{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261020-r01-discrete-dynamical-system-0002", "pilot_number": "R0002-01577", "number": 1577, "family_id": "qf-app20-dynamics-state-update", "title": { "zh-Hant": "第 1577 題：有限離散動力核對", "en": "Question 1577: A Typed Finite State and Its Canonical Residue Update" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261020-r01-discrete-dynamical-system-0002", "batch_id": "batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-discrete-dynamical-system" ], "title_zh": "有限離散動力核對", "stem_zh": "明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":6,\"a\":-6,\"b\":-9,\"x\":1}。請依 [modulus,a,b,x,T(x)] 欄序回答。", "answer": { "kind": "expression", "value": "(6, -6, -9, 1, 3)", "explanation_zh": "應用基礎核對=(6, -6, -9, 1, 3)；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；有限軌道不代表一般動力行為；不宣稱穩定性。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "有限離散動力核對" ] }, "variant": { "family_id": "qf-app20-dynamics-state-update", "seed": 202610201002, "parameters": { "modulus": 6, "a": -6, "b": -9, "x": 1, "domain": "Fin_discrete_dynamics", "operation": "dynamics-state-update", "semantic_template": "qf-app20-dynamics-state-update/{\"a\":-6,\"b\":-9,\"modulus\":6,\"x\":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、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-r01-discrete-dynamical-system-0002", "batch_id": "batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "a25d81f674c02cb63e6e84b97d510ea30f9738a8edeb82f4be778eb7e422d861", "family_id": "qf-app20-dynamics-state-update", "seed": 202610201002, "semantic_template": "qf-app20-dynamics-state-update/{\"a\":-6,\"b\":-9,\"modulus\":6,\"x\":1}" }, "fields": { "title": "A Typed Finite State and Its Canonical Residue Update", "stem": "Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=-6, b=-9. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=1. Give (modulus, a, b, x, T(x)).", "answer": { "kind": "expression", "value": "(6, -6, -9, 1, 3)", "explanation": "Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, -6, -9, 1, 3)." }, "skills": [ "A Typed Finite State and Its Canonical Residue 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": "明示有限狀態自映射。操作=dynamics-state-update；資料={\"modulus\":6,\"a\":-6,\"b\":-9,\"x\":1}。請依 [modulus,a,b,x,T(x)] 欄序回答。", "en": "Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=-6, b=-9. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=1. Give (modulus, a, b, x, T(x))." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "X={0 through 5}; T(x)=(-6x+(-9)) mod 6", "after": "remainder representative always lies in X", "relation": "implies", "rule_id": "finite_total_map", "conditions": [], "explanation": { "zh-Hant": "固定完整有限狀態域與同一更新規則；負整數也取0到n−1的標準餘數。", "en": "Fix the full finite state carrier and the same update rule;negative integers also use the standard remainder from0 through n−1." }, "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": "one step: a·x := mul(-6, 1)", "after": "one step: a·x = -6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先把原係數乘當前狀態，不先把負數截成0。", "en": "Multiply the original coefficient by the current state;do not clip negative values to0." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 one step: a·x，不以小數近似替代。", "en": "Recompute one step: a·x with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-6", "1" ], "output": "-6", "label": "one step: a·x" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "one step: a·x+b := sum(-6, -9)", "after": "one step: a·x+b = -15", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "加上同一常數平移後才取模。", "en": "Add the same offset before reducing modulo n." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 one step: a·x+b，不以小數近似替代。", "en": "Recompute one step: a·x+b with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-6", "-9" ], "output": "-15", "label": "one step: a·x+b" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "reduce -15 modulo 6", "after": "-15=-3·6+3", "relation": "implies", "rule_id": "euclidean_remainder", "conditions": [], "explanation": { "zh-Hant": "此整數等式及0≤餘數