{"abstract":"Even zero requested frames move the body.","category":"Physics integrator stability","checks":8,"contract":"solve(x0, v0, k, m, c, dt, steps): damped spring a=(-k*x-c*v)/m integrated with semi-implicit Euler (velocity first, then position with the new velocity). Each frame of length dt is split into ceil(dt/0.05) equal substeps (at least one). Return [x, v] rounded to 6 decimals after steps frames.","evaluation_group":"w2-physics_integrator_stability-substep-symplectic-euler","failed_approach":"Starting the range at one runs one frame too few.","family":"w2-physics_integrator_stability-substep-symplectic-euler-frame-loop-count","id":"FA-86866","implementations":{"attempt":{"sha256":"2311afed003cf886fd81a6d60e509236af2cd7c9bd16671bbda98d40ad959829","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(x0, v0, k, m, c, dt, steps):\n    n_sub = max(1, math.ceil(dt / 0.05 - 1e-12))\n    h = dt / n_sub\n    x, v = x0, v0\n    for _ in range(1, steps):\n        for _ in range(n_sub):\n            a = (-k * x - c * v) / m\n            v = v + a * h\n            x = x + v * h\n    return [round(x, 6), round(v, 6)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[[[-1.484, 1.712, 27.312, 2.961, 1.788, 0.13, 4], [0.446958, 3.829901]], [[1.133, -1.182, 1.342, 0.781, 1.334, 0.1, 6], [0.491288, -0.912623]], [[-1.578, 0.145, 23.597, 0.588, 1.601, 0.01, 4], [-1.511386, 2.533387]], [[-1.392, -0.154, 8.638, 2.736, 1.406, 0.02, 2], [-1.392808, 0.024118]], [[0.514, 1.143, 25.432, 0.518, 1.496, 0.13, 9], [-0.010702, -0.661591]], [[0.0, 1.0, 4.0, 2.0, 0.5, 0.1, 3], [0.278786, 0.856518]], [[-1.377, 0.559, 2.87, 2.898, 1.015, 0.2, 0], [-1.377, 0.559]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]]], [[[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[-1.526, 0.873, 6.362, 1.828, 1.137, 0.13, 0], [-1.526, 0.873]], [[1.082, 0.058, 37.566, 1.034, 1.55, 0.05, 9], [-0.710961, -1.842409]], [[1.297, -0.319, 39.855, 2.122, 0.103, 0.01, 11], [1.106036, -2.858781]], [[0.764, 1.798, 31.467, 2.606, 1.52, 0.1, 4], [0.561426, -2.076687]], [[-1.669, -1.23, 27.053, 1.533, 0.991, 0.13, 5], [1.161742, 3.128527]], [[1.602, 1.18, 24.517, 1.738, 1.228, 0.05, 12], [-0.636953, -4.398087]], [[0.0, 1.0, 4.0, 2.0, 0.5, 0.1, 3], [0.278786, 0.856518]]], [[[0.718, 1.584, 9.356, 1.657, 0.286, 0.02, 6], [0.869722, 1.022798]], [[0.784, -0.426, 6.875, 0.731, 0.044, 0.1, 14], [-0.132405, 2.308875]], [[-1.544, -1.79, 32.966, 0.813, 1.455, 0.13, 4], [0.965451, -0.257038]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[0.0, 1.0, 4.0, 2.0, 0.5, 0.1, 3], [0.278786, 0.856518]], [[0.044, 1.377, 37.377, 1.154, 1.523, 0.13, 8], [-0.013801, 0.684737]], [[-1.525, 1.375, 3.248, 1.693, 1.607, 0.1, 4], [-0.87326, 1.732745]], [[-1.184, -1.519, 7.411, 1.786, 1.332, 0.04, 9], [-1.302744, 0.571925]]], [[[0.0, 1.0, 4.0, 2.0, 0.5, 0.1, 3], [0.278786, 0.856518]], [[1.818, 0.29, 35.382, 2.651, 1.322, 0.04, 