{"abstract":"Peak energy ratio is inflated whenever the spring is stretched.","category":"Physics integrator stability","checks":8,"contract":"solve(x0, v0, k, m, c, dt, steps): backward Euler for x'=v, m v'=-k x-c v in closed form v1=(v-dt*k*x/m)/(1+dt*c/m+dt^2*k/m), x1=x+dt*v1. Energy E=0.5 m v^2+0.5 k x^2; report the peak E_n/E_0 including step 0 (0 when E_0=0). Return [x, v, peak] rounded to 6.","evaluation_group":"w2-physics_integrator_stability-backward-euler-spring","failed_approach":"Using mass instead of stiffness in the potential mixes units.","family":"w2-physics_integrator_stability-backward-euler-spring-potential-energy-half","id":"FA-87106","implementations":{"attempt":{"sha256":"b68e8ad5c400cc76c4a5ce90cc6f173ca7c63d650fbf9553424ce724a8f6bc7e","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    x, v = x0, v0\n    def energy(p, q):\n        return 0.5 * m * q * q + 0.5 * m * p * p\n    e0 = energy(x, v)\n    peak = 1.0 if e0 > 0 else 0.0\n    for _ in range(steps):\n        v = (v - dt * k * x / m) / (1 + dt * c / m + dt * dt * k / m)\n        x = x + dt * v\n        if e0 > 0:\n            peak = max(peak, energy(x, v) / e0)\n    return [round(x, 6), round(v, 6), round(peak, 6)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[[[0.24, -0.911, 49.012, 1.578, 2.704, 0.1, 5], [-0.09708, -0.084706, 1.0]], [[1.287, -0.056, 59.572, 2.284, 0.95, 0.01, 3], [1.265475, -1.044567, 1.0]], [[-0.665, 1.365, 36.996, 1.759, 0.53, 0.5, 3], [0.035382, -0.128106, 1.0]], [[-0.386, -0.618, 22.711, 1.957, 1.669, 0.05, 3], [-0.405267, 0.106477, 1.0]], [[1.126, 0.46, 23.793, 1.74, 1.365, 0.01, 2], [1.130507, 0.147492, 1.0]], [[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[-0.525, -1.536, 11.54, 2.318, 2.111, 0.1, 6], [-0.633115, 0.624515, 1.0]], [[0.798, 0.799, 72.701, 2.561, 2.07, 0.05, 3], [0.616397, -2.061708, 1.0]]], [[[0.326, 0.466, 61.522, 2.623, 2.291, 0.01, 3], [0.335138, 0.223966, 1.0]], [[-1.894, -0.891, 20.673, 0.599, 1.526, 0.5, 6], [-0.001474, -0.000425, 1.0]], [[-0.309, -1.711, 54.616, 0.839, 0.325, 0.5, 6], [-4.2e-05, 0.000468, 1.0]], [[-1.228, -0.451, 23.36, 2.048, 1.522, 0.01, 2], [-1.232747, -0.166441, 1.0]], [[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[0.614, -0.235, 4.708, 0.51, 0.987, 0.05, 3], [0.517177, -0.816611, 1.0]], [[-1.593, 1.89, 51.296, 0.937, 1.405, 0.01, 5], [-1.382702, 5.629739, 1.0]], [[-1.267, -0.184, 5.424, 1.543, 1.623, 0.01, 7], [-1.267247, 0.128498, 1.0]]], [[[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[1.099, -1.351, 78.259, 0.426, 0.263, 0.05, 3], [-0.156961, -7.876689, 1.0]], [[-1.054, -0.343, 49.43, 2.124, 2.635, 0.05, 1], [-1.014568, 0.788639, 1.0]], [[0.257, 1.826, 18.613, 1.995, 1.806, 0.1, 7], [0.273046, -0.937505, 1.0]], [[-1.814, -0.942, 