{"abstract":"A constant-slope phase keeps the tiny initial step forever and wastes work.","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":"A growth factor of 1.5 still deviates from the contracted cap.","family":"w2-physics_integrator_stability-adaptive-heun-euler-zero-error-growth","id":"FA-87061","implementations":{"attempt":{"sha256":"98db207aef38976f4d26e02be0b9c75039e9a4b85f70a2397004c9d4801dff68","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 - t)\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 = 1.5 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 = [[[[-2.23, 0.286, -1.937, 2.0, 0.1, 0.0001], [-4.208794, 48, 1, 0.056624]], [[-0.603, 0.0, 1.691, 2.0, 0.1, 0.0001], [2.779, 5, 0, 1.0]], [[0.355, 0.0, -0.581, 2.0, 0.3, 0.0001], [-0.807, 3, 0, 2.2]], [[1.314, 3.637, 1.469, 1.0, 0.1, 0.001], [0.428088, 42, 2, 0.054341]], [[-1.56, 0.0, -1.68, 1.5, 0.05, 0.001], [-4.08, 5, 0, 1.5]], [[2.009, 1.907, -1.617, 0.5, 0.3, 0.01], [0.255071, 10, 1, 0.029016]], [[1.996, 1.795, -1.923, 1.0, 0.3, 0.0001], [-0.561722, 156, 3, 0.009874]], [[1.301, 0.0, 1.346, 0.5, 0.1, 0.0001], [1.974, 3, 0, 0.4]]], [[[0.881, 2.451, 0.201, 1.5, 0.7, 0.0001], [0.102256, 121, 3, 0.035311]], [[-0.911, 2.367, 1.851, 1.5, 0.7, 0.0001], [0.733371, 173, 4, 0.019948]], [[-2.2, 0.0, -0.933, 1.0, 0.3, 0.001], [-3.133, 3, 0, 0.2]], [[0.366, 0.277, -0.958, 2.0, 0.3, 0.0001], [-1.260745, 75, 2, 0.009378]], [[0.034, 0.0, -0.536, 1.5, 0.3, 0.001], [-0.77, 3, 0, 1.2]], [[1.324, 1.927, -0.437, 2.0, 0.3, 0.001], [-0.193674, 55, 2, 0.105477]], [[1.164, 0.0, 0.608, 0.5, 0.05, 0.0001], [1.468, 4, 0, 0.3]], [[-2.161, 4.981, 0.821, 2.0, 0.05, 0.0001], [0.164701, 232, 2, 0.153215]]], [[[-2.605, 4.58, 1.505, 0.5, 0.1, 0.01], [0.029147, 18, 1, 0.021233]], [[-0.022, 0.0, -1.06, 2.0, 0.05, 0.001], [-2.142, 6, 0, 0.9]], [[-1.772, 4.412, -1.322, 0.5, 0.05, 0.0001], [-0.461828, 121, 2, 0.007093]], [[-2.296, 0.0, -1.609, 1.5, 0.3, 0.001], [-4.7095, 3, 0, 1.2]], [[-0.931, 0.0, 0.846, 1.0, 0.05, 0.01], [-0.085, 5, 0, 0.5]], [[-0.18, 0.0, 0.811, 2.0, 0.05, 0.0001], [1.442, 6, 0, 0.9]], [[0.886, 0.0, -0.594, 2.0, 0.1, 0.0001], [-0.302, 5, 0, 1.0]], [[0.299, 2.489, -1.677, 0.5, 0.3, 0.001], [-0.393348, 24, 2, 0.029906]]], [[[-2.124, 0.0, -1.31, 0.5, 0.7, 0.01], [-2.779, 1, 0, 1.0]], [[0.551, 2.318, 0.186, 1.5, 0.7, 0.0001], [0.094812, 92, 3, 0.032361]], [[1.022, 0.992, -0.734, 2.0, 0.05, 0.001], [-0.497403, 43, 1, 0.07961]], [[-2.5, 0.0, 0.727, 1.0, 0.05, 0.0001], [-1.773, 5, 0, 0.5]], [[-1.664, 0.608, -1.437, 1.0, 0.05, 0.01], [-1.981366, 5, 0, 0.426033]], [[-2.405, 0.0, -1.998, 1.5, 0.3, 