{"abstract":"The borrowed flat-six chord in C major is labelled bV instead of bVI.","category":"Music interval and transposition theory","checks":8,"contract":"Input [tonic_pc, mode, root_pc, quality] with mode major or minor (natural minor scale 0 2 3 5 7 8 10) and quality maj, min, dim or aug. Diatonic roots use their degree; in minor the leading tone (11 semitones) is degree VII with no prefix. Other roots in major are written b + the degree a semitone above; in minor # + the degree a semitone below. Numerals are uppercase for maj and aug, lowercase for min and dim, with suffix ° for dim and + for aug. Invalid input returns None.","contract_signature":"x","evaluation_group":"w2-music-interval-roman-numeral","failed_approach":"Switching to a sharp prefix with the lower degree labels bVI as #V, contradicting the major-key flat convention.","family":"w2-music-interval-roman-numeral-major-chromatic-reference","id":"FA-81186","implementations":{"attempt":{"sha256":"4f25fe921161d159b3c2390f0e0982a47339788eeee67e64e127920087926e95","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 len(x) != 4:\n        return None\n    tonic, mode, root, quality = x\n    scales = {'major': [0, 2, 4, 5, 7, 9, 11], 'minor': [0, 2, 3, 5, 7, 8, 10]}\n    if mode not in scales or quality not in ('maj', 'min', 'dim', 'aug'):\n        return None\n    if not all(isinstance(v, int) for v in (tonic, root)):\n        return None\n    rel = (root - tonic) % 12\n    scale = scales[mode]\n    names = ['I', 'II', 'III', 'IV', 'V', 'VI', 'VII']\n    prefix = ''\n    if rel in scale:\n        deg = scale.index(rel)\n    elif mode == 'minor' and rel == 11:\n        deg = 6\n    elif mode == 'major':\n        prefix = '#'\n        deg = scale.index((rel - 1) % 12)\n    else:\n        prefix = '#'\n        deg = scale.index((rel - 1) % 12)\n    numeral = names[deg] if quality in ('maj', 'aug') else names[deg].lower()\n    suffix = {'maj': '', 'min': '', 'dim': '°', 'aug': '+'}[quality]\n    return prefix + numeral + suffix\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([0, 'major', 7, 'maj'], 'V'), ([0, 'major', 2, 'min'], 'ii'), ([0, 'major', 11, 'dim'], 'vii°'), ([0, 'major', 8, 'maj'], 'bVI'), ([0, 'major', 1, 'maj'], 'bII'), ([0, 'major', 3, 'maj'], 'bIII'), ([9, 'minor', 4, 'maj'], 'V'), ([9, 'minor', 8, 'dim'], 'vii°')], [([0, 'major', 10, 'maj'], 'bVII'), ([0, 'major', 3, 'maj'], 'bIII'), ([0, 'major', 6, 'dim'], 'bv°'), ([9, 'minor', 4, 'maj'], 'V'), ([9, 'minor', 8, 'dim'], 'vii°'), ([9, 'minor', 0, 'maj'], 'III'), ([9, 'minor', 7, 'maj'], 'VII'), ([9, 'minor', 6, 'min'], '#vi')], [([0, 'major', 8, 'maj'], 'bVI'), ([0, 'major', 1, 'maj'], 'bII'), ([0, 'major', 6, 'dim'], 'bv°'), ([9, 'minor', 7, 'maj'], 'VII'), ([9, 'minor', 6, 'min'], '#vi'), ([9, 'minor', 1, 'maj'], '#III'), ([9, 'minor', 3, 'dim'], '#iv°'), ([0, 'minor', 11, 'dim'], 'vii°')], [([0, 'major', 8, 'maj'], 'bVI'), ([0, 'major', 10, 'maj'], 'bVII'), ([0, 'major', 3, 'maj'], 'bIII'), ([9, 'minor', 3, 'dim'], '#iv°'), ([0, 'minor', 11, 'dim'], 'vii°'), ([0, 'minor', 9, 