9.978, 1.746, 0.712, 0.01, 9], [-1.850539, 0.019999, 1.0]], [[-1.949, -0.338, 47.923, 2.883, 2.062, 0.1, 3], [-0.781938, 5.129576, 1.0]], [[-0.678, -1.597, 66.475, 1.112, 2.962, 0.01, 5], [-0.693206, 0.522371, 1.0]], [[0.787, -0.883, 93.251, 1.633, 2.06, 0.05, 3], [0.262131, -4.192805, 1.0]]], [[[-1.031, -0.971, 69.316, 1.698, 1.184, 0.01, 12], [-0.833114, 3.639745, 1.0]], [[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[0.283, 1.441, 38.896, 1.952, 0.391, 0.05, 8], [0.217372, -1.212269, 1.0]], [[-0.933, 1.78, 8.086, 0.726, 0.672, 0.1, 2], [-0.439694, 2.591455, 1.0]], [[-1.345, -0.611, 81.079, 1.16, 2.627, 0.01, 4], [-1.279534, 2.916997, 1.0]], [[0.48, 0.698, 50.79, 1.825, 2.065, 0.05, 5], [0.243232, -1.683845, 1.0]], [[-0.239, -1.564, 66.752, 1.696, 1.527, 0.01, 3], [-0.278987, -1.213386, 1.0]], [[1.505, -0.392, 75.224, 1.938, 0.736, 0.5, 10], [1e-05, 1.2e-05, 1.0]]], [[[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[-1.508, 1.83, 17.985, 1.33, 1.085, 0.01, 10], [-1.231328, 3.455601, 1.0]], [[-1.039, -1.09, 32.245, 0.581, 1.221, 0.1, 8], [0.030932, -0.820883, 1.0]], [[-1.064, 1.282, 76.595, 2.19, 0.837, 0.05, 8], [0.553758, 2.614546, 1.0]], [[1.155, 1.357, 6.477, 2.373, 1.143, 0.1, 4], [1.316819, -0.14298, 1.0]], [[0.031, 0.742, 81.09, 2.83, 2.129, 0.05, 4], [0.111129, 0.16107, 1.0]], [[0.544, 1.609, 65.688, 1.374, 0.719, 0.01, 10], [0.552951, -1.128422, 1.0]], [[1.48, -1.5, 47.647, 0.462, 0.24, 0.5, 7], [-1.4e-05, 2e-05, 1.0]]]]\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":"fec480bab1f0f1e16b715c1e5d3d520aa403874b5bf3052ed6787a540e2d2ab4","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    x, v = x0, v0\n    def energy(p, q):\n        return 0.5 * m * q * q + k * p * p\n    e0 = energy(x, v)\n    peak = 1.0 if e0 > 0 else 0.0\n    for _ in range(steps):\n        v = (v - dt * k * x / m) / (1 + dt * c / m + dt * dt * k / m)\n        x = x + dt * v\n        if e0 > 0:\n            peak = max(peak, energy(x, v) / e0)\n    return [round(x, 6), round(v, 6), round(peak, 6)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[[[0.24, -0.911, 49.012, 1.578, 2.704, 0.1, 5], [-0.09708, -0.084706, 1.0]], [[1.287, -0.056, 59.572, 2.284, 0.95, 0.01, 3], [1.265475, -1.044567, 1.0]], [[-0.665, 1.365, 36.996, 1.759, 0.53, 0.5, 3], [0.035382, -0.128106, 1.0]], [[-0.386, -0.618, 22.711, 1.957, 1.669, 0.05, 3], [-0.405267, 0.106477, 1.0]], [[1.126, 0.46, 23.793, 1.74, 1.365, 0.01, 2], [1.130507, 0.147492, 1.0]], [[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[-0.525, -1.536, 11.54, 2.318, 2.111, 0.1, 6], [-0.633115, 0.624515, 1.0]], [[0.798, 0.799, 72.701, 2.561, 2.07, 0.05, 3], [0.616397, -2.061708, 1.0]]], [[[0.326, 0.466, 61.522, 2.623, 2.291, 0.01, 3], [0.335138, 0.223966, 