{"abstract":"Dividing all words by the length mishandles constant words with smaller rotation orbits.","category":"Combinatorics","checks":4,"contract":"Nonnegative integer counts. Permutation and combination counts are zero when a requested selection exceeds the available objects. Shape sizes satisfy the preconditions stated below. length is prime and colors>=1; count colorings of a labelled-position cycle up to rotations, retaining reflection distinctions. Exact operational definition: (colors**length+(length-1)*colors)//length","contract_signature":"length, colors","evaluation_group":"model-e301f2e6def4c5e0","failed_approach":"Rotationally equivalent colorings are counted separately.","family":"xn-prime-length-necklace-colorings","id":"FA-5626","implementations":{"attempt":{"sha256":"822e3ef8d431bcb2e2e665d1f900b4a5fe5dd32003dd6b59309b8be5a656345a","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nimport calendar\nimport statistics\nimport itertools\nfrom fractions import Fraction\nfrom datetime import date, datetime, timedelta, timezone\nfrom decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR\n\nN = 1\nobservations = []\ndef solve(length, colors):\n    return colors**length\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (3, 2)', solve(*(3, 2)), 4)\ncheck('fixture 2: (5, 2)', solve(*(5, 2)), 8)\ncheck('fixture 3: (2, 3)', solve(*(2, 3)), 6)\ncheck('fixture 4: (3, 1)', solve(*(3, 1)), 1)\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":"919cf139f81d820ecb217792c5f2560cb81782f581051d15f77b7b51bef591a3","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nimport calendar\nimport statistics\nimport itertools\nfrom fractions import Fraction\nfrom datetime import date, datetime, timedelta, timezone\nfrom decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR\n\nN = 1\nobservations = []\ndef solve(length, colors):\n    return colors**length//length\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (3, 2)', solve(*(3, 2)), 4)\ncheck('fixture 2: (5, 2)', solve(*(5, 2)), 8)\ncheck('fixture 3: (2, 3)', solve(*(2, 3)), 6)\ncheck('fixture 4: (3, 1)', solve(*(3, 1)), 1)\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":" 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":"xn-prime-length-necklace-colorings","generated_at":"2026-09-29T14:37:51.774206+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"Small exact fixtures expose this error without platform timing, external services, or probabilistic observations. Combinatorics results depend on the stated convention.","root_cause":"Dividing all words by the length mishandles constant words with smaller rotation orbits.","sha256":"546935f29573b5749b20a8b147a42c662c4538ce540cce95294e4240876c5322","title":"Prime length necklace colorings · 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":51.212,"exit_code":1,"observations":[{"actual":8,"check":"fixture 1: (3, 2)","expected":4,"passed":false},{"actual":32,"check":"fixture 2: (5, 2)","expected":8,"passed":false},{"actual":9,"check":"fixture 3: (2, 3)","expected":6,"passed":false},{"actual":1,"check":"fixture 4: (3, 1)","expected":1,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (3, 2)\", \"actual\": 8, \"expected\": 4, \"passed\": false}, {\"check\": \"fixture 2: (5, 2)\", \"actual\": 32, \"expected\": 8, \"passed\": false}, {\"check\": \"fixture 3: (2, 3)\", \"actual\": 9, \"expected\": 6, \"passed\": false}, {\"check\": \"fixture 4: (3, 1)\", \"actual\": 1, \"expected\": 1, \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":50.154,"exit_code":1,"observations":[{"actual":2,"check":"fixture 1: (3, 2)","expected":4,"passed":false},{"actual":6,"check":"fixture 2: (5, 2)","expected":8,"passed":false},{"actual":4,"check":"fixture 3: (2, 3)","expected":6,"passed":false},{"actual":0,"check":"fixture 4: (3, 1)","expected":1,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (3, 2)\", \"actual\": 2, \"expected\": 4, \"passed\": false}, {\"check\": \"fixture 2: (5, 2)\", \"actual\": 6, \"expected\": 8, \"passed\": false}, {\"check\": \"fixture 3: (2, 3)\", \"actual\": 4, \"expected\": 6, \"passed\": false}, {\"check\": \"fixture 4: (3, 1)\", \"actual\": 0, \"expected\": 1, \"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."}}