{"abstract":"Badly rejected steps retry too large and cause repeated rejections.","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 0.5 floor prevents the controller from shrinking far enough.","family":"w2-physics_integrator_stability-adaptive-heun-euler-rejection-shrink-rule","id":"FA-87066","implementations":{"attempt":{"sha256":"684d9108f52223523d1c8dbaf71b1583c22d862599e1adc08c4c6fba38cb47c5","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 = 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(0.5, 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 = [[[[-0.696, 4.83, -1.861, 0.5, 0.7, 0.0001], [-0.41309, 64, 3, 0.004672]], [[-0.952, 1.695, -1.777, 2.0, 0.3, 0.01], [-1.043824, 6, 1, 0.806875]], [[1.329, 2.639, 0.535, 1.5, 0.7, 0.01], [0.225936, 17, 2, 0.200122]], [[-0.678, 2.724, 0.55, 1.5, 0.05, 0.01], [0.185448, 16, 0, 0.164424]], [[-0.303, 0.644, 0.316, 1.5, 0.05, 0.01], [0.187358, 8, 0, 0.223575]], [[1.121, 4.954, -0.6, 0.5, 0.3, 0.01], [-0.014719, 14, 2, 0.068659]], [[-0.223, 4.209, -0.57, 1.5, 0.7, 0.01], [-0.136586, 8, 2, 0.375934]], [[-0.603, 0.0, 1.691, 2.0, 0.1, 0.0001], [2.779, 5, 0, 1.0]]], [[[0.248, 2.246, 0.565, 1.5, 0.7, 0.01], [0.249628, 2, 0, 1.166103]], [[2.431, 0.0, 0.71, 1.0, 0.1, 0.01], [3.141, 4, 0, 0.6]], [[-0.907, 0.0, 0.08, 2.0, 0.3, 0.0001], [-0.747, 3, 0, 2.2]], [[0.342, 2.406, 1.149, 1.0, 0.3, 0.001], [0.465139, 15, 2, 0.090253]], [[1.498, 1.599, -1.343, 1.5, 0.7, 0.0001], [-0.627468, 167, 3, 0.017123]], [[2.09, 3.725, -1.778, 1.0, 0.7, 0.0001], [-0.415384, 205, 4, 0.005852]], [[1.324, 1.927, -0.437, 2.0, 0.3, 0.001], [-0.193674, 55, 2, 0.105477]], [[-2.813, 4.676, -0.613, 1.0, 0.1, 0.01], [-0.158154, 23, 1, 0.132111]]], [[[-2.871, 0.0, 0.577, 0.5, 0.3, 0.001], [-2.5825, 2, 0, 0.4]], [[-0.304, 3.173, 1.878, 2.0, 0.1, 0.0001], [0.590274, 148, 2, 0.004498]], [[2.099, 1.458, -0.969, 2.0, 0.7, 0.01], [-0.512845, 21, 2, 0.154279]], [[-0.18, 0.0, 0.811, 2.0, 0.05, 0.0001], [1.442, 6, 0, 0.9]], [[1.402, 2.3, -1.252, 0.5, 0.1, 0.01], [0.073687, 10, 1, 0.066063]], [[1.749, 2.243, 1.405, 1.5, 0.05, 0.01], [0.667087, 15, 0, 0.241935]], [[2.731, 1.096, -1.616, 1.5, 0.1, 0.0001], [-0.661911, 164, 2, 0.005732]], [[-1.619, 0.0, 0.292, 1.0, 0.1, 0.0001], [-1.327, 4, 0, 0.6]]], [[[-2.345, 2.942, -1.888, 2.0, 0.7, 0.01], [-0.64795, 21, 2, 0.218574]], [[0.219, 2.157, 0.867, 1.0, 0.1, 0.01], [0.379141, 6, 0, 0.311548]], [[0.687, 3.619, 0.954, 0.5, 0.05, 0.01], [0.334636, 8, 0, 0.050261]], [[-1.101, 3.233, -1.581, 2.0, 0.1, 0.001], [-0.49013, 42, 2, 0.175182]], [[-2.44, 2.788, -0.813, 1.5, 0.7, 0.01], [-0.326363, 21, 2, 0.098371]], [[-2.521, 1.781, 1.018, 0.5, 0.3, 0.0001], [-0.697788, 83, 3, 0.000494]], [[-2.752, 