{"abstract":"Heavy bodies are over-damped and light ones under-damped relative to the contract.","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":"Dividing the damping term by mass inside the parentheses and again outside double-scales it.","family":"w2-physics_integrator_stability-substep-symplectic-euler-damping-mass-division","id":"FA-86851","implementations":{"attempt":{"sha256":"d0e7adac2473f0f11a1f3fa272b02feacaf79c5878688b099a06f15ee60363c7","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) / 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]], [[-0.104, -1.626, 5.159, 2.304, 0.637, 0.2, 0], [-0.104, -1.626]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[0.086, 0.733, 7.163, 2.307, 1.249, 0.05, 1], [0.12099, 0.699807]], [[1.545, 1.962, 22.255, 1.613, 1.385, 0.02, 0], [1.545, 1.962]], [[1.916, 1.522, 23.049, 0.619, 1.046, 0.2, 2], [-0.971294, -6.107647]], [[-1.343, -1.045, 36.463, 2.947, 1.73, 0.02, 1], [-1.357008, -0.700394]], [[0.567, -0.563, 3.019, 2.331, 1.024, 0.05, 8], [0.30826, -0.689182]]], [[[1.925, -0.173, 39.051, 1.209, 0.422, 0.01, 5], [1.824415, -3.211948]], [[-0.384, 0.483, 39.776, 2.351, 1.463, 0.04, 0], [-0.384, 0.483]], [[-1.087, 0.963, 34.734, 2.546, 1.868, 0.07, 9], [0.718936, 1.786483]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[1.742, -0.907, 38.421, 2.244, 0.819, 0.1, 5], [-1.031195, -5.442883]], [[-1.792, 1.668, 7.957, 1.875, 1.186, 0.04, 9], [-0.822549, 3.240967]], [[-1.669, -1.23, 27.053, 1.533, 0.991, 0.13, 5], [1.161742, 3.128527]], [[-1.46, -0.406, 15.912, 1.138, 0.653, 0.07, 0], [-1.46, -0.406]]], [[[-1.712, -1.9, 9.139, 0.693, 0.61, 0.05, 13], [0.624756, 4.351887]], [[0.6, -0.396, 35.253, 1.711, 1.361, 0.04, 8], [-0.018685, -2.456697]], [[1.207, 0.351, 14.617, 2.861, 0.895, 0.07, 6], [0.802339, -1.906391]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[1.648, -0.117, 24.609, 0.651, 1.732, 0.05, 0], [1.648, -0.117]], [[0.575, -1.191, 6.727, 1.218, 1.658, 0.2, 4], [-0.282948, -0.458231]], [[0.87, 1.182, 19.781, 1.146, 0.3, 0.1, 0], [0.87, 1.182]], [[0.566, 0.949, 26.197, 1.061, 0.815, 0.04, 4], [0.488626, -1.290634]]], [[[1.645, -1.841, 34.562, 1.534, 1.725, 0.2, 6], [0.803376, 1.245092]], [[-1.284, 1.069, 32.194, 0.542, 0.353, 0.05, 11], [0.174523, -8.251404]], [[-0.117, 1.853, 21.165, 2.719, 0.626, 0.07, 12], [0.503218, -0.957084]], [[0.484, 0.239, 12.355, 0.868, 0.3, 0.07, 0], [0.484, 0.239]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[1.074, -0.57, 32.767, 1.613, 0.402, 0.02, 7], [0.767571, -3.27442]], [[0.525, -0.515, 17.156, 2.442, 0.805, 0.05, 2], [0.447863, -0.851821]], [[1.977, 0.462, 36.991, 1.489, 1.033, 0.13, 0], [1.977, 0.462]]], [[[-1.576, -0.203, 24.72, 2.347, 1.363, 0.07, 10], [0.760633, 3.318988]], [[-1.172, -1.957, 14.575, 1.726, 0.305, 0.1, 12], [1.182005, 0.606601]], [[-0.914, 0.601, 8.25, 1.548, 0.798, 0.02, 0], [-0.914, 0.601]], [[-0.078, -0.863, 3.994, 2.622, 1.462, 0.1, 0], [-0.078, -0.863]], [[-1.215, 0.2, 16.12, 1.251, 1.235, 0.02, 10], [-0.872021, 2.751611]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[-0.66, -0.547, 35.572, 2.632, 0.117, 0.05, 8], [-0.1546, 2.305749]], [[1.873, -0.51, 29.384, 1.99, 0.411, 0.2, 16], [1.352031, 