{"abstract":"The augmented triad [0,4,8] returns [8,0,4] and the diminished seventh starts on 9.","category":"Music interval and transposition theory","checks":8,"contract":"Input a list of integers (reduced mod 12, duplicates removed). Return the normal form: among all rotations of the ascending set, choose the one with the smallest span first-to-last, then smallest span first-to-penultimate, and so on toward the second element; a full tie picks the rotation starting on the lowest pitch class. Empty input returns []; non-integer input returns None.","evaluation_group":"w2-music-interval-normal-form","failed_approach":"Breaking ties on the last reduced member prefers rotations ending on 0, which again starts symmetric sets high.","family":"w2-music-interval-normal-form-full-tie-break","id":"FA-81036","implementations":{"attempt":{"sha256":"bbd2696b88ca1e2b380a48f8643b02f4e3c8f0673d1a5f22fa564e8c987c6bc0","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\n\nN = 1\nobservations = []\ndef solve(x):\n    if not isinstance(x, list) or not all(isinstance(v, int) for v in x):\n        return None\n    pcs = sorted(set(v % 12 for v in x))\n    if not pcs:\n        return []\n    n = len(pcs)\n    best = None\n    for r in range(n):\n        rot = pcs[r:] + [p + 12 for p in pcs[:r]]\n        key = [rot[-1] - rot[0]] + [rot[j] - rot[0] for j in range(n - 2, 0, -1)]\n        cand = (key, rot[-1] % 12)\n        if best is None or cand < best[0]:\n            best = (cand, [p % 12 for p in rot])\n    return best[1]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([0, 4, 7], [0, 4, 7]), ([7, 4, 0], [0, 4, 7]), ([11, 0, 4], [11, 0, 4]), ([0, 1, 5, 6, 8], [0, 1, 5, 6, 8]), ([0, 1, 3, 5, 8, 9], [8, 9, 0, 1, 3, 5]), ([0, 4, 8], [0, 4, 8]), ([0, 6], [0, 6]), ([0, 2, 4, 6, 8, 10], [0, 2, 4, 6, 8, 10])], [([0, 1, 5, 6, 8], [0, 1, 5, 6, 8]), ([0, 1, 3, 5, 8, 9], [8, 9, 0, 1, 3, 5]), ([0, 2, 3, 6, 7, 9], [0, 2, 3, 6, 7, 9]), ([0, 1, 2, 5, 6, 7, 9], [0, 1, 2, 5, 6, 7, 9]), ([0, 1, 2, 4, 5, 7, 9, 10], [9, 10, 0, 1, 2, 4, 5, 7]), ([0, 3, 6, 9], [0, 3, 6, 9]), ([0, 2, 4, 6, 8, 10], [0, 2, 4, 6, 8, 10]), ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11])], [([0, 1, 2, 5, 6, 7, 9], [0, 1, 2, 5, 6, 7, 9]), ([0, 1, 2, 4, 5, 7, 9, 10], [9, 10, 0, 1, 2, 4, 5, 7]), ([0, 6], [0, 6]), ([9, 0, 4, 4, 16], [9, 0, 4]), ([2, 5, 9], [2, 5, 9]), ([10, 2, 5], [10, 2, 5]), ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]), ([0, 1, 6, 7], [0, 1, 6, 7])], [([2, 5, 9], [2, 5, 9]), ([10, 2, 5], [10, 2, 5]), ([1, 5, 8], [1, 5, 8]), ([11, 2, 6], [11, 2, 6]), ([0, 1, 4, 6], [0, 1, 4, 6]), ([0, 2, 4, 6, 8, 10], [0, 2, 4, 6, 8, 10]), ([0, 1, 6, 7], [0, 1, 6, 7]), ([1, 4, 7, 10, 1], [1, 4, 7, 10])], [([11, 2, 6], [11, 2, 6]), ([0, 1, 4, 6], [0, 1, 4, 6]), ([0, 1, 3, 7], [0, 1, 3, 7]), ([5], [5]), ([], []), ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]), ([1, 4, 7, 10, 1], [1, 4, 7, 10]), ([0, 1, 4, 5, 8, 9], [0, 1, 4, 5, 8, 9])]]\nfor i, (args, expected) in enumerate(fixtures[N-1]):\n    check(\"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":"1590962b96f647e82d0a9efe7148e0fd27d003e32f98b7fc38d84008888fb455","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\n\nN = 1\nobservations = []\ndef solve(x):\n    if not isinstance(x, list) or not all(isinstance(v, int) for v in x):\n        return None\n    pcs = sorted(set(v % 12 for v in x))\n    if not pcs:\n        return []\n    n = len(pcs)\n    best = None\n    for r in range(n):\n        rot = pcs[r:] + [p + 12 for p in pcs[:r]]\n        key = [rot[-1] - rot[0]] + [rot[j] - rot[0] for j in range(n - 2, 0, -1)]\n        cand = (key, -rot[0])\n        if best is None or cand < best[0]:\n            best = (cand, [p % 12 for p in rot])\n    return best[1]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([0, 4, 7], [0, 4, 7]), ([7, 4, 0], [0, 4, 7]), ([11, 0, 4], [11, 0, 4]), ([0, 1, 5, 6, 8], [0, 1, 5, 6, 8]), ([0, 1, 3, 5, 8, 9], [8, 9, 0, 1, 3, 5]), ([0, 4, 8], [0, 4, 8]), ([0, 6], [0, 6]), ([0, 2, 4, 6, 8, 10], [0, 2, 4, 6, 8, 10])], [([0, 1, 5, 6, 8], [0, 1, 5, 6, 8]), ([0, 1, 3, 5, 8, 9], [8, 9, 0, 1, 3, 5]), ([0, 2, 3, 6, 7, 9], [0, 2, 3, 6, 7, 9]), ([0, 1, 2, 5, 6, 7, 9], [0, 1, 2, 5, 6, 7, 9]), ([0, 1, 2, 4, 5, 7, 9, 10], [9, 10, 0, 1, 2, 4, 5, 7]), ([0, 3, 6, 9], [0, 3, 6, 9]), ([0, 2, 4, 6, 8, 10], [0, 2, 4, 6, 8, 10]), ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11])], [([0, 1, 2, 5, 6, 7, 9], [0, 1, 2, 5, 6, 7, 9]), ([0, 1, 2, 4, 5, 7, 9, 10], [9, 10, 0, 1, 2, 4, 5, 7]), ([0, 6], [0, 6]), ([9, 0, 4, 4, 16], [9, 0, 4]), ([2, 5, 9], [2, 5, 9]), ([10, 2, 5], [10, 2, 5]), ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]), ([0, 1, 6, 7], [0, 1, 6, 7])], [([2, 5, 9], [2, 5, 9]), ([10, 2, 5], [10, 2, 5]), ([1, 5, 8], [1, 5, 8]), ([11, 2, 6], [11, 2, 6]), ([0, 1, 4, 6], [0, 1, 4, 6]), ([0, 2, 4, 6, 8, 10], [0, 2, 4, 6, 8, 10]), ([0, 1, 6, 7], [0, 1, 6, 7]), ([1, 4, 7, 10, 1], [1, 4, 7, 10])], [([11, 2, 6], [11, 2, 6]), ([0, 1, 4, 6], [0, 1, 4, 6]), ([0, 1, 3, 7], [0, 1, 3, 7]), ([5], [5]), ([], []), ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]), ([1, 4, 7, 10, 1], [1, 4, 7, 10]), ([0, 1, 4, 5, 8, 9], [0, 1, 4, 5, 8, 9])]]\nfor i, (args, expected) in enumerate(fixtures[N-1]):\n    check(\"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"},"fixed":{"sha256":"869def97b20cdaa6ae5d737ee8fb818b5c7fa548c3f5f5593e5d9424f39bebbf","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\n\nN = 1\nobservations = []\ndef solve(x):\n    if not isinstance(x, list) or not all(isinstance(v, int) for v in x):\n        return None\n    pcs = sorted(set(v % 12 for v in x))\n    if not pcs:\n        return []\n    n = len(pcs)\n    best = None\n    for r in range(n):\n        rot = pcs[r:] + [p + 12 for p in pcs[:r]]\n        key = [rot[-1] - rot[0]] + [rot[j] - rot[0] for j in range(n - 2, 0, -1)]\n        cand = (key, rot[0])\n        if best is None or cand < best[0]:\n            best = (cand, [p % 12 for p in rot])\n    return best[1]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([0, 4, 7], [0, 4, 7]), ([7, 4, 0], [0, 4, 7]), ([11, 0, 4], [11, 0, 4]), ([0, 1, 5, 6, 8], [0, 1, 5, 6, 8]), ([0, 1, 3, 5, 8, 9], [8, 9, 0, 1, 3, 5]), ([0, 4, 8], [0, 4, 8]), ([0, 6], [0, 6]), ([0, 2, 4, 6, 8, 10], [0, 2, 4, 6, 8, 10])], [([0, 1, 5, 6, 8], [0, 1, 5, 6, 8]), ([0, 1, 3, 5, 8, 9], [8, 9, 0, 1, 3, 5]), ([0, 2, 3, 6, 7, 9], [0, 2, 3, 