0.0001], [-5.402, 3, 0, 1.2]], [[-1.257, 0.0, -1.478, 0.5, 0.1, 0.01], [-1.996, 3, 0, 0.4]], [[-2.419, 2.259, -1.694, 2.0, 0.1, 0.0001], [-0.768126, 161, 2, 0.040422]]], [[[2.915, 0.0, 1.496, 2.0, 0.1, 0.0001], [5.907, 5, 0, 1.0]], [[-1.909, 4.509, 1.302, 1.5, 0.05, 0.0001], [0.286193, 222, 2, 0.001448]], [[2.746, 4.491, -1.815, 2.0, 0.05, 0.001], [-0.403609, 86, 2, 0.189861]], [[-2.167, 0.0, 1.842, 2.0, 0.7, 0.0001], [1.517, 2, 0, 2.6]], [[1.902, 0.0, 1.399, 0.5, 0.1, 0.001], [2.6015, 3, 0, 0.4]], [[1.604, 3.262, 1.026, 1.0, 0.1, 0.0001], [0.363958, 141, 2, 0.004622]], [[-1.375, 1.447, 0.137, 2.0, 0.7, 0.01], [0.011153, 17, 2, 0.010327]], [[-1.008, 0.873, 1.67, 1.0, 0.05, 0.0001], [0.69286, 96, 2, 0.013152]]]]\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":"d5b7824d3324b3249e51127bb3c273ef1b23e70a7325a452740dcde9582d5bdd","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 - t)\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 = 1.0 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 = [[[[-2.23, 0.286, -1.937, 2.0, 0.1, 0.0001], [-4.208794, 48, 1, 0.056624]], [[-0.603, 0.0, 1.691, 2.0, 0.1, 0.0001], [2.779, 5, 0, 1.0]], [[0.355, 0.0, -0.581, 2.0, 0.3, 0.0001], [-0.807, 3, 0, 2.2]], [[1.314, 3.637, 1.469, 1.0, 0.1, 0.001], [0.428088, 42, 2, 0.054341]], [[-1.56, 0.0, -1.68, 1.5, 0.05, 0.001], [-4.08, 5, 0, 1.5]], [[2.009, 1.907, -1.617, 0.5, 0.3, 0.01], [0.255071, 10, 1, 0.029016]], [[1.996, 1.795, -1.923, 1.0, 0.3, 0.0001], [-0.561722, 156, 3, 0.009874]], [[1.301, 0.0, 1.346, 0.5, 0.1, 0.0001], [1.974, 3, 0, 0.4]]], [[[0.881, 2.451, 0.201, 1.5, 0.7, 0.0001], [0.102256, 121, 3, 0.035311]], [[-0.911, 2.367, 1.851, 1.5, 0.7, 0.0001], [0.733371, 173, 4, 0.019948]], [[-2.2, 0.0, -0.933, 1.0, 0.3, 0.001], [-3.133, 3, 0, 0.2]], [[0.366, 0.277, -0.958, 2.0, 0.3, 0.0001], [-1.260745, 75, 2, 0.009378]], [[0.034, 0.0, -0.536, 1.5, 0.3, 0.001], [-0.77, 3, 0, 1.2]], [[1.324, 1.927, -0.437, 2.0, 0.3, 0.001], [-0.193674, 55, 2, 0.105477]], [[1.164, 0.0, 0.608, 0.5, 0.05, 0.0001], [1.468, 4, 0, 0.3]], [[-2.161, 4.981, 0.821, 2.0, 0.05, 0.0001], [0.164701, 232, 2, 0.153215]]], [[[-2.605, 4.58, 1.505, 0.5, 0.1, 0.01], [0.029147, 18, 1, 0.021233]], [[-0.022, 0.0, -1.06, 2.0, 0.05, 0.001], [-2.142, 6, 0, 0.9]], [[-1.772, 4.412, -1.322, 0.5, 0.05, 0.0001], [-0.461828, 121, 2, 0.007093]], [[-2.296, 0.0, -1.609, 1.5, 0.3, 0.001], [-4.7095, 3, 0, 1.2]], [[-0.931, 0.0, 0.846, 1.0, 0.05, 0.01], [-0.085, 5, 0, 0.5]], [[-0.18, 0.0, 0.811, 2.0, 0.05, 0.0001], [1.442, 6, 0, 0.9]], [[0.886, 0.0, -0.594, 2.0, 0.1, 0.0001], [-0.302, 5, 0, 1.0]], [[0.299, 2.489, -1.677, 0.5, 