0.801221]]]]\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":"2971b521803ee055ff13bb69c5384bfa4536029f47fd87eeebfae9508920f20c","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 / m - c * v\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]], [[-0.104, -1.626, 5.159, 2.304, 0.637, 0.2, 0], [-0.104, -1.626]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[0.086, 0.733, 7.163, 2.307, 1.249, 0.05, 1], [0.12099, 0.699807]], [[1.545, 1.962, 22.255, 1.613, 1.385, 0.02, 0], [1.545, 1.962]], [[1.916, 1.522, 23.049, 0.619, 1.046, 0.2, 2], [-0.971294, -6.107647]], [[-1.343, -1.045, 36.463, 2.947, 1.73, 0.02, 1], [-1.357008, -0.700394]], [[0.567, -0.563, 3.019, 2.331, 1.024, 0.05, 8], [0.30826, -0.689182]]], [[[1.925, -0.173, 39.051, 1.209, 0.422, 0.01, 5], [1.824415, -3.211948]], [[-0.384, 0.483, 39.776, 2.351, 1.463, 0.04, 0], [-0.384, 0.483]], [[-1.087, 0.963, 34.734, 2.546, 1.868, 0.07, 9], [0.718936, 1.786483]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[1.742, -0.907, 38.421, 2.244, 0.819, 0.1, 5], [-1.031195, -5.442883]], [[-1.792, 1.668, 7.957, 1.875, 1.186, 0.04, 9], [-0.822549, 3.240967]], [[-1.669, -1.23, 27.053, 1.533, 0.991, 0.13, 5], [1.161742, 3.128527]], [[-1.46, -0.406, 15.912, 1.138, 0.653, 0.07, 0], [-1.46, -0.406]]], [[[-1.712, -1.9, 9.139, 0.693, 0.61, 0.05, 13], [0.624756, 4.351887]], [[0.6, -0.396, 35.253, 1.711, 1.361, 0.04, 8], [-0.018685, -2.456697]], [[1.207, 0.351, 14.617, 2.861, 0.895, 0.07, 6], [0.802339, -1.906391]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[1.648, -0.117, 24.609, 0.651, 1.732, 0.05, 0], [1.648, -0.117]], [[0.575, -1.191, 6.727, 1.218, 1.658, 0.2, 4], [-0.282948, -0.458231]], [[0.87, 1.182, 19.781, 1.146, 0.3, 0.1, 0], [0.87, 1.182]], [[0.566, 0.949, 26.197, 1.061, 0.815, 0.04, 4], [0.488626, -1.290634]]], [[[1.645, -1.841, 34.562, 1.534, 1.725, 0.2, 6], [0.803376, 1.245092]], [[-1.284, 1.069, 32.194, 0.542, 0.353, 0.05, 11], [0.174523, -8.251404]], [[-0.117, 1.853, 21.165, 2.719, 0.626, 0.07, 12], [0.503218, -0.957084]], [[0.484, 0.239, 12.355, 0.868, 0.3, 0.07, 0], [0.484, 0.239]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[1.074, -0.57, 32.767, 1.613, 0.402, 0.02, 7], [0.767571, -3.27442]], [[0.525, -0.515, 17.156, 2.442, 0.805, 0.05, 2], [0.447863, -0.851821]], [[1.977, 0.462, 36.991, 1.489, 1.033, 0.13, 0], [1.977, 0.462]]], [[[-1.576, -0.203, 24.72, 2.347, 1.363, 0.07, 10], [0.760633, 3.318988]], [[-1.172, -1.957, 14.575, 1.726, 0.305, 0.1, 12], [1.182005, 0.606601]], [[-0.914, 0.601, 8.25, 1.548, 0.798, 0.02, 0], [-0.914, 0.601]], [[-0.078, -0.863, 3.994, 2.622, 1.462, 0.1, 0], [-0.078, -0.863]], [[-1.215, 0.2, 16.12, 1.251, 1.235, 0.02, 10], [-0.872021, 2.751611]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[-0.66, -0.547, 35.572, 2.632, 0.117, 0.05, 8], [-0.1546, 2.305749]], [[1.873, -0.51, 29.384, 1.99, 0.411, 0.2, 16], [1.352031, 0.801221]]]]\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":"15926f856752c86fc1eb7b319ba2894e9cbd594d094d3fa3d79ae831ef44dad0","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]], [[-0.104, -1.626, 5.159, 2.304, 0.637, 0.2, 0], [-0.104, -1.626]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[0.086, 0.733, 7.163, 2.307, 1.249, 0.05, 1], [0.12099, 0.699807]], [[1.545, 1.962, 22.255, 1.613, 1.385, 0.02, 0], [1.545, 1.962]], [[1.916, 1.522, 