{"abstract":"The reduction disagrees with its explicit aggregation oracle.","category":"Numerical aggregation","checks":8,"contract":"For paired ordinal observations return sign(tau_b)*tau_b**2 as an exact Fraction string. Every unordered observation pair contributes its coordinate comparison; normalize signed concordance squared by products of non-tied x/y pair counts. Undefined if either coordinate has no varying pairs.","evaluation_group":"s3-na-tie-adjusted-concordance-squared","failed_approach":"Zero association is also a measured value, not the undefined normalization sentinel.","family":"s3-numerical-aggregation-tie-adjusted-concordance-squared-concordance-degenerate-value","id":"FA-13756","implementations":{"attempt":{"sha256":"b32997c9d13adfd59700b77617cd3d9e36308aad8fff17f17d9a1785e09dfa6d","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nfrom fractions import Fraction\nfrom collections import Counter, defaultdict\nimport math\nimport itertools\nN = 1\nobservations = []\ndef solve(points):\n    concordant=discordant=tx=ty=0\n    for i in range(len(points)):\n        for j in range(i+1,len(points)):\n            dx=points[i][0]-points[j][0]\n            dy=points[i][1]-points[j][1]\n            if dx: tx+=1\n            if dy: ty+=1\n            concordant+=int(dx*dy>0)\n            discordant+=int(dx*dy<0)\n    if not tx or not ty: return \"0\"\n    d=concordant-discordant\n    return str(Fraction((1 if d>=0 else -1)*d*d,tx*ty))\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('regression 1', solve(*([(1, 1), (2, 2), (3, 3)],)), '1')\ncheck('regression 2', solve(*([(1, 3), (2, 2), (3, 1)],)), '-1')\ncheck('regression 3', solve(*([(1, 1), (1, 2), (2, 2), (3, 1)],)), '-1/20')\ncheck('regression 4', solve(*([],)), None)\ncheck('regression 5', solve(*([(1, 1), (1, 2)],)), None)\ncheck('regression 6', solve(*([(1, 1), (2, 2), (2, 2), (4, 3)],)), '1')\ncheck('regression 7', solve(*([(1, 2), (3, 1), (4, 4), (2, 5)],)), '0')\ncheck(\"variable discordance direction\",solve([(0,N),(N,0)]),\"-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":"fa3a18caa7829452290c39e118f63b0a66a1dd0e031c60b617a534b5cb9d9b3d","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nfrom fractions import Fraction\nfrom collections import Counter, defaultdict\nimport math\nimport itertools\nN = 1\nobservations = []\ndef solve(points):\n    concordant=discordant=tx=ty=0\n    for i in range(len(points)):\n        for j in range(i+1,len(points)):\n            dx=points[i][0]-points[j][0]\n            dy=points[i][1]-points[j][1]\n            if dx: tx+=1\n            if dy: ty+=1\n            concordant+=int(dx*dy>0)\n            discordant+=int(dx*dy<0)\n    if not tx or not ty: return \"1\"\n    d=concordant-discordant\n    return str(Fraction((1 if d>=0 else -1)*d*d,tx*ty))\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('regression 1', solve(*([(1, 1), (2, 2), (3, 3)],)), '1')\ncheck('regression 2', solve(*([(1, 3), (2, 2), (3, 1)],)), '-1')\ncheck('regression 3', solve(*([(1, 1), (1, 2), (2, 2), (3, 1)],)), '-1/20')\ncheck('regression 4', solve(*([],)), None)\ncheck('regression 5', solve(*([(1, 1), (1, 2)],)), None)\ncheck('regression 6', solve(*([(1, 1), (2, 2), (2, 2), (4, 3)],)), '1')\ncheck('regression 7', solve(*([(1, 2), (3, 1), (4, 4), (2, 5)],)), '0')\ncheck(\"variable discordance direction\",solve([(0,N),(N,0)]),\"-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"},"fixed":{"sha256":"4af632310cc2bf180cf06298cff0fd8a24e43365131b2d9f51d804bc0b9d6e7b","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nfrom fractions import Fraction\nfrom collections import Counter, defaultdict\nimport math\nimport itertools\nN = 1\nobservations = []\ndef solve(points):\n    concordant=discordant=tx=ty=0\n    for i in range(len(points)):\n        for j in range(i+1,len(points)):\n            dx=points[i][0]-points[j][0]\n            dy=points[i][1]-points[j][1]\n            if dx: tx+=1\n            if dy: ty+=1\n            concordant+=int(dx*dy>0)\n            discordant+=int(dx*dy<0)\n    if not tx or not ty: return None\n    d=concordant-discordant\n    return str(Fraction((1 if d>=0 else -1)*d*d,tx*ty))\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('regression 1', solve(*([(1, 1), (2, 2), (3, 3)],)), '1')\ncheck('regression 2', solve(*([(1, 3), (2, 2), (3, 1)],)), '-1')\ncheck('regression 3', solve(*([(1, 1), (1, 2), (2, 2), (3, 1)],)), '-1/20')\ncheck('regression 4', solve(*([],)), None)\ncheck('regression 5', solve(*([(1, 1), (1, 2)],)), None)\ncheck('regression 6', solve(*([(1, 1), (2, 2), (2, 2), (4, 3)],)), '1')\ncheck('regression 7', solve(*([(1, 2), (3, 1), (4, 4), (2, 5)],)), '0')\ncheck(\"variable discordance direction\",solve([(0,N),(N,0)]),\"-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":"Small offline integer/rational inputs only; no performance, statistical inference, or production-library conformance claim. 