{"abstract":"The solver runs twice as many substeps as the contract bound requires.","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":"Using pi/omega exceeds the stability limit.","family":"w2-physics_integrator_stability-spring-network-critical-step-stability-bound-factor","id":"FA-86951","implementations":{"attempt":{"sha256":"6d50e3cefd8208152f69cd1dc6fd75595ad118b34293f2055c1569583b1d966b","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 * math.pi / 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 = [[[[[[181.3, 0.765, 0.718], [113.7, 1.47, 0.0]], 0.8, 0.05, 2], [1, 0.05, 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]], [[[[106.5, 1.63, 0.0]], 0.9, 0.016666666666666666, 2], [1, 0.016667, False]], [[[[148.6, 0.387, 0.848], [287.5, 0.0, 1.337]], 0.5, 0.03333333333333333, 2], [1, 0.033333, False]], [[[[127.0, 1.3, 0.0]], 0.5, 0.03333333333333333, 10], [1, 0.033333, False]], [[[[0.0, 0.133, 0.288], [277.5, 1.393, 0.313], [186.0, 1.259, 0.245], [78.0, 1.034, 0.148]], 0.9, 0.03333333333333333, 1], [1, 0.033333, False]], [[[[383.4, 1.008, 0.724]], 0.9, 0.1, 12], [2, 0.05, False]], [[[[200.1, 1.603, 0.972], [288.5, 1.122, 1.97]], 0.5, 0.1, 8], [3, 0.033333, False]]], [[[[[211.7, 1.064, 0.812], [384.0, 1.693, 1.686]], 0.5, 0.1, 7], [4, 0.025, False]], [[[[95.6, 1.372, 0.756], [95.0, 0.946, 1.265], [396.4, 1.377, 0.0], [218.5, 0.0, 1.301]], 0.5, 0.03333333333333333, 1], [1, 0.033333, False]], [[[[14.3, 0.266, 0.874], [8.7, 1.8, 1.042]], 0.8, 0.1, 10], [1, 0.1, False]], [[[[97.9, 0.409, 0.102], [187.1, 0.0, 1.876], [175.5, 0.772, 1.165]], 0.9, 0.1, 3], [2, 0.05, False]], [[[[0.0, 1.446, 0.0]], 0.8, 0.03333333333333333, 11], [1, 0.033333, False]], [[[[52.7, 0.654, 0.798], [173.5, 1.41, 0.659]], 0.9, 0.1, 3], [2, 0.05, False]], [[[[233.9, 0.0, 0.0], [376.9, 0.322, 1.171], [104.8, 1.006, 0.815]], 0.9, 0.05, 2], [1, 0.05, False]], [[[[205.5, 0.0, 0.24]], 0.8, 0.03333333333333333, 8], [1, 0.033333, False]]], [[[[[281.2, 0.534, 0.0], [35.0, 0.0, 0.0]], 0.5, 0.03333333333333333, 12], [1, 0.033333, False]], [[[[308.1, 0.0, 0.454], [396.0, 0.194, 1.062]], 0.5, 0.05, 10], [2, 0.025, False]], [[[[153.0, 0.822, 1.885]], 0.8, 0.016666666666666666, 11], [1, 0.016667, False]], [[[[349.4, 0.0, 1.034], [394.6, 1.246, 1.597], [212.5, 0.0, 0.0], [153.3, 0.0, 1.362]], 0.8, 0.03333333333333333, 10], [1, 0.033333, False]], [[[[194.8, 0.0, 1.946], [76.0, 0.616, 1.748], [133.0, 0.0, 1.751], [341.8, 1.754, 0.0]], 0.5, 0.05, 8], [2, 0.025, False]], [[[[84.5, 1.172, 0.389], [387.6, 0.894, 0.997], [341.4, 0.801, 0.0]], 0.9, 0.05, 1], [1, 0.05, False]], [[[[139.9, 1.445, 0.0], [19.1, 0.0, 0.157], [93.7, 1.683, 0.761], [185.2, 0.0, 0.0]], 0.9, 0.016666666666666666, 9], [1, 0.016667, False]], [[[[246.0, 0.0, 0.0], [348.1, 1.08, 0.0]], 0.9, 0.1, 7], [2, 0.05, False]]], [[[[[40.5, 0.524, 0.0], [195.2, 0.0, 0.632]], 0.8, 0.016666666666666666, 9], [1, 0.016667, False]], [[[[0.7, 0.0, 1.518], [157.3, 1.84, 0.368], [155.6, 0.0, 0.884], [325.1, 1.537, 0.59]], 0.5, 0.05, 9], [2, 0.025, False]], [[[[112.1, 1.978, 0.662]], 0.9, 0.016666666666666666, 