{"abstract":"The solution is reported for a time later than t_end.","category":"Physics integrator stability","checks":8,"contract":"solve(y0, lam, s, t_end, dt0, tol): integrate y'=-lam*y+s to t_end with the embedded Euler/Heun pair. err=|y_heun-y_euler|/max(1,|y|) at the step start. Accept if err<=tol (propagate Heun); grow h by min(2, 0.9*(tol/err)^(1/2)) (2 when err=0); on reject shrink by max(0.2, 0.9*(tol/err)^(1/2)). Steps are clipped to t_end; at most 500 attempts. Return [y, accepted, rejected, last h] rounded.","contract_signature":"y0, lam, s, t_end, dt0, tol","evaluation_group":"w2-physics_integrator_stability-adaptive-heun-euler","failed_approach":"Clipping to t_end instead of the remainder still allows overshoot late in the run.","family":"w2-physics_integrator_stability-adaptive-heun-euler-end-time-clipping","id":"FA-87051","implementations":{"attempt":{"sha256":"434daa75785d6546d822426c95214b35e0b9f77c7fd845617e67ea49c91b687d","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(y0, lam, s, t_end, dt0, tol):\n    def f(y):\n        return -lam * y + s\n    t, y, h = 0.0, y0, dt0\n    acc = rej = 0\n    while t < t_end - 1e-12 and acc + rej < 500:\n        h = min(h, t_end)\n        k1 = f(y)\n        k2 = f(y + h * k1)\n        y_low = y + h * k1\n        y_high = y + h / 2 * (k1 + k2)\n        err = abs(y_high - y_low) / max(1.0, abs(y))\n        if err <= tol:\n            t += h\n            y = y_high\n            acc += 1\n            grow = 2.0 if err == 0 else min(2.0, 0.9 * (tol / err) ** 0.5)\n            h = h * grow\n        else:\n            rej += 1\n            h = h * max(0.2, 0.9 * (tol / err) ** 0.5)\n    return [round(y, 6), acc, rej, round(h, 6)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[[[0.734, 1.789, 1.259, 1.5, 0.1, 0.001], [0.705988, 9, 0, 0.108235]], [[-0.952, 1.695, -1.777, 2.0, 0.3, 0.01], [-1.043824, 6, 1, 0.806875]], [[1.314, 3.637, 1.469, 1.0, 0.1, 0.001], [0.428088, 42, 2, 0.054341]], [[2.009, 1.907, -1.617, 0.5, 0.3, 0.01], [0.255071, 10, 1, 0.029016]], [[-2.661, 0.0, 0.567, 1.5, 0.05, 0.001], [-1.8105, 5, 0, 1.5]], [[1.665, 1.112, 1.969, 2.0, 0.05, 0.001], [1.758958, 11, 0, 0.396553]], [[1.329, 2.639, 0.535, 1.5, 0.7, 0.01], [0.225936, 17, 2, 0.200122]], [[-2.569, 4.482, -1.811, 1.5, 0.1, 0.001], [-0.406872, 68, 2, 0.019689]]], [[[-2.712, 3.277, 1.673, 1.0, 0.05, 0.01], [0.386621, 23, 1, 0.024826]], [[2.279, 0.391, 1.095, 2.0, 0.3, 0.001], [2.561598, 9, 1, 0.024918]], [[2.431, 0.0, 0.71, 1.0, 0.1, 0.01], [3.141, 4, 0, 0.6]], [[2.09, 3.725, -1.778, 1.0, 0.7, 0.0001], [-0.415384, 205, 4, 0.005852]], [[-2.161, 4.981, 0.821, 2.0, 0.05, 0.0001], [0.164701, 232, 2, 0.153215]], [[-0.62, 4.337, 0.838, 0.5, 0.3, 0.001], [0.100011, 32, 3, 0.010439]], [[2.544, 