{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0047", "pilot_number": "R0048-24647", "number": 24647, "family_id": "qf-mod44-ogf-finite-partial-sum", "title": { "zh-Hant": "第 24647 題：解析普通生成函數", "en": "Question 24647: Support Coverage Is Separate from Accidental Equality at One Point" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0047", "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-finite-partial-sum；資料={\"finite_support_coefficients\":[22,44,66,88],\"x\":[1,2],\"cutoff\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([22, 44, 66, 88], 1/2, 2, [[0, 22, 22], [1, 44, 22], [2, 66, 33/2]], 121/2, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)", "explanation_zh": "Wave6第七層核對=([22, 44, 66, 88], 1/2, 2, [[0, 22, 22], [1, 44, 22], [2, 66, 33/2]], 121/2, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)；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": 3, "skills": [ "解析普通生成函數" ] }, "variant": { "family_id": "qf-mod44-ogf-finite-partial-sum", "seed": 202611133047, "parameters": { "finite_support_coefficients": [ 22, 44, 66, 88 ], "x": [ 1, 2 ], "cutoff": 2, "domain": "Analytic_ordinary_generating_function", "operation": "ogf-finite-partial-sum", "semantic_template": "qf-mod44-ogf-finite-partial-sum/{\"cutoff\":2,\"finite_support_coefficients\":[[22,1],[44,1],[66,1],[88,1]],\"x\":[1,2]}" } }, "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-0047", "batch_id": "batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "944d3bab3a28f833597dda32b014409fc12e2ed6ffdd0063bc4e33066012fd12", "family_id": "qf-mod44-ogf-finite-partial-sum", "seed": 202611133047, "semantic_template": "qf-mod44-ogf-finite-partial-sum/{\"cutoff\":2,\"finite_support_coefficients\":[[22,1],[44,1],[66,1],[88,1]],\"x\":[1,2]}" }, "fields": { "title": "Support Coverage Is Separate from Accidental Equality at One Point", "stem": "Use finite_support_coefficients=[22, 44, 66, 88] 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. Use x=1/2, cutoff=2. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).", "answer": { "kind": "expression", "value": "([22, 44, 66, 88], 1/2, 2, [[0, 22, 22], [1, 44, 22], [2, 66, 33/2]], 121/2, No, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)", "explanation": "List [n,a_n,a_n*x^n] through the cutoff and sum exactly. complete_support_included means every nonzero-support coefficient is included; it is not a pointwise-equality test. At x=0 or through cancellation a shorter sum may have the same value without being the whole polynomial identity. The exact result, retaining the requested order, is ([22, 44, 66, 88], 1/2, 2, [[0, 22, 22], [1, 44, 22], [2, 66, 33/2]], 121/2, No, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)." }, "skills": [ "Support Coverage Is Separate from Accidental Equality at One Point" ], "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-finite-partial-sum；資料={\"finite_support_coefficients\":[22,44,66,88],\"x\":[1,2],\"cutoff\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use finite_support_coefficients=[22, 44, 66, 88] 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. Use x=1/2, cutoff=2. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [ogf-finite-partial-sum]", "after": "derived = [[finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, 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 = [[22, 44, 66, 88]]", "after": "derived = [[22, 44, 66, 88], 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": "(1/2)^0", "after": "OGF ORIGINAL x power0 = 1", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。", "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity." }, "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": { "operation": "integer_power", "inputs": [ "1/2", "0" ], "output": "1", "label": "OGF ORIGINAL x power0" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "OGF term0 coefficient times power := mul(22, 1)", "after": "OGF term0 coefficient times power = 22", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，分母先核非零，不用浮點近似。", "en": "Substitute original roles and established intermediates exactly,checking nonzero denominators without floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 OGF term0 coefficient times power，不以小數近似替代。", "en": "Recompute OGF term0 coefficient times power with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "22", "1" ], "output": "22", "label": "OGF term0 coefficient times power" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "OGF partial sum through0 := sum(0, 22)", "after": "OGF partial sum through0 = 22", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，分母先核非零，不用浮點近似。", "en": "Substitute original roles and established intermediates exactly,checking nonzero denominators without floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 OGF partial sum through0，不以小數近似替代。", "en": "Recompute OGF partial sum through0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "22" ], "output": "22", "label": "OGF partial sum