{"abstract":"A single stiff spring in a soft network is integrated with the soft spring step.","category":"Physics integrator stability","checks":8,"contract":"solve(springs, safety, dt_frame, max_sub): springs are [k, inv_mass_a, inv_mass_b] (inverse mass 0 = static). A spring with k>0 and nonzero summed inverse mass w limits the substep to safety*2/sqrt(k*w). n=ceil(dt_frame/h_min) clamped to [1,max_sub]; return [n, dt_frame/n rounded to 6, whether n had to be clamped down]. No limiting spring gives [1, dt_frame, False].","contract_signature":"springs, safety, dt_frame, max_sub","evaluation_group":"w2-physics_integrator_stability-spring-network-critical-step","failed_approach":"Taking only the first limiting spring ignores later stiffer ones.","family":"w2-physics_integrator_stability-spring-network-critical-step-most-restrictive-spring","id":"FA-86966","implementations":{"attempt":{"sha256":"9e4efd5b474d5c95f79a74263ffbe3815e0aee4abdbded09e202074a515ef172","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(springs, safety, dt_frame, max_sub):\n    hc = None\n    for k, im1, im2 in springs:\n        w = im1 + im2\n        if w == 0 or k <= 0:\n            continue\n        h = safety * 2.0 / math.sqrt(k * w)\n        if hc is None:\n            hc = h\n    if hc is None:\n        return [1, round(dt_frame, 6), False]\n    n = math.ceil(dt_frame / hc - 1e-12)\n    clamped = n > max_sub\n    n = min(max(n, 1), max_sub)\n    return [n, round(dt_frame / n, 6), clamped]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[[[[[274.6, 1.615, 1.293], [279.2, 0.0, 1.072], [36.0, 1.606, 0.0]], 0.8, 0.1, 8], [2, 0.05, False]], [[[[176.6, 0.318, 1.727]], 0.9, 0.1, 1], [1, 0.1, True]], [[[[217.3, 0.0, 0.272], [226.4, 1.256, 1.506], [86.4, 1.936, 1.121]], 0.8, 0.1, 7], [2, 0.05, False]], [[[[51.4, 0.26, 0.0], [194.4, 1.517, 0.0]], 0.5, 0.1, 5], [2, 0.05, False]], [[[[148.8, 1.386, 0.442]], 0.8, 0.016666666666666666, 4], [1, 0.016667, False]], [[[[260.1, 0.0, 1.95]], 0.9, 0.016666666666666666, 1], [1, 0.016667, False]], [[[[299.0, 0.587, 0.527], [101.7, 1.086, 0.0], [96.2, 1.589, 0.676]], 0.9, 0.05, 9], [1, 0.05, False]], [[[[140.9, 1.164, 1.839], [288.3, 1.884, 0.0]], 0.9, 0.016666666666666666, 5], [1, 0.016667, False]]], [[[[[252.4, 0.823, 0.921], [0.0, 0.0, 1.281], [144.4, 0.0, 0.0], [399.0, 0.924, 1.926]], 0.9, 0.03333333333333333, 12], [1, 0.033333, False]], [[[[225.1, 0.0, 1.023]], 0.5, 0.016666666666666666, 4], [1, 0.016667, False]], [[[[239.0, 0.421, 0.398], [322.8, 0.0, 1.917], [189.8, 0.0, 0.0], [229.0, 1.496, 0.0]], 0.5, 0.05, 9], [2, 0.025, False]], [[[[297.9, 0.714, 1.84], [274.1, 0.797, 0.0], [150.2, 0.931, 1.101], [84.3, 0.7, 1.504]], 0.9, 0.016666666666666666, 12], [1, 0.016667, False]], [[[[331.5, 