{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261020-r03-numerical-approximation-error-0063", "pilot_number": "R0002-01688", "number": 1688, "family_id": "qf-app20-error-tolerance-audit", "title": { "zh-Hant": "第 1688 題：精確有理誤差", "en": "Question 1688: Inclusive Absolute-Error Tolerance" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261020-r03-numerical-approximation-error-0063", "batch_id": "batch-20261013-advance-20261020-r03-numerical-approximation-error-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-numerical-approximation-error" ], "title_zh": "精確有理誤差", "stem_zh": "精確有理數。操作=error-tolerance-audit；資料={\"approximation\":[27,3],\"reference\":[4,1],\"tolerance\":[1,7]}。請依 [approximation,reference,absolute_error,tolerance,within_tolerance] 欄序回答。", "answer": { "kind": "expression", "value": "(9, 4, 5, 1/7, 否)", "explanation_zh": "應用基礎核對=(9, 4, 5, 1/7, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；relative error要求reference非零；不混淆截斷誤差與捨入誤差。" }, "difficulty": { "band": "upper_secondary", "level": 2, "skills": [ "精確有理誤差" ] }, "variant": { "family_id": "qf-app20-error-tolerance-audit", "seed": 202610203063, "parameters": { "approximation": [ 27, 3 ], "reference": [ 4, 1 ], "tolerance": [ 1, 7 ], "domain": "Exact_rational_error", "operation": "error-tolerance-audit", "semantic_template": "qf-app20-error-tolerance-audit/{\"approximation\":[27,3],\"reference\":[4,1],\"tolerance\":[1,7]}" } }, "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、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-20261020-r03-numerical-approximation-error-0063", "batch_id": "batch-20261013-advance-20261020-r03-numerical-approximation-error-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "4464dd78d5841336d193193aa6c3bd7d40edb22a9fa60341a05f026eb9c84af6", "family_id": "qf-app20-error-tolerance-audit", "seed": 202610203063, "semantic_template": "qf-app20-error-tolerance-audit/{\"approximation\":[27,3],\"reference\":[4,1],\"tolerance\":[1,7]}" }, "fields": { "title": "Inclusive Absolute-Error Tolerance", "stem": "Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=27/3, reference=4/1, tolerance=1/7≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?).", "answer": { "kind": "expression", "value": "(9, 4, 5, 1/7, No)", "explanation": "The tolerance comparison includes equality; a negative tolerance is not admitted. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 4, 5, 1/7, No)." }, "skills": [ "Inclusive Absolute-Error Tolerance" ], "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": "精確有理數。操作=error-tolerance-audit；資料={\"approximation\":[27,3],\"reference\":[4,1],\"tolerance\":[1,7]}。請依 [approximation,reference,absolute_error,tolerance,within_tolerance] 欄序回答。", "en": "Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. approximation=27/3, reference=4/1, tolerance=1/7≥0. Give (approximation, reference, absolute error, tolerance, error≤tolerance?)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "reference=4", "after": "all values are exact rationals", "relation": "implies", "rule_id": "exact_error_model", "conditions": [], "explanation": { "zh-Hant": "這是指定參考值下的精確誤差運算，不以浮點四捨五入改變等號邊界。", "en": "This is exact error arithmetic against a specified reference,without floating-point rounding that could alter an equality boundary." }, "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": "approximation−reference := sub(9, 4)", "after": "approximation−reference = 5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先計算帶符號誤差，順序固定為近似值減參考值。", "en": "Compute the signed error as approximation minus reference." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 approximation−reference，不以小數近似替代。", "en": "Recompute approximation−reference with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "9", "4" ], "output": "5", "label": "approximation−reference" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "absolute error := mul(1, 5)", "after": "absolute error = 5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依誤差符號取絕對值，結果非負。", "en": "Use the sign to take the absolute value;the result is nonnegative." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 absolute error，不以小數近似替代。", "en": "Recompute absolute error with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "5" ], "output": "5", "label": "absolute error" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "signed error=5", "after": "absolute error=5≥0", "relation": "implies", "rule_id": "absolute_error_definition", "conditions": [], "explanation": { "zh-Hant": "絕對誤差與原量同單位；負參考值也不會使絕對誤差變負。", "en": "Absolute error has the original quantity's units;a negative reference does not make it negative." }, "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": "S005", "uses": [ "G001", "S004" ], "before": "absolute error=5; tolerance=1/7", "after": "absolute error≤tolerance: 否", "relation": "implies", "rule_id": "inclusive_tolerance", "conditions": [], "explanation": { "zh-Hant": "合約採≤而不是<，等於容差的案例仍通過。", "en": "The contract uses≤,not<;an error exactly equal to tolerance passes." }, "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": [ "9", "4", "5", "1/7", false ], "signed_error": "5", "absolute_error": "5" }, "conclusion": { "canonical_value": "(9, 4, 5, 1/7, 否)", "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-20261013-advance-20261020-r03-numerical-approximation-error-foundations", "batch_path": "public/data/questions/batches/batch-20261013-advance-20261020-r03-numerical-approximation-error-foundations/questions.jsonl", "batch_sha256": "9cbfa5de4b68ee867820180889bb56e47266e5a2cdcf12ca61429187c230f424", "original_line": 63, "original_line_sha256": "45647bdde2122298e1e745e6fc79b752c32fa1ca7088ac8e4fdc5bac2da89306", "question_compact_sha256": "4464dd78d5841336d193193aa6c3bd7d40edb22a9fa60341a05f026eb9c84af6", "english_row_compact_sha256": "f5e6ca6dbae655d1a3a3309cbcce58efdead9354556b3d588a017e12f0842f08", "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261020-r03-numerical-approximation-error-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261020-r03-numerical-approximation-error-foundations.en.jsonl" }, "mko_bindings": { "mko-numerical-approximation-error": { "sha256": "ced94d430d6417b9e607be1f09be7234591c3138507fecd536d3d7dc024ecbe0", "english_sha256": "fe552b17f866ceee13097202f2ac4dd54a72ef64059402444b22146b41d82149", "original_path": "public/data/mko/numerical-approximation-error.json", "english_path": "public/data/i18n/editions/mko-numerical-approximation-error.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "e476360038ede38e6f70fbe8c039b14a4f4bd45937ee88050de8370944911f19", "original_candidate_sha256": "d4b9acaa52bb0b9945f98ab9f5ac95a4508498499c95b3abf078abd064fdd627", "original_candidate_line_sha256": "d4b9acaa52bb0b9945f98ab9f5ac95a4508498499c95b3abf078abd064fdd627" }, "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-20261026-004-p3", "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": "340d74dcc6e37df6249594a0b79a9ee2af804337b7847478b3c1342af9722861", "question_first_witness_sha256": "9a06233d190d31a16f13fb03b6c3ceba4cbb748fff6c997941f66ab1888dcca8" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 5, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "a9fddd01b0da42ec22385c06f70d6aeb976ff992046e80f48b656346edc34180", "arithmetic_steps": 2 }, "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." } }