{"abstract":"The exact repeating radix fraction result violates the stated contract at remainder position.","category":"Numerics","checks":8,"contract":"Input [p,q,base], 0<=p<q, q>0, base2..10; return nonrepeating digit list and repeating digit list. Bounds: 2<=q<=19 and 2<=base<=8.","contract_signature":"x","evaluation_group":"s3-numerics-repeating-radix-fraction","failed_approach":"The partial repair max(0,len(digits)-1) still violates the remainder position invariant.","family":"s3-numerics-repeating-radix-fraction-remainder-position","id":"FA-14471","implementations":{"attempt":{"sha256":"1fc0542d7badb9bad3de62573527ac539046869d98fa0cd43c0416e6db8189bc","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    p,q,base=x\n    seen={};digits=[];r=p\n    for _ in range(100):\n     if r==0: return [digits,[]]\n     if r in seen:\n      k=seen[r]\n      return [digits[:k],digits[k:]]\n     seen[r]=max(0,len(digits)-1)\n     t=r*base\n     digits.append(t//q)\n     r=t%q\n    return [digits,[]]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([1, 2, 3], [[], [1]]), ([1, 6, 2], [[0], [0, 1]]), ([0, 2, 2], [[], []]), ([18, 19, 8], [[], [7, 4, 5, 0, 3, 2]]), ([0, 2, 3], [[], []]), ([0, 2, 4], [[], []]), ([0, 2, 5], [[], []]), ([0, 2, 6], [[], []])], [([1, 2, 5], [[], [2]]), ([1, 6, 8], [[1], [2, 5]]), ([0, 2, 2], [[], []]), ([18, 19, 8], [[], [7, 4, 5, 0, 3, 2]]), ([0, 3, 5], [[], []]), ([0, 3, 6], [[], []]), ([0, 3, 7], [[], []]), ([0, 3, 8], [[], []])], [([1, 2, 7], [[], [3]]), ([5, 6, 4], [[3], [1]]), ([0, 2, 2], [[], []]), ([18, 19, 8], [[], [7, 4, 5, 0, 3, 2]]), ([2, 3, 8], [[], [5, 2]]), ([0, 4, 2], [[], []]), ([0, 4, 3], [[], []]), ([0, 4, 4], [[], []])], [([1, 3, 2], [[], [0, 1]]), ([1, 10, 4], [[0], [1, 2]]), ([0, 2, 2], [[], []]), ([18, 19, 8], [[], [7, 4, 5, 0, 3, 2]]), ([2, 4, 4], [[2], []]), ([2, 4, 5], [[], [2]]), ([2, 4, 6], [[3], []]), ([2, 4, 7], [[], [3]])], [([1, 3, 4], [[], [1]]), ([1, 10, 8], [[0], [6, 3, 1, 4]]), ([0, 2, 2], [[], []]), ([18, 19, 8], [[], [7, 4, 5, 0, 3, 2]]), ([0, 5, 7], [[], []]), ([0, 5, 8], [[], []]), ([1, 5, 2], [[], [0, 0, 1, 1]]), ([1, 5, 3], [[], [0, 1, 2, 1]])]]\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":"f60bc1eec21285d0ddb58ab343ae38233d94e6d340628e73d3c023aa27fab461","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    p,q,base=x\n    seen={};digits=[];r=p\n    for _ in range(100):\n     if r==0: return [digits,[]]\n     if r in seen:\n      k=seen[r]\n      return [digits[:k],digits[k:]]\n     seen[r]=len(digits)+1\n     t=r*base\n     digits.append(t//q)\n     r=t%q\n    return [digits,[]]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([1, 2, 3], [[], [1]]), ([1, 6, 2], [[0], [0, 1]]), ([0, 2, 2], [[], []]), ([18, 19, 8], [[], [7, 4, 5, 0, 3, 2]]), ([0, 2, 3], [[], []]), ([0, 2, 4], [[], []]), ([0, 2, 5], [[], []]), ([0, 2, 6], [[], []])], [([1, 2, 5], [[], [2]]), ([1, 6, 8], [[1], [2, 5]]), ([0, 2, 2], [[], []]), ([18, 19, 8], [[], [7, 4, 5, 0, 3, 2]]), ([0, 3, 5], [[], []]), ([0, 3, 6], [[], []]), ([0, 3, 7], [[], []]), ([0, 3, 8], [[], []])], [([1, 2, 7], [[], [3]]), ([5, 6, 4], [[3], [1]]), ([0, 2, 2], [[], []]), ([18, 19, 8], [[], [7, 4, 5, 0, 3, 2]]), ([2, 3, 8], [[], [5, 2]]), ([0, 4, 2], [[], []]), ([0, 4, 3], [[], []]), ([0, 4, 4], [[], []])], [([1, 3, 2], [[], [0, 1]]), ([1, 10, 4], [[0], [1, 2]]), ([0, 2, 2], [[], []]), ([18, 19, 8], [[], [7, 4, 5, 0, 3, 2]]), ([2, 4, 4], [[2], []]), ([2, 4, 5], [[], [2]]), ([2, 4, 6], [[3], []]), ([2, 4, 7], [[], [3]])], [([1, 3, 4], [[], [1]]), ([1, 10, 8], [[0], [6, 3, 1, 4]]), ([0, 2, 2], [[], []]), ([18, 19, 8], [[], [7, 4, 5, 0, 3, 2]]), ([0, 5, 7], [[], []]), ([0, 5, 8], [[], []]), ([1, 