0.435, 0.0], [309.7, 1.309, 1.986]], 0.8, 0.1, 9], [2, 0.05, False]], [[[[208.5, 1.926, 0.106]], 0.9, 0.1, 11], [2, 0.05, False]], [[[[176.2, 1.797, 1.825], [298.2, 0.526, 0.891], [118.3, 0.227, 0.0]], 0.5, 0.1, 2], [2, 0.05, True]], [[[[288.5, 0.0, 0.0], [135.2, 0.764, 0.24], [389.1, 0.45, 1.145]], 0.5, 0.05, 6], [2, 0.025, False]]], [[[[[312.6, 0.0, 1.453]], 0.5, 0.1, 12], [3, 0.033333, False]], [[[[160.0, 1.174, 1.024], [222.4, 0.746, 0.568], [234.1, 0.0, 0.0], [305.9, 0.261, 0.0]], 0.8, 0.1, 5], [2, 0.05, False]], [[[[219.4, 0.47, 0.357], [244.7, 1.869, 0.0], [206.2, 0.734, 1.4]], 0.9, 0.1, 8], [2, 0.05, False]], [[[[8.4, 0.0, 1.944]], 0.9, 0.016666666666666666, 2], [1, 0.016667, False]], [[[[187.2, 0.446, 0.0], [384.2, 0.164, 1.101]], 0.5, 0.1, 6], [3, 0.033333, False]], [[[[0.0, 0.0, 0.802], [39.7, 0.0, 0.859]], 0.5, 0.1, 8], [1, 0.1, False]], [[[[98.4, 0.155, 0.544], [243.2, 0.0, 0.932], [174.9, 0.0, 0.913], [326.4, 1.87, 1.935]], 0.8, 0.05, 8], [2, 0.025, False]], [[[[308.1, 0.0, 0.454], [396.0, 0.194, 1.062]], 0.5, 0.05, 10], [2, 0.025, False]]], [[[[[231.0, 0.0, 1.82], [25.6, 1.068, 0.0], [268.8, 1.477, 0.0]], 0.5, 0.05, 5], [2, 0.025, False]], [[[[112.1, 1.181, 0.0], [364.2, 0.0, 0.0], [122.5, 0.0, 1.265]], 0.5, 0.05, 5], [1, 0.05, False]], [[[[74.9, 1.534, 1.135]], 0.8, 0.016666666666666666, 12], [1, 0.016667, False]], [[[[30.9, 0.794, 0.198], [0.0, 0.658, 0.0], [398.0, 1.934, 0.424]], 0.8, 0.1, 12], [2, 0.05, False]], [[[[385.7, 1.862, 1.408], [57.2, 1.136, 0.0], [93.5, 0.0, 0.0]], 0.5, 0.05, 12], [2, 0.025, False]], [[[[12.1, 0.309, 1.274], [153.5, 0.738, 0.969]], 0.5, 0.1, 12], [2, 0.05, False]], [[[[112.1, 1.978, 0.662]], 0.9, 0.016666666666666666, 10], [1, 0.016667, False]], [[[[103.2, 0.656, 0.0], [364.0, 0.395, 1.113]], 0.8, 0.016666666666666666, 1], [1, 0.016667, False]]], [[[[[25.9, 0.416, 0.0], [156.4, 1.139, 0.483], [279.4, 0.175, 1.863], [0.0, 1.682, 0.782]], 0.9, 0.03333333333333333, 7], [1, 0.033333, False]], [[[[0.0, 1.7, 0.0], [0.0, 0.0, 1.054], [328.3, 0.57, 0.359], [398.1, 1.479, 0.689]], 0.9, 0.1, 12], [2, 0.05, False]], [[[[0.0, 0.0, 0.743], [153.1, 1.007, 0.97]], 0.5, 0.05, 6], [1, 0.05, False]], [[[[97.9, 0.752, 1.178], [382.4, 1.927, 1.693], [68.3, 0.0, 0.0]], 0.9, 0.1, 11], [3, 0.033333, False]], [[[[149.6, 1.802, 0.77], [102.2, 1.828, 0.436]], 0.5, 0.1, 7], [2, 0.05, False]], [[[[0.0, 1.613, 0.0], [46.8, 1.168, 1.601], [255.1, 0.0, 0.0], [397.8, 0.0, 0.649]], 0.8, 0.1, 1], [1, 0.1, True]], [[[[338.0, 0.517, 0.0], [288.8, 0.0, 1.978], [37.6, 1.094, 1.076]], 0.9, 0.05, 11], [1, 0.05, False]], [[[[333.7, 1.663, 0.348], [161.4, 0.0, 0.877]], 0.5, 0.1, 7], [3, 