0.3, 0.001], [-0.393348, 24, 2, 0.029906]]], [[[-2.124, 0.0, -1.31, 0.5, 0.7, 0.01], [-2.779, 1, 0, 1.0]], [[0.551, 2.318, 0.186, 1.5, 0.7, 0.0001], [0.094812, 92, 3, 0.032361]], [[1.022, 0.992, -0.734, 2.0, 0.05, 0.001], [-0.497403, 43, 1, 0.07961]], [[-2.5, 0.0, 0.727, 1.0, 0.05, 0.0001], [-1.773, 5, 0, 0.5]], [[-1.664, 0.608, -1.437, 1.0, 0.05, 0.01], [-1.981366, 5, 0, 0.426033]], [[-2.405, 0.0, -1.998, 1.5, 0.3, 0.0001], [-5.402, 3, 0, 1.2]], [[-1.257, 0.0, -1.478, 0.5, 0.1, 0.01], [-1.996, 3, 0, 0.4]], [[-2.419, 2.259, -1.694, 2.0, 0.1, 0.0001], [-0.768126, 161, 2, 0.040422]]], [[[2.915, 0.0, 1.496, 2.0, 0.1, 0.0001], [5.907, 5, 0, 1.0]], [[-1.909, 4.509, 1.302, 1.5, 0.05, 0.0001], [0.286193, 222, 2, 0.001448]], [[2.746, 4.491, -1.815, 2.0, 0.05, 0.001], [-0.403609, 86, 2, 0.189861]], [[-2.167, 0.0, 1.842, 2.0, 0.7, 0.0001], [1.517, 2, 0, 2.6]], [[1.902, 0.0, 1.399, 0.5, 0.1, 0.001], [2.6015, 3, 0, 0.4]], [[1.604, 3.262, 1.026, 1.0, 0.1, 0.0001], [0.363958, 141, 2, 0.004622]], [[-1.375, 1.447, 0.137, 2.0, 0.7, 0.01], [0.011153, 17, 2, 0.010327]], [[-1.008, 0.873, 1.67, 1.0, 0.05, 0.0001], [0.69286, 96, 2, 0.013152]]]]\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-zero-error-growth","generated_at":"2026-09-29T14:50:55.287598+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 err == 0 branch uses growth factor 1.","sha256":"d116e75071bc95408c7041877014ceed45f5fc0063f6df22c48be12fdf983ffb","title":"Exact steps never grow the step size · 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":41.101,"exit_code":1,"observations":[{"actual":[-4.208794,48,1,0.056624],"check":"case 0","expected":[-4.208794,48,1,0.056624],"passed":true},{"actual":[2.779,6,0,1.021875],"check":"case 1","expected":[2.779,5,0,1.0],"passed":false},{"actual":[-0.807,4,0,0.8625],"check":"case 2","expected":[-0.807,3,0,2.2],"passed":false},{"actual":[0.428088,42,2,0.054341],"check":"case 3","expected":[0.428088,42,2,0.054341],"passed":true},{"actual":[-4.08,7,0,0.691406],"check":"case 4","expected":[-4.08,5,0,1.5],"passed":false},{"actual":[0.255071,10,1,0.029016],"check":"case 5","expected":[0.255071,10,1,0.029016],"passed":true},{"actual":[-0.561722,156,3,0.009874],"check":"case 6","expected":[-0.561722,156,3,0.009874],"passed":true},{"actual":[1.974,4,0,0.0375],"check":"case 7","expected":[1.974,3,0,0.4],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"case 0\", \"actual\": [-4.208794, 48, 1, 0.056624], \"expected\": [-4.208794, 48, 1, 0.056624], \"passed\": true}, {\"check\": \"case 1\", \"actual\": [2.779, 6, 0, 1.021875], \"expected\": [2.779, 5, 0, 