{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261006-advance-20261007-r04-eigenvalue-0097", "pilot_number": "R0062-31522", "number": 31522, "family_id": "qf-eigenvalue-real-count", "title": { "zh-Hant": "第 31522 題：二階方陣的互異實特徵值數", "en": "Question 31522: Number of Distinct Real Eigenvalues of a 2×2 Matrix" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261006-advance-20261007-r04-eigenvalue-0097", "batch_id": "batch-20261006-advance-20261007-r04-eigenvalue-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-eigenvalue" ], "title_zh": "二階方陣的互異實特徵值數", "stem_zh": "實二階方陣，論域 R，M=[[-7, 1], [1, 7]]。只求互異實特徵值的個數 0、1 或2，不求小數根。", "answer": { "kind": "short_text", "value": "2", "explanation_zh": "特徵多項式係數=(1, 0, -50)；判別式=200。產生端由 (a−d)²+4bc 判別實根數；零判別式為一個相異實值，不聲稱可對角化或複數／Jordan 結論。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "二階方陣的互異實特徵值數" ] }, "variant": { "family_id": "qf-eigenvalue-real-count", "seed": 202610074097, "parameters": { "M": [ [ -7, 1 ], [ 1, 7 ] ], "operation": "real_eigenvalue_count", "domain": "reals", "semantic_template": "qf-eigenvalue-real-count/{\"inputs\":[\"0\",\"-50\"],\"operation\":\"real_eigenvalue_count\"}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_integer_rational_arithmetic", "producer_finite_enumeration", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-06", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261006-advance-20261007-r04-eigenvalue-0097", "batch_id": "batch-20261006-advance-20261007-r04-eigenvalue-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "9da74a3a4a3654ee3b6f053847bf401d3e843c38060d410218160a59acc64838", "family_id": "qf-eigenvalue-real-count", "seed": 202610074097, "semantic_template": "qf-eigenvalue-real-count/{\"inputs\":[\"0\",\"-50\"],\"operation\":\"real_eigenvalue_count\"}" }, "fields": { "title": "Number of Distinct Real Eigenvalues of a 2×2 Matrix", "stem": "Over R, the 2×2 matrix is M=[[-7, 1], [1, 7]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.", "answer": { "kind": "short_text", "value": "2", "explanation": "The characteristic polynomial coefficients are (1,0,-50), with discriminant Δ=200. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 2. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here." }, "skills": [ "Number of Distinct Real Eigenvalues of a 2×2 Matrix" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "實二階方陣，論域 R，M=[[-7, 1], [1, 7]]。只求互異實特徵值的個數 0、1 或2，不求小數根。", "en": "Over R, the 2×2 matrix is M=[[-7, 1], [1, 7]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-eigenvalue-real-count, reals, real_eigenvalue_count, no-form-field]", "after": "derived = [[distinct_real_count_STRING]]", "relation": "implies", "rule_id": "R62_EXACT_source_keys_domain_operation_form_AND_output_types", "conditions": [], "explanation": { "zh-Hant": "按原bank核欄位：BASE方程用form及論域，BASE事件／補集無operation，階乘無domain；ext18用local-hyphen，其他legacy用underscore。輸出型別不統一成tuple或fraction。", "en": "Check THIS bank interface:BASE equations use form/domain;BASE events/complement have no operation;factorials have no domain.ext18 uses local hyphens,other legacy operations underscores.Do not force every answer into a tuple or fraction." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。", "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Eigen count ORIGINAL diagonal trace := sum(-7, 7)", "after": "Eigen count ORIGINAL diagonal trace = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Eigen count ORIGINAL diagonal trace，不以小數近似替代。", "en": "Recompute Eigen count ORIGINAL diagonal trace with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-7", "7" ], "output": "0", "label": "Eigen count ORIGINAL diagonal trace" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Eigen count ORIGINAL ad := mul(-7, 7)", "after": "Eigen count ORIGINAL ad = -49", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Eigen count ORIGINAL ad，不以小數近似替代。", "en": "Recompute Eigen count ORIGINAL ad with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-7", "7" ], "output": "-49", "label": "Eigen count ORIGINAL ad" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Eigen count ORIGINAL bc := mul(1, 1)", "after": "Eigen count ORIGINAL bc = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Eigen count ORIGINAL bc，不以小數近似替代。", "en": "Recompute Eigen count ORIGINAL bc with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "Eigen count ORIGINAL bc" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Eigen count ORIGINAL ad minus bc := sub(-49, 1)", "after": "Eigen count ORIGINAL ad minus bc = -50", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Eigen count ORIGINAL ad minus bc，不以小數近似替代。", "en": "Recompute Eigen count ORIGINAL ad minus bc with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-49", "1" ], "output": "-50", "label": "Eigen count ORIGINAL ad minus bc" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Eigen count ORIGINAL negative trace := sub(0, 0)", "after": "Eigen count ORIGINAL negative trace = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Eigen count ORIGINAL negative