10], [1, 0.016667, False]], [[[[20.9, 0.0, 0.0], [250.8, 0.0, 0.0]], 0.9, 0.05, 10], [1, 0.05, False]], [[[[348.9, 0.376, 0.912]], 0.5, 0.05, 6], [2, 0.025, False]], [[[[230.1, 0.0, 1.021], [77.8, 0.488, 1.614], [0.0, 0.798, 1.482]], 0.8, 0.03333333333333333, 5], [1, 0.033333, False]], [[[[295.6, 1.379, 0.496], [26.1, 1.99, 1.706], [6.0, 1.103, 1.549]], 0.8, 0.1, 10], [2, 0.05, False]], [[[[188.7, 1.402, 0.721], [392.3, 0.0, 1.896], [272.9, 0.701, 1.314], [383.2, 1.675, 1.515]], 0.5, 0.03333333333333333, 8], [2, 0.016667, False]]], [[[[[70.2, 1.762, 1.247], [340.8, 0.0, 1.231]], 0.9, 0.1, 12], [2, 0.05, False]], [[[[149.6, 1.802, 0.77], [102.2, 1.828, 0.436]], 0.5, 0.1, 7], [2, 0.05, False]], [[[[227.2, 1.778, 0.152]], 0.5, 0.05, 8], [2, 0.025, False]], [[[[336.1, 0.713, 0.0], [192.3, 1.124, 1.291], [177.3, 0.468, 0.473], [377.9, 0.567, 0.0]], 0.9, 0.03333333333333333, 11], [1, 0.033333, False]], [[[[0.0, 0.0, 1.51], [228.1, 0.0, 0.322], [74.8, 1.176, 1.179], [268.4, 1.044, 1.986]], 0.5, 0.05, 5], [2, 0.025, False]], [[[[333.7, 1.663, 0.348], [161.4, 0.0, 0.877]], 0.5, 0.1, 7], [3, 0.033333, False]], [[[[317.5, 0.0, 0.172]], 0.9, 0.03333333333333333, 3], [1, 0.033333, False]], [[[[140.9, 0.0, 0.527], [20.1, 0.0, 1.755], [0.0, 1.669, 0.955], [122.6, 0.0, 0.656]], 0.9, 0.016666666666666666, 8], [1, 0.016667, 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":"93814adb2c792b83babe903abc4b52a360a77e9150b28f1c64acfb51a4e1087c","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 / 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 = [[[[[[181.3, 0.765, 0.718], [113.7, 1.47, 0.0]], 0.8, 0.05, 2], [1, 0.05, 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]], [[[[106.5, 1.63, 0.0]], 0.9, 0.016666666666666666, 2], [1, 0.016667, False]], [[[[148.6, 0.387, 0.848], [287.5, 0.0, 1.337]], 0.5, 0.03333333333333333, 2], [1, 0.033333, False]], [[[[127.0, 1.3, 0.0]], 0.5, 0.03333333333333333, 10], [1, 0.033333, False]], [[[[0.0, 0.133, 0.288], [277.5, 1.393, 0.313], [186.0, 1.259, 0.245], [78.0, 1.034, 0.148]], 0.9, 0.03333333333333333, 1], [1, 0.033333, False]], [[[[383.4, 1.008, 0.724]], 0.9, 0.1, 12], [2, 0.05, False]], [[[[200.1, 1.603, 0.972], [288.5, 1.122, 1.97]], 0.5, 0.1, 8], [3, 0.033333, False]]], [[[[[211.7, 1.064, 0.812], [384.0, 1.693, 1.686]], 0.5, 0.1, 7], [4, 0.025, False]], [[[[95.6, 1.372, 0.756], [95.0, 0.946, 1.265], [396.4, 1.377, 0.0], [218.5, 0.0, 1.301]], 0.5, 0.03333333333333333, 1], [1, 0.033333, False]], [[[[14.3, 0.266, 0.874], [8.7, 1.8, 1.042]], 0.8, 0.1, 10], [1, 0.1, False]], [[[[97.9, 0.409, 0.102], [187.1, 0.0, 1.876], [175.5, 0.772, 1.165]], 0.9, 0.1, 3], [2, 0.05, False]], [[[[0.0, 1.446, 0.0]], 0.8, 0.03333333333333333, 11], [1, 0.033333, False]], [[[[52.7, 0.654, 0.798], [173.5, 1.41, 0.659]], 0.9, 0.1, 3], [2, 0.05, False]], [[[[233.9, 0.0, 0.0], [376.9, 0.322, 1.171], [104.8, 1.006, 0.815]], 0.9, 0.05, 2], [1, 0.05, False]], [[[[205.5, 0.0, 0.24]], 0.8, 0.03333333333333333, 8], [1, 0.033333, False]]], [[[[[281.2, 0.534, 0.0], [35.0, 0.0, 0.0]], 0.5, 0.03333333333333333, 12], [1, 0.033333, False]], [[[[308.1, 