{"abstract":"The exact integer matrix binary power result violates the stated contract at multiplicative identity seed.","category":"Numerics","checks":8,"contract":"Input [square integer matrix,nonnegative exponent<=12]; return exact matrix power.","evaluation_group":"s3-numerics-integer-matrix-binary-power","failed_approach":"The partial repair [row[:] for row in a] still violates the multiplicative identity seed invariant.","family":"s3-numerics-integer-matrix-binary-power-multiplicative-identity-seed","id":"FA-15466","implementations":{"attempt":{"sha256":"de091ca921b510f8d66613bf561fd0d888786d9f31069a6e9e86855a4455321d","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nimport itertools\nfrom fractions import Fraction\nN = 1\nobservations = []\ndef solve(x):\n    a,e=x;n=len(a)\n    def mul(a,b):return [[sum(a[i][k]*b[k][j] for k in range(n)) for j in range(n)] for i in range(n)]\n    r=[row[:] for row in a]\n    for _ in range(12):\n     if e==0:break\n     if e%2==1:r=mul(r,a)\n     a=mul(a,a)\n     e=e//2\n    return r\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([[[-1]], 0], [[1]]), ([[[-1, 3], [-1, -3]], 8], [[1168, 2688], [-896, -624]]), ([[[-1]], 1], [[-1]]), ([[[-1]], 2], [[1]]), ([[[-1]], 3], [[-1]]), ([[[-1]], 4], [[1]]), ([[[-1]], 5], [[-1]]), ([[[-1]], 6], [[1]])], [([[[-1]], 1], [[-1]]), ([[[-1]], 3], [[-1]]), ([[[-1]], 0], [[1]]), ([[[-1, 3], [-1, -3]], 8], [[1168, 2688], [-896, -624]]), ([[[2]], 8], [[256]]), ([[[2]], 0], [[1]]), ([[[2]], 1], [[2]]), ([[[2]], 2], [[4]])], [([[[-1]], 2], [[1]]), ([[[-1]], 6], [[1]]), ([[[-1]], 0], [[1]]), ([[[-1, 3], [-1, -3]], 8], [[1168, 2688], [-896, -624]]), ([[[1]], 7], [[1]]), ([[[1]], 8], [[1]]), ([[[3]], 0], [[1]]), ([[[3]], 1], [[3]])], [([[[-1]], 3], [[-1]]), ([[[2]], 0], [[1]]), ([[[-1]], 0], [[1]]), ([[[-1, 3], [-1, -3]], 8], [[1168, 2688], [-896, -624]]), ([[[-1]], 6], [[1]]), ([[[-1]], 7], [[-1]]), ([[[-1]], 8], [[1]]), ([[[3]], 0], [[1]])], [([[[-1]], 4], [[1]]), ([[[2]], 3], [[8]]), ([[[-1]], 0], [[1]]), ([[[-1, 3], [-1, -3]], 8], [[1168, 2688], [-896, -624]]), ([[[-3]], 5], [[-243]]), ([[[-3]], 6], [[729]]), ([[[-3]], 7], [[-2187]]), ([[[-3]], 8], [[6561]])]]\nfor i, (args, expected) in enumerate(fixtures[N-1]):\n    check(\"explicit oracle %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":"c1626b18a42dc67a2d314882eb675dfc83b656304c5287a4a8e9c23643168498","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nimport itertools\nfrom fractions import Fraction\nN = 1\nobservations = []\ndef solve(x):\n    a,e=x;n=len(a)\n    def mul(a,b):return [[sum(a[i][k]*b[k][j] for k in range(n)) for j in range(n)] for i in range(n)]\n    r=[[0]*n for _ in range(n)]\n    for _ in range(12):\n     if e==0:break\n     if e%2==1:r=mul(r,a)\n     a=mul(a,a)\n     e=e//2\n    return r\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([[[-1]], 0], [[1]]), ([[[-1, 3], [-1, -3]], 8], [[1168, 2688], [-896, -624]]), ([[[-1]], 1], [[-1]]), ([[[-1]], 2], [[1]]), ([[[-1]], 3], [[-1]]), ([[[-1]], 4], [[1]]), ([[[-1]], 5], [[-1]]), ([[[-1]], 6], [[1]])], [([[[-1]], 1], [[-1]]), ([[[-1]], 3], [[-1]]), ([[[-1]], 0], [[1]]), ([[[-1, 3], [-1, -3]], 8], [[1168, 2688], [-896, -624]]), ([[[2]], 8], [[256]]), ([[[2]], 0], [[1]]), ([[[2]], 1], [[2]]), ([[[2]], 2], [[4]])], [([[[-1]], 2], [[1]]), ([[[-1]], 6], [[1]]), ([[[-1]], 0], [[1]]), ([[[-1, 3], [-1, -3]], 8], [[1168, 2688], [-896, -624]]), ([[[1]], 7], [[1]]), ([[[1]], 8], [[1]]), ([[[3]], 0], [[1]]), ([[[3]], 1], [[3]])], [([[[-1]], 3], [[-1]]), ([[[2]], 0], [[1]]), ([[[-1]], 0], [[1]]), ([[[-1, 3], [-1, -3]], 8], [[1168, 2688], [-896, -624]]), ([[[-1]], 6], [[1]]), ([[[-1]], 7], [[-1]]), ([[[-1]], 8], [[1]]), ([[[3]], 0], [[1]])], [([[[-1]], 4], [[1]]), ([[[2]], 3], [[8]]), ([[[-1]], 0], [[1]]), ([[[-1, 3], [-1, -3]], 8], [[1168, 2688], [-896, -624]]), ([[[-3]], 5], [[-243]]), ([[[-3]], 6], [[729]]), ([[[-3]], 7], [[-2187]]), ([[[-3]], 8], [[6561]])]]\nfor i, (args, expected) in enumerate(fixtures[N-1]):\n    check(\"explicit oracle %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":"beab4e23ad47e54bab607908c54a591b300ce89fa9e6c206b37bb711620ef63a","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nimport itertools\nfrom fractions import Fraction\nN = 1\nobservations = []\ndef solve(x):\n    a,e=x;n=len(a)\n    def mul(a,b):return [[sum(a[i][k]*b[k][j] for k in range(n)) for j in range(n)] for i in range(n)]\n    r=[[int(i==j) for j in range(n)] for i in range(n)]\n    for _ in range(12):\n     if e==0:break\n     if e%2==1:r=mul(r,a)\n     a=mul(a,a)\n     e=e//2\n    return r\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([[[-1]], 0], [[1]]), ([[[-1, 3], [-1, -3]], 8], [[1168, 2688], [-896, -624]]), ([[[-1]], 1], [[-1]]), ([[[-1]], 2], [[1]]), ([[[-1]], 3], [[-1]]), ([[[-1]], 4], [[1]]), ([[[-1]], 5], [[-1]]), ([[[-1]], 6], [[1]])], [([[[-1]], 1], [[-1]]), ([[[-1]], 3], [[-1]]), ([[[-1]], 0], [[1]]), ([[[-1, 3], [-1, -3]], 8], [[1168, 2688], [-896, -624]]), ([[[2]], 8], [[256]]), ([[[2]], 0], [[1]]), ([[[2]], 1], [[2]]), ([[[2]], 2], [[4]])], [([[[-1]], 2], [[1]]), ([[[-1]], 6], [[1]]), ([[[-1]], 0], [[1]]), ([[[-1, 3], [-1, -3]], 8], [[1168, 2688], [-896, -624]]), ([[[1]], 7], [[1]]), ([[[1]], 8], [[1]]), ([[[3]], 0], [[1]]), ([[[3]], 1], [[3]])], [([[[-1]], 3], [[-1]]), ([[[2]], 0], [[1]]), ([[[-1]], 0], [[1]]), ([[[-1, 3], [-1, -3]], 8], [[1168, 2688], [-896, -624]]), ([[[-1]], 6], [[1]]), ([[[-1]], 7], [[-1]]), ([[[-1]], 8], [[1]]), ([[[3]], 0], [[1]])], [([[[-1]], 4], [[1]]), ([[[2]], 3], [[8]]), ([[[-1]], 0], [[1]]), ([[[-1, 3], [-1, -3]], 8], [[1168, 2688], [-896, -624]]), ([[[-3]], 5], [[-243]]), ([[[-3]], 6], [[729]]), ([[[-3]], 7], [[-2187]]), ([[[-3]], 8], [[6561]])]]\nfor i, (args, expected) in enumerate(fixtures[N-1]):\n    check(\"explicit oracle %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. Inputs are restricted to the explicit contract; this is not a production algebra library. 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":"s3-numerics-integer-matrix-binary-power-multiplicative-identity-seed","generated_at":"2026-09-29T14:39:27.036561+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"Exact discrete arithmetic with observable algorithmic state; no floating point approximation is used.","repair":"Use [[int(i==j) for j in range(n)] for i in range(n)] at the multiplicative identity seed step.","root_cause":"The multiplicative identity seed step uses [[0]*n for _ in range(n)] instead of [[int(i==j) for j in range(n)] for i in range(n)].","sha256":"63fc77639070155b58d6b26ba77dbac7d30e3537b9f3aeab82118b1521642f32","title":"Integer matrix binary power: multiplicative identity seed · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":46.498,"exit_code":1,"observations":[{"actual":[[-1]],"check":"explicit oracle 0","expected":[[1]],"passed":false},{"actual":[[-3856,-4560],[1520,-816]],"check":"explicit oracle 1","expected":[[1168,2688],[-896,-624]],"passed":false},{"actual":[[1]],"check":"explicit oracle 2","expected":[[-1]],"passed":false},{"actual":[[-1]],"check":"explicit oracle 3","expected":[[1]],"passed":false},{"actual":[[1]],"check":"explicit oracle 4","expected":[[-1]],"passed":false},{"actual":[[-1]],"check":"explicit oracle 5","expected":[[1]],"passed":false},{"actual":[[1]],"check":"explicit oracle 6","expected":[[-1]],"passed":false},{"actual":[[-1]],"check":"explicit