{"abstract":"Spinning bodies precess about the wrong axis once they are tilted.","category":"Physics integrator stability","checks":8,"contract":"solve(q, omega, dt, steps): orientation quaternion [w,x,y,z], world-frame angular velocity omega. Each step q += 0.5*dt*(0,omega) (x) q (Hamilton product, omega on the left), then renormalize to unit length. Return q with w>=0 (negate all components otherwise), rounded to 6.","contract_signature":"q, omega, dt, steps","evaluation_group":"w2-physics_integrator_stability-quaternion-world-spin","failed_approach":"Multiplying by the conjugate of q still does not express world-frame spin.","family":"w2-physics_integrator_stability-quaternion-world-spin-angular-velocity-frame","id":"FA-87161","implementations":{"attempt":{"sha256":"caf1c84deebb4427a076323d76527f537250d7512e860c7ea5a8c38a3e4add35","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(q, omega, dt, steps):\n    def mul(a, b):\n        aw, ax, ay, az = a\n        bw, bx, by, bz = b\n        return [aw * bw - ax * bx - ay * by - az * bz,\n                aw * bx + ax * bw + ay * bz - az * by,\n                aw * by - ax * bz + ay * bw + az * bx,\n                aw * bz + ax * by - ay * bx + az * bw]\n    w = [0.0, omega[0], omega[1], omega[2]]\n    q = list(q)\n    for _ in range(steps):\n        d = mul(w, [q[0], -q[1], -q[2], -q[3]])\n        q = [q[i] + 0.5 * dt * d[i] for i in range(4)]\n        n = math.sqrt(sum(c * c for c in q))\n        q = [c / n for c in q]\n    if q[0] < 0:\n        q = [-c for c in q]\n    return [round(c, 6) for c in q]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[[[[-0.1116, -0.6859, -0.2807, 0.6621], [-1.367, -3.32, -3.869], 0.05, 1], [0.093483, 0.757592, 0.180891, -0.620156]], [[[0.5719, -0.201, -0.6787, -0.4146], [1.924, -1.133, -2.402], 0.05, 5], [0.357598, -0.191716, -0.546007, -0.732971]], [[[-0.1804, -0.126, 0.6456, -0.7313], [0.559, -1.91, -2.794], 0.05, 1], [0.198164, 0.048329, -0.670767, 0.713069]], [[[0.5784, -0.2766, 0.5787, -0.504], [-2.529, -0.35, -1.088], 0.1, 1], [0.52115, -0.306506, 0.514969, -0.607671]], [[[0.5438, 0.3302, -0.6582, 0.4024], [-2.872, -3.023, 2.126], 0.05, 4], [0.303775, 0.162656, -0.567546, 0.747767]], [[[-0.4336, 0.4529, 0.2639, 0.733], [-3.578, -1.691, -2.103], 0.01, 1], [0.415442, -0.457106, -0.27584, -0.736461]], [[[-0.6434, -0.5565, -0.5031, -0.1523], [-2.902, -3.621, 0.012], 0.05, 6], [0.967882, 0.094358, 0.128105, 0.194656]], [[[0.1693, 0.5991, -0.3007, 0.7225], [-0.846, 0.203, 2.272], 0.1, 5], [0.11186, -0.653252, -0.222024, -0.71516]]], [[[[-0.3472, -0.3736, 0.4295, 0.7453], [-2.315, 1.527, -3.853], 0.01, 8], [0.287249, 0.223901, -0.5265, -0.768215]], [[[0.8268, -0.0559, 0.3935, 0.398], [-0.387, 0.922, 1.986], 0.1, 7], [0.220922, -0.271091, 0.534556, 0.769385]], [[[0.7724, -0.3363, 