0.0, 0.454], [396.0, 0.194, 1.062]], 0.5, 0.05, 10], [2, 0.025, False]], [[[[153.0, 0.822, 1.885]], 0.8, 0.016666666666666666, 11], [1, 0.016667, False]], [[[[349.4, 0.0, 1.034], [394.6, 1.246, 1.597], [212.5, 0.0, 0.0], [153.3, 0.0, 1.362]], 0.8, 0.03333333333333333, 10], [1, 0.033333, False]], [[[[194.8, 0.0, 1.946], [76.0, 0.616, 1.748], [133.0, 0.0, 1.751], [341.8, 1.754, 0.0]], 0.5, 0.05, 8], [2, 0.025, False]], [[[[84.5, 1.172, 0.389], [387.6, 0.894, 0.997], [341.4, 0.801, 0.0]], 0.9, 0.05, 1], [1, 0.05, False]], [[[[139.9, 1.445, 0.0], [19.1, 0.0, 0.157], [93.7, 1.683, 0.761], [185.2, 0.0, 0.0]], 0.9, 0.016666666666666666, 9], [1, 0.016667, False]], [[[[246.0, 0.0, 0.0], [348.1, 1.08, 0.0]], 0.9, 0.1, 7], [2, 0.05, False]]], [[[[[40.5, 0.524, 0.0], [195.2, 0.0, 0.632]], 0.8, 0.016666666666666666, 9], [1, 0.016667, False]], [[[[0.7, 0.0, 1.518], [157.3, 1.84, 0.368], [155.6, 0.0, 0.884], [325.1, 1.537, 0.59]], 0.5, 0.05, 9], [2, 0.025, False]], [[[[112.1, 1.978, 0.662]], 0.9, 0.016666666666666666, 10], [1, 0.016667, False]], [[[[20.9, 0.0, 0.0], [250.8, 0.0, 0.0]], 0.9, 0.05, 10], [1, 0.05, False]], [[[[348.9, 0.376, 0.912]], 0.5, 0.05, 6], [2, 0.025, False]], [[[[230.1, 0.0, 1.021], [77.8, 0.488, 1.614], [0.0, 0.798, 1.482]], 0.8, 0.03333333333333333, 5], [1, 0.033333, False]], [[[[295.6, 1.379, 0.496], [26.1, 1.99, 1.706], [6.0, 1.103, 1.549]], 0.8, 0.1, 10], [2, 0.05, False]], [[[[188.7, 1.402, 0.721], [392.3, 0.0, 1.896], [272.9, 0.701, 1.314], [383.2, 1.675, 1.515]], 0.5, 0.03333333333333333, 8], [2, 0.016667, False]]], [[[[[70.2, 1.762, 1.247], [340.8, 0.0, 1.231]], 0.9, 0.1, 12], [2, 0.05, False]], [[[[149.6, 1.802, 0.77], [102.2, 1.828, 0.436]], 0.5, 0.1, 7], [2, 0.05, False]], [[[[227.2, 1.778, 0.152]], 0.5, 0.05, 8], [2, 0.025, False]], [[[[336.1, 0.713, 0.0], [192.3, 1.124, 1.291], [177.3, 0.468, 0.473], [377.9, 0.567, 0.0]], 0.9, 0.03333333333333333, 11], [1, 0.033333, False]], [[[[0.0, 0.0, 1.51], [228.1, 0.0, 0.322], [74.8, 1.176, 1.179], [268.4, 1.044, 1.986]], 0.5, 0.05, 5], [2, 0.025, False]], [[[[333.7, 1.663, 0.348], [161.4, 0.0, 0.877]], 0.5, 0.1, 7], [3, 0.033333, False]], [[[[317.5, 0.0, 0.172]], 0.9, 0.03333333333333333, 3], [1, 0.033333, False]], [[[[140.9, 0.0, 0.527], [20.1, 0.0, 1.755], [0.0, 1.669, 0.955], [122.6, 0.0, 0.656]], 0.9, 0.016666666666666666, 8], [1, 0.016667, 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-stability-bound-factor","generated_at":"2026-09-29T14:50:54.330998+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 symplectic Euler limit 2/omega is written as 1/omega.","sha256":"306696df897f3db57095197cea6808a05efd5ade9a1c42c9dc160c32e3771686","title":"Critical step uses one over omega instead of two over omega · 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":40.125,"exit_code":1,"observations":[{"actual":[1,0.05,false],"check":"case 0","expected":[1,0.05,false],"passed":true},{"actual":[1,0.05,false],"check":"case 1","expected":[1,0.05,false],"passed":true},{"actual":[1,0.016667,false],"check":"case 2","expected":[1,0.016667,false],"passed":true},{"actual":[1,0.033333,false],"check":"case 