12], [-0.242082, -5.8862]], [[-0.271, 1.727, 34.523, 2.128, 1.589, 0.13, 15], [0.205585, 0.49101]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[-0.56, 1.419, 29.61, 1.024, 1.85, 0.1, 6], [0.298786, -1.014037]], [[1.308, -0.567, 4.917, 1.142, 0.149, 0.02, 14], [0.933908, -1.927012]], [[-0.362, 1.363, 36.638, 0.869, 0.886, 0.2, 7], [0.193298, -0.396669]], [[0.484, 0.239, 12.355, 0.868, 0.3, 0.07, 0], [0.484, 0.239]]], [[[-1.576, -0.203, 24.72, 2.347, 1.363, 0.07, 10], [0.760633, 3.318988]], [[0.0, 1.0, 4.0, 2.0, 0.5, 0.1, 3], [0.278786, 0.856518]], [[-0.232, 1.993, 10.835, 2.4, 1.522, 0.01, 9], [-0.054651, 1.942283]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[-1.68, 0.833, 26.799, 1.025, 1.524, 0.13, 6], [0.548391, -4.033319]], [[-1.38, 1.106, 38.807, 2.361, 0.186, 0.05, 14], [1.397704, 0.597706]], [[0.695, 1.714, 8.192, 1.726, 1.775, 0.02, 2], [0.757467, 1.510568]], [[1.759, -0.492, 32.001, 1.805, 1.669, 0.13, 5], [-1.219998, -1.776701]]]]\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":"d80dba52bccbea2ab1b71d9bc12eeb1f5a16ad3bb1a24ab028e78d8b7d1945c0","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(x0, v0, k, m, c, dt, steps):\n    n_sub = max(1, math.ceil(dt / 0.05 - 1e-12))\n    h = dt / n_sub\n    x, v = x0, v0\n    for _ in range(steps + 1):\n        for _ in range(n_sub):\n            a = (-k * x - c * v) / m\n            v = v + a * h\n            x = x + v * h\n    return [round(x, 6), round(v, 6)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[[[-1.484, 1.712, 27.312, 2.961, 1.788, 0.13, 4], [0.446958, 3.829901]], [[1.133, -1.182, 1.342, 0.781, 1.334, 0.1, 6], [0.491288, -0.912623]], [[-1.578, 0.145, 23.597, 0.588, 1.601, 0.01, 4], [-1.511386, 2.533387]], [[-1.392, -0.154, 8.638, 2.736, 1.406, 0.02, 2], [-1.392808, 0.024118]], [[0.514, 1.143, 25.432, 0.518, 1.496, 0.13, 9], [-0.010702, -0.661591]], [[0.0, 1.0, 4.0, 2.0, 0.5, 0.1, 3], [0.278786, 0.856518]], [[-1.377, 0.559, 2.87, 2.898, 1.015, 0.2, 0], [-1.377, 0.559]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]]], [[[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[-1.526, 0.873, 6.362, 1.828, 1.137, 0.13, 0], [-1.526, 0.873]], [[1.082, 0.058, 37.566, 1.034, 1.55, 0.05, 9], [-0.710961, -1.842409]], [[1.297, -0.319, 39.855, 2.122, 0.103, 0.01, 11], [1.106036, -2.858781]], [[0.764, 1.798, 31.467, 2.606, 1.52, 0.1, 4], [0.561426, -2.076687]], [[-1.669, -1.23, 27.053, 1.533, 0.991, 0.13, 5], [1.161742, 3.128527]], [[1.602, 1.18, 24.517, 1.738, 1.228, 0.05, 12], [-0.636953, -4.398087]], [[0.0, 1.0, 4.0, 2.0, 0.5, 0.1, 3], [0.278786, 0.856518]]], [[[0.718, 1.584, 9.356, 1.657, 0.286, 0.02, 6], [0.869722, 1.022798]], [[0.784, -0.426, 6.875, 0.731, 0.044, 0.1, 14], [-0.132405, 2.308875]], [[-1.544, -1.79, 32.966, 0.813, 1.455, 0.13, 4], [0.965451, -0.257038]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[0.0, 1.0, 4.0, 2.0, 0.5, 0.1, 3], [0.278786, 0.856518]], [[0.044, 