0.033333, False]]]]\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":"44c15d0cd1488432f5a1e202c30149548a979170186b93272ab7f8c7ce4e22bd","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(springs, safety, dt_frame, max_sub):\n    hc = None\n    for k, im1, im2 in springs:\n        w = im1 + im2\n        if w == 0 or k <= 0:\n            continue\n        h = safety * 2.0 / math.sqrt(k * w)\n        if hc is None or h > hc:\n            hc = h\n    if hc is None:\n        return [1, round(dt_frame, 6), False]\n    n = math.ceil(dt_frame / hc - 1e-12)\n    clamped = n > max_sub\n    n = min(max(n, 1), max_sub)\n    return [n, round(dt_frame / n, 6), clamped]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[[[[[274.6, 1.615, 1.293], [279.2, 0.0, 1.072], [36.0, 1.606, 0.0]], 0.8, 0.1, 8], [2, 0.05, False]], [[[[176.6, 0.318, 1.727]], 0.9, 0.1, 1], [1, 0.1, True]], [[[[217.3, 0.0, 0.272], [226.4, 1.256, 1.506], [86.4, 1.936, 1.121]], 0.8, 0.1, 7], [2, 0.05, False]], [[[[51.4, 0.26, 0.0], [194.4, 1.517, 0.0]], 0.5, 0.1, 5], [2, 0.05, False]], [[[[148.8, 1.386, 0.442]], 0.8, 0.016666666666666666, 4], [1, 0.016667, False]], [[[[260.1, 0.0, 1.95]], 0.9, 0.016666666666666666, 1], [1, 0.016667, False]], [[[[299.0, 0.587, 0.527], [101.7, 1.086, 0.0], [96.2, 1.589, 0.676]], 0.9, 0.05, 9], [1, 0.05, False]], [[[[140.9, 1.164, 1.839], [288.3, 1.884, 0.0]], 0.9, 0.016666666666666666, 5], [1, 0.016667, False]]], [[[[[252.4, 0.823, 0.921], [0.0, 0.0, 1.281], [144.4, 0.0, 0.0], [399.0, 0.924, 1.926]], 0.9, 0.03333333333333333, 12], [1, 0.033333, False]], [[[[225.1, 0.0, 1.023]], 0.5, 0.016666666666666666, 4], [1, 0.016667, False]], [[[[239.0, 0.421, 0.398], [322.8, 0.0, 1.917], [189.8, 0.0, 0.0], [229.0, 1.496, 0.0]], 0.5, 0.05, 9], [2, 0.025, False]], [[[[297.9, 0.714, 1.84], [274.1, 0.797, 0.0], [150.2, 0.931, 1.101], [84.3, 0.7, 1.504]], 0.9, 0.016666666666666666, 12], [1, 0.016667, False]], [[[[331.5, 0.435, 0.0], [309.7, 1.309, 1.986]], 0.8, 0.1, 9], [2, 0.05, False]], [[[[208.5, 1.926, 0.106]], 0.9, 0.1, 11], [2, 0.05, False]], [[[[176.2, 1.797, 1.825], [298.2, 0.526, 0.891], [118.3, 0.227, 0.0]], 0.5, 0.1, 2], [2, 0.05, True]], [[[[288.5, 0.0, 0.0], [135.2, 0.764, 0.24], [389.1, 0.45, 1.145]], 0.5, 0.05, 6], [2, 0.025, False]]], [[[[[312.6, 0.0, 1.453]], 0.5, 0.1, 12], [3, 0.033333, False]], [[[[160.0, 1.174, 1.024], [222.4, 0.746, 0.568], [234.1, 0.0, 0.0], [305.9, 0.261, 0.0]], 0.8, 0.1, 5], [2, 0.05, False]], [[[[219.4, 0.47, 0.357], [244.7, 1.869, 0.0], [206.2, 0.734, 1.4]], 0.9, 0.1, 8], [2, 0.05, False]], [[[[8.4, 0.0, 1.944]], 0.9, 0.016666666666666666, 2], [1, 0.016667, False]], [[[[187.2, 0.446, 0.0], [384.2, 0.164, 1.101]], 0.5, 0.1, 6], [3, 0.033333, False]], [[[[0.0, 0.0, 0.802], [39.7, 0.0, 0.859]], 0.5, 0.1, 8], [1, 0.1, False]], [[[[98.4, 0.155, 0.544], [243.2, 0.0, 