through0" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "(1/2)^1", "after": "OGF ORIGINAL x power1 = 1/2", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。", "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity." }, "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": { "operation": "integer_power", "inputs": [ "1/2", "1" ], "output": "1/2", "label": "OGF ORIGINAL x power1" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "OGF term1 coefficient times power := mul(44, 1/2)", "after": "OGF term1 coefficient times power = 22", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，分母先核非零，不用浮點近似。", "en": "Substitute original roles and established intermediates exactly,checking nonzero denominators without floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 OGF term1 coefficient times power，不以小數近似替代。", "en": "Recompute OGF term1 coefficient times power with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "44", "1/2" ], "output": "22", "label": "OGF term1 coefficient times power" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "OGF partial sum through1 := sum(22, 22)", "after": "OGF partial sum through1 = 44", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，分母先核非零，不用浮點近似。", "en": "Substitute original roles and established intermediates exactly,checking nonzero denominators without floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 OGF partial sum through1，不以小數近似替代。", "en": "Recompute OGF partial sum through1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "22", "22" ], "output": "44", "label": "OGF partial sum through1" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "(1/2)^2", "after": "OGF ORIGINAL x power2 = 1/4", "relation": "implies", "rule_id": "integer_power", "conditions": [], "explanation": { "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。", "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity." }, "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": { "operation": "integer_power", "inputs": [ "1/2", "2" ], "output": "1/4", "label": "OGF ORIGINAL x power2" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "OGF term2 coefficient times power := mul(66, 1/4)", "after": "OGF term2 coefficient times power = 33/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，分母先核非零，不用浮點近似。", "en": "Substitute original roles and established intermediates exactly,checking nonzero denominators without floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 OGF term2 coefficient times power，不以小數近似替代。", "en": "Recompute OGF term2 coefficient times power with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "66", "1/4" ], "output": "33/2", "label": "OGF term2 coefficient times power" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "OGF partial sum through2 := sum(44, 33/2)", "after": "OGF partial sum through2 = 121/2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已證中間量精確代值，分母先核非零，不用浮點近似。", "en": "Substitute original roles and established intermediates exactly,checking nonzero denominators without floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 OGF partial sum through2，不以小數近似替代。", "en": "Recompute OGF partial sum through2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "44", "33/2" ], "output": "121/2", "label": "OGF partial sum through2" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "source roles = [[22, 44, 66, 88], 1/2, 2]", "after": "derived = [121/2, 否]", "relation": "implies", "rule_id": "OGF_support_coverage_NOT_accidental_pointwise_equality", "conditions": [], "explanation": { "zh-Hant": "coverage旗標比較最後非零support與cutoff，不測單點值偶然相等；x=0或相消時未含完整support仍可能數值相同。", "en": "The coverage flag compares cutoff with the final nonzero support,not accidental point equality.At x=0 or by cancellation,a shorter sum may have the same value." }, "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": [ [ "22", "44", "66", "88" ], "1/2", 2, [ [ 0, "22", "22" ], [ 1, "44", "22" ], [ 2, "66", "33/2" ] ], "121/2", false, "finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included" ] }, "conclusion": { "canonical_value": "([22, 44, 66, 88], 1/2, 2, [[0, 22, 22], [1, 44, 22], [2, 66, 33/2]], 121/2, 否, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)", "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": 47, "original_line_sha256": "c615de094e715e951df77f0e15909f00f56fdabe9f449c82ce2c8ae0e2b3116a", "question_compact_sha256": "944d3bab3a28f833597dda32b014409fc12e2ed6ffdd0063bc4e33066012fd12", "english_row_compact_sha256": "b920262610f898dfd3cf43b1353bcfa23a4ce3a19a9cee89e64e3ea1033d55c8", "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": "9cb8bafb787a308a7c9c4038e2dfb35995b7ca90459c0a4f3474df4b9c752a3b", "original_candidate_line_sha256": "9cb8bafb787a308a7c9c4038e2dfb35995b7ca90459c0a4f3474df4b9c752a3b" }, "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": 12, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "3bc66c70935e0745250f398daab35f209192829f654e51c82f8ede2aede263a9", "arithmetic_steps": 9 }, "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." } }