{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261031-r04-finite-quotient-ring-0007", "pilot_number": "R0030-15932", "number": 15932, "family_id": "qf-mod31-quotring-classes", "title": { "zh-Hant": "第 15932 題：有限商環", "en": "Question 15932: Build the Quotient from Full Additive Cosets" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261031-r04-finite-quotient-ring-0007", "batch_id": "batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-finite-quotient-ring" ], "title_zh": "有限商環", "stem_zh": "Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":9,\"d\":3}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。", "answer": { "kind": "expression", "value": "([0, 3, 6], [[0, 3, 6], [1, 4, 7], [2, 5, 8]], 3, proper_two_sided_ideal_in_commutative_Zn)", "explanation_zh": "Wave4收束核對=([0, 3, 6], [[0, 3, 6], [1, 4, 7], [2, 5, 8]], 3, proper_two_sided_ideal_in_commutative_Zn)；依完整明示資料重算；加法與乘法皆須與代表元選擇無關；排除I=R的zero ring且不宣稱商環為域。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "有限商環" ] }, "variant": { "family_id": "qf-mod31-quotring-classes", "seed": 202610314007, "parameters": { "n": 9, "d": 3, "domain": "Exact_quotient_of_Zn_by_proper_ideal", "operation": "quotring-classes", "semantic_template": "qf-mod31-quotring-classes/{\"d\":3,\"n\":9}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261031-r04-finite-quotient-ring-0007", "batch_id": "batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "7ce6feaba03c2e52a8148b03ef1740a7d2a7189c9359ce06d61ccc18a984a1a0", "family_id": "qf-mod31-quotring-classes", "seed": 202610314007, "semantic_template": "qf-mod31-quotring-classes/{\"d\":3,\"n\":9}" }, "fields": { "title": "Build the Quotient from Full Additive Cosets", "stem": "Use the parent ring Z_9 with residues 0 through 8 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn).", "answer": { "kind": "expression", "value": "([0, 3, 6], [[0, 3, 6], [1, 4, 7], [2, 5, 8]], 3, proper_two_sided_ideal_in_commutative_Zn)", "explanation": "List all multiples of d as the ideal and then all residue classes modulo d. The quotient elements are full cosets, not a selected representative list. The exact result, retaining the requested order, is ([0, 3, 6], [[0, 3, 6], [1, 4, 7], [2, 5, 8]], 3, proper_two_sided_ideal_in_commutative_Zn)." }, "skills": [ "Build the Quotient from Full Additive Cosets" ], "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": "Z_n與proper雙邊理想的全部加法cosets。操作=quotring-classes；資料={\"n\":9,\"d\":3}。請依 [ideal,cosets,quotient_size,certificate] 欄序回答。", "en": "Use the parent ring Z_9 with residues 0 through 8 and the proper two-sided ideal I of multiples of d=3, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Give (ideal,cosets,quotient_size,proper_two_sided_ideal_in_commutative_Zn)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "quotient ring parent modulus divided by ideal generator := div(9, 3)", "after": "quotient ring parent modulus divided by ideal generator = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確計算，保留索引、符號、分母與運算所在載體。", "en": "Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 quotient ring parent modulus divided by ideal generator，不以小數近似替代。", "en": "Recompute quotient ring parent modulus divided by ideal generator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "9", "3" ], "output": "3", "label": "quotient ring parent modulus divided by ideal generator" } }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "same declared parent Z_9;generator d=3;n/d=3;ideal multiples=[0, 3, 6]", "after": "I=kernel of residue map Z_9 to Z_3;proper two-sided ideal=是;zero ideal allowed when d=n", "relation": "implies", "rule_id": "quotient_ring_actual_parent_proper_two_sided_ideal", "conditions": [], "explanation": { "zh-Hant": "因d整除n，整數代表改加n仍不改mod d，π:Z_n→Z_d良定且保加乘。核I由d倍數組成，加法差封閉且左右乘任意母環元素仍在I。d≥2使1不在I，故proper；d=n可得零理想，不把它排除。", "en": "Since d divides n,changing an integer representative by n does not change mod d.The map π:Z_n→Z_d is well-defined and preserves addition and multiplication.Its kernel consists of multiples of d,closed under differences and multiplication on either side by parent elements.With d≥2 it excludes1 and is proper;d=n gives the allowed zero ideal." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "complete parent residues=[0, 1, 2, 3, 4, 5, 6, 7, 8];same ideal I=[0, 3, 6]", "after": "all additive cosets in representative order0..d-1=[[0, 3, 6], [1, 4, 7], [2, 5, 8]];quotient size=3;no field conclusion", "relation": "implies", "rule_id": "quotient_ring_full_cosets_not_representative_carrier", "conditions": [], "explanation": { "zh-Hant": "每個加法coset列全體同餘mod d的母環元素，按代表0..d−1排序；它們互斥且覆蓋Z_n。商元素是整類，輸出class標籤只是索引，不因有限商環就宣稱域。", "en": "List every parent element congruent mod d in each additive coset,ordered by representatives0 through d−1.They are disjoint and cover Z_n.Quotient elements are whole cosets;class labels are indices only,and a finite quotient ring is not automatically a field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ [ 0, 3, 6 ], [ [ 0, 3, 6 ], [ 1, 4, 7 ], [ 2, 5, 8 ] ], 3, "proper_two_sided_ideal_in_commutative_Zn" ] }, "conclusion": { "canonical_value": "([0, 3, 6], [[0, 3, 6], [1, 4, 7], [2, 5, 8]], 3, proper_two_sided_ideal_in_commutative_Zn)", "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-20261014-advance-20261031-r04-finite-quotient-ring-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations/questions.jsonl", "batch_sha256": "990159f2eda0c7cd420834e0b7b4b5b70e6cde01e33924c6340b2b0b90b0d4f8", "original_line": 7, "original_line_sha256": "519777b94a57e542709f18fc287c2fb39824f7111a15804181b40f6fb5515baa", "question_compact_sha256": "7ce6feaba03c2e52a8148b03ef1740a7d2a7189c9359ce06d61ccc18a984a1a0", "english_row_compact_sha256": "13b0c7ee39d7afc54d5e70c25ca020f0fa9c1fbf4bf43a521b5fbd4f493b3d78", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261031-r04-finite-quotient-ring-foundations.en.jsonl" }, "mko_bindings": { "mko-finite-quotient-ring": { "sha256": "8c2b45b38428b422d01b869c06eaca3a0db6e14e6fafc691ef583d4614d39216", "english_sha256": "b144945c75f4a276442029de263e7ef036043832247b0b951049af0cc80623fa", "original_path": "public/data/mko/finite-quotient-ring.json", "english_path": "public/data/i18n/editions/mko-finite-quotient-ring.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "d85118d74859ec7e59736714f767732c6ec092eaa0bf5279abeb28e4a628e3c3", "original_candidate_sha256": "696cd6775125459b4d85f44bee14cd180028652fd21dc61562649f7d5468b55a", "original_candidate_line_sha256": "696cd6775125459b4d85f44bee14cd180028652fd21dc61562649f7d5468b55a" }, "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-20261026-007-p31", "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": "73bc08066ffb1a81eacb2d4e288d03f522aeefd3949fa0d97da41b484ed737bc", "question_first_witness_sha256": "ac120452a208b257b8270f38351a09dcd832cdbc8d9db6b69933f83e43cc9336" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 3, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "9f58e208ad84828642b489006c320a63d36219fcbe63f6689d4a5b97fc4b9ee4", "arithmetic_steps": 1 }, "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." } }