3.611, 0.037, 1.5, 0.05, 0.0001], [0.021528, 217, 2, 0.032019]], [[0.762, 3.516, -1.514, 2.0, 0.05, 0.01], [-0.428351, 21, 1, 0.517761]]], [[[0.886, 0.0, -0.594, 2.0, 0.1, 0.0001], [-0.302, 5, 0, 1.0]], [[1.412, 4.795, 1.541, 1.0, 0.3, 0.001], [0.330607, 50, 3, 0.043844]], [[-1.429, 4.357, 1.585, 2.0, 0.7, 0.001], [0.363343, 71, 3, 0.130298]], [[-2.866, 0.946, 0.95, 1.0, 0.3, 0.001], [-0.498791, 30, 2, 0.034169]], [[-0.217, 4.32, 1.674, 1.5, 0.1, 0.001], [0.38641, 42, 2, 0.111773]], [[2.311, 3.452, -0.798, 2.0, 0.1, 0.001], [-0.228411, 77, 2, 0.022342]], [[2.099, 1.458, -0.969, 2.0, 0.7, 0.01], [-0.512845, 21, 2, 0.154279]], [[1.267, 2.693, -0.039, 0.5, 0.7, 0.0001], [0.318914, 87, 3, 0.008112]]], [[[-2.363, 2.228, 1.475, 2.0, 0.3, 0.0001], [0.626886, 233, 3, 0.006708]], [[2.284, 0.645, 1.783, 1.0, 0.05, 0.0001], [2.512294, 20, 1, 0.061128]], [[-0.815, 0.978, -0.497, 0.5, 0.3, 0.001], [-0.696428, 7, 1, 0.050506]], [[-1.257, 0.0, -1.478, 0.5, 0.1, 0.01], [-1.996, 3, 0, 0.4]], [[0.219, 2.157, 0.867, 1.0, 0.1, 0.01], [0.379141, 6, 0, 0.311548]], [[-2.405, 0.0, -1.998, 1.5, 0.3, 0.0001], [-5.402, 3, 0, 1.2]], [[0.957, 4.675, -1.869, 1.0, 0.3, 0.0001], [-0.387109, 169, 3, 0.023272]], [[0.741, 0.0, 1.287, 1.5, 0.1, 0.001], [2.6715, 4, 0, 1.6]]], [[[-0.41, 3.365, -1.877, 2.0, 0.3, 0.0001], [-0.557606, 63, 3, 0.216167]], [[1.604, 3.262, 1.026, 1.0, 0.1, 0.0001], [0.363958, 141, 2, 0.004622]], [[-2.568, 3.406, -1.466, 1.0, 0.3, 0.001], [-0.501574, 54, 2, 0.042216]], [[1.532, 3.924, -1.443, 0.5, 0.7, 0.0001], [-0.100654, 134, 4, 0.001341]], [[-1.508, 0.973, -1.003, 1.5, 0.7, 0.001], [-1.141938, 17, 2, 0.103067]], [[-0.96, 2.462, -0.926, 1.5, 0.1, 0.001], [-0.390863, 35, 1, 0.112273]], [[2.555, 2.399, 1.507, 2.0, 0.3, 0.0001], [0.644093, 176, 3, 0.040274]], [[-1.261, 2.268, -0.675, 1.5, 0.3, 0.0001], [-0.329731, 128, 3, 0.007802]]]]\nfor i, (args, expected) in enumerate(fixtures[N-1]):\n    check(\"case %d\" % i, solve(*args), expected)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"},"broken":{"sha256":"d8babf6794b657f78b47294667e5adc06fe2af71b16f67733ad5f4bb9f18d9cc","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(y0, lam, s, t_end, dt0, tol):\n    def f(y):\n        return -lam * y + s\n    t, y, h = 0.0, y0, dt0\n    acc = rej = 0\n    while t < t_end - 1e-12 and acc + rej < 500:\n        k1 = f(y)\n        k2 = f(y + h * k1)\n        y_low = y + h * k1\n        y_high = y + h / 2 * (k1 + k2)\n        err = abs(y_high - y_low) / max(1.0, abs(y))\n        if err <= tol:\n            t += h\n            y = y_high\n            acc += 1\n            grow = 2.0 if err == 0 else min(2.0, 0.9 * (tol / err) ** 0.5)\n            h = h * grow\n        else:\n            rej += 1\n            h = h * max(0.2, 0.9 * (tol / err) ** 0.5)\n    return [round(y, 6), acc, rej, round(h, 6)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[[[0.734, 1.789, 1.259, 1.5, 0.1, 0.001], [0.705988, 9, 0, 0.108235]], [[-0.952, 1.695, -1.777, 2.0, 0.3, 0.01], [-1.043824, 6, 1, 0.806875]], [[1.314, 3.637, 1.469, 1.0, 0.1, 0.001], [0.428088, 42, 2, 0.054341]], [[2.009, 1.907, -1.617, 0.5, 0.3, 0.01], [0.255071, 10, 1, 0.029016]], [[-2.661, 0.0, 0.567, 1.5, 0.05, 0.001], [-1.8105, 5, 0, 1.5]], [[1.665, 1.112, 1.969, 2.0, 0.05, 0.001], [1.758958, 11, 0, 0.396553]], [[1.329, 2.639, 0.535, 1.5, 0.7, 0.01], [0.225936, 17, 2, 0.200122]], [[-2.569, 4.482, -1.811, 1.5, 0.1, 0.001], [-0.406872, 68, 2, 0.019689]]], [[[-2.712, 3.277, 1.673, 1.0, 0.05, 0.01], [0.386621, 23, 1, 0.024826]], [[2.279, 0.391, 1.095, 2.0, 0.3, 0.001], [2.561598, 9, 1, 0.024918]], [[2.431, 0.0, 0.71, 1.0, 0.1, 0.01], [3.141, 4, 0, 0.6]], [[2.09, 3.725, -1.778, 1.0, 0.7, 0.0001], [-0.415384, 205, 4, 0.005852]], [[-2.161, 4.981, 0.821, 2.0, 0.05, 0.0001], [0.164701, 232, 2, 0.153215]], [[-0.62, 4.337, 0.838, 0.5, 0.3, 0.001], [0.100011, 32, 3, 0.010439]], [[2.544, 3.611, 0.037, 1.5, 0.05, 0.0001], [0.021528, 217, 2, 0.032019]], [[0.762, 3.516, -1.514, 2.0, 0.05, 0.01], [-0.428351, 21, 1, 0.517761]]], [[[0.886, 0.0, -0.594, 2.0, 0.1, 0.0001], [-0.302, 5, 0, 1.0]], [[1.412, 4.795, 1.541, 1.0, 0.3, 0.001], [0.330607, 50, 3, 0.043844]], [[-1.429, 4.357, 1.585, 2.0, 0.7, 0.001], [0.363343, 71, 3, 0.130298]], [[-2.866, 0.946, 0.95, 1.0, 0.3, 0.001], [-0.498791, 30, 2, 0.034169]], [[-0.217, 4.32, 1.674, 1.5, 0.1, 0.001], [0.38641, 42, 2, 0.111773]], [[2.311, 3.452, -0.798, 2.0, 0.1, 0.001], [-0.228411, 77, 2, 0.022342]], [[2.099, 1.458, -0.969, 2.0, 0.7, 0.01], [-0.512845, 21, 2, 0.154279]], [[1.267, 2.693, -0.039, 0.5, 0.7, 0.0001], [0.318914, 87, 3, 0.008112]]], [[[-2.363, 2.228, 1.475, 2.0, 0.3, 0.0001], [0.626886, 233, 3, 0.006708]], [[2.284, 0.645, 1.783, 1.0, 0.05, 0.0001], [2.512294, 20, 1, 0.061128]], [[-0.815, 0.978, -0.497, 0.5, 0.3, 0.001], [-0.696428, 7, 1, 0.050506]], [[-1.257, 0.0, -1.478, 0.5, 0.1, 0.01], [-1.996, 3, 0, 0.4]], [[0.219, 2.157, 0.867, 1.0, 0.1, 0.01], [0.379141, 6, 0, 0.311548]], [[-2.405, 0.0, -1.998, 1.5, 0.3, 0.0001], [-5.402, 3, 0, 1.2]], [[0.957, 4.675, -1.869, 1.0, 0.3, 0.0001], [-0.387109, 169, 3, 0.023272]], [[0.741, 0.0, 1.287, 1.5, 0.1, 0.001], [2.6715, 4, 0, 1.6]]], [[[-0.41, 3.365, -1.877, 2.0, 0.3, 0.0001], [-0.557606, 63, 3, 0.216167]], [[1.604, 3.262, 1.026, 1.0, 0.1, 0.0001], [0.363958, 141, 2, 0.004622]], [[-2.568, 3.406, -1.466, 1.0, 0.3, 0.001], [-0.501574, 54, 2, 0.042216]], [[1.532, 3.924, -1.443, 0.5, 0.7, 0.0001], [-0.100654, 134, 4, 0.001341]], [[-1.508, 0.973, -1.003, 1.5, 0.7, 0.001], [-1.141938, 17, 2, 0.103067]], [[-0.96, 2.462, -0.926, 