{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261112-r02-taylor-polynomial-0085", "pilot_number": "R0047-24035", "number": 24035, "family_id": "qf-mod43-taylor-definition-boundary", "title": { "zh-Hant": "第 24035 題：Taylor多項式", "en": "Question 24035: Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261112-r02-taylor-polynomial-0085", "batch_id": "batch-20261014-advance-20261112-r02-taylor-polynomial-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-taylor-polynomial" ], "title_zh": "Taylor多項式", "stem_zh": "固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"wrong_center\",\"derivatives\":[10,20,30],\"center\":-3,\"order\":2,\"alternate_center\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(wrong_center, 2, [10, 20, 30], -3, -2, 否, a_jet_is_bound_to_its_declared_center)", "explanation_zh": "Wave6第六層核對=(wrong_center, 2, [10, 20, 30], -3, -2, 否, a_jet_is_bound_to_its_declared_center)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "Taylor多項式" ] }, "variant": { "family_id": "qf-mod43-taylor-definition-boundary", "seed": 202611122085, "parameters": { "mode": "wrong_center", "derivatives": [ 10, 20, 30 ], "center": -3, "order": 2, "alternate_center": -2, "domain": "Exact_finite_Taylor_polynomial", "operation": "taylor-definition-boundary", "semantic_template": "qf-mod43-taylor-definition-boundary/{\"alternate_center\":-2,\"center\":-3,\"derivatives\":[10,20,30],\"mode\":\"wrong_center\",\"order\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或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-20261112-r02-taylor-polynomial-0085", "batch_id": "batch-20261014-advance-20261112-r02-taylor-polynomial-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "5ca07e3ee244c1184323d0f05efb15dfac2a80b8ad8c84592530c558f6a61864", "family_id": "qf-mod43-taylor-definition-boundary", "seed": 202611122085, "semantic_template": "qf-mod43-taylor-definition-boundary/{\"alternate_center\":-2,\"center\":-3,\"derivatives\":[10,20,30],\"mode\":\"wrong_center\",\"order\":2}" }, "fields": { "title": "Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries", "stem": "Use mode=wrong_center, supplied derivatives=[10, 20, 30], center=-3, order=2, alternate_center=-2. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary).", "answer": { "kind": "expression", "value": "(wrong_center, 2, [10, 20, 30], -3, -2, No, a_jet_is_bound_to_its_declared_center)", "explanation": "zero through n means orders0 through n, not zero derivative values. missing_derivative lacks an order; wrong_center changes the jet binding; missing_factorial omits k! scaling; remainder_claim tries to infer an unavailable error/series conclusion. The fixed No refuses that unsupported claim without rewriting the data. The exact result, retaining the requested order, is (wrong_center, 2, [10, 20, 30], -3, -2, No, a_jet_is_bound_to_its_declared_center)." }, "skills": [ "Missing Orders, Wrong Centers and Remainder Claims Are Separate Taylor Boundaries" ], "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": "固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-definition-boundary；資料={\"mode\":\"wrong_center\",\"derivatives\":[10,20,30],\"center\":-3,\"order\":2,\"alternate_center\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use mode=wrong_center, supplied derivatives=[10, 20, 30], center=-3, order=2, alternate_center=-2. Give (mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [taylor-definition-boundary]", "after": "derived = [[mode, order, derivatives, center, alternate_center, unsupported_definition_or_remainder_claim, boundary]]", "relation": "implies", "rule_id": "source_ordered_fields_without_JS", "conditions": [], "explanation": { "zh-Hant": "本頁列完整來源欄序，不需JavaScript；gradient boundary依原mode選6／8／7欄，不補欄也不由答案猜mode。", "en": "List the source field order without JavaScript.Gradient boundary uses its original mode to choose6,8 or7 fields;no padding or answer-derived mode." }, "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 = [wrong_center, 2, [10, 20, 30], -3, -2]", "after": "derived = [[0, 1, 2], 3, 否, a_jet_is_bound_to_its_declared_center]", "relation": "implies", "rule_id": "Taylor_definition_boundary_keeps_actual_jet_and_center", "conditions": [], "explanation": { "zh-Hant": "missing_derivative缺的是0至n各階資料（含f(a)），不是要求每個值都為0；wrong_center不能把同一jet直接換基底中心，即使常數特例可能相同。missing_factorial仍須除k!，即使低階或零值可能偶然不變。remainder_claim缺更高階控制／解析性，No不等於所有特例必不相等。", "en": "Missing data means every derivative order0..n,including f(a),not that values must vanish.A center-bound jet cannot simply be shifted,even if a constant special case agrees.Division by k! is required despite accidental low-order equality.A polynomial alone supplies neither a remainder bound nor analyticity;No does not deny special equality." }, "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": [ "wrong_center", 2, [ "10", "20", "30" ], "-3", "-2", false, "a_jet_is_bound_to_its_declared_center" ] }, "conclusion": { "canonical_value": "(wrong_center, 2, [10, 20, 30], -3, -2, 否, a_jet_is_bound_to_its_declared_center)", "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-20261112-r02-taylor-polynomial-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261112-r02-taylor-polynomial-foundations/questions.jsonl", "batch_sha256": "c67ac0dd673d664623d9707d94303285afad1472da9173be45b762e8e2059407", "original_line": 85, "original_line_sha256": "9e5394308da223c5f8d05fceaa0d44193d4d3627e3bca0ec1a341e99733fca08", "question_compact_sha256": "5ca07e3ee244c1184323d0f05efb15dfac2a80b8ad8c84592530c558f6a61864", "english_row_compact_sha256": "b4410ed05c1e1643e384feb8e142775288e42e2ac43bfd758b91dbe45df2c25b", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261112-r02-taylor-polynomial-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261112-r02-taylor-polynomial-foundations.en.jsonl" }, "mko_bindings": { "mko-taylor-polynomial": { "sha256": "ee8f291872fc7e5a97cdbe468fffe5f7e4e3857373415ca1a65e5ab2b61a53a1", "english_sha256": "5009d7ebab6a826e0e1ca6a349eadfed04ba911cc721a2b103d2e4e14229974f", "original_path": "public/data/mko/taylor-polynomial.json", "english_path": "public/data/i18n/editions/mko-taylor-polynomial.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "b3aad195dad743751e4c8a4a6ce44f091c3824da1e52c21e9266d4ccd1dc1b68", "original_candidate_sha256": "0d00171253348f6ad35b6dd0b6efeffdfcf7e2584e973b65602f99bab59cad75", "original_candidate_line_sha256": "0d00171253348f6ad35b6dd0b6efeffdfcf7e2584e973b65602f99bab59cad75" }, "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-p48", "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": "9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386", "question_first_witness_sha256": "eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50b" }, "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": "f2deab13057360bec6e59d6c180fc9e3d486e3768d2c32a386642d4f2af158ca", "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." } }