2.034, 0.926, 1.5, 0.7, 0.001], [0.303269, 65, 3, 0.023728]], [[0.741, 0.0, 1.287, 1.5, 0.1, 0.001], [2.6715, 4, 0, 1.6]]], [[[-1.507, 4.389, 1.938, 2.0, 0.3, 0.01], [0.439365, 26, 2, 0.506375]], [[-0.306, 1.726, 0.514, 0.5, 0.3, 0.0001], [0.043044, 44, 3, 0.012094]], [[-1.261, 2.268, -0.675, 1.5, 0.3, 0.0001], [-0.329731, 128, 3, 0.007802]], [[1.532, 3.924, -1.443, 0.5, 0.7, 0.0001], [-0.100654, 134, 4, 0.001341]], [[2.878, 0.0, 1.169, 1.5, 0.7, 0.0001], [4.6315, 2, 0, 1.6]], [[-1.151, 1.35, -1.656, 0.5, 0.1, 0.001], [-1.188015, 5, 0, 0.046485]], [[1.917, 0.404, 1.543, 1.5, 0.1, 0.01], [2.779594, 6, 0, 0.239185]], [[2.057, 4.405, 0.797, 1.5, 0.1, 0.01], [0.185265, 23, 1, 0.310434]]]]\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":"df196a32c0c1b264abe196ed187709195efa2d489f1030688acf6ffff38386f9","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 = 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 * 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 = [[[[-0.696, 4.83, -1.861, 0.5, 0.7, 0.0001], [-0.41309, 64, 3, 0.004672]], [[-0.952, 1.695, -1.777, 2.0, 0.3, 0.01], [-1.043824, 6, 1, 0.806875]], [[1.329, 2.639, 0.535, 1.5, 0.7, 0.01], [0.225936, 17, 2, 0.200122]], [[-0.678, 2.724, 0.55, 1.5, 0.05, 0.01], [0.185448, 16, 0, 0.164424]], [[-0.303, 0.644, 0.316, 1.5, 0.05, 0.01], [0.187358, 8, 0, 0.223575]], [[1.121, 4.954, -0.6, 0.5, 0.3, 0.01], [-0.014719, 14, 2, 0.068659]], [[-0.223, 4.209, -0.57, 1.5, 0.7, 0.01], [-0.136586, 8, 2, 0.375934]], [[-0.603, 0.0, 1.691, 2.0, 0.1, 0.0001], [2.779, 5, 0, 1.0]]], [[[0.248, 2.246, 0.565, 1.5, 0.7, 0.01], [0.249628, 2, 0, 1.166103]], [[2.431, 0.0, 0.71, 1.0, 0.1, 0.01], [3.141, 4, 0, 0.6]], [[-0.907, 0.0, 0.08, 2.0, 0.3, 0.0001], [-0.747, 3, 0, 2.2]], [[0.342, 2.406, 1.149, 1.0, 0.3, 0.001], [0.465139, 15, 2, 0.090253]], [[1.498, 1.599, -1.343, 1.5, 0.7, 0.0001], [-0.627468, 167, 3, 0.017123]], [[2.09, 3.725, -1.778, 1.0, 0.7, 0.0001], [-0.415384, 205, 4, 0.005852]], [[1.324, 1.927, -0.437, 2.0, 0.3, 0.001], [-0.193674, 55, 2, 0.105477]], [[-2.813, 4.676, -0.613, 1.0, 0.1, 0.01], [-0.158154, 23, 1, 0.132111]]], [[[-2.871, 0.0, 0.577, 0.5, 0.3, 0.001], [-2.5825, 2, 0, 0.4]], [[-0.304, 3.173, 1.878, 2.0, 0.1, 0.0001], [0.590274, 148, 2, 0.004498]], [[2.099, 1.458, -0.969, 2.0, 0.7, 0.01], [-0.512845, 21, 2, 0.154279]], [[-0.18, 0.0, 0.811, 2.0, 0.05, 0.0001], [1.442, 6, 0, 0.9]], [[1.402, 2.3, -1.252, 0.5, 0.1, 0.01], [0.073687, 10, 1, 0.066063]], [[1.749, 2.243, 1.405, 1.5, 0.05, 0.01], [0.667087, 15, 0, 0.241935]], [[2.731, 1.096, -1.616, 1.5, 0.1, 0.0001], [-0.661911, 164, 2, 0.005732]], [[-1.619, 0.0, 0.292, 1.0, 0.1, 0.0001], [-1.327, 4, 0, 0.6]]], [[[-2.345, 2.942, -1.888, 