1.5, 0.1, 0.001], [-0.390863, 35, 1, 0.112273]], [[2.555, 2.399, 1.507, 2.0, 0.3, 0.0001], [0.644093, 176, 3, 0.040274]], [[-1.261, 2.268, -0.675, 1.5, 0.3, 0.0001], [-0.329731, 128, 3, 0.007802]]]]\nfor i, (args, expected) in enumerate(fixtures[N-1]):\n    check(\"case %d\" % i, solve(*args), expected)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"}},"limitations":"A deterministic bounded teaching model with stipulated toy conventions and rounded float output; not a production physics engine or a proof of numerical stability. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.","method":"Deterministic executable model with adversarial boundary fixtures.","provenance":{"created_by":"Failure Map","dependencies":"Python standard library","family":"w2-physics_integrator_stability-adaptive-heun-euler-end-time-clipping","generated_at":"2026-09-29T14:50:55.287868+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"Game and robotics physics loops depend on integrator update order, step control and stabilization terms; a wrong decision point turns a stable simulation into drifting or exploding motion.","root_cause":"The step is never clipped to t_end - t.","sha256":"28b803ddf697aa1818bca642c668c67744ae9579a7a9a79d02fa355d31b23a0c","title":"Adaptive steps overshoot the requested end time · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verified":true,"visibility":"public","verification":{"attempt":{"elapsed_ms":46.457,"exit_code":1,"observations":[{"actual":[0.705144,9,0,0.452604],"check":"case 0","expected":[0.705988,9,0,0.108235],"passed":false},{"actual":[-1.044234,6,1,0.841589],"check":"case 1","expected":[-1.043824,6,1,0.806875],"passed":false},{"actual":[0.425311,42,2,0.067739],"check":"case 2","expected":[0.428088,42,2,0.054341],"passed":false},{"actual":[0.164836,10,1,0.062678],"check":"case 3","expected":[0.255071,10,1,0.029016],"passed":false},{"actual":[-1.78215,5,0,1.6],"check":"case 4","expected":[-1.8105,5,0,1.5],"passed":false},{"actual":[1.760516,11,0,0.396553],"check":"case 5","expected":[1.758958,11,0,0.396553],"passed":false},{"actual":[0.220382,17,2,0.277955],"check":"case 6","expected":[0.225936,17,2,0.200122],"passed":false},{"actual":[-0.405776,68,2,0.165688],"check":"case 7","expected":[-0.406872,68,2,0.019689],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"case 0\", \"actual\": [0.705144, 9, 0, 0.452604], \"expected\": [0.705988, 9, 0, 0.108235], \"passed\": false}, {\"check\": \"case 1\", \"actual\": [-1.044234, 6, 1, 0.841589], \"expected\": [-1.043824, 6, 1, 0.806875], \"passed\": false}, {\"check\": \"case 2\", \"actual\": [0.425311, 42, 2, 0.067739], \"expected\": [0.428088, 42, 2, 0.054341], \"passed\": false}, {\"check\": \"case 3\", \"actual\": [0.164836, 10, 