{"abstract":"A 30 degree release swings as if released from 30 radians.","category":"Physics integrator stability","checks":8,"contract":"solve(theta_deg, omega, g, L, dt, steps): planar pendulum theta''=-(g/L)*sin(theta), initial angle in degrees, omega in rad/s, integrated with velocity Verlet carrying the acceleration between steps. Return [angle in degrees, omega, energy per unit mass 0.5*L^2*omega^2 + g*L*(1-cos theta)], rounded to 6.","evaluation_group":"w2-physics_integrator_stability-pendulum-velocity-verlet","failed_approach":"Multiplying by pi/360 halves the angle.","family":"w2-physics_integrator_stability-pendulum-velocity-verlet-angle-unit-conversion","id":"FA-86886","implementations":{"attempt":{"sha256":"359ccec6952eb26e9ec0a3ddac6d01a74c8ee758da350b874112c1672646548f","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(theta_deg, omega, g, L, dt, steps):\n    th = theta_deg * math.pi / 360\n    w = omega\n    def acc(t):\n        return -(g / L) * math.sin(t)\n    a0 = acc(th)\n    for _ in range(steps):\n        th = th + w * dt + 0.5 * a0 * dt * dt\n        a1 = acc(th)\n        w = w + 0.5 * (a0 + a1) * dt\n        a0 = a1\n    energy = 0.5 * L * L * w * w + g * L * (1 - math.cos(th))\n    return [round(math.degrees(th), 6), round(w, 6), round(energy, 6)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[[[-90.84, -2.902, 9.81, 1.0, 0.01, 5], [-98.45263, -2.413579, 14.16467]], [[0.0, 0.0, 9.81, 1.0, 0.01, 5], [0.0, 0.0, 0.0]], [[47.0, -0.897, 1.62, 1.061, 0.05, 10], [14.725067, -1.294022, 0.998958]], [[87.81, 0.31, 3.71, 1.235, 0.01, 11], [88.722943, -0.020319, 4.480049]], [[-129.15, 2.597, 9.81, 1.034, 0.02, 2], [-122.853785, 2.903609, 20.153379]], [[90.0, 0.0, 9.81, 2.0, 0.05, 1], [89.648705, -0.245248, 19.619998]], [[63.66, 2.606, 3.71, 0.985, 0.02, 8], [84.9478, 2.028054, 5.327806]], [[140.32, -0.763, 9.81, 2.845, 0.05, 10], [99.827729, -2.171297, 51.752981]]], [[[132.37, 1.164, 3.71, 1.897, 0.05, 8], [153.323241, 0.70366, 14.217487]], [[0.0, 0.0, 9.81, 1.0, 0.01, 5], [0.0, 0.0, 0.0]], [[-148.24, 0.711, 1.62, 1.415, 0.1, 3], [-134.302426, 0.923966, 4.748011]], [[112.84, -2.85, 1.62, 0.918, 0.05, 18], [-61.55966, -3.341714, 5.484281]], [[50.01, 1.967, 9.81, 2.774, 0.01, 17], [66.748485, 1.454635, 24.611383]], [[-83.79, 0.173, 3.71, 1.845, 0.02, 8], [-80.740326, 0.491946, 6.15544]], [[90.0, 0.0, 9.81, 2.0, 0.05, 1], [89.648705, -0.245248, 19.619998]], [[132.44, 0.741, 3.71, 2.113, 0.1, 16], [120.242008, -1.072974, 14.357572]]], [[[-132.91, 1.906, 3.71, 1.161, 0.05, 10], [-57.95129, 3.371139, 9.680955]], [[-124.01, -2.0, 9.81, 0.956, 0.01, 22], [-138.432846, -0.351896, 16.451635]], [[90.0, 0.0, 9.81, 2.0, 0.05, 1], [89.648705, -0.245248, 19.619998]], [[-87.9, -2.746, 9.81, 0.669, 0.05, 22], [97.511749, 1.623222, 8.010482]], [[-147.17, 0.123, 9.81, 2.31, 0.1, 15], [54.162842, 3.475648, 41.623788]], [[-23.64, 0.163, 9.81, 2.99, 