{"abstract":"The implicit solve pushes the mass away from rest.","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":"Dropping the mass division scales the spring force wrongly.","family":"w2-physics_integrator_stability-backward-euler-spring-implicit-force-numerator","id":"FA-87121","implementations":{"attempt":{"sha256":"22e47d1dedf793536a2a1645f2056507fd4adfe08ad8f27f2f004ba30aa56a12","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) / (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 = [[[[0.191, 1.945, 34.85, 2.123, 1.81, 0.5, 2], [0.064229, -0.321773, 1.0]], [[-1.795, -0.563, 97.971, 1.205, 2.366, 0.01, 5], [-1.618428, 6.062195, 1.0]], [[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[-0.221, -1.064, 98.514, 1.675, 2.729, 0.05, 6], [-0.014738, 1.136191, 1.0]], [[1.261, 0.83, 48.017, 0.99, 1.646, 0.1, 5], [-0.320762, -0.982525, 1.0]], [[-1.173, 0.504, 23.099, 0.515, 1.355, 0.01, 5], [-1.077545, 2.77492, 1.0]], [[-0.792, 1.489, 8.768, 2.262, 2.81, 0.01, 2], [-0.761883, 1.511237, 1.0]], [[-1.37, -0.31, 86.176, 2.116, 2.597, 0.05, 3], [-0.794438, 5.334862, 1.0]]], [[[0.194, -0.92, 25.988, 2.712, 0.886, 0.1, 8], [-0.209628, 0.163927, 1.0]], [[1.723, -0.188, 11.545, 1.405, 2.146, 0.01, 2], [1.715173, -0.458297, 1.0]], [[-0.645, 1.055, 48.194, 1.466, 2.383, 0.01, 5], [-0.565671, 1.91353, 1.0]], [[0.421, 0.744, 85.01, 1.471, 2.008, 0.05, 3], [0.24471, -1.858224, 1.0]], [[0.285, 0.068, 25.144, 2.544, 0.635, 0.05, 8], [0.107212, -0.715572, 1.0]], [[-0.368, -0.287, 23.934, 2.087, 1.05, 0.05, 2], [-0.364916, 0.133775, 1.0]], [[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[-0.519, -0.968, 62.61, 2.039, 0.152, 0.1, 6], [0.217976, 0.56902, 1.0]]], [[[-0.504, 1.004, 53.161, 1.37, 0.128, 0.01, 2], [-0.478281, 1.378125, 1.0]], [[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[-1.291, 0.844, 51.11, 0.463, 2.435, 0.1, 3], [0.082694, 2.804545, 1.0]], [[-1.724, -0.957, 58.152, 0.412, 2.15, 0.01, 9], [-0.983354, 13.094621, 1.0]], [[1.295, 1.182, 85.267, 2.457, 0.514, 0.5, 4], [0.000372, 0.08085, 1.0]], [[-1.201, -0.558, 89.531, 1.586, 0.182, 0.1, 5], [0.386107, -0.013533, 1.0]], [[-0.643, 1.644, 71.947, 1.526, 2.926, 0.01, 9], [-0.398185, 3.378185, 1.0]], [[-1.096, -0.407, 20.232, 0.596, 1.678, 0.1, 9], [0.100413, -0.614384, 1.0]]], [[[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[-0.969, 1.013, 41.406, 1.249, 0.992, 0.1, 10], [-0.075433, -0.873216, 1.0]], [[0.283, 1.441, 38.896, 1.952, 0.391, 0.05, 8], [0.217372, -1.212269, 1.0]], [[0.48, 0.698, 50.79, 1.825, 2.065, 0.05, 5], [0.243232, -1.683845, 1.0]], [[1.258, 0.72, 24.592, 2.071, 0.163, 0.5, 3], [-0.155797, -0.107234, 1.0]], [[-1.031, -0.971, 69.316, 1.698, 1.184, 0.01, 12], [-0.833114, 3.639745, 1.0]], [[0.777, 1.994, 57.363, 2.434, 0.585, 0.01, 3], [0.82519, 1.409422, 1.0]], [[-0.659, 0.567, 60.81, 2.3, 1.191, 0.5, 12], [-0.0, 1.4e-05, 1.0]]], [[[-1.133, 1.016, 46.913, 2.142, 2.129, 0.05, 9], [0.363941, 2.748968, 1.0]], [[0.539, 1.387, 93.276, 0.984, 1.857, 