5, 2], [[], [0, 0, 1, 1]]), ([1, 5, 3], [[], [0, 1, 2, 1]])]]\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-repeating-radix-fraction-remainder-position","generated_at":"2026-09-29T14:39:17.185044+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 remainder position step uses len(digits)+1 instead of len(digits).","sha256":"bd66071fdaec75502168769229b08153b65930453636ed3db176cdc965692327","title":"Repeating radix fraction: remainder position · 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":47.246,"exit_code":1,"observations":[{"actual":[[],[1]],"check":"explicit oracle 0","expected":[[],[1]],"passed":true},{"actual":[[],[0,0,1]],"check":"explicit oracle 1","expected":[[0],[0,1]],"passed":false},{"actual":[[],[]],"check":"explicit oracle 2","expected":[[],[]],"passed":true},{"actual":[[],[7,4,5,0,3,2]],"check":"explicit oracle 3","expected":[[],[7,4,5,0,3,2]],"passed":true},{"actual":[[],[]],"check":"explicit oracle 4","expected":[[],[]],"passed":true},{"actual":[[],[]],"check":"explicit oracle 5","expected":[[],[]],"passed":true},{"actual":[[],[]],"check":"explicit oracle 6","expected":[[],[]],"passed":true},{"actual":[[],[]],"check":"explicit oracle 7","expected":[[],[]],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [[], [1]], \"expected\": [[], [1]], \"passed\": true}, {\"check\": \"explicit oracle 1\", \"actual\": [[], [0, 0, 1]], \"expected\": [[0], [0, 1]], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [[], []], \"expected\": [[], []], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [[], [7, 4, 5, 0, 3, 2]], \"expected\": [[], [7, 4, 5, 0, 3, 2]], \"passed\": true}, {\"check\": \"explicit oracle 4\", \"actual\": [[], []], \"expected\": [[], []], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [[], []], \"expected\": [[], []], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [[], []], \"expected\": [[], []], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [[], []], \"expected\": [[], []], \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":44.487,"exit_code":1,"observations":[{"actual":[[1],[]],"check":"explicit oracle 0","expected":[[],[1]],"passed":false},{"actual":[[0,0],[1]],"check":"explicit oracle 1","expected":[[0],[0,1]],"passed":false},{"actual":[[],[]],"check":"explicit oracle 2","expected":[[],[]],"passed":true},{"actual":[[7],[4,5,0,3,2]],"check":"explicit oracle 3","expected":[[],[7,4,5,0,3,2]],"passed":false},{"actual":[[],[]],"check":"explicit oracle 4","expected":[[],[]],"passed":true},{"actual":[[],[]],"check":"explicit oracle 5","expected":[[],[]],"passed":true},{"actual":[[],[]],"check":"explicit oracle 6","expected":[[],[]],"passed":true},{"actual":[[],[]],"check":"explicit oracle 7","expected":[[],[]],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"explicit oracle 0\", \"actual\": [[1], []], \"expected\": [[], [1]], \"passed\": false}, {\"check\": \"explicit oracle 1\", \"actual\": [[0, 0], [1]], \"expected\": [[0], [0, 1]], \"passed\": false}, {\"check\": \"explicit oracle 2\", \"actual\": [[], []], \"expected\": [[], []], \"passed\": true}, {\"check\": \"explicit oracle 3\", \"actual\": [[7], [4, 5, 0, 3, 2]], \"expected\": [[], [7, 4, 5, 0, 3, 2]], \"passed\": false}, {\"check\": \"explicit oracle 4\", \"actual\": [[], []], \"expected\": [[], []], \"passed\": true}, {\"check\": \"explicit oracle 5\", \"actual\": [[], []], \"expected\": [[], []], \"passed\": true}, {\"check\": \"explicit oracle 6\", \"actual\": [[], []], \"expected\": [[], []], \"passed\": true}, {\"check\": \"explicit oracle 7\", \"actual\": [[], []], \"expected\": [[], []], \"passed\": true}], \"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."}}