'min'], '#vi'), ([0, 'minor', 4, 'maj'], '#III'), ([0, 'minor', 3, 'aug'], 'III+')], [([0, 'major', 10, 'maj'], 'bVII'), ([0, 'major', 1, 'maj'], 'bII'), ([0, 'major', 6, 'dim'], 'bv°'), ([0, 'minor', 4, 'maj'], '#III'), ([0, 'minor', 3, 'aug'], 'III+'), ([7, 'major', 2, 'maj'], 'V'), ([7, 'major', 0, 'maj'], 'IV'), ([2, 'major', 1, 'dim'], 'vii°')]]\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":"7054939575c4affdab33bbab113a80d1aa0194c178fc0a4d235e748f2376f4a6","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 len(x) != 4:\n        return None\n    tonic, mode, root, quality = x\n    scales = {'major': [0, 2, 4, 5, 7, 9, 11], 'minor': [0, 2, 3, 5, 7, 8, 10]}\n    if mode not in scales or quality not in ('maj', 'min', 'dim', 'aug'):\n        return None\n    if not all(isinstance(v, int) for v in (tonic, root)):\n        return None\n    rel = (root - tonic) % 12\n    scale = scales[mode]\n    names = ['I', 'II', 'III', 'IV', 'V', 'VI', 'VII']\n    prefix = ''\n    if rel in scale:\n        deg = scale.index(rel)\n    elif mode == 'minor' and rel == 11:\n        deg = 6\n    elif mode == 'major':\n        prefix = 'b'\n        deg = scale.index((rel - 1) % 12)\n    else:\n        prefix = '#'\n        deg = scale.index((rel - 1) % 12)\n    numeral = names[deg] if quality in ('maj', 'aug') else names[deg].lower()\n    suffix = {'maj': '', 'min': '', 'dim': '°', 'aug': '+'}[quality]\n    return prefix + numeral + suffix\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[([0, 'major', 7, 'maj'], 'V'), ([0, 'major', 2, 'min'], 'ii'), ([0, 'major', 11, 'dim'], 'vii°'), ([0, 'major', 8, 'maj'], 'bVI'), ([0, 'major', 1, 'maj'], 'bII'), ([0, 'major', 3, 'maj'], 'bIII'), ([9, 'minor', 4, 'maj'], 'V'), ([9, 'minor', 8, 'dim'], 'vii°')], [([0, 'major', 10, 'maj'], 'bVII'), ([0, 'major', 3, 'maj'], 'bIII'), ([0, 'major', 6, 'dim'], 'bv°'), ([9, 'minor', 4, 'maj'], 'V'), ([9, 'minor', 8, 'dim'], 'vii°'), ([9, 'minor', 0, 'maj'], 'III'), ([9, 'minor', 7, 'maj'], 'VII'), ([9, 'minor', 6, 'min'], '#vi')], [([0, 'major', 8, 'maj'], 'bVI'), ([0, 'major', 1, 'maj'], 'bII'), ([0, 'major', 6, 'dim'], 'bv°'), ([9, 'minor', 7, 'maj'], 'VII'), ([9, 'minor', 6, 'min'], '#vi'), ([9, 'minor', 1, 'maj'], '#III'), ([9, 'minor', 3, 'dim'], '#iv°'), ([0, 'minor', 11, 'dim'], 'vii°')], [([0, 'major', 8, 'maj'], 'bVI'), ([0, 'major', 10, 'maj'], 'bVII'), ([0, 'major', 3, 'maj'], 'bIII'), ([9, 'minor', 3, 'dim'], '#iv°'), ([0, 'minor', 11, 'dim'], 'vii°'), ([0, 'minor', 9, 'min'], '#vi'), ([0, 'minor', 4, 'maj'], '#III'), ([0, 'minor', 3, 'aug'], 'III+')], [([0, 'major', 10, 'maj'], 'bVII'), ([0, 'major', 1, 'maj'], 'bII'), ([0, 'major', 6, 'dim'], 'bv°'), ([0, 'minor', 4, 'maj'], '#III'), ([0, 'minor', 3, 'aug'], 'III+'), ([7, 'major', 2, 'maj'], 'V'), ([7, 'major', 0, 'maj'], 'IV'), ([2, 'major', 1, 'dim'], 'vii°')]]\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-roman-numeral-major-chromatic-reference","generated_at":"2026-09-29T14:50:00.609072+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"Harmonic analysis and ear-training tools