0.05, 7], [-16.088501, 0.565375, 2.577636]], [[-65.1, -0.228, 1.62, 2.64, 0.05, 2], [-66.246499, -0.172077, 2.657281]], [[0.0, 0.0, 9.81, 1.0, 0.01, 5], [0.0, 0.0, 0.0]]], [[[-106.5, -1.792, 9.81, 2.402, 0.1, 18], [21.607042, 3.609021, 39.230431]], [[-73.59, 1.271, 9.81, 0.814, 0.1, 5], [24.673646, 4.040094, 6.136624]], [[7.34, 1.721, 9.81, 1.693, 0.01, 8], [15.045893, 1.630818, 4.380856]], [[37.92, 1.544, 1.62, 2.753, 0.02, 25], [78.875105, 1.297176, 9.975805]], [[129.71, 0.223, 1.62, 2.683, 0.01, 21], [131.813176, 0.127103, 7.302408]], [[0.0, 0.0, 9.81, 1.0, 0.01, 5], [0.0, 0.0, 0.0]], [[-113.01, 0.456, 3.71, 2.862, 0.02, 8], [-107.945309, 0.65005, 15.620154]], [[90.0, 0.0, 9.81, 2.0, 0.05, 1], [89.648705, -0.245248, 19.619998]]], [[[-16.99, -0.173, 3.71, 2.908, 0.05, 16], [-17.529938, 0.150997, 0.597434]], [[89.14, -1.757, 9.81, 1.734, 0.05, 20], [-85.551502, -1.947875, 21.395293]], [[0.0, 0.0, 9.81, 1.0, 0.01, 5], [0.0, 0.0, 0.0]], [[90.0, 0.0, 9.81, 2.0, 0.05, 1], [89.648705, -0.245248, 19.619998]], [[94.96, 0.484, 9.81, 1.629, 0.1, 17], [-95.192476, 0.430719, 17.672903]], [[116.79, -2.258, 1.62, 2.493, 0.1, 6], [32.851266, -2.601404, 21.675418]], [[-87.63, 2.437, 9.81, 2.308, 0.01, 7], [-77.262703, 2.731595, 37.522955]], [[-54.71, -2.248, 1.62, 1.318, 0.02, 30], [-120.050645, -1.550048, 5.291224]]]]\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":"8cce3ab98993af2df7d5186de7cb4ad9bb315c4dd0ff4fa9210f7ff4e05e7f48","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(theta_deg, omega, g, L, dt, steps):\n    th = theta_deg\n    w = omega\n    def acc(t):\n        return -(g / L) * math.sin(t)\n    a0 = acc(th)\n    for _ in range(steps):\n        th = th + w * dt + 0.5 * a0 * dt * dt\n        a1 = acc(th)\n        w = w + 0.5 * (a0 + a1) * dt\n        a0 = a1\n    energy = 0.5 * L * L * w * w + g * L * (1 - math.cos(th))\n    return [round(math.degrees(th), 6), round(w, 6), round(energy, 6)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[[[-90.84, -2.902, 9.81, 1.0, 0.01, 5], [-98.45263, -2.413579, 14.16467]], [[0.0, 0.0, 9.81, 1.0, 0.01, 5], [0.0, 0.0, 0.0]], [[47.0, -0.897, 1.62, 1.061, 0.05, 10], [14.725067, -1.294022, 0.998958]], [[87.81, 0.31, 3.71, 1.235, 0.01, 11], [88.722943, -0.020319, 4.480049]], [[-129.15, 2.597, 9.81, 1.034, 0.02, 2], [-122.853785, 2.903609, 20.153379]], [[90.0, 0.0, 9.81, 2.0, 0.05, 1], [89.648705, -0.245248, 19.619998]], [[63.66, 2.606, 3.71, 0.985, 0.02, 8], [84.9478, 2.028054, 5.327806]], [[140.32, -0.763, 9.81, 2.845, 0.05, 10], [99.827729, -2.171297, 51.752981]]], [[[132.37, 1.164, 3.71, 1.897, 0.05, 8], [153.323241, 0.70366, 14.217487]], [[0.0, 0.0, 9.81, 1.0, 0.01, 5], [0.0, 0.0, 0.0]], [[-148.24, 0.711, 1.62, 1.415, 0.1, 3], [-134.302426, 0.923966, 4.748011]], [[112.84, -2.85, 1.62, 0.918, 0.05, 18], [-61.55966, -3.341714, 5.484281]], [[50.01, 1.967, 9.81, 2.774, 0.01, 17], [66.748485, 1.454635, 