6, 7, 9]), ([0, 1, 2, 5, 6, 7, 9], [0, 1, 2, 5, 6, 7, 9]), ([0, 1, 2, 4, 5, 7, 9, 10], [9, 10, 0, 1, 2, 4, 5, 7]), ([0, 3, 6, 9], [0, 3, 6, 9]), ([0, 2, 4, 6, 8, 10], [0, 2, 4, 6, 8, 10]), ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11])], [([0, 1, 2, 5, 6, 7, 9], [0, 1, 2, 5, 6, 7, 9]), ([0, 1, 2, 4, 5, 7, 9, 10], [9, 10, 0, 1, 2, 4, 5, 7]), ([0, 6], [0, 6]), ([9, 0, 4, 4, 16], [9, 0, 4]), ([2, 5, 9], [2, 5, 9]), ([10, 2, 5], [10, 2, 5]), ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]), ([0, 1, 6, 7], [0, 1, 6, 7])], [([2, 5, 9], [2, 5, 9]), ([10, 2, 5], [10, 2, 5]), ([1, 5, 8], [1, 5, 8]), ([11, 2, 6], [11, 2, 6]), ([0, 1, 4, 6], [0, 1, 4, 6]), ([0, 2, 4, 6, 8, 10], [0, 2, 4, 6, 8, 10]), ([0, 1, 6, 7], [0, 1, 6, 7]), ([1, 4, 7, 10, 1], [1, 4, 7, 10])], [([11, 2, 6], [11, 2, 6]), ([0, 1, 4, 6], [0, 1, 4, 6]), ([0, 1, 3, 7], [0, 1, 3, 7]), ([5], [5]), ([], []), ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]), ([1, 4, 7, 10, 1], [1, 4, 7, 10]), ([0, 1, 4, 5, 8, 9], [0, 1, 4, 5, 8, 9])]]\nfor i, (args, expected) in enumerate(fixtures[N-1]):\n    check(\"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 with a stipulated toy contract; it is not a complete music notation or theory engine. 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":"w2-music-interval-normal-form-full-tie-break","generated_at":"2026-09-29T14:49:59.295913+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"Post-tonal analysis tools use normal form to compare and catalogue pitch-class sets.","repair":"Restore the full tie break step so that it reads `cand = (key, rot[0])`.","root_cause":"The final tie-break prefers the largest starting pitch class.","sha256":"a89588b2559df7015fa92d530c00c3557fda6257426982c263e08c8a658ccefa","title":"Pitch-class set normal form (right-packed): symmetric sets start on the highest pitch class · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":42.395,"exit_code":1,"observations":[{"actual":[0,4,7],"check":"oracle 0","expected":[0,4,7],"passed":true},{"actual":[0,4,7],"check":"oracle 1","expected":[0,4,7],"passed":true},{"actual":[11,0,4],"check":"oracle 2","expected":[11,0,4],"passed":true},{"actual":[0,1,5,6,8],"check":"oracle 3","expected":[0,1,5,6,8],"passed":true},{"actual":[8,9,0,1,3,5],"check":"oracle 4","expected":[8,9,0,1,3,5],"passed":true},{"actual":[4,8,0],"check":"oracle 5","expected":[0,4,8],"passed":false},{"actual":[6,0],"check":"oracle 6","expected":[0,6],"passed":false},{"actual":[2,4,6,8,10,0],"check":"oracle 7","expected":[0,2,4,6,8,10],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"oracle 0\", \"actual\": [0, 4, 7], \"expected\": [0, 4, 7], \"passed\": true}, {\"check\": \"oracle 1\", \"actual\": [0, 4, 7], \"expected\": [0, 4, 7], \"passed\": true}, {\"check\": \"oracle 2\", \"actual\": [11, 0, 4], \"expected\": [11, 0, 4], \"passed\": true}, {\"check\": \"oracle 3\", \"actual\": [0, 1, 5, 6, 8], \"expected\": [0, 1, 5, 6, 8], \"passed\": true}, {\"check\": \"oracle 4\", \"actual\": [8, 9, 0, 1, 3, 5], \"expected\": [8, 9, 0, 1, 3, 5], \"passed\": true}, {\"check\": \"oracle 5\", \"actual\": [4, 8, 0], \"expected\": [0, 4, 8], \"passed\": false}, {\"check\": \"oracle 6\", \"actual\": [6, 