{"abstract":"After the first step the integrator keeps using the initial-angle acceleration.","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.","contract_signature":"theta_deg, omega, g, L, dt, steps","evaluation_group":"w2-physics_integrator_stability-pendulum-velocity-verlet","failed_approach":"Recomputing a0 from an extrapolated previous angle is not the acceleration at the current angle.","family":"w2-physics_integrator_stability-pendulum-velocity-verlet-acceleration-carry","id":"FA-86881","implementations":{"attempt":{"sha256":"df29db479b780d7e671e889b547142e625f2a0b79ba53aac7b2b6e5aae58b36e","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 = acc(th - w * dt)\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 = [[[[45.28, -1.957, 9.81, 0.998, 0.1, 14], [7.145368, 3.058241, 4.733766]], [[0.0, 0.0, 9.81, 1.0, 0.01, 5], [0.0, 0.0, 0.0]], [[-88.3, 0.349, 3.71, 1.329, 0.02, 13], [-77.714345, 1.069639, 4.891833]], [[116.98, -2.664, 3.71, 2.568, 0.1, 10], [-63.542799, -3.110777, 37.190477]], [[149.67, 1.436, 1.62, 2.272, 0.05, 9], [185.401488, 1.368665, 12.179767]], [[90.0, 0.0, 9.81, 2.0, 0.05, 1], [89.648705, -0.245248, 19.619998]], [[-46.19, -1.108, 9.81, 1.086, 0.1, 13], [51.096139, -0.291005, 4.012934]], [[-46.16, 2.086, 3.71, 0.917, 0.1, 1], [-33.372114, 2.343182, 2.869399]]], [[[50.01, 1.967, 9.81, 2.774, 0.01, 17], [66.748485, 1.454635, 24.611383]], [[0.0, 0.0, 9.81, 1.0, 0.01, 5], [0.0, 0.0, 0.0]], [[-71.06, 1.343, 9.81, 1.769, 0.05, 2], [-61.883545, 1.851333, 14.538443]], [[39.61, 2.112, 9.81, 1.387, 0.02, 3], [46.385577, 1.822399, 7.415255]], [[55.82, -1.801, 1.62, 2.039, 0.02, 1], [53.748674, -1.81398, 8.190126]], [[90.0, 0.0, 9.81, 2.0, 0.05, 1], [89.648705, -0.245248, 19.619998]], [[-73.6, -2.997, 9.81, 1.448, 0.01, 6], [-83.223165, -2.599131, 19.610793]], [[-12.71, 2.5, 1.62, 2.44, 0.05, 9], [51.203117, 2.405701, 18.704048]]], [[[100.85, -0.237, 1.62, 0.605, 0.1, 1], [98.738703, -0.500821, 1.174909]], [[90.0, 0.0, 9.81, 2.0, 0.05, 1], [89.648705, -0.245248, 19.619998]], [[71.19, 0.807, 9.81, 1.157, 0.1, 14], [-71.100761, 0.82223, 8.1263]], [[86.31, -0.863, 1.62, 2.852, 0.1, 6], [51.020408, -1.177026, 7.348228]], [[-110.76, -2.814, 3.71, 2.007, 0.02, 2], [-117.130885, -2.74648, 26.03361]], [[-87.9, -2.746, 9.81, 0.669, 0.05, 22], [97.511749, 1.623222, 8.010482]], [[0.0, 0.0, 9.81, 1.0, 0.01, 5], [0.0, 0.0, 0.0]], [[-133.47, 2.786, 9.81, 0.714, 0.01, 8], [-118.744784, 3.66775, 13.801778]]], [[[148.74, 0.509, 3.71, 1.117, 0.05, 7], [153.321028, -0.038026, 7.847849]], [[-82.24, 1.27, 3.71, 0.87, 0.02, 1], [-80.736269, 1.354341, 3.402275]], [[88.81, -0.345, 3.71, 1.561, 0.01, 14], [84.709379, -0.677201, 5.816049]], [[-54.29, 1.6, 9.81, 1.365, 0.05, 24], [60.323988, -1.135897, 7.963031]], [[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]], [[93.61, 1.047, 3.71, 1.718, 0.05, 12], [107.880879, -0.204741, 8.392642]], [[12.62, 0.495, 9.81, 1.593, 0.01, 11], [15.238893, 0.330964, 0.688467]]], [[[90.0, 0.0, 9.81, 2.0, 0.05, 1], [89.648705, -0.245248, 19.619998]], [[101.22, 0.152, 3.71, 1.491, 0.02, 6], [101.25883, -0.140698, 6.633614]], [[64.12, 1.072, 9.81, 2.267, 0.1, 5], [65.863237, -0.95491, 15.488411]], [[-113.82, 1.15, 9.81, 2.142, 0.01, 4], [-110.991123, 1.319308, 32.533389]], [[0.0, 0.0, 9.81, 1.0, 0.01, 5], [0.0, 0.0, 0.0]], [[-44.62, 1.36, 9.81, 1.251, 0.05, 21], [52.156799, -0.560005, 4.988612]], [[66.17, -1.096, 3.71, 2.374, 0.1, 1], [59.480851, -1.23479, 8.631365]], [[89.14, -1.757, 9.81, 1.734, 0.05, 20], [-85.551502, -1.947875, 21.395293]]]]\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":"88ef12b167e01031733bb7bcbc0343c612920c83e99127c2ee65233f3b19a4e2","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    