{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261021-r05-bisection-method-0074", "pilot_number": "R0003-02049", "number": 2049, "family_id": "qf-app21-bisection-finite-trace", "title": { "zh-Hant": "第 2049 題：有限二分bracket", "en": "Question 2049: A Finite Trace Stops at the First Declared Zero Midpoint" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261021-r05-bisection-method-0074", "batch_id": "batch-20261013-advance-20261021-r05-bisection-method-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-bisection-method" ], "title_zh": "有限二分bracket", "stem_zh": "已提供的符號 bracket 資料。操作=bisection-finite-trace；資料={\"left\":[23,1],\"right\":[27,1],\"left_sign\":-1,\"right_sign\":1,\"mid_signs\":[1,1,1,-1]}。請依 [initial_left,initial_right,left_sign,right_sign,mid_signs,rows,final_left,final_right] 欄序回答。", "answer": { "kind": "expression", "value": "(23, 27, -1, 1, [1, 1, 1, -1], [[23, 27, 25, 1, 23, 25], [23, 25, 24, 1, 23, 24], [23, 24, 47/2, 1, 23, 47/2], [23, 47/2, 93/4, -1, 93/4, 47/2]], 93/4, 47/2)", "explanation_zh": "應用擴展核對=(23, 27, -1, 1, [1, 1, 1, -1], [[23, 27, 25, 1, 23, 25], [23, 25, 24, 1, 23, 24], [23, 24, 47/2, 1, 23, 47/2], [23, 47/2, 93/4, -1, 93/4, 47/2]], 93/4, 47/2)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不由異號自行推出根存在；只核對有限更新與寬度。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "有限二分bracket" ] }, "variant": { "family_id": "qf-app21-bisection-finite-trace", "seed": 202610215074, "parameters": { "left": [ 23, 1 ], "right": [ 27, 1 ], "left_sign": -1, "right_sign": 1, "mid_signs": [ 1, 1, 1, -1 ], "domain": "Declared_sign_bracket_updates", "operation": "bisection-finite-trace", "semantic_template": "qf-app21-bisection-finite-trace/{\"left\":[23,1],\"left_sign\":-1,\"mid_signs\":[1,1,1,-1],\"right\":[27,1],\"right_sign\":1}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-13", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261013-advance-20261021-r05-bisection-method-0074", "batch_id": "batch-20261013-advance-20261021-r05-bisection-method-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "66178e5f0145920a84354d0a0c56ca5d2ba52cd5768cc9adc4f0e5e781a76425", "family_id": "qf-app21-bisection-finite-trace", "seed": 202610215074, "semantic_template": "qf-app21-bisection-finite-trace/{\"left\":[23,1],\"left_sign\":-1,\"mid_signs\":[1,1,1,-1],\"right\":[27,1],\"right_sign\":1}" }, "fields": { "title": "A Finite Trace Stops at the First Declared Zero Midpoint", "stem": "Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=23/1, right=27/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1, -1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.", "answer": { "kind": "expression", "value": "(23, 27, -1, 1, [1, 1, 1, -1], [[23, 27, 25, 1, 23, 25], [23, 25, 24, 1, 23, 24], [23, 24, 47/2, 1, 23, 47/2], [23, 47/2, 93/4, -1, 93/4, 47/2]], 93/4, 47/2)", "explanation": "Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (23, 27, -1, 1, [1, 1, 1, -1], [[23, 27, 25, 1, 23, 25], [23, 25, 24, 1, 23, 24], [23, 24, 47/2, 1, 23, 47/2], [23, 47/2, 93/4, -1, 93/4, 47/2]], 93/4, 47/2)." }, "skills": [ "A Finite Trace Stops at the First Declared Zero Midpoint" ], "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": "已提供的符號 bracket 資料。操作=bisection-finite-trace；資料={\"left\":[23,1],\"right\":[27,1],\"left_sign\":-1,\"right_sign\":1,\"mid_signs\":[1,1,1,-1]}。請依 [initial_left,initial_right,left_sign,right_sign,mid_signs,rows,final_left,final_right] 欄序回答。", "en": "Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=23/1, right=27/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1, -1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "left=23