trace，不以小數近似替代。", "en": "Recompute Eigen count ORIGINAL negative trace with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "0" ], "output": "0", "label": "Eigen count ORIGINAL negative trace" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Eigen count ORIGINAL trace squared := mul(0, 0)", "after": "Eigen count ORIGINAL trace squared = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Eigen count ORIGINAL trace squared，不以小數近似替代。", "en": "Recompute Eigen count ORIGINAL trace squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "Eigen count ORIGINAL trace squared" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Eigen count ORIGINAL four times determinant := mul(4, -50)", "after": "Eigen count ORIGINAL four times determinant = -200", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Eigen count ORIGINAL four times determinant，不以小數近似替代。", "en": "Recompute Eigen count ORIGINAL four times determinant with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "-50" ], "output": "-200", "label": "Eigen count ORIGINAL four times determinant" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Eigen count ORIGINAL characteristic discriminant := sub(0, -200)", "after": "Eigen count ORIGINAL characteristic discriminant = 200", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。", "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Eigen count ORIGINAL characteristic discriminant，不以小數近似替代。", "en": "Recompute Eigen count ORIGINAL characteristic discriminant with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "-200" ], "output": "200", "label": "Eigen count ORIGINAL characteristic discriminant" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "source roles = [[[-7, 1], [1, 7]]]", "after": "derived = [[1, 0, -50], 200]", "relation": "implies", "rule_id": "Eigen_COUNT_characteristic_polynomial_ALL_coefficients", "conditions": [], "explanation": { "zh-Hant": "由det(λI−M)=(λ−a)(λ−d)−bc得到λ²−(a+d)λ+(ad−bc)，再算判別式。係數與判別式都是衍生精確值，不重套SAFE。", "en": "Expand det(λI−M)=(λ−a)(λ−d)−bc to obtain λ²−(a+d)λ+(ad−bc),then its discriminant.Coefficients/discriminant are exact derived values without a RAW SAFE recap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。", "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate." } }, "arithmetic": null }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "source roles = [200]", "after": "derived = [2]", "relation": "implies", "rule_id": "Eigen_COUNT_DISTINCT_REAL_roots_by_sign_NOT_square_or_multiplicity", "conditions": [], "explanation": { "zh-Hant": "實係數二次式判別式負／零／正對應0／1／2個相異實根。正但非完全平方仍有2個實根；重根只算1，不推論可對角化、Jordan結構或複數譜。", "en": "A real quadratic has0/1/2 DISTINCT real roots for negative/zero/positive discriminant.Positive nonsquare values still give2;count a repeated root ONCE.No diagonalizability,Jordan or complex-spectrum claim follows." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。", "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate." } }, "arithmetic": null } ], "math_model": { "result": "2" }, "conclusion": { "canonical_value": "2", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261006-advance-20261007-r04-eigenvalue-foundations", "batch_path": "public/data/questions/batches/batch-20261006-advance-20261007-r04-eigenvalue-foundations/questions.jsonl", "batch_sha256": "d81d816856425974bbc3c6986818908fe3b9c0210bc8ab9f4c988e781227b637", "original_line": 97, "original_line_sha256": "78e5b693b92f5573628ca997020a5dccf81a208e2aea7a180a1044cc04656372", "question_compact_sha256": "9da74a3a4a3654ee3b6f053847bf401d3e843c38060d410218160a59acc64838", "english_row_compact_sha256": "7ec2aeabc32dabb8ca83b8c37b990ca344d9ec67e876587340dd49de566743f4", "english_manifest_path": "public/data/i18n/editions/batch-20261006-advance-20261007-r04-eigenvalue-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261006-advance-20261007-r04-eigenvalue-foundations.en.jsonl" }, "mko_bindings": { "mko-eigenvalue": { "sha256": "5e1e1186120be48f1e86332bf1d679143e44074e2b3e9f59c5c219cb76a99e36", "english_sha256": "e48363ac57349147d31e8427dc2281ab02ae34f1abaea817c89e3e0f431fe5ed", "original_path": "public/data/mko/eigenvalue.json", "english_path": "public/data/i18n/editions/mko-eigenvalue.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "59d384ca3d958b27165f4295ff96efb155e7b12d3fb70bcdd0ed68148513c05a", "original_candidate_sha256": "12ad9e8cb2b83c6c3189acc74108feb70cab21fe3d014b388f1d1c33ae43a713", "original_candidate_line_sha256": "12ad9e8cb2b83c6c3189acc74108feb70cab21fe3d014b388f1d1c33ae43a713" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-011-p63", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "963099ffc9f754069fdce6792949a85ed350472465eaceb3038fa8011cf672bd", "question_first_witness_sha256": "d7655324c5d11528e1af90b757384cebd3313e19dc5f6f5b9b8ea7624fb06f6f" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 11, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "aa86099942602f4fc1bcb4d40ec76aa6302a784c2857cba5a4767eefc77a4a0a", "arithmetic_steps": 8 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }