{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0006", "pilot_number": "R0048-24606", "number": 24606, "family_id": "qf-mod44-ogf-coefficient-extraction", "title": { "zh-Hant": "第 24606 題：解析普通生成函數", "en": "Question 24606: Coefficient Extraction Uses the Declared Complete Zero Tail" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0006", "batch_id": "batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-analytic-ordinary-generating-function" ], "title_zh": "解析普通生成函數", "stem_zh": "A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[6,12,18,24],\"index\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([6, 12, 18, 24], a_n=0_after_the_list, 5, 0, coefficient_extraction_uses_the_complete_finite_support_law)", "explanation_zh": "Wave6第七層核對=([6, 12, 18, 24], a_n=0_after_the_list, 5, 0, coefficient_extraction_uses_the_complete_finite_support_law)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "解析普通生成函數" ] }, "variant": { "family_id": "qf-mod44-ogf-coefficient-extraction", "seed": 202611133006, "parameters": { "finite_support_coefficients": [ 6, 12, 18, 24 ], "index": 5, "domain": "Analytic_ordinary_generating_function", "operation": "ogf-coefficient-extraction", "semantic_template": "qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[6,1],[12,1],[18,1],[24,1]],\"index\":5}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-15", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0006", "batch_id": "batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "24b3132686220cb7a34fe37ad109e2571eb79dc57aad63defaddcbd94386093d", "family_id": "qf-mod44-ogf-coefficient-extraction", "seed": 202611133006, "semantic_template": "qf-mod44-ogf-coefficient-extraction/{\"finite_support_coefficients\":[[6,1],[12,1],[18,1],[24,1]],\"index\":5}" }, "fields": { "title": "Coefficient Extraction Uses the Declared Complete Zero Tail", "stem": "Use finite_support_coefficients=[6, 12, 18, 24] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=5. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).", "answer": { "kind": "expression", "value": "([6, 12, 18, 24], a_n=0_after_the_list, 5, 0, coefficient_extraction_uses_the_complete_finite_support_law)", "explanation": "Return the indexed coefficient or0 beyond support. Ordinary generating coefficients are a_n, not a_n/n! of an exponential generating function. A finite list with an explicit zero tail differs from an unknown continuation. The exact result, retaining the requested order, is ([6, 12, 18, 24], a_n=0_after_the_list, 5, 0, coefficient_extraction_uses_the_complete_finite_support_law)." }, "skills": [ "Coefficient Extraction Uses the Declared Complete Zero Tail" ], "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": "A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-coefficient-extraction；資料={\"finite_support_coefficients\":[6,12,18,24],\"index\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use finite_support_coefficients=[6, 12, 18, 24] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=5. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [ogf-coefficient-extraction]", "after": "derived = [[finite_support_coefficients, zero_tail_law, index, coefficient, certificate]]", "relation": "implies", "rule_id": "source_ordered_fields_without_JS", "conditions": [], "explanation": { "zh-Hant": "直接列來源完整欄序，不需JavaScript；原題、布林旗標、數學物件與非聲稱保持分層。", "en": "List the full source field order without JavaScript,keeping original data,Boolean flags,mathematical objects and nonclaims distinct." }, "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": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [[6, 12, 18, 24]]", "after": "derived = [[6, 12, 18, 24], a_n=0 for EVERY n beyond this finite list]", "relation": "implies", "rule_id": "OGF_complete_finite_support_zero_tail_NOT_unknown_prefix", "conditions": [], "explanation": { "zh-Hant": "只修canonical尾零，不改原參數；至少保留一個係數。這是整條sequence含零尾的定義，不是觀察到的有限prefix，也不是EGF的a_n/n!。", "en": "Trim only canonical trailing zeros without changing inputs,retaining at least one coefficient.This defines the entire sequence with its zero tail,not an observed prefix or the EGF coefficient a_n/n!." }, "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": "source roles = [[6, 12, 18, 24], 5]", "after": "derived = [0]", "relation": "implies", "rule_id": "OGF_exact_index_or_declared_zero_tail", "conditions": [], "explanation": { "zh-Hant": "索引在support內直接取a_k，超出則依已宣告全零尾取0。", "en": "Read a_k inside support;outside it use the declared complete zero tail." }, "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": [ [ "6", "12", "18", "24" ], "a_n=0_after_the_list", 5, "0", "coefficient_extraction_uses_the_complete_finite_support_law" ] }, "conclusion": { "canonical_value": "([6, 12, 18, 24], a_n=0_after_the_list, 5, 0, coefficient_extraction_uses_the_complete_finite_support_law)", "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-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations/questions.jsonl", "batch_sha256": "528d8bb148daa9519f9f78e207714bd01d13b8fc7e44ae1d64c36fc87228ff44", "original_line": 6, "original_line_sha256": "0958a7224bd5454f33a987b1640a1ab284470e57f15fc45b7f8827ec3196ff5c", "question_compact_sha256": "24b3132686220cb7a34fe37ad109e2571eb79dc57aad63defaddcbd94386093d", "english_row_compact_sha256": "6181ba6f71bd2d202fbb322bddbc4d4e90f1a24147ec820cc4b10dfc4ec605e1", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations.en.jsonl" }, "mko_bindings": { "mko-analytic-ordinary-generating-function": { "sha256": "bb78af651663a5b612aa987ab6c1ce997dd023e539de44fa7304c0b7363cb94c", "english_sha256": "a1f2435f6e237f8605cfd2cb349ded6e3b6d8d24b3bcfd2a84fb13e2862ff835", "original_path": "public/data/mko/analytic-ordinary-generating-function.json", "english_path": "public/data/i18n/editions/mko-analytic-ordinary-generating-function.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "7f1649723787c6d7cf1abe85636fd3740f2c3cc1784071db7e38069361c81fd4", "original_candidate_sha256": "38603da94e2a8e747aaf05ab68be4f25f3925f4b4975ede18078f7a5e2ecf7e3", "original_candidate_line_sha256": "38603da94e2a8e747aaf05ab68be4f25f3925f4b4975ede18078f7a5e2ecf7e3" }, "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-20261026-009-p49", "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": "049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0", "question_first_witness_sha256": "0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9" }, "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": "db972b02e01f1fa37fc209dec0c96d9afae6869decdf05728ffd6c0050d8581d", "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." } }