{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261020-r01-discrete-dynamical-system-0026",
  "pilot_number": "R0002-01551",
  "number": 1551,
  "family_id": "qf-app20-dynamics-orbit-prefix",
  "title": {
    "zh-Hant": "第 1551 題：有限離散動力核對",
    "en": "Question 1551: A Fixed-Length Discrete Orbit Prefix"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261020-r01-discrete-dynamical-system-0026",
    "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-orbit-prefix；資料={\"modulus\":5,\"a\":1,\"b\":-7,\"x0\":0,\"steps\":0}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(5, 1, -7, 0, 0, [0])",
      "explanation_zh": "應用基礎核對=(5, 1, -7, 0, 0, [0])；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-orbit-prefix",
      "seed": 202610201026,
      "parameters": {
        "modulus": 5,
        "a": 1,
        "b": -7,
        "x0": 0,
        "steps": 0,
        "domain": "Fin_discrete_dynamics",
        "operation": "dynamics-orbit-prefix",
        "semantic_template": "qf-app20-dynamics-orbit-prefix/{\"a\":1,\"b\":-7,\"modulus\":5,\"steps\":0,\"x0\":0}"
      }
    },
    "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-0026",
    "batch_id": "batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "cd842cc1849a3d83a60a464ef772cee877fbd83ed63657ecee1785b13d60752b",
      "family_id": "qf-app20-dynamics-orbit-prefix",
      "seed": 202610201026,
      "semantic_template": "qf-app20-dynamics-orbit-prefix/{\"a\":1,\"b\":-7,\"modulus\":5,\"steps\":0,\"x0\":0}"
    },
    "fields": {
      "title": "A Fixed-Length Discrete Orbit Prefix",
      "stem": "Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=1, b=-7. 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. x0=0, updates=0. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].",
      "answer": {
        "kind": "expression",
        "value": "(5, 1, -7, 0, 0, [0])",
        "explanation": "Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 1, -7, 0, 0, [0])."
      },
      "skills": [
        "A Fixed-Length Discrete Orbit Prefix"
      ],
      "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-orbit-prefix；資料={\"modulus\":5,\"a\":1,\"b\":-7,\"x0\":0,\"steps\":0}。請依 [modulus,a,b,x0,steps,orbit] 欄序回答。",
        "en": "Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=1, b=-7. 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. x0=0, updates=0. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0]."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "X={0 through 4}; T(x)=(1x+(-7)) mod 5",
      "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": "x₀=0; transitions=0",
      "after": "initial orbit prefix=[0]",
      "relation": "implies",
      "rule_id": "orbit_initial_state",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先保留初始狀態；steps=0仍有一個已知狀態，不是空表。",
        "en": "Retain the initial state;steps=0 still has one known state,not an empty list."
      },
      "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": "computed transitions=0",
      "after": "orbit=[0]; number of entries=1",
      "relation": "implies",
      "rule_id": "orbit_prefix_length",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每次從前一步結果套同一T，所以k次轉移給k+1個狀態。",
        "en": "Apply the same T to the preceding result; k transitions give k+1 states."
      },
      "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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "a finite requested prefix",
      "after": "no unrequested least-period claim",
      "relation": "implies",
      "rule_id": "finite_prefix_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此輸出只列指定前綴；不把少量軌道值當作未核的最小週期證明。",
        "en": "This output is the requested prefix;it is not an unverified least-period proof from a few orbit 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
    }
  ],
  "math_model": {
    "result": [
      5,
      1,
      -7,
      0,
      0,
      [
        0
      ]
    ],
    "evaluations": []
  },
  "conclusion": {
    "canonical_value": "(5, 1, -7, 0, 0, [0])",
    "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-r01-discrete-dynamical-system-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations/questions.jsonl",
      "batch_sha256": "5f5f7ffde34f79b3a160c843b058e6c24e8dd5fda6cb219b2588cfe96e5c846e",
      "original_line": 26,
      "original_line_sha256": "a36350fce17d8f69be7a46f9f28d9398ef391c2c273173cd1cc49d5e3fddd965",
      "question_compact_sha256": "cd842cc1849a3d83a60a464ef772cee877fbd83ed63657ecee1785b13d60752b",
      "english_row_compact_sha256": "50b5fa659a79af1099223ca7a5d10e9b23fe2f32427559e21e3d53f18bb79000",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-discrete-dynamical-system": {
        "sha256": "83db6a4ce40522862935c10fd58632bc890dbcf7e2d2277368dad51ee0b72837",
        "english_sha256": "8513c6a6ed28df30295d27ab05a1d1c2a008427aceaf56ba9bfa33819c320e29",
        "original_path": "public/data/mko/discrete-dynamical-system.json",
        "english_path": "public/data/i18n/editions/mko-discrete-dynamical-system.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "e476360038ede38e6f70fbe8c039b14a4f4bd45937ee88050de8370944911f19",
    "original_candidate_sha256": "82f33fed0c9e4320ad36c3441a27105bdfb2f292fb54b7eeacf22bf9ab90c098",
    "original_candidate_line_sha256": "82f33fed0c9e4320ad36c3441a27105bdfb2f292fb54b7eeacf22bf9ab90c098"
  },
  "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": "ad5dc0b7911b440dc9bebb7f5fc9e3435e656c81e5794a30716520f90b2d8da7",
      "arithmetic_steps": 0
    },
    "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."
  }
}