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 = [[[[45.28, -1.957, 9.81, 0.998, 0.1, 14], [7.145368, 3.058241, 4.733766]], [[0.0, 0.0, 9.81, 1.0, 0.01, 5], [0.0, 0.0, 0.0]], [[-88.3, 0.349, 3.71, 1.329, 0.02, 13], [-77.714345, 1.069639, 4.891833]], [[116.98, -2.664, 3.71, 2.568, 0.1, 10], [-63.542799, -3.110777, 37.190477]], [[149.67, 1.436, 1.62, 2.272, 0.05, 9], [185.401488, 1.368665, 12.179767]], [[90.0, 0.0, 9.81, 2.0, 0.05, 1], [89.648705, -0.245248, 19.619998]], [[-46.19, -1.108, 9.81, 1.086, 0.1, 13], [51.096139, -0.291005, 4.012934]], [[-46.16, 2.086, 3.71, 0.917, 0.1, 1], [-33.372114, 2.343182, 2.869399]]], [[[50.01, 1.967, 9.81, 2.774, 0.01, 17], [66.748485, 1.454635, 24.611383]], [[0.0, 0.0, 9.81, 1.0, 0.01, 5], [0.0, 0.0, 0.0]], [[-71.06, 1.343, 9.81, 1.769, 0.05, 2], [-61.883545, 1.851333, 14.538443]], [[39.61, 2.112, 9.81, 1.387, 0.02, 3], [46.385577, 1.822399, 7.415255]], [[55.82, -1.801, 1.62, 2.039, 0.02, 1], [53.748674, -1.81398, 8.190126]], [[90.0, 0.0, 9.81, 2.0, 0.05, 1], [89.648705, -0.245248, 19.619998]], [[-73.6, -2.997, 9.81, 1.448, 0.01, 6], [-83.223165, -2.599131, 19.610793]], [[-12.71, 2.5, 1.62, 2.44, 0.05, 9], [51.203117, 2.405701, 18.704048]]], [[[100.85, -0.237, 1.62, 0.605, 0.1, 1], [98.738703, -0.500821, 1.174909]], [[90.0, 0.0, 9.81, 2.0, 0.05, 1], [89.648705, -0.245248, 19.619998]], [[71.19, 0.807, 9.81, 1.157, 0.1, 14], [-71.100761, 0.82223, 8.1263]], [[86.31, -0.863, 1.62, 2.852, 0.1, 6], [51.020408, -1.177026, 7.348228]], [[-110.76, -2.814, 3.71, 2.007, 0.02, 2], [-117.130885, -2.74648, 26.03361]], [[-87.9, -2.746, 9.81, 0.669, 0.05, 22], [97.511749, 1.623222, 8.010482]], [[0.0, 0.0, 9.81, 1.0, 0.01, 5], [0.0, 0.0, 0.0]], [[-133.47, 2.786, 9.81, 0.714, 0.01, 8], [-118.744784, 3.66775, 13.801778]]], [[[148.74, 0.509, 3.71, 1.117, 0.05, 7], [153.321028, -0.038026, 7.847849]], [[-82.24, 1.27, 3.71, 0.87, 0.02, 1], [-80.736269, 1.354341, 3.402275]], [[88.81, -0.345, 3.71, 1.561, 0.01, 14], [84.709379, -0.677201, 5.816049]], [[-54.29, 1.6, 9.81, 1.365, 0.05, 24], [60.323988, -1.135897, 7.963031]], [[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]], [[93.61, 1.047, 3.71, 1.718, 0.05, 12], [107.880879, -0.204741, 8.392642]], [[12.62, 0.495, 9.81, 1.593, 0.01, 11], [15.238893, 0.330964, 0.688467]]], [[[90.0, 0.0, 9.81, 2.0, 0.05, 1], [89.648705, -0.245248, 19.619998]], [[101.22, 0.152, 3.71, 1.491, 0.02, 6], [101.25883, -0.140698, 6.633614]], [[64.12, 1.072, 9.81, 2.267, 0.1, 5], [65.863237, -0.95491, 15.488411]], [[-113.82, 1.15, 9.81, 2.142, 0.01, 4], [-110.991123, 1.319308, 32.533389]], [[0.0, 0.0, 9.81, 1.0, 0.01, 5], [0.0, 0.0, 0.0]], [[-44.62, 1.36, 9.81, 1.251, 0.05, 21], [52.156799, -0.560005, 4.988612]], [[66.17, -1.096, 3.71, 2.374, 0.1, 1], [59.480851, -1.23479, 8.631365]], [[89.14, -1.757, 9.81, 1.734, 0.05, 20], [-85.551502, -1.947875, 21.395293]]]]\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-acceleration-carry","generated_at":"2026-09-29T14:50:53.698313+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 a0 = a1 hand-off at the end of the loop is missing.","sha256":"cb67c7c4e315b90e9f137148d81084942e57c778d9670918d9848cbf990ec4bf","title":"Verlet never carries the new acceleration into the next step · 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":41.799,"exit_code":1,"observations":[{"actual":[-0.587999,3.994733,7.947577],"check":"case 0","expected":[7.145368,3.058241,4.733766],"passed":false},{"actual":[0.0,0.0,0.0],"check":"case 1","expected":[0.0,0.0,0.0],"passed":true},{"actual":[-77.711835,1.070164,4.892614],"check":"case 