0], \"expected\": [0, 6], \"passed\": false}, {\"check\": \"oracle 7\", \"actual\": [2, 4, 6, 8, 10, 0], \"expected\": [0, 2, 4, 6, 8, 10], \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":40.807,"exit_code":1,"observations":[{"actual":[0,4,7],"check":"oracle 0","expected":[0,4,7],"passed":true},{"actual":[0,4,7],"check":"oracle 1","expected":[0,4,7],"passed":true},{"actual":[11,0,4],"check":"oracle 2","expected":[11,0,4],"passed":true},{"actual":[0,1,5,6,8],"check":"oracle 3","expected":[0,1,5,6,8],"passed":true},{"actual":[8,9,0,1,3,5],"check":"oracle 4","expected":[8,9,0,1,3,5],"passed":true},{"actual":[8,0,4],"check":"oracle 5","expected":[0,4,8],"passed":false},{"actual":[6,0],"check":"oracle 6","expected":[0,6],"passed":false},{"actual":[10,0,2,4,6,8],"check":"oracle 7","expected":[0,2,4,6,8,10],"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"oracle 0\", \"actual\": [0, 4, 7], \"expected\": [0, 4, 7], \"passed\": true}, {\"check\": \"oracle 1\", \"actual\": [0, 4, 7], \"expected\": [0, 4, 7], \"passed\": true}, {\"check\": \"oracle 2\", \"actual\": [11, 0, 4], \"expected\": [11, 0, 4], \"passed\": true}, {\"check\": \"oracle 3\", \"actual\": [0, 1, 5, 6, 8], \"expected\": [0, 1, 5, 6, 8], \"passed\": true}, {\"check\": \"oracle 4\", \"actual\": [8, 9, 0, 1, 3, 5], \"expected\": [8, 9, 0, 1, 3, 5], \"passed\": true}, {\"check\": \"oracle 5\", \"actual\": [8, 0, 4], \"expected\": [0, 4, 8], \"passed\": false}, {\"check\": \"oracle 6\", \"actual\": [6, 0], \"expected\": [0, 6], \"passed\": false}, {\"check\": \"oracle 7\", \"actual\": [10, 0, 2, 4, 6, 8], \"expected\": [0, 2, 4, 6, 8, 10], \"passed\": false}], \"passed\": false}\n"},"fixed":{"elapsed_ms":41.332,"exit_code":0,"observations":[{"actual":[0,4,7],"check":"oracle 0","expected":[0,4,7],"passed":true},{"actual":[0,4,7],"check":"oracle 1","expected":[0,4,7],"passed":true},{"actual":[11,0,4],"check":"oracle 2","expected":[11,0,4],"passed":true},{"actual":[0,1,5,6,8],"check":"oracle 3","expected":[0,1,5,6,8],"passed":true},{"actual":[8,9,0,1,3,5],"check":"oracle 4","expected":[8,9,0,1,3,5],"passed":true},{"actual":[0,4,8],"check":"oracle 5","expected":[0,4,8],"passed":true},{"actual":[0,6],"check":"oracle 6","expected":[0,6],"passed":true},{"actual":[0,2,4,6,8,10],"check":"oracle 7","expected":[0,2,4,6,8,10],"passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"oracle 0\", \"actual\": [0, 4, 7], \"expected\": [0, 4, 7], \"passed\": true}, {\"check\": \"oracle 1\", \"actual\": [0, 4, 7], \"expected\": [0, 4, 7], \"passed\": true}, {\"check\": \"oracle 2\", \"actual\": [11, 0, 4], \"expected\": [11, 0, 4], \"passed\": true}, {\"check\": \"oracle 3\", \"actual\": [0, 1, 5, 6, 8], \"expected\": [0, 1, 5, 6, 8], \"passed\": true}, {\"check\": \"oracle 4\", \"actual\": [8, 9, 0, 1, 3, 5], \"expected\": [8, 9, 0, 1, 3, 5], \"passed\": true}, {\"check\": \"oracle 5\", \"actual\": [0, 4, 8], \"expected\": [0, 4, 8], \"passed\": true}, {\"check\": \"oracle 6\", \"actual\": [0, 6], \"expected\": [0, 6], \"passed\": true}, {\"check\": \"oracle 7\", \"actual\": [0, 2, 4, 6, 8, 10], \"expected\": [0, 2, 4, 6, 8, 10], \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}