23.049, 0.619, 1.046, 0.2, 2], [-0.971294, -6.107647]], [[-1.343, -1.045, 36.463, 2.947, 1.73, 0.02, 1], [-1.357008, -0.700394]], [[0.567, -0.563, 3.019, 2.331, 1.024, 0.05, 8], [0.30826, -0.689182]]], [[[1.925, -0.173, 39.051, 1.209, 0.422, 0.01, 5], [1.824415, -3.211948]], [[-0.384, 0.483, 39.776, 2.351, 1.463, 0.04, 0], [-0.384, 0.483]], [[-1.087, 0.963, 34.734, 2.546, 1.868, 0.07, 9], [0.718936, 1.786483]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[1.742, -0.907, 38.421, 2.244, 0.819, 0.1, 5], [-1.031195, -5.442883]], [[-1.792, 1.668, 7.957, 1.875, 1.186, 0.04, 9], [-0.822549, 3.240967]], [[-1.669, -1.23, 27.053, 1.533, 0.991, 0.13, 5], [1.161742, 3.128527]], [[-1.46, -0.406, 15.912, 1.138, 0.653, 0.07, 0], [-1.46, -0.406]]], [[[-1.712, -1.9, 9.139, 0.693, 0.61, 0.05, 13], [0.624756, 4.351887]], [[0.6, -0.396, 35.253, 1.711, 1.361, 0.04, 8], [-0.018685, -2.456697]], [[1.207, 0.351, 14.617, 2.861, 0.895, 0.07, 6], [0.802339, -1.906391]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[1.648, -0.117, 24.609, 0.651, 1.732, 0.05, 0], [1.648, -0.117]], [[0.575, -1.191, 6.727, 1.218, 1.658, 0.2, 4], [-0.282948, -0.458231]], [[0.87, 1.182, 19.781, 1.146, 0.3, 0.1, 0], [0.87, 1.182]], [[0.566, 0.949, 26.197, 1.061, 0.815, 0.04, 4], [0.488626, -1.290634]]], [[[1.645, -1.841, 34.562, 1.534, 1.725, 0.2, 6], [0.803376, 1.245092]], [[-1.284, 1.069, 32.194, 0.542, 0.353, 0.05, 11], [0.174523, -8.251404]], [[-0.117, 1.853, 21.165, 2.719, 0.626, 0.07, 12], [0.503218, -0.957084]], [[0.484, 0.239, 12.355, 0.868, 0.3, 0.07, 0], [0.484, 0.239]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[1.074, -0.57, 32.767, 1.613, 0.402, 0.02, 7], [0.767571, -3.27442]], [[0.525, -0.515, 17.156, 2.442, 0.805, 0.05, 2], [0.447863, -0.851821]], [[1.977, 0.462, 36.991, 1.489, 1.033, 0.13, 0], [1.977, 0.462]]], [[[-1.576, -0.203, 24.72, 2.347, 1.363, 0.07, 10], [0.760633, 3.318988]], [[-1.172, -1.957, 14.575, 1.726, 0.305, 0.1, 12], [1.182005, 0.606601]], [[-0.914, 0.601, 8.25, 1.548, 0.798, 0.02, 0], [-0.914, 0.601]], [[-0.078, -0.863, 3.994, 2.622, 1.462, 0.1, 0], [-0.078, -0.863]], [[-1.215, 0.2, 16.12, 1.251, 1.235, 0.02, 10], [-0.872021, 2.751611]], [[1.0, 0.0, 10.0, 1.0, 0.0, 0.05, 0], [1.0, 0.0]], [[-0.66, -0.547, 35.572, 2.632, 0.117, 0.05, 8], [-0.1546, 2.305749]], [[1.873, -0.51, 29.384, 1.99, 0.411, 0.2, 16], [1.352031, 0.801221]]]]\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-damping-mass-division","generated_at":"2026-09-29T14:50:53.367702+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":"Divide the whole force (-k*x - c*v) by the mass.","root_cause":"Only the spring term is divided by mass because of operator precedence; c*v is used directly as acceleration.","sha256":"5d0b334a70eccb3cf65622c851e662d2dcc098b0b5b6ea5a5fe688b5c7fbeb7f","title":"Viscous damping force is applied as an acceleration without dividing by mass · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":47.821,"exit_code":1,"observations":[{"actual":[0.595139,4.326906],"check":"case 0","expected":[0.446958,3.829901],"passed":false},{"actual":[-0.104,-1.626],"check":"case 1","expected":[-0.104,-1.626],"passed":true},{"actual":[1.0,0.0],"check":"case 2","expected":[1.0,0.0],"passed":true},{"actual":[0.121552,0.711048],"check":"case 