1.0]], [[-1.894, -0.891, 20.673, 0.599, 1.526, 0.5, 6], [-0.001474, -0.000425, 1.0]], [[-0.309, -1.711, 54.616, 0.839, 0.325, 0.5, 6], [-4.2e-05, 0.000468, 1.0]], [[-1.228, -0.451, 23.36, 2.048, 1.522, 0.01, 2], [-1.232747, -0.166441, 1.0]], [[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[0.614, -0.235, 4.708, 0.51, 0.987, 0.05, 3], [0.517177, -0.816611, 1.0]], [[-1.593, 1.89, 51.296, 0.937, 1.405, 0.01, 5], [-1.382702, 5.629739, 1.0]], [[-1.267, -0.184, 5.424, 1.543, 1.623, 0.01, 7], [-1.267247, 0.128498, 1.0]]], [[[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[1.099, -1.351, 78.259, 0.426, 0.263, 0.05, 3], [-0.156961, -7.876689, 1.0]], [[-1.054, -0.343, 49.43, 2.124, 2.635, 0.05, 1], [-1.014568, 0.788639, 1.0]], [[0.257, 1.826, 18.613, 1.995, 1.806, 0.1, 7], [0.273046, -0.937505, 1.0]], [[-1.814, -0.942, 9.978, 1.746, 0.712, 0.01, 9], [-1.850539, 0.019999, 1.0]], [[-1.949, -0.338, 47.923, 2.883, 2.062, 0.1, 3], [-0.781938, 5.129576, 1.0]], [[-0.678, -1.597, 66.475, 1.112, 2.962, 0.01, 5], [-0.693206, 0.522371, 1.0]], [[0.787, -0.883, 93.251, 1.633, 2.06, 0.05, 3], [0.262131, -4.192805, 1.0]]], [[[-1.031, -0.971, 69.316, 1.698, 1.184, 0.01, 12], [-0.833114, 3.639745, 1.0]], [[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[0.283, 1.441, 38.896, 1.952, 0.391, 0.05, 8], [0.217372, -1.212269, 1.0]], [[-0.933, 1.78, 8.086, 0.726, 0.672, 0.1, 2], [-0.439694, 2.591455, 1.0]], [[-1.345, -0.611, 81.079, 1.16, 2.627, 0.01, 4], [-1.279534, 2.916997, 1.0]], [[0.48, 0.698, 50.79, 1.825, 2.065, 0.05, 5], [0.243232, -1.683845, 1.0]], [[-0.239, -1.564, 66.752, 1.696, 1.527, 0.01, 3], [-0.278987, -1.213386, 1.0]], [[1.505, -0.392, 75.224, 1.938, 0.736, 0.5, 10], [1e-05, 1.2e-05, 1.0]]], [[[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[-1.508, 1.83, 17.985, 1.33, 1.085, 0.01, 10], [-1.231328, 3.455601, 1.0]], [[-1.039, -1.09, 32.245, 0.581, 1.221, 0.1, 8], [0.030932, -0.820883, 1.0]], [[-1.064, 1.282, 76.595, 2.19, 0.837, 0.05, 8], [0.553758, 2.614546, 1.0]], [[1.155, 1.357, 6.477, 2.373, 1.143, 0.1, 4], [1.316819, -0.14298, 1.0]], [[0.031, 0.742, 81.09, 2.83, 2.129, 0.05, 4], [0.111129, 0.16107, 1.0]], [[0.544, 1.609, 65.688, 1.374, 0.719, 0.01, 10], [0.552951, -1.128422, 1.0]], [[1.48, -1.5, 47.647, 0.462, 0.24, 0.5, 7], [-1.4e-05, 2e-05, 1.0]]]]\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":"e73b1710a4c237c0759349e06b9bfcb656efb966e10d1a704cf40fd4e40522ae","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    x, v = x0, v0\n    def energy(p, q):\n        return 0.5 * m * q * q + 0.5 * k * p * p\n    e0 = energy(x, v)\n    peak = 1.0 if e0 > 0 else 0.0\n    for _ in range(steps):\n        v = (v - dt * k * x / m) / (1 + dt * c / m + dt * dt * k / m)\n        x = x + dt * v\n        