0.5, 7], [-4e-06, -4.4e-05, 1.0]], [[-1.397, -0.537, 14.554, 2.905, 0.576, 0.1, 10], [0.392502, 2.064847, 1.0]], [[0.537, 1.491, 17.106, 2.255, 2.958, 0.05, 7], [0.657535, -0.41846, 1.0]], [[1.48, -1.5, 47.647, 0.462, 0.24, 0.5, 7], [-1.4e-05, 2e-05, 1.0]], [[0.544, 1.609, 65.688, 1.374, 0.719, 0.01, 10], [0.552951, -1.128422, 1.0]], [[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[-1.039, -1.09, 32.245, 0.581, 1.221, 0.1, 8], [0.030932, -0.820883, 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":"f4ad7c0d042431d933ea6e39cd9df07864f37d3658b479f27bec3fac252ae290","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 = [[[[0.191, 1.945, 34.85, 2.123, 1.81, 0.5, 2], [0.064229, -0.321773, 1.0]], [[-1.795, -0.563, 97.971, 1.205, 2.366, 0.01, 5], [-1.618428, 6.062195, 1.0]], [[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[-0.221, -1.064, 98.514, 1.675, 2.729, 0.05, 6], [-0.014738, 1.136191, 1.0]], [[1.261, 0.83, 48.017, 0.99, 1.646, 0.1, 5], [-0.320762, -0.982525, 1.0]], [[-1.173, 0.504, 23.099, 0.515, 1.355, 0.01, 5], [-1.077545, 2.77492, 1.0]], [[-0.792, 1.489, 8.768, 2.262, 2.81, 0.01, 2], [-0.761883, 1.511237, 1.0]], [[-1.37, -0.31, 86.176, 2.116, 2.597, 0.05, 3], [-0.794438, 5.334862, 1.0]]], [[[0.194, -0.92, 25.988, 2.712, 0.886, 0.1, 8], [-0.209628, 0.163927, 1.0]], [[1.723, -0.188, 11.545, 1.405, 2.146, 0.01, 2], [1.715173, -0.458297, 1.0]], [[-0.645, 1.055, 48.194, 1.466, 2.383, 0.01, 5], [-0.565671, 1.91353, 1.0]], [[0.421, 0.744, 85.01, 1.471, 2.008, 0.05, 3], [0.24471, -1.858224, 1.0]], [[0.285, 0.068, 25.144, 2.544, 0.635, 0.05, 8], [0.107212, -0.715572, 1.0]], [[-0.368, -0.287, 23.934, 2.087, 1.05, 0.05, 2], [-0.364916, 0.133775, 1.0]], [[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[-0.519, -0.968, 62.61, 2.039, 0.152, 0.1, 6], [0.217976, 0.56902, 1.0]]], [[[-0.504, 1.004, 53.161, 1.37, 0.128, 0.01, 2], [-0.478281, 1.378125, 1.0]], [[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[-1.291, 0.844, 51.11, 0.463, 2.435, 0.1, 3], [0.082694, 2.804545, 1.0]], [[-1.724, -0.957, 58.152, 0.412, 2.15, 0.01, 9], [-0.983354, 13.094621, 1.0]], [[1.295, 1.182, 85.267, 2.457, 0.514, 0.5, 4], [0.000372, 0.08085, 1.0]], [[-1.201, -0.558, 89.531, 1.586, 0.182, 0.1, 5], [0.386107, -0.013533, 1.0]], [[-0.643, 1.644, 71.947, 1.526, 2.926, 0.01, 9], [-0.398185, 3.378185, 1.0]], [[-1.096, -0.407, 20.232, 0.596, 1.678, 0.1, 9], [0.100413, -0.614384, 1.0]]], [[[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[-0.969, 1.013, 41.406, 1.249, 0.992, 0.1, 10], [-0.075433, -0.873216, 1.0]], [[0.283, 1.441, 38.896, 1.952, 0.391, 0.05, 8], [0.217372, -1.212269, 1.0]], [[0.48, 0.698, 50.79, 1.825, 2.065, 0.05, 5], [0.243232, -1.683845, 1.0]], [[1.258, 0.72, 24.592, 2.071, 0.163, 0.5, 3], [-0.155797, -0.107234, 1.0]], [[-1.031, -0.971, 69.316, 1.698, 1.184, 0.01, 12], [-0.833114, 3.639745, 1.0]], [[0.777, 1.994, 57.363, 2.434, 0.585, 0.01, 3], [0.82519, 1.409422, 1.0]], [[-0.659, 0.567, 60.81, 2.3, 1.191, 0.5, 12], [-0.0, 