{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261109-r05-binomial-theorem-0095", "pilot_number": "P1000-0470", "number": 470, "family_id": "qf-mod40-binomial-exponent-boundary", "title": { "zh-Hant": "第 470 題：二項式定理", "en": "Question 470: Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261109-r05-binomial-theorem-0095", "batch_id": "batch-20261014-advance-20261109-r05-binomial-theorem-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-binomial-theorem" ], "title_zh": "二項式定理", "stem_zh": "commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(-20, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)", "explanation_zh": "Wave6第三層核對=(-20, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)；依完整明示資料獨立重算；coefficients exact且k、n-k exponents不可錯置；n=0給constant one；不套用negative/noninteger exponent或noncommuting variables；combinatorial witness選k個y factors。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "二項式定理" ] }, "variant": { "family_id": "qf-mod40-binomial-exponent-boundary", "seed": 202611095095, "parameters": { "exponent": -20, "domain": "Exact_commutative_binomial_theorem", "operation": "binomial-exponent-boundary", "semantic_template": "qf-mod40-binomial-exponent-boundary/{\"exponent\":-20}" } }, "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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r05-binomial-theorem-0095", "batch_id": "batch-20261014-advance-20261109-r05-binomial-theorem-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "dee03ed03aa4162580cd6f6fb7399959b30f986d24c42968f8af7ff8ee0aea9f", "family_id": "qf-mod40-binomial-exponent-boundary", "seed": 202611095095, "semantic_template": "qf-mod40-binomial-exponent-boundary/{\"exponent\":-20}" }, "fields": { "title": "Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem", "stem": "Use boundary exponent=-20. Give (exponent, finite_polynomial_claim, boundary).", "answer": { "kind": "expression", "value": "(-20, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)", "explanation": "The fixed No withholds the natural-exponent finite binomial-polynomial claim. It is not a statement that other series expansions cannot exist; those need separate hypotheses and convergence conventions. The exact result, retaining the requested order, is (-20, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)." }, "skills": [ "Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem" ], "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": "commutative coefficient setting與n in natural including zero。操作=binomial-exponent-boundary；資料={\"exponent\":-20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use boundary exponent=-20. Give (exponent, finite_polynomial_claim, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "exponent=-20", "after": "finite polynomial formula requires a nonnegative integer n", "relation": "implies", "rule_id": "natural_exponent", "conditions": [], "explanation": { "zh-Hant": "先讀出本定理的指數域；有限個因子要求非負整數。", "en": "The finite-product theorem requires a nonnegative integer exponent." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "exponent=-20", "after": "not a permitted finite-product exponent", "relation": "implies", "rule_id": "exponent_rejection", "conditions": [], "explanation": { "zh-Hant": "負數或非整數不在這個有限展開的宣告域。", "en": "A negative or noninteger exponent is outside this finite expansion's declared domain." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "finite product with n factors", "after": "cannot enumerate a negative/noninteger number of factors", "relation": "implies", "rule_id": "finite_factor_count", "conditions": [], "explanation": { "zh-Hant": "所以原本 k=0 到 n 的有限係數列不能直接套用。", "en": "Consequently the finite coefficient list k=0 through n cannot be applied directly." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "this finite theorem is inapplicable", "after": "no claim about a different generalized binomial series", "relation": "implies", "rule_id": "scope_not_global_disproof", "conditions": [], "explanation": { "zh-Hant": "這不是否定其他收斂條件下的廣義級數；也不省略乘法可交換的條件。", "en": "This does not disprove generalized series under other convergence hypotheses, and it does not drop commutativity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "係數先在整數中計數；不在一般係數環中除以可能不可逆的階乘。", "en": "Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring." } }, "arithmetic": null } ], "math_model": { "result": [ -20, false, "finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication" ] }, "conclusion": { "canonical_value": "(-20, 否, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)", "explanation": { "zh-Hant": "上述完整代值與條件核對得到原題答案；欄位次序及真假標記保持原始合約。", "en": "The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings." } }, "definition_note": { "zh-Hant": "有限二項展開只適用於非負整數指數；本題只否定該有限多項式主張，不否定其他級數。", "en": "The finite binomial expansion needs a nonnegative integer exponent. Only that finite-polynomial claim is withheld; other series are not disproved." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261014-advance-20261109-r05-binomial-theorem-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261109-r05-binomial-theorem-foundations/questions.jsonl", "batch_sha256": "f83302c1353681ecb1209ee1ccbef0d266f15fb9415191ed9289eb329e1175cc", "original_line": 95, "original_line_sha256": "9b6f1589010e9227321b61e2b62a97ffbe0b87ca82d33898030d7263a1da318b", "question_compact_sha256": "dee03ed03aa4162580cd6f6fb7399959b30f986d24c42968f8af7ff8ee0aea9f", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261109-r05-binomial-theorem-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261109-r05-binomial-theorem-foundations.en.jsonl", "english_row_compact_sha256": "d0b49d750688deab5f69db301b49b4ed05a6ad866ce4a661fa355bed81f08f49" }, "mko_bindings": { "mko-binomial-theorem": { "sha256": "c8fefe3783fc4f7633b4b22b8cf149d96a683900f181e3b42c690409ce90e5b7", "english_sha256": "1086dfe7c31f4e6bf8934681a0a080b50c14cb828f1a8b17e59143ec2cf06c83", "original_path": "public/data/mko/binomial-theorem.json", "english_path": "public/data/i18n/editions/mko-binomial-theorem.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "afa7c0e9f7084cc5929124dcffe9929d5db7def32e40047056d19ea25535873e", "original_candidate_sha256": "9b14b96912cce05d5beb84b8f6a19229b211a10caf404d8c5dca13e36eed12f7", "original_candidate_line_sha256": "9b14b96912cce05d5beb84b8f6a19229b211a10caf404d8c5dca13e36eed12f7" }, "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": "來源綁定的教學推導候選；精確算術及有限列舉不是獨立AI審查或一般形式化證明。原題、答案、MKO、公式、AST及歷史保持不變。", "en": "Source-bound pedagogical derivation candidate: exact arithmetic and finite enumeration are not independent AI 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-20261026-003-01", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "Thirty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct sentence shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full answer tuples, model results 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": "6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772", "question_first_witness_sha256": "77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 4, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "c99eec935c0add0c472c0dad0cd4ec4109b196a4f8d9cce03cf4143d1010dc39", "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." } }