oracle 7","expected":[[1]],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [[-1]], \"expected\": [[1]], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [[-3856, -4560], [1520, -816]], \"expected\": [[1168, 2688], [-896, -624]], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [[1]], \"expected\": [[-1]], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [[-1]], \"expected\": [[1]], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [[1]], \"expected\": [[-1]], \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": [[-1]], \"expected\": [[1]], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [[1]], \"expected\": [[-1]], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [[-1]], \"expected\": [[1]], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":44.858,"exit_code":1,"observations":[{"actual":[[0]],"check":"explicit oracle 0","expected":[[1]],"passed":false},{"actual":[[0,0],[0,0]],"check":"explicit oracle 1","expected":[[1168,2688],[-896,-624]],"passed":false},{"actual":[[0]],"check":"explicit oracle 2","expected":[[-1]],"passed":false},{"actual":[[0]],"check":"explicit oracle 3","expected":[[1]],"passed":false},{"actual":[[0]],"check":"explicit oracle 4","expected":[[-1]],"passed":false},{"actual":[[0]],"check":"explicit oracle 5","expected":[[1]],"passed":false},{"actual":[[0]],"check":"explicit oracle 6","expected":[[-1]],"passed":false},{"actual":[[0]],"check":"explicit oracle 7","expected":[[1]],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [[0]], \"expected\": [[1]], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [[0, 0], [0, 0]], \"expected\": [[1168, 2688], [-896, -624]], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [[0]], \"expected\": [[-1]], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [[0]], \"expected\": [[1]], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [[0]], \"expected\": [[-1]], \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": [[0]], \"expected\": [[1]], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [[0]], \"expected\": [[-1]], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [[0]], \"expected\": [[1]], \"passed\": false}], \"passed\": false}\n"},"fixed":{"elapsed_ms":44.82,"exit_code":0,"observations":[{"actual":[[1]],"check":"explicit oracle 0","expected":[[1]],"passed":true},{"actual":[[1168,2688],[-896,-624]],"check":"explicit oracle 1","expected":[[1168,2688],[-896,-624]],"passed":true},{"actual":[[-1]],"check":"explicit oracle 2","expected":[[-1]],"passed":true},{"actual":[[1]],"check":"explicit oracle 3","expected":[[1]],"passed":true},{"actual":[[-1]],"check":"explicit oracle 4","expected":[[-1]],"passed":true},{"actual":[[1]],"check":"explicit oracle 5","expected":[[1]],"passed":true},{"actual":[[-1]],"check":"explicit oracle 6","expected":[[-1]],"passed":true},{"actual":[[1]],"check":"explicit oracle 7","expected":[[1]],"passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [[1]], \"expected\": [[1]], \"passed\": true}, {\"check\": \"explicit oracle 1\", \"actual\": [[1168, 2688], [-896, -624]], \"expected\": [[1168, 2688], [-896, -624]], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [[-1]], \"expected\": [[-1]], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [[1]], \"expected\": [[1]], \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": [[-1]], \"expected\": [[-1]], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [[1]], \"expected\": [[1]], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [[-1]], \"expected\": [[-1]], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [[1]], \"expected\": [[1]], \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}