1.4e-05, 1.0]]], [[[-1.133, 1.016, 46.913, 2.142, 2.129, 0.05, 9], [0.363941, 2.748968, 1.0]], [[0.539, 1.387, 93.276, 0.984, 1.857, 0.5, 7], [-4e-06, -4.4e-05, 1.0]], [[-1.397, -0.537, 14.554, 2.905, 0.576, 0.1, 10], [0.392502, 2.064847, 1.0]], [[0.537, 1.491, 17.106, 2.255, 2.958, 0.05, 7], [0.657535, -0.41846, 1.0]], [[1.48, -1.5, 47.647, 0.462, 0.24, 0.5, 7], [-1.4e-05, 2e-05, 1.0]], [[0.544, 1.609, 65.688, 1.374, 0.719, 0.01, 10], [0.552951, -1.128422, 1.0]], [[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[-1.039, -1.09, 32.245, 0.581, 1.221, 0.1, 8], [0.030932, -0.820883, 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":"440f1e36add67f97885b3fcb18f14d16717ec6a13a966c8d69e63afed96a0b08","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 = [[[[0.191, 1.945, 34.85, 2.123, 1.81, 0.5, 2], [0.064229, -0.321773, 1.0]], [[-1.795, -0.563, 97.971, 1.205, 2.366, 0.01, 5], [-1.618428, 6.062195, 1.0]], [[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[-0.221, -1.064, 98.514, 1.675, 2.729, 0.05, 6], [-0.014738, 1.136191, 1.0]], [[1.261, 0.83, 48.017, 0.99, 1.646, 0.1, 5], [-0.320762, -0.982525, 1.0]], [[-1.173, 0.504, 23.099, 0.515, 1.355, 0.01, 5], [-1.077545, 2.77492, 1.0]], [[-0.792, 1.489, 8.768, 2.262, 2.81, 0.01, 2], [-0.761883, 1.511237, 1.0]], [[-1.37, -0.31, 86.176, 2.116, 2.597, 0.05, 3], [-0.794438, 5.334862, 1.0]]], [[[0.194, -0.92, 25.988, 2.712, 0.886, 0.1, 8], [-0.209628, 0.163927, 1.0]], [[1.723, -0.188, 11.545, 1.405, 2.146, 0.01, 2], [1.715173, -0.458297, 1.0]], [[-0.645, 1.055, 48.194, 1.466, 2.383, 0.01, 5], [-0.565671, 1.91353, 1.0]], [[0.421, 0.744, 85.01, 1.471, 2.008, 0.05, 3], [0.24471, -1.858224, 1.0]], [[0.285, 0.068, 25.144, 2.544, 0.635, 0.05, 8], [0.107212, -0.715572, 1.0]], [[-0.368, -0.287, 23.934, 2.087, 1.05, 0.05, 2], [-0.364916, 0.133775, 1.0]], [[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[-0.519, -0.968, 62.61, 2.039, 0.152, 0.1, 6], [0.217976, 0.56902, 1.0]]], [[[-0.504, 1.004, 53.161, 1.37, 0.128, 0.01, 2], [-0.478281, 1.378125, 1.0]], [[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[-1.291, 0.844, 51.11, 0.463, 2.435, 0.1, 3], [0.082694, 2.804545, 1.0]], [[-1.724, -0.957, 58.152, 0.412, 2.15, 0.01, 9], [-0.983354, 13.094621, 1.0]], [[1.295, 1.182, 85.267, 2.457, 0.514, 0.5, 4], [0.000372, 0.08085, 1.0]], [[-1.201, -0.558, 89.531, 1.586, 0.182, 0.1, 5], [0.386107, -0.013533, 1.0]], [[-0.643, 1.644, 71.947, 1.526, 2.926, 0.01, 9], [-0.398185, 3.378185, 1.0]], [[-1.096, -0.407, 20.232, 0.596, 1.678, 0.1, 9], [0.100413, -0.614384, 1.0]]], [[[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[-0.969, 1.013, 41.406, 1.249, 0.992, 0.1, 10], [-0.075433, -0.873216, 1.0]], [[0.283, 1.441, 38.896, 1.952, 0.391, 0.05, 8], [0.217372, -1.212269, 1.0]], [[0.48, 0.698, 50.79, 1.825, 2.065, 0.05, 5], [0.243232, -1.683845, 1.0]], [[1.258, 0.72, 24.592, 2.071, 0.163, 0.5, 3], [-0.155797, -0.107234, 1.0]], [[-1.031, -0.971, 69.316, 1.698, 1.184, 0.01, 12], [-0.833114, 3.639745, 