2.0, 0.7, 0.01], [-0.64795, 21, 2, 0.218574]], [[0.219, 2.157, 0.867, 1.0, 0.1, 0.01], [0.379141, 6, 0, 0.311548]], [[0.687, 3.619, 0.954, 0.5, 0.05, 0.01], [0.334636, 8, 0, 0.050261]], [[-1.101, 3.233, -1.581, 2.0, 0.1, 0.001], [-0.49013, 42, 2, 0.175182]], [[-2.44, 2.788, -0.813, 1.5, 0.7, 0.01], [-0.326363, 21, 2, 0.098371]], [[-2.521, 1.781, 1.018, 0.5, 0.3, 0.0001], [-0.697788, 83, 3, 0.000494]], [[-2.752, 2.034, 0.926, 1.5, 0.7, 0.001], [0.303269, 65, 3, 0.023728]], [[0.741, 0.0, 1.287, 1.5, 0.1, 0.001], [2.6715, 4, 0, 1.6]]], [[[-1.507, 4.389, 1.938, 2.0, 0.3, 0.01], [0.439365, 26, 2, 0.506375]], [[-0.306, 1.726, 0.514, 0.5, 0.3, 0.0001], [0.043044, 44, 3, 0.012094]], [[-1.261, 2.268, -0.675, 1.5, 0.3, 0.0001], [-0.329731, 128, 3, 0.007802]], [[1.532, 3.924, -1.443, 0.5, 0.7, 0.0001], [-0.100654, 134, 4, 0.001341]], [[2.878, 0.0, 1.169, 1.5, 0.7, 0.0001], [4.6315, 2, 0, 1.6]], [[-1.151, 1.35, -1.656, 0.5, 0.1, 0.001], [-1.188015, 5, 0, 0.046485]], [[1.917, 0.404, 1.543, 1.5, 0.1, 0.01], [2.779594, 6, 0, 0.239185]], [[2.057, 4.405, 0.797, 1.5, 0.1, 0.01], [0.185265, 23, 1, 0.310434]]]]\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-rejection-shrink-rule","generated_at":"2026-09-29T14:50:55.368961+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 reject branch multiplies h by 0.5 ignoring the error ratio.","sha256":"22966c956e369f2eb834bd1fac9d8d63b6dbeadd6e64a69716a98fd6877c9c4e","title":"Rejected steps are always halved · 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":38.509,"exit_code":1,"observations":[{"actual":[-0.41309,64,7,0.004672],"check":"case 0","expected":[-0.41309,64,3,0.004672],"passed":false},{"actual":[-1.043824,6,1,0.806875],"check":"case 1","expected":[-1.043824,6,1,0.806875],"passed":true},{"actual":[0.225936,17,4,0.200122],"check":"case 2","expected":[0.225936,17,2,0.200122],"passed":false},{"actual":[0.185448,16,0,0.164424],"check":"case 3","expected":[0.185448,16,0,0.164424],"passed":true},{"actual":[0.187358,8,0,0.223575],"check":"case 4","expected":[0.187358,8,0,0.223575],"passed":true},{"actual":[-0.014719,14,4,0.068659],"check":"case 5","expected":[-0.014719,14,2,0.068659],"passed":false},{"actual":[-0.136586,8,3,0.375934],"check":"case 6","expected":[-0.136586,8,2,0.375934],"passed":false},{"actual":[2.779,5,0,1.0],"check":"case 7","expected":[2.779,5,0,1.0],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"case 0\", \"actual\": [-0.41309, 64, 7, 0.004672], \"expected\": [-0.41309, 64, 3, 0.004672], \"passed\": false}, {\"check\": \"case 1\", \"actual\": [-1.043824, 6, 1, 0.806875], \"expected\": [-1.043824, 6, 1, 0.806875], \"passed\": true}, {\"check\": \"case 2\", \"actual\": [0.225936, 