch-4subset-3dist-n16/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/verifier"):
+        with open("/logs/verifier/details.json", "w") as f:
+            json.dump(det, f)
+    print(reward)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/tasks/optimize-finch-4subset-3dist-n16/tests/test.sh b/tasks/optimize-finch-4subset-3dist-n16/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/optimize-finch-4subset-3dist-n16/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt
diff --git a/tasks/optimize-fs-cap-set-n6/environment/Dockerfile b/tasks/optimize-fs-cap-set-n6/environment/Dockerfile
new file mode 100644
index 0000000000000000000000000000000000000000..06d0365485bd1fc9437a07fafe38cb8333e9ac65
--- /dev/null
+++ b/tasks/optimize-fs-cap-set-n6/environment/Dockerfile
@@ -0,0 +1,7 @@
+FROM python:3.11.16-slim
+RUN apt-get update && apt-get install -y --no-install-recommends tmux curl git procps build-essential \
+    && rm -rf /var/lib/apt/lists/*
+RUN pip install --no-cache-dir numpy==2.4.6 scipy==1.17.1 sympy==1.14.0 networkx==3.6.1 \
+    python-sat==1.9.dev15 z3-solver==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-fs-cap-set-n6/environment/instance.json b/tasks/optimize-fs-cap-set-n6/environment/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..3cf759ee95c50fae350ed40ac589c483058fddec
--- /dev/null
+++ b/tasks/optimize-fs-cap-set-n6/environment/instance.json
@@ -0,0 +1 @@
+{"n": 6, "direction": "maximize", "family": "fs_cap_set", "subset": "optimize"}
\ No newline at end of file
diff --git a/tasks/optimize-fs-cap-set-n6/solution/solve.sh b/tasks/optimize-fs-cap-set-n6/solution/solve.sh
new file mode 100644
index 0000000000000000000000000000000000000000..6bb7d6dcbe18af4e5884b1d965a9bb853554b400
--- /dev/null
+++ b/tasks/optimize-fs-cap-set-n6/solution/solve.sh
@@ -0,0 +1,5 @@
+#!/bin/bash
+# oracle: a known valid construction (planted witness or published solution)
+cat > /workdir/answer.json <<'RLMATH_EOF'
+{"vectors": [[1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 2, 2], [1, 1, 1, 2, 1, 2], [1, 1, 1, 2, 2, 1], [1, 1, 2, 1, 1, 2], [1, 1, 2, 1, 2, 1], [1, 1, 2, 2, 1, 1], [1, 1, 2, 2, 2, 2], [1, 2, 1, 1, 1, 2], [1, 2, 1, 1, 2, 1], [1, 2, 1, 2, 1, 1], [1, 2, 1, 2, 2, 2], [1, 2, 2, 1, 1, 1], [1, 2, 2, 1, 2, 2], [1, 2, 2, 2, 1, 2], [1, 2, 2, 2, 2, 1], [2, 