{"abstract":"The exact colex combination rank result violates the stated contract at binomial lower index.","category":"Numerics","checks":8,"contract":"Input strictly increasing selected nonnegative integers; zero-based colex rank=sum choose(a_i,i+1).","contract_signature":"x","evaluation_group":"s3-numerics-colex-combination-rank","failed_approach":"The partial repair len(a)-i still violates the binomial lower index invariant.","family":"s3-numerics-colex-combination-rank-binomial-lower-index","id":"FA-15151","implementations":{"attempt":{"sha256":"75035d13a1d2e26784575fbe9b8663fd0e72e46fbf8799de21d214d408e455d5","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=x;rank=0\n    for i in range(len(a)):\n     top=a[i];bottom=len(a)-i\n     term=math.comb(top,bottom) if 0<=bottom<=top else 0\n     rank=rank+term\n    return rank\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([0], 0), ([0, 1], 0), ([4, 5, 6, 7, 8], 125), ([0], 0), ([1], 1), ([0], 0), ([1], 1), ([2], 2)], [([0], 0), ([0, 1, 2], 0), ([0], 0), ([4, 5, 6, 7, 8], 125), ([0, 3], 3), ([1, 2], 2), ([1, 3], 4), ([2, 3], 5)], [([0, 1], 0), ([1, 3], 4), ([0], 0), ([4, 5, 6, 7, 8], 125), ([0, 4], 6), ([1, 2], 2), ([1, 3], 4), ([1, 4], 7)], [([0], 0), ([0, 1, 3], 1), ([0], 0), ([4, 5, 6, 7, 8], 125), ([0, 1, 2, 3], 0), ([0, 1, 2, 4], 1), ([0, 1, 3, 4], 2), ([0, 2, 3, 4], 3)], [([2], 2), ([0, 1, 2, 3], 0), ([0], 0), ([4, 5, 6, 7, 8], 125), ([1, 2], 2), ([1, 3], 4), ([1, 4], 7), ([1, 5], 11)]]\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":"0d4d87db1b05d10a7c7243df12289480bc7a56a1a8ae7254d91d9720f1858eed","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=x;rank=0\n    for i in range(len(a)):\n     top=a[i];bottom=i\n     term=math.comb(top,bottom) if 0<=bottom<=top else 0\n     rank=rank+term\n    return rank\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([0], 0), ([0, 1], 0), ([4, 5, 6, 7, 8], 125), ([0], 0), ([1], 1), ([0], 0), ([1], 1), ([2], 2)], [([0], 0), ([0, 1, 2], 0), ([0], 0), ([4, 5, 6, 7, 8], 125), ([0, 3], 3), ([1, 2], 2), ([1, 3], 4), ([2, 3], 5)], [([0, 1], 0), ([1, 3], 4), ([0], 0), ([4, 5, 6, 7, 8], 125), ([0, 4], 6), ([1, 2], 2), ([1, 3], 4), ([1, 4], 7)], [([0], 0), ([0, 1, 3], 1), ([0], 0), ([4, 5, 6, 7, 8], 125), ([0, 1, 2, 3], 0), ([0, 1, 2, 4], 1), ([0, 1, 3, 4], 2), ([0, 2, 3, 4], 3)], [([2], 2), ([0, 1, 2, 3], 0), ([0], 0), ([4, 5, 6, 7, 8], 125), ([1, 2], 2), ([1, 3], 4), ([1, 4], 7), ([1, 5], 11)]]\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-colex-combination-rank-binomial-lower-index","generated_at":"2026-09-29T14:39:23.925740+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 binomial lower index step uses i instead of i+1.","sha256":"8218ec17686fff0750f50c8fec1f2ff14fcbe94d8531b27feb95e225fe6fd324","title":"Colex combination rank: binomial lower index · 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":45.676,"exit_code":1,"observations":[{"actual":0,"check":"explicit oracle 0","expected":0,"passed":true},{"actual":1,"check":"explicit oracle 1","expected":0,"passed":false},{"actual":54,"check":"explicit oracle 2","expected":125,"passed":false},{"actual":0,"check":"explicit oracle 3","expected":0,"passed":true},{"actual":1,"check":"explicit oracle 4","expected":1,"passed":true},{"actual":0,"check":"explicit oracle 5","expected":0,"passed":true},{"actual":1,"check":"explicit oracle 6","expected":1,"passed":true},{"actual":2,"check":"explicit oracle 7","expected":2,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 1\", \"actual\": 1, \"expected\": 0, \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": 54, \"expected\": 125, \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": 2, \"expected\": 2, \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":43.329,"exit_code":1,"observations":[{"actual":1,"check":"explicit oracle 0","expected":0,"passed":false},{"actual":2,"check":"explicit oracle 1","expected":0,"passed":false},{"actual":126,"check":"explicit oracle 2","expected":125,"passed":false},{"actual":1,"check":"explicit oracle 3","expected":0,"passed":false},{"actual":1,"check":"explicit oracle 4","expected":1,"passed":true},{"actual":1,"check":"explicit oracle 5","expected":0,"passed":false},{"actual":1,"check":"explicit oracle 6","expected":1,"passed":true},{"actual":1,"check":"explicit oracle 7","expected":2,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": 1, \"expected\": 0, \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": 2, \"expected\": 0, \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": 126, \"expected\": 125, \"passed\": false}, {\"check\": \"explicit oracle 3\", \"actual\": 1, \"expected\": 0, \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": 1, \"expected\": 0, \"passed\": false}, {\"check\": \"explicit oracle 6\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": 1, \"expected\": 2, \"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."}}