{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261006-advance-20261007-r04-eigenvalue-0035", "pilot_number": "R0061-31460", "number": 31460, "family_id": "qf-eigenvalue-diagonal-set", "title": { "zh-Hant": "第 31460 題：對角矩陣互異實特徵值集合", "en": "Question 31460: Distinct Real Eigenvalues of a Diagonal Matrix" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261006-advance-20261007-r04-eigenvalue-0035", "batch_id": "batch-20261006-advance-20261007-r04-eigenvalue-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-eigenvalue" ], "title_zh": "對角矩陣互異實特徵值集合", "stem_zh": "實二階方陣，論域 R，diag(-3,10)。求互異實特徵值集合，遞增列出且重複值只列一次。", "answer": { "kind": "expression", "value": "{-3, 10}", "explanation_zh": "互異特徵值={-3, 10}。對角兩值都列入，若相同只留下單元素；不混用代數重數。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "對角矩陣互異實特徵值集合" ] }, "variant": { "family_id": "qf-eigenvalue-diagonal-set", "seed": 202610074035, "parameters": { "a": -3, "d": 10, "operation": "diagonal_eigenvalues", "domain": "reals", "semantic_template": "qf-eigenvalue-diagonal-set/{\"inputs\":[\"-3\",\"10\"],\"operation\":\"diagonal_eigenvalues\"}" } }, "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-0035", "batch_id": "batch-20261006-advance-20261007-r04-eigenvalue-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "1afa72952785c2bd1a5d445fee3ed33f803cfe9981c01c5f447a7811e6fe3bf1", "family_id": "qf-eigenvalue-diagonal-set", "seed": 202610074035, "semantic_template": "qf-eigenvalue-diagonal-set/{\"inputs\":[\"-3\",\"10\"],\"operation\":\"diagonal_eigenvalues\"}" }, "fields": { "title": "Distinct Real Eigenvalues of a Diagonal Matrix", "stem": "Over R, the 2×2 matrix is diag(-3,10). Give its distinct real eigenvalues as an increasing set with each value once.", "answer": { "kind": "expression", "value": "{-3, 10}", "explanation": "The two diagonal values give {-3, 10}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count." }, "skills": [ "Distinct Real Eigenvalues of a Diagonal 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，diag(-3,10)。求互異實特徵值集合，遞增列出且重複值只列一次。", "en": "Over R, the 2×2 matrix is diag(-3,10). Give its distinct real eigenvalues as an increasing set with each value once." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-eigenvalue-diagonal-set, diagonal_eigenvalues, domain, reals]", "after": "derived = [[distinct_numeric_sorted_set]]", "relation": "implies", "rule_id": "R61_SOURCE_exact_keys_operation_domain_OR_frame_and_output_roles", "conditions": [], "explanation": { "zh-Hant": "先核本題欄位與來源協議：distance只用完整座標frame，其他用domain；離散運算local-hyphen，舊bank用underscore。輸出欄序及型別依題族，保留原始中英資料與狀態。", "en": "Check THIS source interface:distance uses ONLY its complete coordinate frame;other families use a domain.Discrete operations use local hyphens,legacy banks underscores.Retain family-specific ordered output types and original bilingual data/status." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。", "en": "Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [-3, 10]", "after": "derived = [[-3, 10], {-3, 10}]", "relation": "implies", "rule_id": "Eigen_DIAGONAL_characteristic_factors_DISTINCT_numeric_set", "conditions": [], "explanation": { "zh-Hant": "diag(a,d)的特徵多項式為(a−λ)(d−λ)，根為a、d。這裡要數值排序的相異集合，重根只留一次，0不代表空集合；不是代數重數列表。", "en": "For diag(a,d),the characteristic factors are(a−λ)(d−λ),so the roots are a,d.Return the DISTINCT numerically sorted set,deduplicating equal roots.Zero is not an empty set;this is not a multiplicity list." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。", "en": "Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof." } }, "arithmetic": null } ], "math_model": { "result": "{-3, 10}" }, "conclusion": { "canonical_value": "{-3, 10}", "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": 35, "original_line_sha256": "231c13f9a9137fd6d6b18942362d994456c4336527a71c07ee0c0fd804225d7c", "question_compact_sha256": "1afa72952785c2bd1a5d445fee3ed33f803cfe9981c01c5f447a7811e6fe3bf1", "english_row_compact_sha256": "0fb182bd5094425802656b65eaa76e88358a8f6b5ae281d7d4fa12ab2a117198", "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": "912db26eb6c90aa6e256f4356cb5ee94cd0a352f04f667876ae6739f8181a17c", "original_candidate_sha256": "af894e0e7a382cebd4b6c96ccbee66ac79b6c3b565e580a2684889581f7d8453", "original_candidate_line_sha256": "af894e0e7a382cebd4b6c96ccbee66ac79b6c3b565e580a2684889581f7d8453" }, "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-p62", "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": "5403317467ff7c6b74c18ae7211530d0156ebc67d63fc00a3e07b81a201d3d69", "question_first_witness_sha256": "c2b5022aa77f803cd3e1ba9ce162a707da0281aa5bb5289111a97de51fe8c495" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 2, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "232514ec23b09bbc95ed14aa9ac674d5f2365be5a59b4def80d7c800f73248cc", "arithmetic_steps": 0 }, "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." } }