0.1754, -0.5095], [-1.034, -3.868, 2.146], 0.05, 4], [0.831714, -0.225919, -0.249805, -0.441373]], [[[0.6623, -0.6015, -0.0252, 0.4461], [1.478, -3.402, -2.007], 0.05, 6], [0.772394, -0.568265, -0.258655, -0.116536]], [[[-0.0856, 0.1732, -0.7636, 0.6161], [-0.55, 3.798, 0.546], 0.01, 4], [0.032223, -0.228697, 0.759147, -0.608568]], [[[-0.4028, 0.5372, 0.3775, -0.6377], [-2.487, -2.209, -0.784], 0.05, 1], [0.359741, -0.602665, -0.348298, 0.621346]], [[[0.4376, -0.0327, 0.5947, -0.6736], [2.933, -2.297, 1.467], 0.01, 6], [0.507773, 0.026136, 0.618032, -0.5996]], [[[0.0263, -0.6798, 0.6943, 0.2347], [-2.851, -1.309, 1.331], 0.01, 4], [0.000571, 0.704341, -0.687306, -0.177522]]], [[[[-0.3438, -0.8064, 0.2793, -0.3918], [0.487, -0.414, 0.563], 0.05, 6], [0.231973, 0.824132, -0.258947, 0.447149]], [[[-0.3511, 0.4777, 0.5685, -0.5704], [2.489, 3.438, -0.071], 0.1, 2], [0.625498, -0.169104, -0.536584, 0.540586]], [[[0.5702, -0.6235, -0.2795, 0.4562], [-0.18, -0.187, 0.167], 0.1, 4], [0.520981, -0.650508, -0.304658, 0.461087]], [[[-0.5755, -0.1729, 0.1896, -0.7765], [-1.564, 1.954, -3.317], 0.01, 5], [0.652658, 0.171677, -0.144528, 0.723655]], [[[-0.3086, -0.4487, -0.811, 0.2137], [-0.353, -3.189, 2.11], 0.01, 10], [0.461721, 0.383888, 0.790513, -0.12055]], [[[-0.3137, -0.499, 0.6659, 0.4573], [0.29, 3.395, 1.611], 0.1, 5], [0.794564, 0.215018, 0.028732, -0.567106]], [[[-0.4413, 0.7728, 0.4297, 0.1528], [1.432, -3.081, -3.746], 0.01, 3], [0.42821, -0.778191, -0.402193, -0.222026]], [[[-0.571, 0.1155, 0.5121, 0.6312], [2.085, -1.83, -1.285], 0.01, 1], [0.563386, -0.107047, -0.509932, -0.641176]]], [[[[0.5728, 0.3899, 0.4262, -0.5816], [-2.116, -3.996, -1.282], 0.05, 1], [0.613137, 0.428408, 0.323485, -0.579559]], [[[0.2873, -0.6346, 0.6402, -0.324], [3.029, -0.324, 1.16], 0.1, 8], [0.817944, -0.105726, 0.220751, 0.520633]], [[[0.2652, 0.6436, 0.4121, 0.588], [-2.665, 2.022, 0.007], 0.1, 11], [0.187909, -0.02187, 0.507266, -0.840769]], [[[0.2307, 0.3931, 0.7631, -0.4581], [2.174, 2.113, 3.444], 0.05, 10], [0.080385, 0.450009, -0.879659, -0.131262]], [[[0.3499, -0.381, 0.1815, -0.8363], [-2.42, 1.295, -0.715], 0.05, 10], [0.136518, 0.702138, 0.159756, 0.680326]], [[[0.346, -0.6937, -0.611, 0.1602], [-2.952, -1.698, -3.004], 0.1, 5], [0.363368, 0.923626, -0.121437, 0.011516]], [[[0.3265, 0.31, 0.1591, -0.8786], [-1.35, 2.136, -0.561], 0.01, 12], [0.297825, 0.173125, 0.117593, -0.931397]], [[[0.1401, -0.0998, 0.7197, 0.6726], [-3.242, -3.373, -3.717], 0.01, 11], [0.381235, -0.097044, 0.793491, 0.464343]]], [[[[-0.6115, -0.3532, -0.6797, -0.1982], [3.428, -1.888, 3.841], 0.05, 13], [0.244774, 0.22911, 0.220548, -0.915944]], [[[0.2915, -0.4846, 0.5466, 0.6176], [-2.376, 3.234, 3.373], 0.1, 13], [0.105141, -0.495302, 0.564739, 0.651683]], [[[-0.2117, -0.7256, 