24.611383]], [[-83.79, 0.173, 3.71, 1.845, 0.02, 8], [-80.740326, 0.491946, 6.15544]], [[90.0, 0.0, 9.81, 2.0, 0.05, 1], [89.648705, -0.245248, 19.619998]], [[132.44, 0.741, 3.71, 2.113, 0.1, 16], [120.242008, -1.072974, 14.357572]]], [[[-132.91, 1.906, 3.71, 1.161, 0.05, 10], [-57.95129, 3.371139, 9.680955]], [[-124.01, -2.0, 9.81, 0.956, 0.01, 22], [-138.432846, -0.351896, 16.451635]], [[90.0, 0.0, 9.81, 2.0, 0.05, 1], [89.648705, -0.245248, 19.619998]], [[-87.9, -2.746, 9.81, 0.669, 0.05, 22], [97.511749, 1.623222, 8.010482]], [[-147.17, 0.123, 9.81, 2.31, 0.1, 15], [54.162842, 3.475648, 41.623788]], [[-23.64, 0.163, 9.81, 2.99, 0.05, 7], [-16.088501, 0.565375, 2.577636]], [[-65.1, -0.228, 1.62, 2.64, 0.05, 2], [-66.246499, -0.172077, 2.657281]], [[0.0, 0.0, 9.81, 1.0, 0.01, 5], [0.0, 0.0, 0.0]]], [[[-106.5, -1.792, 9.81, 2.402, 0.1, 18], [21.607042, 3.609021, 39.230431]], [[-73.59, 1.271, 9.81, 0.814, 0.1, 5], [24.673646, 4.040094, 6.136624]], [[7.34, 1.721, 9.81, 1.693, 0.01, 8], [15.045893, 1.630818, 4.380856]], [[37.92, 1.544, 1.62, 2.753, 0.02, 25], [78.875105, 1.297176, 9.975805]], [[129.71, 0.223, 1.62, 2.683, 0.01, 21], [131.813176, 0.127103, 7.302408]], [[0.0, 0.0, 9.81, 1.0, 0.01, 5], [0.0, 0.0, 0.0]], [[-113.01, 0.456, 3.71, 2.862, 0.02, 8], [-107.945309, 0.65005, 15.620154]], [[90.0, 0.0, 9.81, 2.0, 0.05, 1], [89.648705, -0.245248, 19.619998]]], [[[-16.99, -0.173, 3.71, 2.908, 0.05, 16], [-17.529938, 0.150997, 0.597434]], [[89.14, -1.757, 9.81, 1.734, 0.05, 20], [-85.551502, -1.947875, 21.395293]], [[0.0, 0.0, 9.81, 1.0, 0.01, 5], [0.0, 0.0, 0.0]], [[90.0, 0.0, 9.81, 2.0, 0.05, 1], [89.648705, -0.245248, 19.619998]], [[94.96, 0.484, 9.81, 1.629, 0.1, 17], [-95.192476, 0.430719, 17.672903]], [[116.79, -2.258, 1.62, 2.493, 0.1, 6], [32.851266, -2.601404, 21.675418]], [[-87.63, 2.437, 9.81, 2.308, 0.01, 7], [-77.262703, 2.731595, 37.522955]], [[-54.71, -2.248, 1.62, 1.318, 0.02, 30], [-120.050645, -1.550048, 5.291224]]]]\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"},"fixed":{"sha256":"ed5f2c0c454a5d21780163940089151f1e3373cfb05f5d458d1cd9c93018cab1","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(theta_deg, omega, g, L, dt, steps):\n    th = math.radians(theta_deg)\n    w = omega\n    def acc(t):\n        return -(g / L) * math.sin(t)\n    a0 = acc(th)\n    for _ in range(steps):\n        th = th + w * dt + 0.5 * a0 * dt * dt\n        a1 = acc(th)\n        w = w + 0.5 * (a0 + a1) * dt\n        a0 = a1\n    energy = 0.5 * L * L * w * w + g * L * (1 - math.cos(th))\n    return [round(math.degrees(th), 6), round(w, 6), round(energy, 6)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[[[-90.84, -2.902, 9.81, 1.0, 0.01, 5], [-98.45263, -2.413579, 14.16467]], [[0.0, 0.0, 9.81, 1.0, 0.01, 5], [0.0, 0.0, 0.0]], [[47.0, -0.897, 1.62, 1.061, 0.05, 10], [14.725067, -1.294022, 0.998958]], [[87.81, 0.31, 3.71, 1.235, 0.01, 