2","expected":[-77.714345,1.069639,4.891833],"passed":false},{"actual":[-65.541177,-3.220973,39.791117],"check":"case 3","expected":[-63.542799,-3.110777,37.190477],"passed":false},{"actual":[185.28061,1.359047,12.112771],"check":"case 4","expected":[185.401488,1.368665,12.179767],"passed":false},{"actual":[89.648705,-0.245248,19.619998],"check":"case 5","expected":[89.648705,-0.245248,19.619998],"passed":true},{"actual":[64.639095,-0.157647,6.105154],"check":"case 6","expected":[51.096139,-0.291005,4.012934],"passed":false},{"actual":[-33.372114,2.343182,2.869399],"check":"case 7","expected":[-33.372114,2.343182,2.869399],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"case 0\", \"actual\": [-0.587999, 3.994733, 7.947577], \"expected\": [7.145368, 3.058241, 4.733766], \"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\": [-77.711835, 1.070164, 4.892614], \"expected\": [-77.714345, 1.069639, 4.891833], \"passed\": false}, {\"check\": \"case 3\", \"actual\": [-65.541177, -3.220973, 39.791117], \"expected\": [-63.542799, -3.110777, 37.190477], \"passed\": false}, {\"check\": \"case 4\", \"actual\": [185.28061, 1.359047, 12.112771], \"expected\": [185.401488, 1.368665, 12.179767], \"passed\": false}, {\"check\": \"case 5\", \"actual\": [89.648705, -0.245248, 19.619998], \"expected\": [89.648705, -0.245248, 19.619998], \"passed\": true}, {\"check\": \"case 6\", \"actual\": [64.639095, -0.157647, 6.105154], \"expected\": [51.096139, -0.291005, 4.012934], \"passed\": false}, {\"check\": \"case 7\", \"actual\": [-33.372114, 2.343182, 2.869399], \"expected\": [-33.372114, 2.343182, 2.869399], \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":38.985,"exit_code":1,"observations":[{"actual":[-246.405975,-5.404692,28.255999],"check":"case 0","expected":[7.145368,3.058241,4.733766],"passed":false},{"actual":[0.0,0.0,0.0],"check":"case 1","expected":[0.0,0.0,0.0],"passed":true},{"actual":[-77.705781,1.071748,4.895102],"check":"case 2","expected":[-77.714345,1.069639,4.891833],"passed":false},{"actual":[-68.028743,-3.461247,45.465284],"check":"case 3","expected":[-63.542799,-3.110777,37.190477],"passed":false},{"actual":[185.002272,1.32632,11.88755],"check":"case 4","expected":[185.401488,1.368665,12.179767],"passed":false},{"actual":[89.648705,-0.245248,19.619998],"check":"case 5","expected":[89.648705,-0.245248,19.619998],"passed":true},{"actual":[113.1036,2.720061,19.19712],"check":"case 6","expected":[51.096139,-0.291005,4.012934],"passed":false},{"actual":[-33.372114,2.343182,2.869399],"check":"case 7","expected":[-33.372114,2.343182,2.869399],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"case 0\", \"actual\": [-246.405975, -5.404692, 28.255999], \"expected\": [7.145368, 3.058241, 4.733766], \"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\": [-77.705781, 1.071748, 4.895102], \"expected\": [-77.714345, 1.069639, 4.891833], \"passed\": false}, {\"check\": \"case 3\", \"actual\": [-68.028743, -3.461247, 45.465284], \"expected\": [-63.542799, -3.110777, 37.190477], \"passed\": false}, {\"check\": \"case 4\", \"actual\": [185.002272, 1.32632, 11.88755], \"expected\": [185.401488, 1.368665, 12.179767], \"passed\": false}, {\"check\": \"case 5\", \"actual\": [89.648705, -0.245248, 19.619998], \"expected\": [89.648705, -0.245248, 19.619998], \"passed\": true}, {\"check\": \"case 6\", \"actual\": [113.1036, 2.720061, 19.19712], \"expected\": [51.096139, -0.291005, 4.012934], \"passed\": false}, {\"check\": \"case 7\", \"actual\": [-33.372114, 2.343182, 2.869399], \"expected\": [-33.372114, 2.343182, 2.869399], \"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."}}