if e0 > 0:\n            peak = max(peak, energy(x, v) / e0)\n    return [round(x, 6), round(v, 6), round(peak, 6)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[[[0.24, -0.911, 49.012, 1.578, 2.704, 0.1, 5], [-0.09708, -0.084706, 1.0]], [[1.287, -0.056, 59.572, 2.284, 0.95, 0.01, 3], [1.265475, -1.044567, 1.0]], [[-0.665, 1.365, 36.996, 1.759, 0.53, 0.5, 3], [0.035382, -0.128106, 1.0]], [[-0.386, -0.618, 22.711, 1.957, 1.669, 0.05, 3], [-0.405267, 0.106477, 1.0]], [[1.126, 0.46, 23.793, 1.74, 1.365, 0.01, 2], [1.130507, 0.147492, 1.0]], [[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[-0.525, -1.536, 11.54, 2.318, 2.111, 0.1, 6], [-0.633115, 0.624515, 1.0]], [[0.798, 0.799, 72.701, 2.561, 2.07, 0.05, 3], [0.616397, -2.061708, 1.0]]], [[[0.326, 0.466, 61.522, 2.623, 2.291, 0.01, 3], [0.335138, 0.223966, 1.0]], [[-1.894, -0.891, 20.673, 0.599, 1.526, 0.5, 6], [-0.001474, -0.000425, 1.0]], [[-0.309, -1.711, 54.616, 0.839, 0.325, 0.5, 6], [-4.2e-05, 0.000468, 1.0]], [[-1.228, -0.451, 23.36, 2.048, 1.522, 0.01, 2], [-1.232747, -0.166441, 1.0]], [[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[0.614, -0.235, 4.708, 0.51, 0.987, 0.05, 3], [0.517177, -0.816611, 1.0]], [[-1.593, 1.89, 51.296, 0.937, 1.405, 0.01, 5], [-1.382702, 5.629739, 1.0]], [[-1.267, -0.184, 5.424, 1.543, 1.623, 0.01, 7], [-1.267247, 0.128498, 1.0]]], [[[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[1.099, -1.351, 78.259, 0.426, 0.263, 0.05, 3], [-0.156961, -7.876689, 1.0]], [[-1.054, -0.343, 49.43, 2.124, 2.635, 0.05, 1], [-1.014568, 0.788639, 1.0]], [[0.257, 1.826, 18.613, 1.995, 1.806, 0.1, 7], [0.273046, -0.937505, 1.0]], [[-1.814, -0.942, 9.978, 1.746, 0.712, 0.01, 9], [-1.850539, 0.019999, 1.0]], [[-1.949, -0.338, 47.923, 2.883, 2.062, 0.1, 3], [-0.781938, 5.129576, 1.0]], [[-0.678, -1.597, 66.475, 1.112, 2.962, 0.01, 5], [-0.693206, 0.522371, 1.0]], [[0.787, -0.883, 93.251, 1.633, 2.06, 0.05, 3], [0.262131, -4.192805, 1.0]]], [[[-1.031, -0.971, 69.316, 1.698, 1.184, 0.01, 12], [-0.833114, 3.639745, 1.0]], [[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[0.283, 1.441, 38.896, 1.952, 0.391, 0.05, 8], [0.217372, -1.212269, 1.0]], [[-0.933, 1.78, 8.086, 0.726, 0.672, 0.1, 2], [-0.439694, 2.591455, 1.0]], [[-1.345, -0.611, 81.079, 1.16, 2.627, 0.01, 4], [-1.279534, 2.916997, 1.0]], [[0.48, 0.698, 50.79, 1.825, 2.065, 0.05, 5], [0.243232, -1.683845, 1.0]], [[-0.239, -1.564, 66.752, 1.696, 1.527, 0.01, 3], [-0.278987, -1.213386, 1.0]], [[1.505, -0.392, 75.224, 1.938, 0.736, 0.5, 10], [1e-05, 1.2e-05, 1.0]]], [[[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[-1.508, 1.83, 17.985, 1.33, 1.085, 0.01, 10], [-1.231328, 3.455601, 1.0]], [[-1.039, -1.09, 32.245, 0.581, 1.221, 0.1, 8], [0.030932, -0.820883, 