{"abstract":"The exact finite field discrete transform result violates the stated contract at character exponent.","category":"Numerics","checks":8,"contract":"Input [values,p,w], len(values)=n and w has exact multiplicative order n modulo prime p; forward transform sum values[j]*w^(j*k) modulo p.","contract_signature":"x","evaluation_group":"s3-numerics-finite-field-discrete-transform","failed_approach":"The partial repair pow(w,-j*k,p) still violates the character exponent invariant.","family":"s3-numerics-finite-field-discrete-transform-character-exponent","id":"FA-15726","implementations":{"attempt":{"sha256":"b963e17e187cfeedc441727f7e59517ea7570116a57c187bcebc268ce1b5240a","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    values,p,w=x;n=len(values);out=[]\n    for k in range(n):\n     total=0\n     for j in range(n):\n      factor=pow(w,-j*k,p)\n      total=(total+values[j]*factor)%p\n     out.append(total)\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([[0, 3], 5, 4], [3, 2]), ([[0, 5, 4], 7, 2], [2, 5, 0]), ([[10, 15, 12, 4, 14, 0, 16, 12], 17, 2], [15, 14, 9, 14, 4, 14, 0, 10]), ([[2, 1], 5, 4], [3, 1]), ([[2, 1], 5, 4], [3, 1]), ([[3, 0], 5, 4], [3, 3]), ([[0, 2], 5, 4], [2, 3]), ([[1, 3], 5, 4], [4, 3])], [([[2, 1], 5, 4], [3, 1]), ([[4, 1, 3], 7, 2], [1, 4, 0]), ([[0, 3], 5, 4], [3, 2]), ([[10, 15, 12, 4, 14, 0, 16, 12], 17, 2], [15, 14, 9, 14, 4, 14, 0, 10]), ([[0, 2], 5, 4], [2, 3]), ([[1, 1], 5, 4], [2, 0]), ([[4, 3], 5, 4], [2, 1]), ([[2, 0], 5, 4], [2, 2])], [([[2, 1], 5, 4], [3, 1]), ([[2, 5, 0], 7, 2], [0, 5, 1]), ([[0, 3], 5, 4], [3, 2]), ([[10, 15, 12, 4, 14, 0, 16, 12], 17, 2], [15, 14, 9, 14, 4, 14, 0, 10]), ([[4, 0], 5, 4], [4, 4]), ([[1, 4], 5, 4], [0, 2]), ([[1, 2], 5, 4], [3, 4]), ([[0, 3], 5, 4], [3, 2])], [([[3, 0], 5, 4], [3, 3]), ([[1, 6, 1], 7, 2], [1, 3, 6]), ([[0, 3], 5, 4], [3, 2]), ([[10, 15, 12, 4, 14, 0, 16, 12], 17, 2], [15, 14, 9, 14, 4, 14, 0, 10]), ([[4, 3], 5, 4], [2, 1]), ([[4, 1], 5, 4], [0, 3]), ([[1, 0], 5, 4], [1, 1]), ([[3, 3], 5, 4], [1, 0])], [([[0, 2], 5, 4], [2, 3]), ([[3, 0, 2], 7, 2], [5, 4, 0]), ([[0, 3], 5, 4], [3, 2]), ([[10, 15, 12, 4, 14, 0, 16, 12], 17, 2], [15, 14, 9, 14, 4, 14, 0, 10]), ([[2, 5, 0], 7, 2], [0, 5, 1]), ([[0, 0, 4], 7, 2], [4, 2, 1]), ([[5, 5, 2], 7, 2], [5, 2, 1]), ([[1, 6, 1], 7, 2], [1, 3, 6])]]\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":"0c9c37d1dd491cd153433546c3b069fca664574a3ce18937e22c878f249e2c86","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    values,p,w=x;n=len(values);out=[]\n    for k in range(n):\n     total=0\n     for j in range(n):\n      factor=pow(w,j+k,p)\n      total=(total+values[j]*factor)%p\n     out.append(total)\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([[0, 3], 5, 4], [3, 2]), ([[0, 5, 4], 7, 2], [2, 5, 0]), ([[10, 15, 12, 4, 14, 0, 16, 12], 17, 2], [15, 14, 9, 14, 4, 14, 0, 10]), ([[2, 1], 5, 4], [3, 1]), ([[2, 1], 5, 4], [3, 1]), ([[3, 0], 5, 4], [3, 3]), ([[0, 2], 5, 4], [2, 3]), ([[1, 3], 5, 4], [4, 3])], [([[2, 1], 5, 4], [3, 1]), ([[4, 1, 3], 7, 2], [1, 4, 0]), ([[0, 3], 5, 4], [3, 2]), ([[10, 15, 12, 4, 14, 0, 16, 12], 17, 2], [15, 14, 9, 14, 4, 14, 0, 10]), ([[0, 2], 5, 4], [2, 3]), ([[1, 1], 5, 4], [2, 0]), ([[4, 3], 5, 4], [2, 1]), ([[2, 0], 5, 4], [2, 2])], [([[2, 1], 5, 4], [3, 1]), ([[2, 5, 0], 7, 2], [0, 5, 1]), ([[0, 3], 5, 4], [3, 2]), ([[10, 15, 12, 4, 14, 0, 16, 12], 17, 2], [15, 14, 9, 14, 4, 14, 0, 10]), ([[4, 0], 5, 4], [4, 4]), ([[1, 4], 5, 4], [0, 2]), ([[1, 2], 5, 4], [3, 4]), ([[0, 3], 5, 4], [3, 2])], [([[3, 0], 5, 4], [3, 3]), ([[1, 6, 1], 7, 2], [1, 3, 6]), ([[0, 3], 5, 4], [3, 2]), ([[10, 15, 12, 4, 14, 0, 16, 12], 17, 2], [15, 14, 9, 14, 4, 14, 0, 10]), ([[4, 3], 5, 4], [2, 1]), ([[4, 1], 5, 4], [0, 3]), ([[1, 