0.617, -0.2191], [-3.406, 0.968, 2.951], 0.1, 9], [0.373544, -0.073188, 0.877043, 0.293093]], [[[-0.4693, 0.5928, -0.5163, -0.4022], [3.917, 1.937, -1.401], 0.05, 6], [0.62339, 0.001817, 0.422365, 0.658019]], [[[0.5273, 0.4224, -0.4746, 0.5641], [-2.08, 3.628, 3.896], 0.01, 13], [0.516939, 0.571342, -0.141768, 0.621485]], [[[-0.447, -0.6364, 0.627, 0.0463], [3.512, -0.033, 2.944], 0.05, 9], [0.164268, -0.967385, -0.052794, 0.185459]], [[[0.0332, -0.1691, -0.7704, -0.6138], [2.17, 2.479, 1.5], 0.1, 9], [0.881211, -0.074251, 0.351735, -0.30698]], [[[0.1785, -0.1228, -0.7653, 0.6061], [2.105, 0.281, 2.09], 0.01, 8], [0.145584, -0.036317, -0.819027, 0.553788]]]]\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":"4205aa4d315cfb4874321baf67cce70757f7c6ae08f16ee61af3639daf8b13d1","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(q, omega, dt, steps):\n    def mul(a, b):\n        aw, ax, ay, az = a\n        bw, bx, by, bz = b\n        return [aw * bw - ax * bx - ay * by - az * bz,\n                aw * bx + ax * bw + ay * bz - az * by,\n                aw * by - ax * bz + ay * bw + az * bx,\n                aw * bz + ax * by - ay * bx + az * bw]\n    w = [0.0, omega[0], omega[1], omega[2]]\n    q = list(q)\n    for _ in range(steps):\n        d = mul(q, w)\n        q = [q[i] + 0.5 * dt * d[i] for i in range(4)]\n        n = math.sqrt(sum(c * c for c in q))\n        q = [c / n for c in q]\n    if q[0] < 0:\n        q = [-c for c in q]\n    return [round(c, 6) for c in q]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[[[[-0.1116, -0.6859, -0.2807, 0.6621], [-1.367, -3.32, -3.869], 0.05, 1], [0.093483, 0.757592, 0.180891, -0.620156]], [[[0.5719, -0.201, -0.6787, -0.4146], [1.924, -1.133, -2.402], 0.05, 5], [0.357598, -0.191716, -0.546007, -0.732971]], [[[-0.1804, -0.126, 0.6456, -0.7313], [0.559, -1.91, -2.794], 0.05, 1], [0.198164, 0.048329, -0.670767, 0.713069]], [[[0.5784, -0.2766, 0.5787, -0.504], [-2.529, -0.35, -1.088], 0.1, 1], [0.52115, -0.306506, 0.514969, -0.607671]], [[[0.5438, 0.3302, -0.6582, 0.4024], [-2.872, -3.023, 2.126], 0.05, 4], [0.303775, 0.162656, -0.567546, 0.747767]], [[[-0.4336, 0.4529, 0.2639, 0.733], [-3.578, -1.691, -2.103], 0.01, 1], [0.415442, -0.457106, -0.27584, -0.736461]], [[[-0.6434, -0.5565, -0.5031, -0.1523], [-2.902, -3.621, 0.012], 0.05, 6], [0.967882, 0.094358, 0.128105, 0.194656]], [[[0.1693, 0.5991, -0.3007, 0.7225], [-0.846, 0.203, 2.272], 0.1, 5], [0.11186, -0.653252, -0.222024, -0.71516]]], [[[[-0.3472, -0.3736, 0.4295, 0.7453], [-2.315, 1.527, -3.853], 0.01, 8], [0.287249, 0.223901, -0.5265, -0.768215]], [[[0.8268, -0.0559, 0.3935, 0.398], [-0.387, 0.922, 1.986], 0.1, 7], [0.220922, -0.271091, 0.534556, 0.769385]], [[[0.7724, -0.3363, 0.1754, -0.5095], [-1.034, -3.868, 2.146], 0.05, 