{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261021-r03-orbit-fixed-periodic-0053", "pilot_number": "R0004-02528", "number": 2528, "family_id": "qf-app21-orbit-least-period", "title": { "zh-Hant": "第 2528 題：有限軌道與週期", "en": "Question 2528: First Repeat, Transient Length and the Starting Point’s Period" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261021-r03-orbit-fixed-periodic-0053", "batch_id": "batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-orbit-fixed-periodic" ], "title_zh": "有限軌道與週期", "stem_zh": "完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":7,\"a\":-1,\"b\":-6,\"x\":5}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。", "answer": { "kind": "expression", "value": "(7, -1, -6, 5, [5, 3], 5, 0, 2)", "explanation_zh": "應用擴展核對=(7, -1, -6, 5, [5, 3], 5, 0, 2)；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；固定點與最小正週期分開；只分類指定有限模型。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "有限軌道與週期" ] }, "variant": { "family_id": "qf-app21-orbit-least-period", "seed": 202610213053, "parameters": { "modulus": 7, "a": -1, "b": -6, "x": 5, "domain": "Finite_orbit_periodicity", "operation": "orbit-least-period", "semantic_template": "qf-app21-orbit-least-period/{\"a\":-1,\"b\":-6,\"modulus\":7,\"x\":5}" } }, "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-r03-orbit-fixed-periodic-0053", "batch_id": "batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "9f915e6db702a8e04259af656bf13e2be823e1344b88bd88ed3bde44f78c39ef", "family_id": "qf-app21-orbit-least-period", "seed": 202610213053, "semantic_template": "qf-app21-orbit-least-period/{\"a\":-1,\"b\":-6,\"modulus\":7,\"x\":5}" }, "fields": { "title": "First Repeat, Transient Length and the Starting Point’s Period", "stem": "Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=-1, b=-6. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=5. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).", "answer": { "kind": "expression", "value": "(7, -1, -6, 5, [5, 3], 5, 0, 2)", "explanation": "Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, -1, -6, 5, [5, 3], 5, 0, 2)." }, "skills": [ "First Repeat, Transient Length and the Starting Point’s Period" ], "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": "完整有限狀態自映射。操作=orbit-least-period；資料={\"modulus\":7,\"a\":-1,\"b\":-6,\"x\":5}。請依 [modulus,a,b,x,trace,repeat,cycle_start,least_period_or_nonperiodic] 欄序回答。", "en": "Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=-1, b=-6. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=5. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "n=7; a=-1; b=-6", "after": "S={0,...,6}; T(z)=(a·z+b) mod n", "relation": "implies", "rule_id": "finite_self_map", "conditions": [], "explanation": { "zh-Hant": "餘數取0到n−1，故每個狀態都有唯一後繼且仍在同一有限集合；負數輸入也不取負餘數。", "en": "Remainders lie in0,...,n−1,so every state has one successor in the same finite set;negative dividends do not produce negative canonical remainders." }, "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": "start=5", "after": "record each state before its first successor calculation", "relation": "implies", "rule_id": "orbit_initial_state", "conditions": [], "explanation": { "zh-Hant": "從指定點本身開始，不把第一次更新值誤記為初始狀態。", "en": "Start with the specified point itself,not its first successor." }, "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": "least period step1 product := mul(-1, 5)", "after": "least period step1 product = -5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先將目前狀態乘以原始係數a，保留其正負號。", "en": "Multiply the current state by the original coefficient a,retaining its sign." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 least period step1 product，不以小數近似替代。", "en": "Recompute least period step1 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "5" ], "output": "-5", "label": "least period step1 product" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "least period step1 affine value := sum(-5, -6)", "after": "least period step1 affine value = -11", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再加上原始常數b，尚未把此整數當成狀態。", "en": "Then add the original constant b;this integer is not yet the canonical state." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 least period step1 affine value，不以小數近似替代。", "en": "Recompute least period step1 affine value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-5", "-6" ], "output": "-11", "label": "least period step1 affine value" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Euclidean division of -11 by 7", "after": "-11=-2·7+3", "relation": "implies", "rule_id": "orbit_successor", "conditions": [], "explanation": { "zh-Hant": "用整除商與非負餘數驗算轉移；n>0且0≤餘數0 and0≤remainder0且0≤餘數0 and0≤remainder