0], 5, 4], [1, 1]), ([[3, 3], 5, 4], [1, 0])], [([[0, 2], 5, 4], [2, 3]), ([[3, 0, 2], 7, 2], [5, 4, 0]), ([[0, 3], 5, 4], [3, 2]), ([[10, 15, 12, 4, 14, 0, 16, 12], 17, 2], [15, 14, 9, 14, 4, 14, 0, 10]), ([[2, 5, 0], 7, 2], [0, 5, 1]), ([[0, 0, 4], 7, 2], [4, 2, 1]), ([[5, 5, 2], 7, 2], [5, 2, 1]), ([[1, 6, 1], 7, 2], [1, 3, 6])]]\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-finite-field-discrete-transform-character-exponent","generated_at":"2026-09-29T14:39:29.477995+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.","root_cause":"The character exponent step uses pow(w,j+k,p) instead of pow(w,j*k,p).","sha256":"fbf35ba4f17d68b28bb027f45b57756854998db58e63f7ae6b71d773254aacb5","title":"Finite field discrete transform: character exponent · 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":43.907,"exit_code":1,"observations":[{"actual":[3,2],"check":"explicit oracle 0","expected":[3,2],"passed":true},{"actual":[2,0,5],"check":"explicit oracle 1","expected":[2,5,0],"passed":false},{"actual":[15,10,0,14,4,14,9,14],"check":"explicit oracle 2","expected":[15,14,9,14,4,14,0,10],"passed":false},{"actual":[3,1],"check":"explicit oracle 3","expected":[3,1],"passed":true},{"actual":[3,1],"check":"explicit oracle 4","expected":[3,1],"passed":true},{"actual":[3,3],"check":"explicit oracle 5","expected":[3,3],"passed":true},{"actual":[2,3],"check":"explicit oracle 6","expected":[2,3],"passed":true},{"actual":[4,3],"check":"explicit oracle 7","expected":[4,3],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [3, 2], \"expected\": [3, 2], \"passed\": true}, {\"check\": \"explicit oracle 1\", \"actual\": [2, 0, 5], \"expected\": [2, 5, 0], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [15, 10, 0, 14, 4, 14, 9, 14], \"expected\": [15, 14, 9, 14, 4, 14, 0, 10], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [3, 1], \"expected\": [3, 1], \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": [3, 1], \"expected\": [3, 1], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [3, 3], \"expected\": [3, 3], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [2, 3], \"expected\": [2, 3], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [4, 3], \"expected\": [4, 3], \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":44.609,"exit_code":1,"observations":[{"actual":[2,3],"check":"explicit oracle 0","expected":[3,2],"passed":false},{"actual":[5,3,6],"check":"explicit oracle 1","expected":[2,5,0],"passed":false},{"actual":[14,11,5,10,3,6,12,7],"check":"explicit oracle 2","expected":[15,14,9,14,4,14,0,10],"passed":false},{"actual":[1,4],"check":"explicit oracle 3","expected":[3,1],"passed":false},{"actual":[1,4],"check":"explicit oracle 4","expected":[3,1],"passed":false},{"actual":[3,2],"check":"explicit oracle 5","expected":[3,3],"passed":false},{"actual":[3,2],"check":"explicit oracle 6","expected":[2,3],"passed":false},{"actual":[3,2],"check":"explicit oracle 7","expected":[4,3],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [2, 3], \"expected\": [3, 2], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [5, 3, 6], \"expected\": [2, 5, 0], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [14, 11, 5, 10, 3, 6, 12, 7], \"expected\": [15, 14, 9, 14, 4, 14, 0, 10], \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": [1, 4], \"expected\": [3, 1], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [1, 4], \"expected\": [3, 1], \"passed\": false}, {\"check\": \"explicit oracle 5\", \"actual\": [3, 2], \"expected\": [3, 3], \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": [3, 2], \"expected\": [2, 3], \"passed\": false}, {\"check\": \"explicit oracle 7\", \"actual\": [3, 2], \"expected\": [4, 3], \"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."}}