17, 4, 0.200122], \"expected\": [0.225936, 17, 2, 0.200122], \"passed\": false}, {\"check\": \"case 3\", \"actual\": [0.185448, 16, 0, 0.164424], \"expected\": [0.185448, 16, 0, 0.164424], \"passed\": true}, {\"check\": \"case 4\", \"actual\": [0.187358, 8, 0, 0.223575], \"expected\": [0.187358, 8, 0, 0.223575], \"passed\": true}, {\"check\": \"case 5\", \"actual\": [-0.014719, 14, 4, 0.068659], \"expected\": [-0.014719, 14, 2, 0.068659], \"passed\": false}, {\"check\": \"case 6\", \"actual\": [-0.136586, 8, 3, 0.375934], \"expected\": [-0.136586, 8, 2, 0.375934], \"passed\": false}, {\"check\": \"case 7\", \"actual\": [2.779, 5, 0, 1.0], \"expected\": [2.779, 5, 0, 1.0], \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":43.407,"exit_code":1,"observations":[{"actual":[-0.41309,64,7,0.009813],"check":"case 0","expected":[-0.41309,64,3,0.004672],"passed":false},{"actual":[-1.043398,7,1,0.064187],"check":"case 1","expected":[-1.043824,6,1,0.806875],"passed":false},{"actual":[0.225932,17,4,0.264651],"check":"case 2","expected":[0.225936,17,2,0.200122],"passed":false},{"actual":[0.185448,16,0,0.164424],"check":"case 3","expected":[0.185448,16,0,0.164424],"passed":true},{"actual":[0.187358,8,0,0.223575],"check":"case 4","expected":[0.187358,8,0,0.223575],"passed":true},{"actual":[-0.014772,15,4,0.02169],"check":"case 5","expected":[-0.014719,14,2,0.068659],"passed":false},{"actual":[-0.136567,8,3,0.483913],"check":"case 6","expected":[-0.136586,8,2,0.375934],"passed":false},{"actual":[2.779,5,0,1.0],"check":"case 7","expected":[2.779,5,0,1.0],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"case 0\", \"actual\": [-0.41309, 64, 7, 0.009813], \"expected\": [-0.41309, 64, 3, 0.004672], \"passed\": false}, {\"check\": \"case 1\", \"actual\": [-1.043398, 7, 1, 0.064187], \"expected\": [-1.043824, 6, 1, 0.806875], \"passed\": false}, {\"check\": \"case 2\", \"actual\": [0.225932, 17, 4, 0.264651], \"expected\": [0.225936, 17, 2, 0.200122], \"passed\": false}, {\"check\": \"case 3\", \"actual\": [0.185448, 16, 0, 0.164424], \"expected\": [0.185448, 16, 0, 0.164424], \"passed\": true}, {\"check\": \"case 4\", \"actual\": [0.187358, 8, 0, 0.223575], \"expected\": [0.187358, 8, 0, 0.223575], \"passed\": true}, {\"check\": \"case 5\", \"actual\": [-0.014772, 15, 4, 0.02169], \"expected\": [-0.014719, 14, 2, 0.068659], \"passed\": false}, {\"check\": \"case 6\", \"actual\": [-0.136567, 8, 3, 0.483913], \"expected\": [-0.136586, 8, 2, 0.375934], \"passed\": false}, {\"check\": \"case 7\", \"actual\": [2.779, 5, 0, 1.0], \"expected\": [2.779, 5, 0, 1.0], \"passed\": true}], \"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."}}