1.0]], [[-1.064, 1.282, 76.595, 2.19, 0.837, 0.05, 8], [0.553758, 2.614546, 1.0]], [[1.155, 1.357, 6.477, 2.373, 1.143, 0.1, 4], [1.316819, -0.14298, 1.0]], [[0.031, 0.742, 81.09, 2.83, 2.129, 0.05, 4], [0.111129, 0.16107, 1.0]], [[0.544, 1.609, 65.688, 1.374, 0.719, 0.01, 10], [0.552951, -1.128422, 1.0]], [[1.48, -1.5, 47.647, 0.462, 0.24, 0.5, 7], [-1.4e-05, 2e-05, 1.0]]]]\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-backward-euler-spring-potential-energy-half","generated_at":"2026-09-29T14:50:55.747954+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":"Use 0.5*k*x^2.","root_cause":"Potential energy is k*x^2.","sha256":"97e61e5fc7105fea514e2736e2c5733bd93bb0abdb2ed3e0e4e97e287bfccde6","title":"Energy monitor doubles the spring potential · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":38.959,"exit_code":1,"observations":[{"actual":[-0.09708,-0.084706,1.42619],"check":"case 0","expected":[-0.09708,-0.084706,1.0],"passed":false},{"actual":[1.265475,-1.044567,1.6225],"check":"case 1","expected":[1.265475,-1.044567,1.0],"passed":false},{"actual":[0.035382,-0.128106,1.0],"check":"case 2","expected":[0.035382,-0.128106,1.0],"passed":true},{"actual":[-0.405267,0.106477,1.0],"check":"case 3","expected":[-0.405267,0.106477,1.0],"passed":true},{"actual":[1.130507,0.147492,1.0],"check":"case 4","expected":[1.130507,0.147492,1.0],"passed":true},{"actual":[0.0,0.0,0.0],"check":"case 5","expected":[0.0,0.0,0.0],"passed":true},{"actual":[-0.633115,0.624515,1.0],"check":"case 6","expected":[-0.633115,0.624515,1.0],"passed":true},{"actual":[0.616397,-2.061708,3.631246],"check":"case 7","expected":[0.616397,-2.061708,1.0],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"case 0\", \"actual\": [-0.09708, -0.084706, 1.42619], \"expected\": [-0.09708, -0.084706, 1.0], \"passed\": false}, {\"check\": \"case 1\", \"actual\": [1.265475, -1.044567, 1.6225], \"expected\": [1.265475, -1.044567, 1.0], \"passed\": false}, {\"check\": \"case 2\", \"actual\": [0.035382, -0.128106, 1.0], \"expected\": [0.035382, -0.128106, 1.0], \"passed\": true}, {\"check\": \"case 3\", \"actual\": [-0.405267, 0.106477, 1.0], \"expected\": [-0.405267, 0.106477, 1.0], \"passed\": true}, {\"check\": \"case 4\", \"actual\": [1.130507, 0.147492, 1.0], \"expected\": [1.130507, 0.147492, 1.0], \"passed\": true}, {\"check\": \"case 5\", \"actual\": [0.0, 0.0, 0.0], \"expected\": [0.0, 0.0, 0.0], \"passed\": true}, {\"check\": \"case 6\", \"actual\": [-0.633115, 0.624515, 1.0], \"expected\": [-0.633115, 0.624515, 1.0], \"passed\": true}, {\"check\": \"case 7\", \"actual\": [0.616397, -2.061708, 3.631246], \"expected\": [0.616397, -2.061708, 1.0], \"passed\": false}], \"passed\": 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