3","expected":[1,0.033333,false],"passed":true},{"actual":[1,0.033333,false],"check":"case 4","expected":[1,0.033333,false],"passed":true},{"actual":[1,0.033333,false],"check":"case 5","expected":[1,0.033333,false],"passed":true},{"actual":[1,0.1,false],"check":"case 6","expected":[2,0.05,false],"passed":false},{"actual":[2,0.05,false],"check":"case 7","expected":[3,0.033333,false],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"case 0\", \"actual\": [1, 0.05, false], \"expected\": [1, 0.05, false], \"passed\": true}, {\"check\": \"case 1\", \"actual\": [1, 0.05, false], \"expected\": [1, 0.05, false], \"passed\": true}, {\"check\": \"case 2\", \"actual\": [1, 0.016667, false], \"expected\": [1, 0.016667, false], \"passed\": true}, {\"check\": \"case 3\", \"actual\": [1, 0.033333, false], \"expected\": [1, 0.033333, false], \"passed\": true}, {\"check\": \"case 4\", \"actual\": [1, 0.033333, false], \"expected\": [1, 0.033333, false], \"passed\": true}, {\"check\": \"case 5\", \"actual\": [1, 0.033333, false], \"expected\": [1, 0.033333, false], \"passed\": true}, {\"check\": \"case 6\", \"actual\": [1, 0.1, false], \"expected\": [2, 0.05, false], \"passed\": false}, {\"check\": \"case 7\", \"actual\": [2, 0.05, false], \"expected\": [3, 0.033333, false], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":41.666,"exit_code":1,"observations":[{"actual":[2,0.025,false],"check":"case 0","expected":[1,0.05,false],"passed":false},{"actual":[2,0.025,false],"check":"case 1","expected":[1,0.05,false],"passed":false},{"actual":[1,0.016667,false],"check":"case 2","expected":[1,0.016667,false],"passed":true},{"actual":[2,0.016667,false],"check":"case 3","expected":[1,0.033333,false],"passed":false},{"actual":[1,0.033333,false],"check":"case 4","expected":[1,0.033333,false],"passed":true},{"actual":[1,0.033333,false],"check":"case 5","expected":[1,0.033333,false],"passed":true},{"actual":[3,0.033333,false],"check":"case 6","expected":[2,0.05,false],"passed":false},{"actual":[6,0.016667,false],"check":"case 7","expected":[3,0.033333,false],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"case 0\", \"actual\": [2, 0.025, false], \"expected\": [1, 0.05, false], \"passed\": false}, {\"check\": \"case 1\", \"actual\": [2, 0.025, false], \"expected\": [1, 0.05, false], \"passed\": false}, {\"check\": \"case 2\", \"actual\": [1, 0.016667, false], \"expected\": [1, 0.016667, false], \"passed\": true}, {\"check\": \"case 3\", \"actual\": [2, 0.016667, false], \"expected\": [1, 0.033333, false], \"passed\": false}, {\"check\": \"case 4\", \"actual\": [1, 0.033333, false], \"expected\": [1, 0.033333, false], \"passed\": true}, {\"check\": \"case 5\", \"actual\": [1, 0.033333, false], \"expected\": [1, 0.033333, false], \"passed\": true}, {\"check\": \"case 6\", \"actual\": [3, 0.033333, false], \"expected\": [2, 0.05, false], \"passed\": false}, {\"check\": \"case 7\", \"actual\": [6, 0.016667, false], \"expected\": [3, 0.033333, false], \"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."}}