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-numerical-aggregation-tie-adjusted-concordance-squared-concordance-degenerate-value","generated_at":"2026-09-29T14:39:10.314103+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"Exact bounded examples isolate a reduction defect without floating-point or external-service effects.","repair":"Preserve the tie adjusted concordance squared contract at the identified reduction decision.","root_cause":"A constant coordinate is assigned a measured perfect association.","sha256":"68670f71088722baeed0a85d6f89a0af1f86c2fe1e01098da9485912f4cd60a3","title":"Tie adjusted concordance squared: A constant coordinate is assigned a measured perfect association. · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":44.744,"exit_code":1,"observations":[{"actual":"1","check":"regression 1","expected":"1","passed":true},{"actual":"-1","check":"regression 2","expected":"-1","passed":true},{"actual":"-1/20","check":"regression 3","expected":"-1/20","passed":true},{"actual":"0","check":"regression 4","expected":null,"passed":false},{"actual":"0","check":"regression 5","expected":null,"passed":false},{"actual":"1","check":"regression 6","expected":"1","passed":true},{"actual":"0","check":"regression 7","expected":"0","passed":true},{"actual":"-1","check":"variable discordance direction","expected":"-1","passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression 1\", \"actual\": \"1\", \"expected\": \"1\", \"passed\": true}, {\"check\": \"regression 2\", \"actual\": \"-1\", \"expected\": \"-1\", \"passed\": true}, {\"check\": \"regression 3\", \"actual\": \"-1/20\", \"expected\": \"-1/20\", \"passed\": true}, {\"check\": \"regression 4\", \"actual\": \"0\", \"expected\": null, \"passed\": false}, {\"check\": \"regression 5\", \"actual\": \"0\", \"expected\": null, \"passed\": false}, {\"check\": \"regression 6\", \"actual\": \"1\", \"expected\": \"1\", \"passed\": true}, {\"check\": \"regression 7\", \"actual\": \"0\", \"expected\": \"0\", \"passed\": true}, {\"check\": \"variable discordance direction\", \"actual\": \"-1\", \"expected\": \"-1\", \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":42.218,"exit_code":1,"observations":[{"actual":"1","check":"regression 1","expected":"1","passed":true},{"actual":"-1","check":"regression 2","expected":"-1","passed":true},{"actual":"-1/20","check":"regression 3","expected":"-1/20","passed":true},{"actual":"1","check":"regression 4","expected":null,"passed":false},{"actual":"1","check":"regression 5","expected":null,"passed":false},{"actual":"1","check":"regression 6","expected":"1","passed":true},{"actual":"0","check":"regression 7","expected":"0","passed":true},{"actual":"-1","check":"variable discordance direction","expected":"-1","passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression 1\", \"actual\": \"1\", \"expected\": \"1\", \"passed\": true}, {\"check\": \"regression 2\", \"actual\": \"-1\", \"expected\": \"-1\", \"passed\": true}, {\"check\": \"regression 3\", \"actual\": \"-1/20\", \"expected\": \"-1/20\", \"passed\": true}, {\"check\": \"regression 4\", \"actual\": \"1\", \"expected\": null, \"passed\": false}, {\"check\": \"regression 5\", \"actual\": \"1\", \"expected\": null, \"passed\": false}, {\"check\": \"regression 6\", \"actual\": \"1\", \"expected\": \"1\", \"passed\": true}, {\"check\": \"regression 7\", \"actual\": \"0\", \"expected\": \"0\", \"passed\": true}, {\"check\": \"variable discordance direction\", \"actual\": \"-1\", \"expected\": \"-1\", \"passed\": true}], \"passed\": false}\n"},"fixed":{"elapsed_ms":44.762,"exit_code":0,"observations":[{"actual":"1","check":"regression 1","expected":"1","passed":true},{"actual":"-1","check":"regression 2","expected":"-1","passed":true},{"actual":"-1/20","check":"regression 3","expected":"-1/20","passed":true},{"actual":null,"check":"regression 4","expected":null,"passed":true},{"actual":null,"check":"regression 5","expected":null,"passed":true},{"actual":"1","check":"regression 6","expected":"1","passed":true},{"actual":"0","check":"regression 7","expected":"0","passed":true},{"actual":"-1","check":"variable discordance direction","expected":"-1","passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression 1\", \"actual\": \"1\", \"expected\": \"1\", \"passed\": true}, {\"check\": \"regression 2\", \"actual\": \"-1\", \"expected\": \"-1\", \"passed\": true}, {\"check\": \"regression 3\", \"actual\": \"-1/20\", \"expected\": \"-1/20\", \"passed\": true}, {\"check\": \"regression 4\", \"actual\": null, \"expected\": null, \"passed\": true}, {\"check\": \"regression 5\", \"actual\": null, \"expected\": null, \"passed\": true}, {\"check\": \"regression 6\", \"actual\": \"1\", \"expected\": \"1\", \"passed\": true}, {\"check\": \"regression 7\", \"actual\": \"0\", \"expected\": \"0\", \"passed\": true}, {\"check\": \"variable discordance direction\", \"actual\": \"-1\", \"expected\": \"-1\", \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}