{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261021-r05-bisection-method-0043", "pilot_number": "R0003-02068", "number": 2068, "family_id": "qf-app21-bisection-single-update", "title": { "zh-Hant": "第 2068 題：有限二分bracket", "en": "Question 2068: One Sign-Bracket Update with Explicit Endpoint-Zero Priority" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261021-r05-bisection-method-0043", "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-single-update；資料={\"left\":[17,1],\"right\":[21,1],\"left_sign\":-1,\"mid_sign\":0,\"right_sign\":1}。請依 [left,right,left_sign,mid_sign,right_sign,midpoint,next_bracket] 欄序回答。", "answer": { "kind": "expression", "value": "(17, 21, -1, 0, 1, 19, [19, 19, 0, 0])", "explanation_zh": "應用擴展核對=(17, 21, -1, 0, 1, 19, [19, 19, 0, 0])；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-single-update", "seed": 202610215043, "parameters": { "left": [ 17, 1 ], "right": [ 21, 1 ], "left_sign": -1, "mid_sign": 0, "right_sign": 1, "domain": "Declared_sign_bracket_updates", "operation": "bisection-single-update", "semantic_template": "qf-app21-bisection-single-update/{\"left\":[17,1],\"left_sign\":-1,\"mid_sign\":0,\"right\":[21,1],\"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-0043", "batch_id": "batch-20261013-advance-20261021-r05-bisection-method-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "9852edffbc65028b0db1fa466ee8be6236b006b887607605780a5f900fba0a6d", "family_id": "qf-app21-bisection-single-update", "seed": 202610215043, "semantic_template": "qf-app21-bisection-single-update/{\"left\":[17,1],\"left_sign\":-1,\"mid_sign\":0,\"right\":[21,1],\"right_sign\":1}" }, "fields": { "title": "One Sign-Bracket Update with Explicit Endpoint-Zero Priority", "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. left=17/1, right=21/1, signs left=-1, midpoint=0, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.", "answer": { "kind": "expression", "value": "(17, 21, -1, 0, 1, 19, [19, 19, 0, 0])", "explanation": "Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (17, 21, -1, 0, 1, 19, [19, 19, 0, 0])." }, "skills": [ "One Sign-Bracket Update with Explicit Endpoint-Zero Priority" ], "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-single-update；資料={\"left\":[17,1],\"right\":[21,1],\"left_sign\":-1,\"mid_sign\":0,\"right_sign\":1}。請依 [left,right,left_sign,mid_sign,right_sign,midpoint,next_bracket] 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. left=17/1, right=21/1, signs left=-1, midpoint=0, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "left=17