{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261021-r04-initial-value-problem-0019", "pilot_number": "R0003-02244", "number": 2244, "family_id": "qf-app21-ivp-first-order-initial-audit", "title": { "zh-Hant": "第 2244 題：多項式初值候選", "en": "Question 2244: An Initial Value Check Is Not an ODE Check" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261021-r04-initial-value-problem-0019", "batch_id": "batch-20261013-advance-20261021-r04-initial-value-problem-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-initial-value-problem" ], "title_zh": "多項式初值候選", "stem_zh": "整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[8,1,-1],\"x0\":1,\"y0\":8}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。", "answer": { "kind": "expression", "value": "([8, 1, -1], 1, 8, 8, 是)", "explanation_zh": "應用擴展核對=([8, 1, -1], 1, 8, 8, 是)；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；不宣稱解存在或唯一；有限網格殘差不是全區間證明。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "多項式初值候選" ] }, "variant": { "family_id": "qf-app21-ivp-first-order-initial-audit", "seed": 202610214019, "parameters": { "coefficients": [ 8, 1, -1 ], "x0": 1, "y0": 8, "domain": "Polynomial_candidate_IVP", "operation": "ivp-first-order-initial-audit", "semantic_template": "qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[8,1,-1],\"x0\":1,\"y0\":8}" } }, "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-r04-initial-value-problem-0019", "batch_id": "batch-20261013-advance-20261021-r04-initial-value-problem-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "8e491cd47cf61dbc835376c5da1beee64b16c06ab9bc432ce976315e14630ca5", "family_id": "qf-app21-ivp-first-order-initial-audit", "seed": 202610214019, "semantic_template": "qf-app21-ivp-first-order-initial-audit/{\"coefficients\":[8,1,-1],\"x0\":1,\"y0\":8}" }, "fields": { "title": "An Initial Value Check Is Not an ODE Check", "stem": "Use the integer polynomial candidate with ascending-power coefficients=[8, 1, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=1, required y0=8. Give (coefficients,x0,y(x0),required y0,match?).", "answer": { "kind": "expression", "value": "([8, 1, -1], 1, 8, 8, Yes)", "explanation": "Substitute the initial point into the candidate. This checks initial data only and does not check the differential equation. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is ([8, 1, -1], 1, 8, 8, Yes)." }, "skills": [ "An Initial Value Check Is Not an ODE Check" ], "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": "整係數多項式候選與明示有限網格。操作=ivp-first-order-initial-audit；資料={\"coefficients\":[8,1,-1],\"x0\":1,\"y0\":8}。請依 [coefficients,x0,y_x0,expected_y,match] 欄序回答。", "en": "Use the integer polynomial candidate with ascending-power coefficients=[8, 1, -1]. Initial data and ODE residuals are separate obligations. Neither a finite grid nor an initial-value match proves existence, uniqueness, a general solution or an identity on an entire interval. x0=1, required y0=8. Give (coefficients,x0,y(x0),required y0,match?)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "candidate polynomial ascending coefficients=[8, 1, -1]", "after": "initial point=1; required y₀=8", "relation": "implies", "rule_id": "candidate_IVP_data", "conditions": [], "explanation": { "zh-Hant": "保留候選多項式及初始資料；網格點不一定包含初始點，需分開代入。", "en": "Retain the polynomial and initial data;the residual grid need not contain the initial point,so evaluate it separately." }, "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": "initial y ascending coefficients=[8, 1, -1]", "after": "evaluate at x=1", "relation": "implies", "rule_id": "polynomial_evaluation_order", "conditions": [], "explanation": { "zh-Hant": "升冪係數用Horner由最高非零項向常數項求值；略去零最高項不改函數，也不改原始係數清單。", "en": "Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original 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": "initial y Horner multiply before degree1 := mul(-1, 1)", "after": "initial y Horner multiply before degree1 = -1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先乘同一代入點。", "en": "Multiply by the same evaluation point first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 initial y Horner multiply before degree1，不以小數近似替代。", "en": "Recompute initial y Horner multiply before degree1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "1" ], "output": "-1", "label": "initial y Horner multiply before degree1" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "initial y Horner add degree1 := sum(-1, 1)", "after": "initial y Horner add degree1 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再加此位置的完整係數。", "en": "Then add the full coefficient at this position." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 initial y Horner add degree1，不以小數近似替代。", "en": "Recompute initial y Horner add degree1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-1", "1" ], "output": "0", "label": "initial y Horner add degree1" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "initial y Horner multiply before degree0 := mul(0, 1)", "after": "initial y Horner multiply before degree0 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "先乘同一代入點。", "en": "Multiply by the same evaluation point first." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 initial y Horner multiply before degree0，不以小數近似替代。", "en": "Recompute initial y Horner multiply before degree0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "1" ], "output": "0", "label": "initial y Horner multiply before degree0" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "initial y Horner add degree0 := sum(0, 8)", "after": "initial y Horner add degree0 = 8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "再加此位置的完整係數。", "en": "Then add the full coefficient at this position." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 initial y Horner add degree0，不以小數近似替代。", "en": "Recompute initial y Horner add degree0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "8" ], "output": "8", "label": "initial y Horner add degree0" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "y(1)=8; required=8", "after": "value condition=是", "relation": "implies", "rule_id": "initial_value_condition", "conditions": [], "explanation": { "zh-Hant": "直接比較初始點的實際函數值，不借用其他網格點。", "en": "Compare the actual function value at the initial point,not another grid point." }, "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": "S008", "uses": [ "G001", "S007" ], "before": "initial data agree or disagree", "after": "no differential-equation assertion from initial data alone", "relation": "implies", "rule_id": "initial_only_scope", "conditions": [], "explanation": { "zh-Hant": "即使初始資料符合，也還沒有核對微分方程，更不是存在唯一性定理。", "en": "Matching initial data do not by themselves check the differential equation or prove an existence-uniqueness theorem." }, "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": [ [ 8, 1, -1 ], 1, "8", 8, true ], "claim_scope": "initial_data_only" }, "conclusion": { "canonical_value": "([8, 1, -1], 1, 8, 8, 是)", "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-20261021-r04-initial-value-problem-foundations", "batch_path": "public/data/questions/batches/batch-20261013-advance-20261021-r04-initial-value-problem-foundations/questions.jsonl", "batch_sha256": "510511592eeb41e0d6eb15d17354e824e2e9a97ffe1195123370f8395c6d3673", "original_line": 19, "original_line_sha256": "9427dd2bac85bf4f53e13c21bbcc195951ba243e3733ce5d7844b80c22dc7177", "question_compact_sha256": "8e491cd47cf61dbc835376c5da1beee64b16c06ab9bc432ce976315e14630ca5", "english_row_compact_sha256": "248eb0e46f1f771a546e79f28bb8a97d375cff7613d599ef5cc25d4955d3a7a4", "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261021-r04-initial-value-problem-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261021-r04-initial-value-problem-foundations.en.jsonl" }, "mko_bindings": { "mko-initial-value-problem": { "sha256": "84ab0a91c84b74df8cfa9a3a8033b7ad83825d55e89f1d0ca847ed81f85e9925", "english_sha256": "9fff0f0029d4033fa0555d7c1431a25e5fe58386793836198e0241c073506dc0", "original_path": "public/data/mko/initial-value-problem.json", "english_path": "public/data/i18n/editions/mko-initial-value-problem.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "09b39eb68bdeb3e1e4277f359de31284852356add54b99a43cf1e38a33d69d15", "original_candidate_sha256": "1efb3caef61c420c9d34de4a057a1a4b3b0d7bcd0ec5215e1b55cea4ca57766e", "original_candidate_line_sha256": "1efb3caef61c420c9d34de4a057a1a4b3b0d7bcd0ec5215e1b55cea4ca57766e" }, "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-p4", "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": "5dbc70ebd5f74b23384c93b63a2b1d6519e206c79f22ae16a6c2d7483cea783f", "question_first_witness_sha256": "cf121c8dd2fec13aa118b04b049ee39494f1d6d9ecdc13514cafb78c0fc021fa" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 8, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "58b7a49621d2b32c42edc62ae8d1be5793ff19c1951d7c7d91a7df06d06fc51a", "arithmetic_steps": 4 }, "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." } }