11], [88.722943, -0.020319, 4.480049]], [[-129.15, 2.597, 9.81, 1.034, 0.02, 2], [-122.853785, 2.903609, 20.153379]], [[90.0, 0.0, 9.81, 2.0, 0.05, 1], [89.648705, -0.245248, 19.619998]], [[63.66, 2.606, 3.71, 0.985, 0.02, 8], [84.9478, 2.028054, 5.327806]], [[140.32, -0.763, 9.81, 2.845, 0.05, 10], [99.827729, -2.171297, 51.752981]]], [[[132.37, 1.164, 3.71, 1.897, 0.05, 8], [153.323241, 0.70366, 14.217487]], [[0.0, 0.0, 9.81, 1.0, 0.01, 5], [0.0, 0.0, 0.0]], [[-148.24, 0.711, 1.62, 1.415, 0.1, 3], [-134.302426, 0.923966, 4.748011]], [[112.84, -2.85, 1.62, 0.918, 0.05, 18], [-61.55966, -3.341714, 5.484281]], [[50.01, 1.967, 9.81, 2.774, 0.01, 17], [66.748485, 1.454635, 24.611383]], [[-83.79, 0.173, 3.71, 1.845, 0.02, 8], [-80.740326, 0.491946, 6.15544]], [[90.0, 0.0, 9.81, 2.0, 0.05, 1], [89.648705, -0.245248, 19.619998]], [[132.44, 0.741, 3.71, 2.113, 0.1, 16], [120.242008, -1.072974, 14.357572]]], [[[-132.91, 1.906, 3.71, 1.161, 0.05, 10], [-57.95129, 3.371139, 9.680955]], [[-124.01, -2.0, 9.81, 0.956, 0.01, 22], [-138.432846, -0.351896, 16.451635]], [[90.0, 0.0, 9.81, 2.0, 0.05, 1], [89.648705, -0.245248, 19.619998]], [[-87.9, -2.746, 9.81, 0.669, 0.05, 22], [97.511749, 1.623222, 8.010482]], [[-147.17, 0.123, 9.81, 2.31, 0.1, 15], [54.162842, 3.475648, 41.623788]], [[-23.64, 0.163, 9.81, 2.99, 0.05, 7], [-16.088501, 0.565375, 2.577636]], [[-65.1, -0.228, 1.62, 2.64, 0.05, 2], [-66.246499, -0.172077, 2.657281]], [[0.0, 0.0, 9.81, 1.0, 0.01, 5], [0.0, 0.0, 0.0]]], [[[-106.5, -1.792, 9.81, 2.402, 0.1, 18], [21.607042, 3.609021, 39.230431]], [[-73.59, 1.271, 9.81, 0.814, 0.1, 5], [24.673646, 4.040094, 6.136624]], [[7.34, 1.721, 9.81, 1.693, 0.01, 8], [15.045893, 1.630818, 4.380856]], [[37.92, 1.544, 1.62, 2.753, 0.02, 25], [78.875105, 1.297176, 9.975805]], [[129.71, 0.223, 1.62, 2.683, 0.01, 21], [131.813176, 0.127103, 7.302408]], [[0.0, 0.0, 9.81, 1.0, 0.01, 5], [0.0, 0.0, 0.0]], [[-113.01, 0.456, 3.71, 2.862, 0.02, 8], [-107.945309, 0.65005, 15.620154]], [[90.0, 0.0, 9.81, 2.0, 0.05, 1], [89.648705, -0.245248, 19.619998]]], [[[-16.99, -0.173, 3.71, 2.908, 0.05, 16], [-17.529938, 0.150997, 0.597434]], [[89.14, -1.757, 9.81, 1.734, 0.05, 20], [-85.551502, -1.947875, 21.395293]], [[0.0, 0.0, 9.81, 1.0, 0.01, 5], [0.0, 0.0, 0.0]], [[90.0, 0.0, 9.81, 2.0, 0.05, 1], [89.648705, -0.245248, 19.619998]], [[94.96, 0.484, 9.81, 1.629, 0.1, 17], [-95.192476, 0.430719, 17.672903]], [[116.79, -2.258, 1.62, 2.493, 0.1, 6], [32.851266, -2.601404, 21.675418]], [[-87.63, 2.437, 9.81, 2.308, 0.01, 7], [-77.262703, 2.731595, 37.522955]], [[-54.71, -2.248, 1.62, 1.318, 0.02, 30], [-120.050645, -1.550048, 5.291224]]]]\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-pendulum-velocity-verlet-angle-unit-conversion","generated_at":"2026-09-29T14:50:53.778673+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.","repair":"Convert with math.radians before integrating.","root_cause":"The