1, 1, 1, 1, 2], [2, 1, 1, 1, 2, 1], [2, 1, 1, 2, 1, 1], [2, 1, 1, 2, 2, 2], [2, 1, 2, 1, 1, 1], [2, 1, 2, 1, 2, 2], [2, 1, 2, 2, 1, 2], [2, 1, 2, 2, 2, 1], [2, 2, 1, 1, 1, 1], [2, 2, 1, 1, 2, 2], [2, 2, 1, 2, 1, 2], [2, 2, 1, 2, 2, 1], [2, 2, 2, 1, 1, 2], [2, 2, 2, 1, 2, 1], [2, 2, 2, 2, 1, 1], [2, 2, 2, 2, 2, 2], [0, 0, 1, 1, 0, 1], [0, 0, 1, 1, 0, 2], [0, 0, 1, 1, 1, 0], [0, 0, 1, 1, 2, 0], [0, 0, 1, 2, 0, 1], [0, 0, 1, 2, 0, 2], [0, 0, 1, 2, 1, 0], [0, 0, 1, 2, 2, 0], [0, 0, 2, 1, 0, 1], [0, 0, 2, 1, 0, 2], [0, 0, 2, 1, 1, 0], [0, 0, 2, 1, 2, 0], [0, 0, 2, 2, 0, 1], [0, 0, 2, 2, 0, 2], [0, 0, 2, 2, 1, 0], [0, 0, 2, 2, 2, 0], [0, 1, 0, 0, 1, 1], [0, 1, 0, 0, 1, 2], [0, 1, 0, 0, 2, 1], [0, 1, 0, 0, 2, 2], [0, 1, 0, 1, 1, 0], [0, 1, 0, 1, 2, 0], [0, 1, 0, 2, 1, 0], [0, 1, 0, 2, 2, 0], [0, 1, 1, 0, 0, 1], [0, 1, 1, 0, 0, 2], [0, 1, 2, 0, 0, 1], [0, 1, 2, 0, 0, 2], [0, 2, 0, 0, 1, 1], [0, 2, 0, 0, 1, 2], [0, 2, 0, 0, 2, 1], [0, 2, 0, 0, 2, 2], [0, 2, 0, 1, 1, 0], [0, 2, 0, 1, 2, 0], [0, 2, 0, 2, 1, 0], [0, 2, 0, 2, 2, 0], [0, 2, 1, 0, 0, 1], [0, 2, 1, 0, 0, 2], [0, 2, 2, 0, 0, 1], [0, 2, 2, 0, 0, 2], [1, 0, 0, 0, 1, 1], [1, 0, 0, 0, 1, 2], [1, 0, 0, 0, 2, 1], [1, 0, 0, 0, 2, 2], [1, 0, 0, 1, 0, 1], [1, 0, 0, 1, 0, 2], [1, 0, 0, 2, 0, 1], [1, 0, 0, 2, 0, 2], [1, 0, 1, 0, 1, 0], [1, 0, 1, 0, 2, 0], [1, 0, 2, 0, 1, 0], [1, 0, 2, 0, 2, 0], [1, 1, 0, 1, 0, 0], [1, 1, 0, 2, 0, 0], [1, 1, 1, 0, 0, 0], [1, 1, 2, 0, 0, 0], [1, 2, 0, 1, 0, 0], [1, 2, 0, 2, 0, 0], [1, 2, 1, 0, 0, 0], [1, 2, 2, 0, 0, 0], [2, 0, 0, 0, 1, 1], [2, 0, 0, 0, 1, 2], [2, 0, 0, 0, 2, 1], [2, 0, 0, 0, 2, 2], [2, 0, 0, 1, 0, 1], [2, 0, 0, 1, 0, 2], [2, 0, 0, 2, 0, 1], [2, 0, 0, 2, 0, 2], [2, 0, 1, 0, 1, 0], [2, 0, 1, 0, 2, 0], [2, 0, 2, 0, 1, 0], [2, 0, 2, 0, 2, 0], [2, 1, 0, 1, 0, 0], [2, 1, 0, 2, 0, 0], [2, 1, 1, 0, 0, 0], [2, 1, 2, 0, 0, 0], [2, 2, 0, 1, 0, 0], [2, 2, 0, 2, 0, 0], [2, 2, 1, 0, 0, 0], [2, 2, 2, 0, 0, 0]]}
+RLMATH_EOF
diff --git a/tasks/optimize-fs-cap-set-n6/tests/check.py b/tasks/optimize-fs-cap-set-n6/tests/check.py
new file mode 100644
index 0000000000000000000000000000000000000000..e0b4309ffc9678bb5a2f8e30718868e5c68b1874
--- /dev/null
+++ b/tasks/optimize-fs-cap-set-n6/tests/check.py
@@ -0,0 +1,21 @@
+"""Cap set in F_3^n: no three distinct vectors x, y, z with x + y + z = 0 (mod 3). Score = size."""