1, 0.062678], \"expected\": [0.255071, 10, 1, 0.029016], \"passed\": false}, {\"check\": \"case 4\", \"actual\": [-1.78215, 5, 0, 1.6], \"expected\": [-1.8105, 5, 0, 1.5], \"passed\": false}, {\"check\": \"case 5\", \"actual\": [1.760516, 11, 0, 0.396553], \"expected\": [1.758958, 11, 0, 0.396553], \"passed\": false}, {\"check\": \"case 6\", \"actual\": [0.220382, 17, 2, 0.277955], \"expected\": [0.225936, 17, 2, 0.200122], \"passed\": false}, {\"check\": \"case 7\", \"actual\": [-0.405776, 68, 2, 0.165688], \"expected\": [-0.406872, 68, 2, 0.019689], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":41.599,"exit_code":1,"observations":[{"actual":[0.705144,9,0,0.452604],"check":"case 0","expected":[0.705988,9,0,0.108235],"passed":false},{"actual":[-1.044234,6,1,0.841589],"check":"case 1","expected":[-1.043824,6,1,0.806875],"passed":false},{"actual":[0.425311,42,2,0.067739],"check":"case 2","expected":[0.428088,42,2,0.054341],"passed":false},{"actual":[0.164836,10,1,0.062678],"check":"case 3","expected":[0.255071,10,1,0.029016],"passed":false},{"actual":[-1.78215,5,0,1.6],"check":"case 4","expected":[-1.8105,5,0,1.5],"passed":false},{"actual":[1.760516,11,0,0.396553],"check":"case 5","expected":[1.758958,11,0,0.396553],"passed":false},{"actual":[0.220382,17,2,0.277955],"check":"case 6","expected":[0.225936,17,2,0.200122],"passed":false},{"actual":[-0.405776,68,2,0.165688],"check":"case 7","expected":[-0.406872,68,2,0.019689],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"case 0\", \"actual\": [0.705144, 9, 0, 0.452604], \"expected\": [0.705988, 9, 0, 0.108235], \"passed\": false}, {\"check\": \"case 1\", \"actual\": [-1.044234, 6, 1, 0.841589], \"expected\": [-1.043824, 6, 1, 0.806875], \"passed\": false}, {\"check\": \"case 2\", \"actual\": [0.425311, 42, 2, 0.067739], \"expected\": [0.428088, 42, 2, 0.054341], \"passed\": false}, {\"check\": \"case 3\", \"actual\": [0.164836, 10, 1, 0.062678], \"expected\": [0.255071, 10, 1, 0.029016], \"passed\": false}, {\"check\": \"case 4\", \"actual\": [-1.78215, 5, 0, 1.6], \"expected\": [-1.8105, 5, 0, 1.5], \"passed\": false}, {\"check\": \"case 5\", \"actual\": [1.760516, 11, 0, 0.396553], \"expected\": [1.758958, 11, 0, 0.396553], \"passed\": false}, {\"check\": \"case 6\", \"actual\": [0.220382, 17, 2, 0.277955], \"expected\": [0.225936, 17, 2, 0.200122], \"passed\": false}, {\"check\": \"case 7\", \"actual\": [-0.405776, 68, 2, 0.165688], \"expected\": [-0.406872, 68, 2, 0.019689], \"passed\": false}], \"passed\": false}\n"}},"member_only":{"stages":["fixed"],"fields":["implementations.fixed","verification.fixed","harness","repair"],"note":"The verified repair, its recorded checks, the repair description, and the scoring harness are available to members."}}