0.932], [174.9, 0.0, 0.913], [326.4, 1.87, 1.935]], 0.8, 0.05, 8], [2, 0.025, False]], [[[[308.1, 0.0, 0.454], [396.0, 0.194, 1.062]], 0.5, 0.05, 10], [2, 0.025, False]]], [[[[[231.0, 0.0, 1.82], [25.6, 1.068, 0.0], [268.8, 1.477, 0.0]], 0.5, 0.05, 5], [2, 0.025, False]], [[[[112.1, 1.181, 0.0], [364.2, 0.0, 0.0], [122.5, 0.0, 1.265]], 0.5, 0.05, 5], [1, 0.05, False]], [[[[74.9, 1.534, 1.135]], 0.8, 0.016666666666666666, 12], [1, 0.016667, False]], [[[[30.9, 0.794, 0.198], [0.0, 0.658, 0.0], [398.0, 1.934, 0.424]], 0.8, 0.1, 12], [2, 0.05, False]], [[[[385.7, 1.862, 1.408], [57.2, 1.136, 0.0], [93.5, 0.0, 0.0]], 0.5, 0.05, 12], [2, 0.025, False]], [[[[12.1, 0.309, 1.274], [153.5, 0.738, 0.969]], 0.5, 0.1, 12], [2, 0.05, False]], [[[[112.1, 1.978, 0.662]], 0.9, 0.016666666666666666, 10], [1, 0.016667, False]], [[[[103.2, 0.656, 0.0], [364.0, 0.395, 1.113]], 0.8, 0.016666666666666666, 1], [1, 0.016667, False]]], [[[[[25.9, 0.416, 0.0], [156.4, 1.139, 0.483], [279.4, 0.175, 1.863], [0.0, 1.682, 0.782]], 0.9, 0.03333333333333333, 7], [1, 0.033333, False]], [[[[0.0, 1.7, 0.0], [0.0, 0.0, 1.054], [328.3, 0.57, 0.359], [398.1, 1.479, 0.689]], 0.9, 0.1, 12], [2, 0.05, False]], [[[[0.0, 0.0, 0.743], [153.1, 1.007, 0.97]], 0.5, 0.05, 6], [1, 0.05, False]], [[[[97.9, 0.752, 1.178], [382.4, 1.927, 1.693], [68.3, 0.0, 0.0]], 0.9, 0.1, 11], [3, 0.033333, False]], [[[[149.6, 1.802, 0.77], [102.2, 1.828, 0.436]], 0.5, 0.1, 7], [2, 0.05, False]], [[[[0.0, 1.613, 0.0], [46.8, 1.168, 1.601], [255.1, 0.0, 0.0], [397.8, 0.0, 0.649]], 0.8, 0.1, 1], [1, 0.1, True]], [[[[338.0, 0.517, 0.0], [288.8, 0.0, 1.978], [37.6, 1.094, 1.076]], 0.9, 0.05, 11], [1, 0.05, False]], [[[[333.7, 1.663, 0.348], [161.4, 0.0, 0.877]], 0.5, 0.1, 7], [3, 0.033333, False]]]]\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-spring-network-critical-step-most-restrictive-spring","generated_at":"2026-09-29T14:50:54.453885+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 minimum search uses h > hc.","sha256":"99f21d0ba89a4155dae6ee0334c41e1cfc2f4492a1d8fa6efa550c17d505f395","title":"Critical step keeps the loosest spring rather than the stiffest · 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":41.477,"exit_code":1,"observations":[{"actual":[2,0.05,false],"check":"case 0","expected":[2,0.05,false],"passed":true},{"actual":[1,0.1,true],"check":"case 1","expected":[1,0.1,true],"passed":true},{"actual":[1,0.1,false],"check":"case 2","expected":[2,0.05,false],"passed":false},{"actual":[1,0.1,false],"check":"case 3","expected":[2,0.05,false],"passed":false},{"actual":[1,0.016667,false],"check":"case 4","expected":[1,0.016667,false],"passed":true},{"actual":[1,0.016667,false],"check":"case 5","expected":[1,0.016667,false],"passed":true},{"actual":[1,0.05,false],"check":"case 