1.377, 37.377, 1.154, 1.523, 0.13, 8], [-0.013801, 0.684737]], [[-1.525, 1.375, 3.248, 1.693, 1.607, 0.1, 4], [-0.87326, 1.732745]], [[-1.184, -1.519, 7.411, 1.786, 1.332, 0.04, 9], [-1.302744, 0.571925]]], [[[0.0, 1.0, 4.0, 2.0, 0.5, 0.1, 3], [0.278786, 0.856518]], [[1.818, 0.29, 35.382, 2.651, 1.322, 0.04, 12], [-0.242082, -5.8862]], [[-0.271, 1.727, 34.523, 2.128, 1.589, 0.13, 15], [0.205585, 0.49101]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[-0.56, 1.419, 29.61, 1.024, 1.85, 0.1, 6], [0.298786, -1.014037]], [[1.308, -0.567, 4.917, 1.142, 0.149, 0.02, 14], [0.933908, -1.927012]], [[-0.362, 1.363, 36.638, 0.869, 0.886, 0.2, 7], [0.193298, -0.396669]], [[0.484, 0.239, 12.355, 0.868, 0.3, 0.07, 0], [0.484, 0.239]]], [[[-1.576, -0.203, 24.72, 2.347, 1.363, 0.07, 10], [0.760633, 3.318988]], [[0.0, 1.0, 4.0, 2.0, 0.5, 0.1, 3], [0.278786, 0.856518]], [[-0.232, 1.993, 10.835, 2.4, 1.522, 0.01, 9], [-0.054651, 1.942283]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[-1.68, 0.833, 26.799, 1.025, 1.524, 0.13, 6], [0.548391, -4.033319]], [[-1.38, 1.106, 38.807, 2.361, 0.186, 0.05, 14], [1.397704, 0.597706]], [[0.695, 1.714, 8.192, 1.726, 1.775, 0.02, 2], [0.757467, 1.510568]], [[1.759, -0.492, 32.001, 1.805, 1.669, 0.13, 5], [-1.219998, -1.776701]]]]\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":"c24f638da77269b6879e30af6cbe467eee10eb6d351b62e4b45608743fc668bd","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(x0, v0, k, m, c, dt, steps):\n    n_sub = max(1, math.ceil(dt / 0.05 - 1e-12))\n    h = dt / n_sub\n    x, v = x0, v0\n    for _ in range(steps):\n        for _ in range(n_sub):\n            a = (-k * x - c * v) / m\n            v = v + a * h\n            x = x + v * h\n    return [round(x, 6), round(v, 6)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[[[-1.484, 1.712, 27.312, 2.961, 1.788, 0.13, 4], [0.446958, 3.829901]], [[1.133, -1.182, 1.342, 0.781, 1.334, 0.1, 6], [0.491288, -0.912623]], [[-1.578, 0.145, 23.597, 0.588, 1.601, 0.01, 4], [-1.511386, 2.533387]], [[-1.392, -0.154, 8.638, 2.736, 1.406, 0.02, 2], [-1.392808, 0.024118]], [[0.514, 1.143, 25.432, 0.518, 1.496, 0.13, 9], [-0.010702, -0.661591]], [[0.0, 1.0, 4.0, 2.0, 0.5, 0.1, 3], [0.278786, 0.856518]], [[-1.377, 0.559, 2.87, 2.898, 1.015, 0.2, 0], [-1.377, 0.559]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]]], [[[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[-1.526, 0.873, 6.362, 1.828, 1.137, 0.13, 0], [-1.526, 0.873]], [[1.082, 0.058, 37.566, 1.034, 1.55, 0.05, 9], [-0.710961, -1.842409]], [[1.297, -0.319, 39.855, 2.122, 0.103, 0.01, 11], [1.106036, -2.858781]], [[0.764, 1.798, 31.467, 2.606, 1.52, 0.1, 4], [0.561426, -2.076687]], [[-1.669, -1.23, 27.053, 1.533, 0.991, 0.13, 5], [1.161742, 3.128527]], [[1.602, 1.18, 24.517, 1.738, 1.228, 0.05, 12], [-0.636953, -4.398087]], [[0.0, 1.0, 4.0, 2.0, 0.5, 0.1, 3], [0.278786, 0.856518]]], [[[0.718, 1.584, 9.356, 