4], [0.831714, -0.225919, -0.249805, -0.441373]], [[[0.6623, -0.6015, -0.0252, 0.4461], [1.478, -3.402, -2.007], 0.05, 6], [0.772394, -0.568265, -0.258655, -0.116536]], [[[-0.0856, 0.1732, -0.7636, 0.6161], [-0.55, 3.798, 0.546], 0.01, 4], [0.032223, -0.228697, 0.759147, -0.608568]], [[[-0.4028, 0.5372, 0.3775, -0.6377], [-2.487, -2.209, -0.784], 0.05, 1], [0.359741, -0.602665, -0.348298, 0.621346]], [[[0.4376, -0.0327, 0.5947, -0.6736], [2.933, -2.297, 1.467], 0.01, 6], [0.507773, 0.026136, 0.618032, -0.5996]], [[[0.0263, -0.6798, 0.6943, 0.2347], [-2.851, -1.309, 1.331], 0.01, 4], [0.000571, 0.704341, -0.687306, -0.177522]]], [[[[-0.3438, -0.8064, 0.2793, -0.3918], [0.487, -0.414, 0.563], 0.05, 6], [0.231973, 0.824132, -0.258947, 0.447149]], [[[-0.3511, 0.4777, 0.5685, -0.5704], [2.489, 3.438, -0.071], 0.1, 2], [0.625498, -0.169104, -0.536584, 0.540586]], [[[0.5702, -0.6235, -0.2795, 0.4562], [-0.18, -0.187, 0.167], 0.1, 4], [0.520981, -0.650508, -0.304658, 0.461087]], [[[-0.5755, -0.1729, 0.1896, -0.7765], [-1.564, 1.954, -3.317], 0.01, 5], [0.652658, 0.171677, -0.144528, 0.723655]], [[[-0.3086, -0.4487, -0.811, 0.2137], [-0.353, -3.189, 2.11], 0.01, 10], [0.461721, 0.383888, 0.790513, -0.12055]], [[[-0.3137, -0.499, 0.6659, 0.4573], [0.29, 3.395, 1.611], 0.1, 5], [0.794564, 0.215018, 0.028732, -0.567106]], [[[-0.4413, 0.7728, 0.4297, 0.1528], [1.432, -3.081, -3.746], 0.01, 3], [0.42821, -0.778191, -0.402193, -0.222026]], [[[-0.571, 0.1155, 0.5121, 0.6312], [2.085, -1.83, -1.285], 0.01, 1], [0.563386, -0.107047, -0.509932, -0.641176]]], [[[[0.5728, 0.3899, 0.4262, -0.5816], [-2.116, -3.996, -1.282], 0.05, 1], [0.613137, 0.428408, 0.323485, -0.579559]], [[[0.2873, -0.6346, 0.6402, -0.324], [3.029, -0.324, 1.16], 0.1, 8], [0.817944, -0.105726, 0.220751, 0.520633]], [[[0.2652, 0.6436, 0.4121, 0.588], [-2.665, 2.022, 0.007], 0.1, 11], [0.187909, -0.02187, 0.507266, -0.840769]], [[[0.2307, 0.3931, 0.7631, -0.4581], [2.174, 2.113, 3.444], 0.05, 10], [0.080385, 0.450009, -0.879659, -0.131262]], [[[0.3499, -0.381, 0.1815, -0.8363], [-2.42, 1.295, -0.715], 0.05, 10], [0.136518, 0.702138, 0.159756, 0.680326]], [[[0.346, -0.6937, -0.611, 0.1602], [-2.952, -1.698, -3.004], 0.1, 5], [0.363368, 0.923626, -0.121437, 0.011516]], [[[0.3265, 0.31, 0.1591, -0.8786], [-1.35, 2.136, -0.561], 0.01, 12], [0.297825, 0.173125, 0.117593, -0.931397]], [[[0.1401, -0.0998, 0.7197, 0.6726], [-3.242, -3.373, -3.717], 0.01, 11], [0.381235, -0.097044, 0.793491, 0.464343]]], [[[[-0.6115, -0.3532, -0.6797, -0.1982], [3.428, -1.888, 3.841], 0.05, 13], [0.244774, 0.22911, 0.220548, -0.915944]], [[[0.2915, -0.4846, 0.5466, 0.6176], [-2.376, 3.234, 3.373], 0.1, 13], [0.105141, -0.495302, 0.564739, 0.651683]], [[[-0.2117, -0.7256, 0.617, -0.2191], [-3.406, 0.968, 2.951], 0.1, 9], [0.373544, -0.073188, 0.877043, 0.293093]], [[[-0.4693, 