{"abstract":"Stiff tethers to the world explode because they never shrink the substep.","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":"Skipping when the first endpoint is static still ignores half of the anchored springs.","family":"w2-physics_integrator_stability-spring-network-critical-step-world-anchored-springs","id":"FA-86936","implementations":{"attempt":{"sha256":"49a724ab413af88b03c5e543d15b29721d34a64870a64be1b5da929b58d9890a","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 im1 == 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 = [[[[[[39.9, 0.852, 0.112]], 0.9, 0.03333333333333333, 8], [1, 0.033333, False]], [[[[167.6, 0.0, 1.769], [238.7, 0.0, 1.824], [101.3, 0.949, 0.0], [328.1, 0.0, 1.65]], 0.9, 0.1, 10], [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]], [[[[172.3, 1.054, 0.0]], 0.5, 0.1, 12], [2, 0.05, False]], [[[[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]], [[[[106.5, 1.63, 0.0]], 0.9, 0.016666666666666666, 2], [1, 0.016667, False]], [[[[0.0, 0.185, 0.0], [369.7, 0.0, 1.891], [271.6, 0.712, 0.0], [0.0, 0.0, 0.0]], 0.9, 0.05, 5], [1, 0.05, False]], [[[[62.5, 0.699, 0.764], [271.2, 0.743, 0.0]], 0.5, 0.1, 1], [1, 0.1, True]]], [[[[[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]], [[[[329.8, 0.384, 1.012]], 0.5, 0.03333333333333333, 12], [1, 0.033333, False]], [[[[225.1, 0.0, 1.023]], 0.5, 0.016666666666666666, 4], [1, 0.016667, False]], [[[[254.1, 0.546, 0.0], [148.9, 0.748, 0.0], [149.5, 0.0, 0.0]], 0.5, 0.05, 1], [1, 0.05, False]], [[[[246.1, 0.0, 1.235]], 0.9, 0.1, 5], [1, 0.1, False]], [[[[68.2, 0.0, 1.77], [298.4, 1.344, 1.263]], 0.8, 0.05, 2], [1, 0.05, False]], [[[[397.6, 1.504, 0.0], [245.5, 1.813, 0.706], [252.0, 0.608, 0.796]], 0.5, 0.1, 9], [3, 0.033333, False]], [[[[356.6, 1.026, 0.0], [52.1, 1.266, 1.354]], 0.8, 0.03333333333333333, 12], [1, 0.033333, False]]], [[[[[80.8, 0.0, 1.835], [0.0, 0.224, 0.0], [41.8, 0.0, 0.0]], 0.8, 0.1, 12], [1, 0.1, False]], [[[[246.0, 0.0, 0.0], [348.1, 1.08, 0.0]], 0.9, 0.1, 7], [2, 0.05, False]], [[[[0.0, 0.686, 0.27]], 0.9, 0.05, 11], [1, 0.05, False]], [[[[308.4, 0.84, 0.0], [0.0, 0.0, 0.663]], 0.5, 0.1, 2], [2, 0.05, False]], [[[[120.7, 0.983, 1.303]], 0.9, 0.1, 9], [1, 0.1, False]], [[[[312.6, 0.0, 1.453]], 0.5, 0.1, 12], [3, 0.033333, False]], [[[[0.0, 0.0, 0.245]], 0.8, 0.1, 6], [1, 0.1, False]], [[[[41.7, 1.531, 1.755], [0.0, 0.0, 0.0], [0.0, 0.0, 1.05], [315.0, 0.0, 1.852]], 0.5, 0.05, 11], [2, 0.025, False]]], [[[[[255.8, 0.0, 0.123], [277.1, 0.0, 0.983], [226.4, 0.861, 0.0], [168.2, 0.0, 0.0]], 0.5, 0.016666666666666666, 12], [1, 0.016667, False]], [[[[112.1, 1.978, 0.662]], 0.9, 0.016666666666666666, 10], [1, 0.016667, False]], [[[[218.1, 0.0, 0.251], [45.1, 0.422, 1.694]], 0.8, 0.1, 7], [1, 0.1, False]], [[[[295.3, 1.026, 0.168], [142.6, 1.553, 1.729], [194.6, 0.909, 1.875], [131.0, 0.781, 0.0]], 0.8, 0.05, 11], [1, 0.05, 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]], [[[[320.1, 0.0, 1.766], [257.0, 0.0, 0.206]], 0.9, 0.1, 3], [2, 0.05, False]], [[[[363.6, 0.0, 1.256], [386.5, 0.409, 0.0]], 0.5, 0.1, 9], [3, 0.033333, False]], [[[[0.0, 1.821, 0.0], [0.0, 0.0, 1.079], [397.4, 0.0, 1.153]], 0.8, 0.1, 4], [2, 0.05, False]]], [[[[[392.2, 0.719, 0.0], [179.3, 0.0, 0.0], [31.9, 0.866, 0.407]], 0.8, 0.03333333333333333, 7], [1, 0.033333, False]], [[[[0.0, 0.215, 0.0], [136.7, 0.0, 1.56]], 0.5, 0.05, 5], [1, 0.05, False]], [[[[70.2, 1.762, 1.247], [340.8, 