label chords relative to the key.","root_cause":"Non-diatonic major-key roots are referenced to the degree a semitone below while still using the flat prefix.","sha256":"a9ae2b1889b5f330b7fef0341d9c2efacf30757977f7c2a424f1530d034dab03","title":"Roman numeral label for a chord in a key: flat prefix attached to the degree below · 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":39.971,"exit_code":1,"observations":[{"actual":"V","check":"oracle 0","expected":"V","passed":true},{"actual":"ii","check":"oracle 1","expected":"ii","passed":true},{"actual":"vii°","check":"oracle 2","expected":"vii°","passed":true},{"actual":"#V","check":"oracle 3","expected":"bVI","passed":false},{"actual":"#I","check":"oracle 4","expected":"bII","passed":false},{"actual":"#II","check":"oracle 5","expected":"bIII","passed":false},{"actual":"V","check":"oracle 6","expected":"V","passed":true},{"actual":"vii°","check":"oracle 7","expected":"vii°","passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"oracle 0\", \"actual\": \"V\", \"expected\": \"V\", \"passed\": true}, {\"check\": \"oracle 1\", \"actual\": \"ii\", \"expected\": \"ii\", \"passed\": true}, {\"check\": \"oracle 2\", \"actual\": \"vii°\", \"expected\": \"vii°\", \"passed\": true}, {\"check\": \"oracle 3\", \"actual\": \"#V\", \"expected\": \"bVI\", \"passed\": false}, {\"check\": \"oracle 4\", \"actual\": \"#I\", \"expected\": \"bII\", \"passed\": false}, {\"check\": \"oracle 5\", \"actual\": \"#II\", \"expected\": \"bIII\", \"passed\": false}, {\"check\": \"oracle 6\", \"actual\": \"V\", \"expected\": \"V\", \"passed\": true}, {\"check\": \"oracle 7\", \"actual\": \"vii°\", \"expected\": \"vii°\", \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":40.153,"exit_code":1,"observations":[{"actual":"V","check":"oracle 0","expected":"V","passed":true},{"actual":"ii","check":"oracle 1","expected":"ii","passed":true},{"actual":"vii°","check":"oracle 2","expected":"vii°","passed":true},{"actual":"bV","check":"oracle 3","expected":"bVI","passed":false},{"actual":"bI","check":"oracle 4","expected":"bII","passed":false},{"actual":"bII","check":"oracle 5","expected":"bIII","passed":false},{"actual":"V","check":"oracle 6","expected":"V","passed":true},{"actual":"vii°","check":"oracle 7","expected":"vii°","passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"oracle 0\", \"actual\": \"V\", \"expected\": \"V\", \"passed\": true}, {\"check\": \"oracle 1\", \"actual\": \"ii\", \"expected\": \"ii\", \"passed\": true}, {\"check\": \"oracle 2\", \"actual\": \"vii°\", \"expected\": \"vii°\", \"passed\": true}, {\"check\": \"oracle 3\", \"actual\": \"bV\", \"expected\": \"bVI\", \"passed\": false}, {\"check\": \"oracle 4\", \"actual\": \"bI\", \"expected\": \"bII\", \"passed\": false}, {\"check\": \"oracle 5\", \"actual\": \"bII\", \"expected\": \"bIII\", \"passed\": false}, {\"check\": \"oracle 6\", \"actual\": \"V\", \"expected\": \"V\", \"passed\": true}, {\"check\": \"oracle 7\", \"actual\": \"vii°\", \"expected\": \"vii°\", \"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."}}