1.0]], [[0.777, 1.994, 57.363, 2.434, 0.585, 0.01, 3], [0.82519, 1.409422, 1.0]], [[-0.659, 0.567, 60.81, 2.3, 1.191, 0.5, 12], [-0.0, 1.4e-05, 1.0]]], [[[-1.133, 1.016, 46.913, 2.142, 2.129, 0.05, 9], [0.363941, 2.748968, 1.0]], [[0.539, 1.387, 93.276, 0.984, 1.857, 0.5, 7], [-4e-06, -4.4e-05, 1.0]], [[-1.397, -0.537, 14.554, 2.905, 0.576, 0.1, 10], [0.392502, 2.064847, 1.0]], [[0.537, 1.491, 17.106, 2.255, 2.958, 0.05, 7], [0.657535, -0.41846, 1.0]], [[1.48, -1.5, 47.647, 0.462, 0.24, 0.5, 7], [-1.4e-05, 2e-05, 1.0]], [[0.544, 1.609, 65.688, 1.374, 0.719, 0.01, 10], [0.552951, -1.128422, 1.0]], [[0.0, 0.0, 5.0, 1.0, 0.5, 0.1, 3], [0.0, 0.0, 0.0]], [[-1.039, -1.09, 32.245, 0.581, 1.221, 0.1, 8], [0.030932, -0.820883, 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-implicit-force-numerator","generated_at":"2026-09-29T14:50:55.915442+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 v-dt*k*x/m.","root_cause":"The numerator is v+dt*k*x/m.","sha256":"fec893d6fa92a93a00e3450bfd5253f1367375ad603b5b60aa76ccaf89d3c074","title":"Backward Euler numerator applies the spring force with the wrong sign · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":39.448,"exit_code":1,"observations":[{"actual":[-0.060561,-0.253006,1.0],"check":"case 0","expected":[0.064229,-0.321773,1.0],"passed":false},{"actual":[-1.577839,7.353168,1.0],"check":"case 1","expected":[-1.618428,6.062195,1.0],"passed":false},{"actual":[0.0,0.0,0.0],"check":"case 2","expected":[0.0,0.0,0.0],"passed":true},{"actual":[0.092256,0.947269,1.0],"check":"case 3","expected":[-0.014738,1.136191,1.0],"passed":false},{"actual":[-0.317004,-1.020761,1.0],"check":"case 4","expected":[-0.320762,-0.982525,1.0],"passed":false},{"actual":[-1.112486,1.649625,1.0],"check":"case 5","expected":[-1.077545,2.77492,1.0],"passed":false},{"actual":[-0.760748,1.586501,1.024043],"check":"case 6","expected":[-0.761883,1.511237,1.0],"passed":false},{"actual":[-0.223925,9.794906,1.280236],"check":"case 7","expected":[-0.794438,5.334862,1.0],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"case 0\", \"actual\": [-0.060561, -0.253006, 1.0], \"expected\": [0.064229, -0.321773, 1.0], \"passed\": false}, {\"check\": \"case 1\", \"actual\": [-1.577839, 7.353168, 1.0], \"expected\": [-1.618428, 6.062195, 1.0], \"passed\": false}, {\"check\": \"case 2\", \"actual\": [0.0, 0.0, 0.0], \"expected\": [0.0, 0.0, 0.0], \"passed\": true}, {\"check\": \"case 3\", \"actual\": [0.092256, 0.947269, 1.0], \"expected\": [-0.014738, 1.136191, 1.0], \"passed\": false}, {\"check\": \"case 4\", \"actual\": [-0.317004, -1.020761, 1.0], \"expected\": [-0.320762, -0.982525, 1.0], \"passed\": false}, {\"check\": \"case 5\", \"actual\": [-1.112486, 1.649625, 1.0], \"expected\": [-1.077545, 2.77492, 1.0], \"passed\": false}, {\"check\": \"case 6\", \"actual\": [-0.760748, 1.586501, 1.024043], \"expected\": [-0.761883, 1.511237, 1.0], \"passed\": false}, {\"check\": \"case 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