initial angle is used directly without converting degrees to radians.","sha256":"ca0b924779e8824d98084dcbeb280f5daa9ad76ed852f68b3905ac340b37cede","title":"Pendulum treats the degree input as radians · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":40.435,"exit_code":1,"observations":[{"actual":[-53.211794,-2.52988,7.135343],"check":"case 0","expected":[-98.45263,-2.413579,14.16467],"passed":false},{"actual":[0.0,0.0,0.0],"check":"case 1","expected":[0.0,0.0,0.0],"passed":true},{"actual":[-4.909826,-1.022943,0.595291],"check":"case 2","expected":[14.725067,-1.294022,0.998958],"passed":false},{"actual":[45.129824,0.077824,1.353961],"check":"case 3","expected":[88.722943,-0.020319,4.480049],"passed":false},{"actual":[-58.23565,2.930105,9.393336],"check":"case 4","expected":[-122.853785,2.903609,20.153379],"passed":false},{"actual":[44.751597,-0.173041,5.746434],"check":"case 5","expected":[89.648705,-0.245248,19.619998],"passed":false},{"actual":[53.974473,2.196017,3.844511],"check":"case 6","expected":[84.9478,2.028054,5.327806],"passed":false},{"actual":[27.339813,-2.088209,20.76501],"check":"case 7","expected":[99.827729,-2.171297,51.752981],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"case 0\", \"actual\": [-53.211794, -2.52988, 7.135343], \"expected\": [-98.45263, -2.413579, 14.16467], \"passed\": false}, {\"check\": \"case 1\", \"actual\": [0.0, 0.0, 0.0], \"expected\": [0.0, 0.0, 0.0], \"passed\": true}, {\"check\": \"case 2\", \"actual\": [-4.909826, -1.022943, 0.595291], \"expected\": [14.725067, -1.294022, 0.998958], \"passed\": false}, {\"check\": \"case 3\", \"actual\": [45.129824, 0.077824, 1.353961], \"expected\": [88.722943, -0.020319, 4.480049], \"passed\": false}, {\"check\": \"case 4\", \"actual\": [-58.23565, 2.930105, 9.393336], \"expected\": [-122.853785, 2.903609, 20.153379], \"passed\": false}, {\"check\": \"case 5\", \"actual\": [44.751597, -0.173041, 5.746434], \"expected\": [89.648705, -0.245248, 19.619998], \"passed\": false}, {\"check\": \"case 6\", \"actual\": [53.974473, 2.196017, 3.844511], \"expected\": [84.9478, 2.028054, 5.327806], \"passed\": false}, {\"check\": \"case 7\", \"actual\": [27.339813, -2.088209, 20.76501], \"expected\": [99.827729, -2.171297, 51.752981], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":38.427,"exit_code":1,"observations":[{"actual":[-5212.908837,-2.807238,23.485257],"check":"case 0","expected":[-98.45263,-2.413579,14.16467],"passed":false},{"actual":[0.0,0.0,0.0],"check":"case 1","expected":[0.0,0.0,0.0],"passed":true},{"actual":[2664.219414,-1.165293,3.877547],"check":"case 2","expected":[14.725067,-1.294022,0.998958],"passed":false},{"actual":[5033.244457,0.355016,0.127928],"check":"case 3","expected":[88.722943,-0.020319,4.480049],"passed":false},{"actual":[-7393.934398,2.487216,23.295608],"check":"case 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