1.657, 0.286, 0.02, 6], [0.869722, 1.022798]], [[0.784, -0.426, 6.875, 0.731, 0.044, 0.1, 14], [-0.132405, 2.308875]], [[-1.544, -1.79, 32.966, 0.813, 1.455, 0.13, 4], [0.965451, -0.257038]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[0.0, 1.0, 4.0, 2.0, 0.5, 0.1, 3], [0.278786, 0.856518]], [[0.044, 1.377, 37.377, 1.154, 1.523, 0.13, 8], [-0.013801, 0.684737]], [[-1.525, 1.375, 3.248, 1.693, 1.607, 0.1, 4], [-0.87326, 1.732745]], [[-1.184, -1.519, 7.411, 1.786, 1.332, 0.04, 9], [-1.302744, 0.571925]]], [[[0.0, 1.0, 4.0, 2.0, 0.5, 0.1, 3], [0.278786, 0.856518]], [[1.818, 0.29, 35.382, 2.651, 1.322, 0.04, 12], [-0.242082, -5.8862]], [[-0.271, 1.727, 34.523, 2.128, 1.589, 0.13, 15], [0.205585, 0.49101]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[-0.56, 1.419, 29.61, 1.024, 1.85, 0.1, 6], [0.298786, -1.014037]], [[1.308, -0.567, 4.917, 1.142, 0.149, 0.02, 14], [0.933908, -1.927012]], [[-0.362, 1.363, 36.638, 0.869, 0.886, 0.2, 7], [0.193298, -0.396669]], [[0.484, 0.239, 12.355, 0.868, 0.3, 0.07, 0], [0.484, 0.239]]], [[[-1.576, -0.203, 24.72, 2.347, 1.363, 0.07, 10], [0.760633, 3.318988]], [[0.0, 1.0, 4.0, 2.0, 0.5, 0.1, 3], [0.278786, 0.856518]], [[-0.232, 1.993, 10.835, 2.4, 1.522, 0.01, 9], [-0.054651, 1.942283]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[-1.68, 0.833, 26.799, 1.025, 1.524, 0.13, 6], [0.548391, -4.033319]], [[-1.38, 1.106, 38.807, 2.361, 0.186, 0.05, 14], [1.397704, 0.597706]], [[0.695, 1.714, 8.192, 1.726, 1.775, 0.02, 2], [0.757467, 1.510568]], [[1.759, -0.492, 32.001, 1.805, 1.669, 0.13, 5], [-1.219998, -1.776701]]]]\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-substep-symplectic-euler-frame-loop-count","generated_at":"2026-09-29T14:50:53.460123+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":"Iterate exactly steps frames.","root_cause":"The outer loop iterates range(steps + 1).","sha256":"d720cd11c91c60405be7db202e1cf75801fdf1055d28efca68c674071fcf6520","title":"Frame loop integrates one frame too many · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":42.421,"exit_code":1,"observations":[{"actual":[-0.076434,4.281738],"check":"case 0","expected":[0.446958,3.829901],"passed":false},{"actual":[0.58429,-0.980985],"check":"case 1","expected":[0.491288,-0.912623],"passed":false},{"actual":[-1.53672,1.970335],"check":"case 2","expected":[-1.511386,2.533387],"passed":false},{"actual":[-1.39329,-0.064522],"check":"case 3","expected":[-1.392808,0.024118],"passed":false},{"actual":[0.07899,-0.603109],"check":"case 4","expected":[-0.010702,-0.661591],"passed":false},{"actual":[0.191397,0.921919],"check":"case 5","expected":[0.278786,0.856518],"passed":false},{"actual":[-1.377,0.559],"check":"case 6","expected":[-1.377,0.559],"passed":true},{"actual":[1.0,0.0],"check":"case 7","expected":[1.0,0.0],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"case 0\", \"actual\": [-0.076434, 4.281738], \"expected\": [0.446958, 