3","expected":[0.12099,0.699807],"passed":false},{"actual":[1.545,1.962],"check":"case 4","expected":[1.545,1.962],"passed":true},{"actual":[-0.704987,-5.053863],"check":"case 5","expected":[-0.971294,-6.107647],"passed":false},{"actual":[-1.35717,-0.7085],"check":"case 6","expected":[-1.357008,-0.700394],"passed":false},{"actual":[0.294996,-0.748576],"check":"case 7","expected":[0.30826,-0.689182],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"case 0\", \"actual\": [0.595139, 4.326906], \"expected\": [0.446958, 3.829901], \"passed\": false}, {\"check\": \"case 1\", \"actual\": [-0.104, -1.626], \"expected\": [-0.104, -1.626], \"passed\": true}, {\"check\": \"case 2\", \"actual\": [1.0, 0.0], \"expected\": [1.0, 0.0], \"passed\": true}, {\"check\": \"case 3\", \"actual\": [0.121552, 0.711048], \"expected\": [0.12099, 0.699807], \"passed\": false}, {\"check\": \"case 4\", \"actual\": [1.545, 1.962], \"expected\": [1.545, 1.962], \"passed\": true}, {\"check\": \"case 5\", \"actual\": [-0.704987, -5.053863], \"expected\": [-0.971294, -6.107647], \"passed\": false}, {\"check\": \"case 6\", \"actual\": [-1.35717, -0.7085], \"expected\": [-1.357008, -0.700394], \"passed\": false}, {\"check\": \"case 7\", \"actual\": [0.294996, -0.748576], \"expected\": [0.30826, -0.689182], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":47.11,"exit_code":1,"observations":[{"actual":[0.089549,2.75242],"check":"case 0","expected":[0.446958,3.829901],"passed":false},{"actual":[-0.104,-1.626],"check":"case 1","expected":[-0.104,-1.626],"passed":true},{"actual":[1.0,0.0],"check":"case 2","expected":[1.0,0.0],"passed":true},{"actual":[0.119694,0.673873],"check":"case 3","expected":[0.12099,0.699807],"passed":false},{"actual":[1.545,1.962],"check":"case 4","expected":[1.545,1.962],"passed":true},{"actual":[-1.158401,-6.921323],"check":"case 5","expected":[-0.971294,-6.107647],"passed":false},{"actual":[-1.35653,-0.676506],"check":"case 6","expected":[-1.357008,-0.700394],"passed":false},{"actual":[0.336433,-0.568618],"check":"case 7","expected":[0.30826,-0.689182],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"case 0\", \"actual\": [0.089549, 2.75242], \"expected\": [0.446958, 3.829901], \"passed\": false}, {\"check\": \"case 1\", \"actual\": [-0.104, -1.626], \"expected\": [-0.104, -1.626], \"passed\": true}, {\"check\": \"case 2\", \"actual\": [1.0, 0.0], \"expected\": [1.0, 0.0], \"passed\": true}, {\"check\": \"case 3\", \"actual\": [0.119694, 0.673873], \"expected\": [0.12099, 0.699807], \"passed\": false}, {\"check\": \"case 4\", \"actual\": [1.545, 1.962], \"expected\": [1.545, 1.962], \"passed\": true}, {\"check\": \"case 5\", \"actual\": [-1.158401, -6.921323], \"expected\": [-0.971294, -6.107647], \"passed\": false}, {\"check\": \"case 6\", \"actual\": [-1.35653, -0.676506], \"expected\": [-1.357008, -0.700394], \"passed\": false}, {\"check\": \"case 7\", \"actual\": [0.336433, -0.568618], \"expected\": [0.30826, -0.689182], \"passed\": false}], \"passed\": false}\n"},"fixed":{"elapsed_ms":38.888,"exit_code":0,"observations":[{"actual":[0.446958,3.829901],"check":"case 0","expected":[0.446958,3.829901],"passed":true},{"actual":[-0.104,-1.626],"check":"case 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