+from run import Invalid, int_list
+
+
+def check(inst, ans):
+    n = inst["n"]
+    vecs = ans.get("vectors")
+    if not isinstance(vecs, list) or not vecs or len(vecs) > 3 ** n:
+        raise Invalid(f"vectors must be a non-empty list of at most 3^{n} vectors")
+    pts = [tuple(int_list(v, n, 0, 2, "vector")) for v in vecs]
+    if len(set(pts)) != len(pts):
+        raise Invalid("duplicate vectors")
+    # incremental check (as in the FunSearch notebook's is_cap_set): a point may not be the third point of a
+    # line through two earlier points; O(c^2 n) and stops at the first violation
+    blocked = set()
+    for i, p in enumerate(pts):
+        if p in blocked:
+            raise Invalid(f"vector {list(p)} completes a line with two earlier vectors")
+        for q in pts[:i]:
+            blocked.add(tuple((-a - b) % 3 for a, b in zip(p, q)))
+    return float(len(pts)), {"size": len(pts)}
diff --git a/tasks/optimize-fs-cap-set-n6/tests/instance.json b/tasks/optimize-fs-cap-set-n6/tests/instance.json
new file mode 100644
index 0000000000000000000000000000000000000000..4058f6c0324301ae49e20bc0c652ef9184a8a3b9
--- /dev/null
+++ b/tasks/optimize-fs-cap-set-n6/tests/instance.json
@@ -0,0 +1 @@
+{"n": 6, "direction": "maximize", "best_known": 112, "baseline": 64, "family": "fs_cap_set", "subset": "optimize"}
\ No newline at end of file
diff --git a/tasks/optimize-fs-cap-set-n6/tests/run.py b/tasks/optimize-fs-cap-set-n6/tests/run.py
new file mode 100644
index 0000000000000000000000000000000000000000..9f6db6107aa1f8132aeb1abdcf9bbf100792e48a
--- /dev/null
+++ b/tasks/optimize-fs-cap-set-n6/tests/run.py
@@ -0,0 +1,127 @@
+"""rlmath runtime grader. Copied into every task as /tests/run.py next to /tests/check.py.
+
+Deterministic; no LLM in the reward path. Runs with python3 + numpy/scipy/sympy/networkx only.
+
+    python3 /tests/run.py /tests/instance.json /workdir/answer.json
+
+Prints the reward in [0, 1] and writes /logs/verifier/details.json when that directory exists.
+
+check.py must define  check(inst, ans) -> (value, extra_dict)  and raise Invalid for a bad answer.
+  construct : value is ignored; a returned check means the witness is valid -> reward 1.0
+  optimize  : value is the raw objective; reward = 0.1 + 0.9 * progress, where
+              progress = clip((s - baseline) / (best_known - baseline), 0, 1)       (maximize)
+                         clip((baseline - s) / (baseline - best_known), 0, 1)       (minimize)
+              inst must carry "direction", "baseline", "best_known".
