{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261021-r05-bisection-method-0077", "pilot_number": "R0003-02077", "number": 2077, "family_id": "qf-app21-bisection-width-after-steps", "title": { "zh-Hant": "第 2077 題：有限二分bracket", "en": "Question 2077: Geometric Width after a Declared Number of Halvings" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261021-r05-bisection-method-0077", "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-width-after-steps；資料={\"left\":[-2,1],\"right\":[6,1],\"steps\":1}。請依 [left,right,steps,width] 欄序回答。", "answer": { "kind": "expression", "value": "(-2, 6, 1, 4)", "explanation_zh": "應用擴展核對=(-2, 6, 1, 4)；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": 1, "skills": [ "有限二分bracket" ] }, "variant": { "family_id": "qf-app21-bisection-width-after-steps", "seed": 202610215077, "parameters": { "left": [ -2, 1 ], "right": [ 6, 1 ], "steps": 1, "domain": "Declared_sign_bracket_updates", "operation": "bisection-width-after-steps", "semantic_template": "qf-app21-bisection-width-after-steps/{\"left\":[-2,1],\"right\":[6,1],\"steps\":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-0077", "batch_id": "batch-20261013-advance-20261021-r05-bisection-method-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "4f133342c4ed1238885eb0dbf3576465cfa9a1735bdbfca5e93fc4c7014c27a7", "family_id": "qf-app21-bisection-width-after-steps", "seed": 202610215077, "semantic_template": "qf-app21-bisection-width-after-steps/{\"left\":[-2,1],\"right\":[6,1],\"steps\":1}" }, "fields": { "title": "Geometric Width after a Declared Number of Halvings", "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=-2/1, right=6/1, halvings=1≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.", "answer": { "kind": "expression", "value": "(-2, 6, 1, 4)", "explanation": "Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-2, 6, 1, 4)." }, "skills": [ "Geometric Width after a Declared Number of Halvings" ], "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-width-after-steps；資料={\"left\":[-2,1],\"right\":[6,1],\"steps\":1}。請依 [left,right,steps,width] 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=-2/1, right=6/1, halvings=1≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "left=-2