0.0, 1.231]], 0.9, 0.1, 12], [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]], [[[[40.2, 0.248, 0.0], [60.6, 0.0, 1.539], [90.0, 0.532, 0.877], [0.0, 0.241, 1.358]], 0.5, 0.03333333333333333, 7], [1, 0.033333, False]], [[[[34.9, 1.893, 0.883], [94.6, 1.187, 0.0], [6.2, 0.0, 0.66]], 0.5, 0.1, 10], [2, 0.05, False]], [[[[0.0, 1.18, 0.594], [62.4, 0.0, 0.0], [347.8, 1.292, 0.0], [365.4, 0.457, 0.0]], 0.5, 0.016666666666666666, 3], [1, 0.016667, False]], [[[[235.0, 0.0, 1.683], [0.0, 1.505, 0.0]], 0.5, 0.1, 10], [2, 0.05, 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":"e7b05990f76287e9c656e19d5574659335928e26bf0266ee31b641c5bae45415","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 im1 == 0 or im2 == 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 = [[[[[[39.9, 0.852, 0.112]], 0.9, 0.03333333333333333, 8], [1, 0.033333, False]], [[[[167.6, 0.0, 1.769], [238.7, 0.0, 1.824], [101.3, 0.949, 0.0], [328.1, 0.0, 1.65]], 0.9, 0.1, 10], [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]], [[[[172.3, 1.054, 0.0]], 0.5, 0.1, 12], [2, 0.05, False]], [[[[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]], [[[[106.5, 1.63, 0.0]], 0.9, 0.016666666666666666, 2], [1, 0.016667, False]], [[[[0.0, 0.185, 0.0], [369.7, 0.0, 1.891], [271.6, 0.712, 0.0], [0.0, 0.0, 0.0]], 0.9, 0.05, 5], [1, 0.05, False]], [[[[62.5, 0.699, 0.764], [271.2, 0.743, 0.0]], 0.5, 0.1, 1], [1, 0.1, True]]], [[[[[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]], [[[[329.8, 0.384, 1.012]], 0.5, 0.03333333333333333, 12], [1, 0.033333, False]], [[[[225.1, 0.0, 1.023]], 0.5, 0.016666666666666666, 4], [1, 0.016667, False]], [[[[254.1, 0.546, 0.0], [148.9, 0.748, 0.0], [149.5, 0.0, 0.0]], 0.5, 0.05, 1], [1, 0.05, False]], [[[[246.1, 0.0, 1.235]], 0.9, 0.1, 5], [1, 0.1, False]], [[[[68.2, 0.0, 1.77], [298.4, 1.344, 1.263]], 0.8, 0.05, 2], [1, 0.05, False]], [[[[397.6, 1.504, 0.0], [245.5, 1.813, 0.706], [252.0, 0.608, 0.796]], 0.5, 0.1, 9], [3, 0.033333, False]], [[[[356.6, 1.026, 0.0], [52.1, 1.266, 1.354]], 0.8, 0.03333333333333333, 12], [1, 0.033333, False]]], [[[[[80.8, 0.0, 1.835], [0.0, 0.224, 0.0], [41.8, 0.0, 0.0]], 0.8, 0.1, 12], [1, 0.1, False]], [[[[246.0, 0.0, 0.0], [348.1, 1.08, 0.0]], 0.9, 0.1, 7], [2, 0.05, False]], [[[[0.0, 0.686, 0.27]], 0.9, 0.05, 11], [1, 0.05, False]], [[[[308.4, 0.84, 0.0], [0.0, 0.0, 0.663]], 0.5, 0.1, 2], [2, 0.05, False]], [[[[120.7, 0.983, 1.303]], 0.9, 0.1, 9], [1, 0.1, False]], [[[[312.6, 0.0, 1.453]], 0.5, 0.1, 12], [3, 0.033333, False]], [[[[0.0, 0.0, 0.245]], 0.8, 0.1, 6], [1, 0.1, False]], [[[[41.7, 1.531, 1.755], [0.0, 0.0, 0.0], [0.0, 0.0, 1.05], [315.0, 0.0, 1.852]], 0.5, 0.05, 11], [2, 0.025, False]]], [[[[[255.8, 0.0, 0.123], [277.1, 0.0, 0.983], [226.4, 0.861, 0.0], [168.2, 0.0, 0.0]], 0.5, 0.016666666666666666, 12], [1, 0.016667, False]], [[[[112.1, 1.978, 0.662]], 0.9, 0.016666666666666666, 10], [1, 0.016667, False]], [[[[218.1, 0.0, 0.251], [45.1, 0.422, 1.694]], 0.8, 0.1, 7], [1, 0.1, False]], [[[[295.3, 1.026, 0.168], [142.6, 1.553, 1.729], [194.6, 0.909, 1.875], [131.0, 0.781, 0.0]], 0.8, 0.05, 11], [1, 0.05, 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]], [[[[320.1, 0.0, 1.766], [257.0, 0.0, 0.206]], 0.9, 0.1, 3], [2, 0.05, False]], [[[[363.6, 0.0, 1.256], [386.5, 0.409, 0.0]], 0.5, 0.1, 9], [3, 0.033333, False]], [[[[0.0, 1.821, 0.0], [0.0, 0.0, 1.079], [397.4, 0.0, 1.153]], 0.8, 0.1, 4], [2, 0.05, False]]], [[[[[392.2, 0.719, 