+"""
+import json
+import math
+import os
+import sys
+
+VALID_FLOOR = 0.1
+
+
+class Invalid(Exception):
+    pass
+
+
+# ------------------------------------------------------------ input helpers
+
+def int_list(v, n=None, lo=None, hi=None, name="list"):
+    if not isinstance(v, list):
+        raise Invalid(f"{name} must be a list")
+    if n is not None and len(v) != n:
+        raise Invalid(f"{name} must have length {n}, got {len(v)}")
+    for x in v:
+        if isinstance(x, bool) or not isinstance(x, int):
+            raise Invalid(f"{name} must contain integers only")
+        if (lo is not None and x < lo) or (hi is not None and x > hi):
+            raise Invalid(f"{name} entry {x} out of range [{lo}, {hi}]")
+    return v
+
+
+def as_int(x, name="value", lo=None, hi=None):
+    return int_list([x], 1, lo, hi, name)[0]
+
+
+def num(x, name="value"):
+    if isinstance(x, bool) or not isinstance(x, (int, float)):
+        raise Invalid(f"{name} must be a number")
+    if not math.isfinite(x):
+        raise Invalid(f"{name} is not finite")
+    return float(x)
+
+
+def float_matrix(v, rows=None, cols=None, name="matrix"):
+    if not isinstance(v, list) or (rows is not None and len(v) != rows):
+        raise Invalid(f"{name} must be a list of {rows} rows")
+    out = []
+    for r in v:
+        if not isinstance(r, list) or (cols is not None and len(r) != cols):
+            raise Invalid(f"each row of {name} must have {cols} numbers")
+        out.append([num(x, name) for x in r])
+    return out
+
+
+def float_list(v, n=None, name="list"):
+    if not isinstance(v, list) or (n is not None and len(v) != n):
+        raise Invalid(f"{name} must be a list of length {n}")
+    return [num(x, name) for x in v]
+
+
+# ------------------------------------------------------------------ reward
+
+def optimize_reward(inst, s):
+    b, best = inst["baseline"], inst["best_known"]
+    if inst.get("progress_scale") == "log":
+        # for non-negative losses spanning orders of magnitude (e.g. overlap loss -> 0);
+        # progress is measured in log10(value + floor)
+        f = inst.get("log_floor", 1e-3)
+        b, best, s = (math.log10(max(x, 0.0) + f) for x in (b, best, s))
+    if inst["direction"] == "maximize":
+        p = (s - b) / (best - b)
+    else:
+        p = (b - s) / (b - best)
+    return VALID_FLOOR + (1 - VALID_FLOOR) * max(0.0, min(1.0, p))
+
+
+def grade(inst, ans, check):
+    try:
+        if not isinstance(ans, dict):
+            raise Invalid("answer must be a JSON object")
+        value, extra = check(inst, ans)
+        if inst.get("subset") == "optimize":
+            s = float(value)
+            if not math.isfinite(s):
+                raise Invalid("non-finite score")
+            return optimize_reward(inst, s), {"valid": True, "raw_score": s, **(extra or {})}
+        return 1.0, {"valid": True, **(extra or {})}
+    except Invalid as e:
+        return 0.0, {"valid": False, "error": str(e)}
+    except Exception as e:  # any malformed answer
+        return 0.0, {"valid": False, "error": f"{type(e).__name__}: {e}"}
+
+
+def main():
+    here = os.path.dirname(os.path.abspath(__file__))
+    sys.path.insert(0, here)
+    # import ourselves as `run` so check.py's `from run import Invalid` is the same class
+    import run as R  # noqa: E402
+    import check as family  # noqa: E402
+
+    inst = json.load(open(sys.argv[1]))
+    try:
+        ans = json.load(open(sys.argv[2]))
+    except Exception as e:
+        reward, det = 0.0, {"valid": False, "error": f"unreadable answer: {e}"}
+    else:
+        reward, det = R.grade(inst, ans, family.check)
+    det["reward"] = reward
+    if os.path.isdir("/logs/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-fs-cap-set-n6/tests/test.sh b/tasks/optimize-fs-cap-set-n6/tests/test.sh
new file mode 100644
index 0000000000000000000000000000000000000000..648521fa89684f62943a573f74ce3cb75cd3ae29
--- /dev/null
+++ b/tasks/optimize-fs-cap-set-n6/tests/test.sh
@@ -0,0 +1,9 @@
+#!/bin/bash
+mkdir -p /logs/verifier
+if [ ! -s /workdir/answer.json ]; then
+  echo '{"valid": false, "error": "no /workdir/answer.json"}' > /logs/verifier/details.json
+  echo 0 > /logs/verifier/reward.txt
+  exit 0
+fi
+timeout 290 python3 /tests/run.py /tests/instance.json /workdir/answer.json \
+  > /logs/verifier/reward.txt 2> /logs/verifier/grader.log || echo 0 > /logs/verifier/reward.txt