0.5928, -0.5163, -0.4022], [3.917, 1.937, -1.401], 0.05, 6], [0.62339, 0.001817, 0.422365, 0.658019]], [[[0.5273, 0.4224, -0.4746, 0.5641], [-2.08, 3.628, 3.896], 0.01, 13], [0.516939, 0.571342, -0.141768, 0.621485]], [[[-0.447, -0.6364, 0.627, 0.0463], [3.512, -0.033, 2.944], 0.05, 9], [0.164268, -0.967385, -0.052794, 0.185459]], [[[0.0332, -0.1691, -0.7704, -0.6138], [2.17, 2.479, 1.5], 0.1, 9], [0.881211, -0.074251, 0.351735, -0.30698]], [[[0.1785, -0.1228, -0.7653, 0.6061], [2.105, 0.281, 2.09], 0.01, 8], [0.145584, -0.036317, -0.819027, 0.553788]]]]\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-quaternion-world-spin-angular-velocity-frame","generated_at":"2026-09-29T14:50:56.264686+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 derivative uses q(x)omega, the body-frame convention.","sha256":"4d41ed9d8100a4e80480c51b6fde23a1c7d188071c75386fa6334712deaea7ca","title":"World angular velocity is applied on the body side of the product · 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":44.111,"exit_code":1,"observations":[{"actual":[0.127308,0.592555,0.355948,-0.711317],"check":"case 0","expected":[0.093483,0.757592,0.180891,-0.620156],"passed":false},{"actual":[0.620567,0.101272,-0.705695,-0.326552],"check":"case 1","expected":[0.357598,-0.191716,-0.546007,-0.732971],"passed":false},{"actual":[0.162392,0.209161,-0.637098,0.723869],"check":"case 2","expected":[0.198164,0.048329,-0.670767,0.713069],"passed":false},{"actual":[0.590636,-0.365283,0.578083,-0.428412],"check":"case 3","expected":[0.52115,-0.306506,0.514969,-0.607671],"passed":false},{"actual":[0.605121,0.066584,-0.772712,0.179754],"check":"case 4","expected":[0.303775,0.162656,-0.567546,0.747767],"passed":false},{"actual":[0.444616,-0.456861,-0.255183,-0.726964],"check":"case 5","expected":[0.415442,-0.457106,-0.27584,-0.736461],"passed":false},{"actual":[0.449778,0.797989,0.392778,0.081481],"check":"case 6","expected":[0.967882,0.094358,0.128105,0.194656],"passed":false},{"actual":[0.393316,0.190007,-0.59746,0.672489],"check":"case 7","expected":[0.11186,-0.653252,-0.222024,-0.71516],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"case 0\", \"actual\": [0.127308, 0.592555, 0.355948, -0.711317], \"expected\": [0.093483, 0.757592, 0.180891, -0.620156], \"passed\": false}, {\"check\": \"case 1\", \"actual\": [0.620567, 0.101272, -0.705695, -0.326552], \"expected\": [0.357598, -0.191716, -0.546007, -0.732971], \"passed\": false}, {\"check\": \"case 2\", \"actual\": [0.162392, 0.209161, -0.637098, 0.723869], \"expected\": [0.198164, 0.048329, -0.670767, 0.713069], \"passed\": false}, {\"check\": \"case 3\", \"actual\": [0.590636, -0.365283, 0.578083, -0.428412], \"expected\": [0.52115, -0.306506, 0.514969, -0.607671], 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