6","expected":[1,0.05,false],"passed":true},{"actual":[1,0.016667,false],"check":"case 7","expected":[1,0.016667,false],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"case 0\", \"actual\": [2, 0.05, false], \"expected\": [2, 0.05, false], \"passed\": true}, {\"check\": \"case 1\", \"actual\": [1, 0.1, true], \"expected\": [1, 0.1, true], \"passed\": true}, {\"check\": \"case 2\", \"actual\": [1, 0.1, false], \"expected\": [2, 0.05, false], \"passed\": false}, {\"check\": \"case 3\", \"actual\": [1, 0.1, false], \"expected\": [2, 0.05, false], \"passed\": false}, {\"check\": \"case 4\", \"actual\": [1, 0.016667, false], \"expected\": [1, 0.016667, false], \"passed\": true}, {\"check\": \"case 5\", \"actual\": [1, 0.016667, false], \"expected\": [1, 0.016667, false], \"passed\": true}, {\"check\": \"case 6\", \"actual\": [1, 0.05, false], \"expected\": [1, 0.05, false], \"passed\": true}, {\"check\": \"case 7\", \"actual\": [1, 0.016667, false], \"expected\": [1, 0.016667, false], \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":40.14,"exit_code":1,"observations":[{"actual":[1,0.1,false],"check":"case 0","expected":[2,0.05,false],"passed":false},{"actual":[1,0.1,true],"check":"case 1","expected":[1,0.1,true],"passed":true},{"actual":[1,0.1,false],"check":"case 2","expected":[2,0.05,false],"passed":false},{"actual":[1,0.1,false],"check":"case 3","expected":[2,0.05,false],"passed":false},{"actual":[1,0.016667,false],"check":"case 4","expected":[1,0.016667,false],"passed":true},{"actual":[1,0.016667,false],"check":"case 5","expected":[1,0.016667,false],"passed":true},{"actual":[1,0.05,false],"check":"case 6","expected":[1,0.05,false],"passed":true},{"actual":[1,0.016667,false],"check":"case 7","expected":[1,0.016667,false],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"case 0\", \"actual\": [1, 0.1, false], \"expected\": [2, 0.05, false], \"passed\": false}, {\"check\": \"case 1\", \"actual\": [1, 0.1, true], \"expected\": [1, 0.1, true], \"passed\": true}, {\"check\": \"case 2\", \"actual\": [1, 0.1, false], \"expected\": [2, 0.05, false], \"passed\": false}, {\"check\": \"case 3\", \"actual\": [1, 0.1, false], \"expected\": [2, 0.05, false], \"passed\": false}, {\"check\": \"case 4\", \"actual\": [1, 0.016667, false], \"expected\": [1, 0.016667, false], \"passed\": true}, {\"check\": \"case 5\", \"actual\": [1, 0.016667, false], \"expected\": [1, 0.016667, false], \"passed\": true}, {\"check\": \"case 6\", \"actual\": [1, 0.05, false], \"expected\": [1, 0.05, false], \"passed\": true}, {\"check\": \"case 7\", \"actual\": [1, 0.016667, false], \"expected\": [1, 0.016667, false], \"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."}}