{"abstract":"The exact integer matrix binary power result violates the stated contract at base power squaring.","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 mul(a,r) still violates the base power squaring invariant.","family":"s3-numerics-integer-matrix-binary-power-base-power-squaring","id":"FA-15481","implementations":{"attempt":{"sha256":"7484706bffe6489d433e0f3d903311f8aa69ac0200cb0e5fec1d88e8538b580b","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,r)\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]], 2], [[1]]), ([[[-1]], 0], [[1]]), ([[[-1, 3], [-1, -3]], 8], [[1168, 2688], [-896, -624]]), ([[[-1]], 1], [[-1]]), ([[[-1]], 3], [[-1]]), ([[[-1]], 4], [[1]]), ([[[-1]], 5], [[-1]]), ([[[-1]], 6], [[1]])], [([[[-1]], 3], [[-1]]), ([[[-1]], 6], [[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]], 4], [[1]]), ([[[2]], 2], [[4]]), ([[[-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]], 5], [[-1]]), ([[[2]], 6], [[64]]), ([[[-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]], 8], [[1]]), ([[[2]], 2], [[4]]), ([[[-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":"396bbe5dacc7be07ab90caa073111aebb032cbe29fb29efcd148959593822bdc","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=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]], 2], [[1]]), ([[[-1]], 0], [[1]]), ([[[-1, 3], [-1, -3]], 8], [[1168, 2688], [-896, -624]]), ([[[-1]], 1], [[-1]]), ([[[-1]], 3], [[-1]]), ([[[-1]], 4], [[1]]), ([[[-1]], 5], [[-1]]), ([[[-1]], 6], [[1]])], [([[[-1]], 3], [[-1]]), ([[[-1]], 6], [[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]], 4], [[1]]), ([[[2]], 2], [[4]]), ([[[-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]], 5], [[-1]]), ([[[2]], 6], [[64]]), ([[[-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]], 8], [[1]]), ([[[2]], 2], [[4]]), ([[[-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":"760aba23d25b55fbbb576cc7b95196e086c2496a34cdf917e7efb00ae9f360be","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]], 2], [[1]]), ([[[-1]], 0], [[1]]), ([[[-1, 3], [-1, -3]], 8], [[1168, 2688], [-896, -624]]), ([[[-1]], 1], [[-1]]), ([[[-1]], 3], [[-1]]), ([[[-1]], 4], [[1]]), ([[[-1]], 5], [[-1]]), ([[[-1]], 6], [[1]])], [([[[-1]], 3], [[-1]]), ([[[-1]], 6], [[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]], 4], [[1]]), ([[[2]], 2], [[4]]), ([[[-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]], 5], [[-1]]), ([[[2]], 6], [[64]]), ([[[-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]], 8], [[1]]), ([[[2]], 2], [[4]]), ([[[-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-base-power-squaring","generated_at":"2026-09-29T14:39:27.362489+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 mul(a,a) at the base power squaring step.","root_cause":"The base power squaring step uses a instead of mul(a,a).","sha256":"777d1b593cfc586873d8dcdbc183a4d52c1fbace48cacb258661b9819ee7b5ee","title":"Integer matrix binary power: base power squaring · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":43.389,"exit_code":1,"observations":[{"actual":[[-1]],"check":"explicit oracle 0","expected":[[1]],"passed":false},{"actual":[[1]],"check":"explicit oracle 1","expected":[[1]],"passed":true},{"actual":[[-1,3],[-1,-3]],"check":"explicit oracle 2","expected":[[1168,2688],[-896,-624]],"passed":false},{"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":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\": [[1]], \"expected\": [[1]], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [[-1, 3], [-1, -3]], \"expected\": [[1168, 2688], [-896, -624]], \"passed\": false}, {\"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\": 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":42.727,"exit_code":1,"observations":[{"actual":[[-1]],"check":"explicit oracle 0","expected":[[1]],"passed":false},{"actual":[[1]],"check":"explicit oracle 1","expected":[[1]],"passed":true},{"actual":[[-1,3],[-1,-3]],"check":"explicit oracle 2","expected":[[1168,2688],[-896,-624]],"passed":false},{"actual":[[-1]],"check":"explicit oracle 3","expected":[[-1]],"passed":true},{"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":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [[-1]], \"expected\": [[1]], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [[1]], \"expected\": [[1]], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [[-1, 3], [-1, -3]], \"expected\": [[1168, 2688], [-896, -624]], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [[-1]], \"expected\": [[-1]], \"passed\": true}, {\"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\": true}], \"passed\": false}\n"},"fixed":{"elapsed_ms":44.611,"exit_code":0,"observations":[{"actual":[[1]],"check":"explicit oracle 0","expected":[[1]],"passed":true},{"actual":[[1]],"check":"explicit oracle 1","expected":[[1]],"passed":true},{"actual":[[1168,2688],[-896,-624]],"check":"explicit oracle 2","expected":[[1168,2688],[-896,-624]],"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\": [[1]], \"expected\": [[1]], \"passed\": true}, {\"check\": \"explicit oracle 2\", \"actual\": [[1168, 2688], [-896, -624]], \"expected\": [[1168, 2688], [-896, -624]], \"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"}