{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261020-r06-discrete-exponential-growth-0015", "pilot_number": "R0002-01790", "number": 1790, "family_id": "qf-app20-growth-recurrence-step", "title": { "zh-Hant": "第 1790 題：離散固定因子模型", "en": "Question 1790: One Exact Fixed-Factor Discrete Update" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261020-r06-discrete-exponential-growth-0015", "batch_id": "batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-discrete-exponential-growth" ], "title_zh": "離散固定因子模型", "stem_zh": "固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[6,5],\"factor\":[1,4]}。請依 [current,factor,next] 欄序回答。", "answer": { "kind": "expression", "value": "(6/5, 1/4, 3/10)", "explanation_zh": "應用基礎核對=(6/5, 1/4, 3/10)；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；不是連續時間模型；不是經驗預測。" }, "difficulty": { "band": "upper_secondary", "level": 2, "skills": [ "離散固定因子模型" ] }, "variant": { "family_id": "qf-app20-growth-recurrence-step", "seed": 202610206015, "parameters": { "current": [ 6, 5 ], "factor": [ 1, 4 ], "domain": "Exact_discrete_exponential_model", "operation": "growth-recurrence-step", "semantic_template": "qf-app20-growth-recurrence-step/{\"current\":[6,5],\"factor\":[1,4]}" } }, "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-r06-discrete-exponential-growth-0015", "batch_id": "batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "14b89ce7ad524b295ab9888caf078eba229a55254a1cfc06c14371d8166e7b54", "family_id": "qf-app20-growth-recurrence-step", "seed": 202610206015, "semantic_template": "qf-app20-growth-recurrence-step/{\"current\":[6,5],\"factor\":[1,4]}" }, "fields": { "title": "One Exact Fixed-Factor Discrete Update", "stem": "Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. 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. current=6/5, factor=1/4. Give (current, factor, next=current*factor).", "answer": { "kind": "expression", "value": "(6/5, 1/4, 3/10)", "explanation": "Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6/5, 1/4, 3/10)." }, "skills": [ "One Exact Fixed-Factor Discrete Update" ], "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": "固定時間步與精確固定因子。操作=growth-recurrence-step；資料={\"current\":[6,5],\"factor\":[1,4]}。請依 [current,factor,next] 欄序回答。", "en": "Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. 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. current=6/5, factor=1/4. Give (current, factor, next=current*factor)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "factor q=1/4≥0", "after": "X_(k+1)=q·X_k", "relation": "implies", "rule_id": "fixed_factor_recurrence", "conditions": [], "explanation": { "zh-Hant": "固定同一因子；這是題目明示的離散模型，不從少量數值猜測一般規律。", "en": "Use the same factor in the declared discrete model,rather than infer a general law from a few values." }, "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": "current=6/5; q=1/4", "after": "substitute into one recurrence step", "relation": "implies", "rule_id": "current_vs_initial", "conditions": [], "explanation": { "zh-Hant": "當前值已給定，不額外假設它是正數或初始值。", "en": "The current value is given;do not assume it is positive or the initial value." }, "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": "S003", "uses": [ "G001", "S002" ], "before": "next value := mul(6/5, 1/4)", "after": "next value = 3/10", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "完整乘同一有理因子。", "en": "Multiply by the same full rational factor." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 next value，不以小數近似替代。", "en": "Recompute next value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "6/5", "1/4" ], "output": "3/10", "label": "next value" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "X_k=6/5", "after": "X_(k+1)=3/10", "relation": "implies", "rule_id": "single_step_scope", "conditions": [], "explanation": { "zh-Hant": "這一個等式不宣稱未給定模型的全域成長或收斂。", "en": "This single equality is not a global-growth or convergence claim about an unspecified model." }, "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": [ "6/5", "1/4", "3/10" ], "recurrence_values": [ "6/5", "3/10" ] }, "conclusion": { "canonical_value": "(6/5, 1/4, 3/10)", "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-r06-discrete-exponential-growth-foundations", "batch_path": "public/data/questions/batches/batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations/questions.jsonl", "batch_sha256": "c2e2d6e56d2e17262678fe2d2a402a35aab9b8fbdd2c94544b16fe6f43d25a81", "original_line": 15, "original_line_sha256": "478ac251d0f85262657eafa106c8bc91344953b10190640d01864c65a7661f48", "question_compact_sha256": "14b89ce7ad524b295ab9888caf078eba229a55254a1cfc06c14371d8166e7b54", "english_row_compact_sha256": "adce49811ba067f9b9b2b93c171a6ac7b8dd77a5154b9172dc8f0cbf74cded0f", "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations.en.jsonl" }, "mko_bindings": { "mko-discrete-exponential-growth": { "sha256": "81d7d953b56dc3b4f26d1051ae74ae30b298a2c6085c59c5f06439b68ea8e780", "english_sha256": "e3b7db219c9df5e1a4a5757566489aba9a73916a7d659377948d93c20b4c77b0", "original_path": "public/data/mko/discrete-exponential-growth.json", "english_path": "public/data/i18n/editions/mko-discrete-exponential-growth.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "e476360038ede38e6f70fbe8c039b14a4f4bd45937ee88050de8370944911f19", "original_candidate_sha256": "fa56a1c5531b4675b375714b56add995ecee477b2cec62da182cc7e3a7bd8b34", "original_candidate_line_sha256": "fa56a1c5531b4675b375714b56add995ecee477b2cec62da182cc7e3a7bd8b34" }, "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": 4, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "ce9d6bf0ff06d05270cc791bb3899f432d1568002618b0f6504a68baa9a4ef2a", "arithmetic_steps": 1 }, "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." } }