0.0], [179.3, 0.0, 0.0], [31.9, 0.866, 0.407]], 0.8, 0.03333333333333333, 7], [1, 0.033333, False]], [[[[0.0, 0.215, 0.0], [136.7, 0.0, 1.56]], 0.5, 0.05, 5], [1, 0.05, False]], [[[[70.2, 1.762, 1.247], [340.8, 0.0, 1.231]], 0.9, 0.1, 12], [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]], [[[[40.2, 0.248, 0.0], [60.6, 0.0, 1.539], [90.0, 0.532, 0.877], [0.0, 0.241, 1.358]], 0.5, 0.03333333333333333, 7], [1, 0.033333, False]], [[[[34.9, 1.893, 0.883], [94.6, 1.187, 0.0], [6.2, 0.0, 0.66]], 0.5, 0.1, 10], [2, 0.05, False]], [[[[0.0, 1.18, 0.594], [62.4, 0.0, 0.0], [347.8, 1.292, 0.0], [365.4, 0.457, 0.0]], 0.5, 0.016666666666666666, 3], [1, 0.016667, False]], [[[[235.0, 0.0, 1.683], [0.0, 1.505, 0.0]], 0.5, 0.1, 10], [2, 0.05, 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-world-anchored-springs","generated_at":"2026-09-29T14:50:54.088433+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":"Any spring with a static endpoint is skipped instead of only springs with two static endpoints.","sha256":"70fe0317f272e8b131e3d39c622354cec5ca0803e5ec31f4f91f80d6af69a238","title":"Springs attached to a static anchor are ignored when limiting the step · 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":44.997,"exit_code":1,"observations":[{"actual":[1,0.033333,false],"check":"case 0","expected":[1,0.033333,false],"passed":true},{"actual":[1,0.1,false],"check":"case 1","expected":[2,0.05,false],"passed":false},{"actual":[2,0.05,false],"check":"case 2","expected":[2,0.05,false],"passed":true},{"actual":[2,0.05,false],"check":"case 3","expected":[2,0.05,false],"passed":true},{"actual":[2,0.05,false],"check":"case 4","expected":[2,0.05,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":[1,0.1,true],"check":"case 7","expected":[1,0.1,true],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"case 0\", \"actual\": [1, 0.033333, false], \"expected\": [1, 0.033333, false], \"passed\": true}, {\"check\": \"case 1\", \"actual\": [1, 0.1, false], \"expected\": [2, 0.05, false], \"passed\": false}, {\"check\": \"case 2\", \"actual\": [2, 0.05, false], \"expected\": [2, 0.05, false], \"passed\": true}, {\"check\": \"case 3\", \"actual\": [2, 0.05, false], \"expected\": [2, 0.05, false], \"passed\": true}, {\"check\": \"case 4\", \"actual\": [2, 0.05, false], \"expected\": [2, 0.05, 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, 0.1, true], \"expected\": [1, 0.1, true], \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":39.878,"exit_code":1,"observations":[{"actual":[1,0.033333,false],"check":"case 0","expected":[1,0.033333,false],"passed":true},{"actual":[1,0.1,false],"check":"case 1","expected":[2,0.05,false],"passed":false},{"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":[2,0.05,false],"check":"case 4","expected":[2,0.05,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":[1,0.1,false],"check":"case 7","expected":[1,0.1,true],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"case 0\", \"actual\": [1, 0.033333, false], \"expected\": [1, 0.033333, false], \"passed\": true}, {\"check\": \"case 1\", \"actual\": [1, 0.1, false], \"expected\": [2, 0.05, false], \"passed\": false}, {\"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\": [2, 0.05, false], \"expected\": [2, 0.05, 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, 0.1, false], \"expected\": [1, 0.1, true], \"passed\": false}], \"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."}}