3.829901], \"passed\": false}, {\"check\": \"case 1\", \"actual\": [0.58429, -0.980985], \"expected\": [0.491288, -0.912623], \"passed\": false}, {\"check\": \"case 2\", \"actual\": [-1.53672, 1.970335], \"expected\": [-1.511386, 2.533387], \"passed\": false}, {\"check\": \"case 3\", \"actual\": [-1.39329, -0.064522], \"expected\": [-1.392808, 0.024118], \"passed\": false}, {\"check\": \"case 4\", \"actual\": [0.07899, -0.603109], \"expected\": [-0.010702, -0.661591], \"passed\": false}, {\"check\": \"case 5\", \"actual\": [0.191397, 0.921919], \"expected\": [0.278786, 0.856518], \"passed\": false}, {\"check\": \"case 6\", \"actual\": [-1.377, 0.559], \"expected\": [-1.377, 0.559], \"passed\": true}, {\"check\": \"case 7\", \"actual\": [1.0, 0.0], \"expected\": [1.0, 0.0], \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":39.751,"exit_code":1,"observations":[{"actual":[0.863255,2.837862],"check":"case 0","expected":[0.446958,3.829901],"passed":false},{"actual":[0.405421,-0.840444],"check":"case 1","expected":[0.491288,-0.912623],"passed":false},{"actual":[-1.480676,3.070942],"check":"case 2","expected":[-1.511386,2.533387],"passed":false},{"actual":[-1.390572,0.111817],"check":"case 3","expected":[-1.392808,0.024118],"passed":false},{"actual":[-0.063446,-0.248749],"check":"case 4","expected":[-0.010702,-0.661591],"passed":false},{"actual":[0.358469,0.775741],"check":"case 5","expected":[0.278786,0.856518],"passed":false},{"actual":[-1.237301,0.776909],"check":"case 6","expected":[-1.377,0.559],"passed":false},{"actual":[0.975,-0.5],"check":"case 7","expected":[1.0,0.0],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"case 0\", \"actual\": [0.863255, 2.837862], \"expected\": [0.446958, 3.829901], \"passed\": false}, {\"check\": \"case 1\", \"actual\": [0.405421, -0.840444], \"expected\": [0.491288, -0.912623], \"passed\": false}, {\"check\": \"case 2\", \"actual\": [-1.480676, 3.070942], \"expected\": [-1.511386, 2.533387], \"passed\": false}, {\"check\": \"case 3\", \"actual\": [-1.390572, 0.111817], \"expected\": [-1.392808, 0.024118], \"passed\": false}, {\"check\": \"case 4\", \"actual\": [-0.063446, -0.248749], \"expected\": [-0.010702, -0.661591], \"passed\": false}, {\"check\": \"case 5\", \"actual\": [0.358469, 0.775741], \"expected\": [0.278786, 0.856518], \"passed\": false}, {\"check\": \"case 6\", \"actual\": [-1.237301, 0.776909], \"expected\": [-1.377, 0.559], \"passed\": false}, {\"check\": \"case 7\", \"actual\": [0.975, -0.5], \"expected\": [1.0, 0.0], \"passed\": false}], \"passed\": false}\n"},"fixed":{"elapsed_ms":40.322,"exit_code":0,"observations":[{"actual":[0.446958,3.829901],"check":"case 0","expected":[0.446958,3.829901],"passed":true},{"actual":[0.491288,-0.912623],"check":"case 1","expected":[0.491288,-0.912623],"passed":true},{"actual":[-1.511386,2.533387],"check":"case 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