1.0], \"passed\": false}, {\"check\": \"case 2\", \"actual\": [-0.807, 4, 0, 0.8625], \"expected\": [-0.807, 3, 0, 2.2], \"passed\": false}, {\"check\": \"case 3\", \"actual\": [0.428088, 42, 2, 0.054341], \"expected\": [0.428088, 42, 2, 0.054341], \"passed\": true}, {\"check\": \"case 4\", \"actual\": [-4.08, 7, 0, 0.691406], \"expected\": [-4.08, 5, 0, 1.5], \"passed\": false}, {\"check\": \"case 5\", \"actual\": [0.255071, 10, 1, 0.029016], \"expected\": [0.255071, 10, 1, 0.029016], \"passed\": true}, {\"check\": \"case 6\", \"actual\": [-0.561722, 156, 3, 0.009874], \"expected\": [-0.561722, 156, 3, 0.009874], \"passed\": true}, {\"check\": \"case 7\", \"actual\": [1.974, 4, 0, 0.0375], \"expected\": [1.974, 3, 0, 0.4], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":40.362,"exit_code":1,"observations":[{"actual":[-4.208794,48,1,0.056624],"check":"case 0","expected":[-4.208794,48,1,0.056624],"passed":true},{"actual":[2.779,20,0,0.1],"check":"case 1","expected":[2.779,5,0,1.0],"passed":false},{"actual":[-0.807,7,0,0.2],"check":"case 2","expected":[-0.807,3,0,2.2],"passed":false},{"actual":[0.428088,42,2,0.054341],"check":"case 3","expected":[0.428088,42,2,0.054341],"passed":true},{"actual":[-4.08,30,0,0.05],"check":"case 4","expected":[-4.08,5,0,1.5],"passed":false},{"actual":[0.255071,10,1,0.029016],"check":"case 5","expected":[0.255071,10,1,0.029016],"passed":true},{"actual":[-0.561722,156,3,0.009874],"check":"case 6","expected":[-0.561722,156,3,0.009874],"passed":true},{"actual":[1.974,5,0,0.1],"check":"case 7","expected":[1.974,3,0,0.4],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"case 0\", \"actual\": [-4.208794, 48, 1, 0.056624], \"expected\": [-4.208794, 48, 1, 0.056624], \"passed\": true}, {\"check\": \"case 1\", \"actual\": [2.779, 20, 0, 0.1], \"expected\": [2.779, 5, 0, 1.0], \"passed\": false}, {\"check\": \"case 2\", \"actual\": [-0.807, 7, 0, 0.2], \"expected\": [-0.807, 3, 0, 2.2], \"passed\": false}, {\"check\": \"case 3\", \"actual\": [0.428088, 42, 2, 0.054341], \"expected\": [0.428088, 42, 2, 0.054341], \"passed\": true}, {\"check\": \"case 4\", \"actual\": [-4.08, 30, 0, 0.05], \"expected\": [-4.08, 5, 0, 1.5], \"passed\": false}, {\"check\": \"case 5\", \"actual\": [0.255071, 10, 1, 0.029016], \"expected\": [0.255071, 10, 1, 0.029016], \"passed\": true}, {\"check\": \"case 6\", \"actual\": [-0.561722, 156, 3, 0.009874], \"expected\": [-0.561722, 156, 3, 0.009874], \"passed\": true}, {\"check\": \"case 7\", \"actual\": [1.974, 5, 0, 0.1], \"expected\": [1.974, 3, 0, 0.4], \"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."}}