{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261020-r05-dimensional-analysis-0056", "pilot_number": "R0001-01456", "number": 1456, "family_id": "qf-app20-dimension-power", "title": { "zh-Hant": "第 1456 題：量綱指數向量", "en": "Question 1456: Integer Power of a Dimension Vector" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261020-r05-dimensional-analysis-0056", "batch_id": "batch-20261013-advance-20261020-r05-dimensional-analysis-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-dimensional-analysis" ], "title_zh": "量綱指數向量", "stem_zh": "基本量綱指數向量[L,M,T]。操作=dimension-power；資料={\"base\":[2,-1,0],\"exponent\":2}。請依 [base,exponent,result] 欄序回答。", "answer": { "kind": "expression", "value": "([2, -1, 0], 2, [4, -2, 0])", "explanation_zh": "應用基礎核對=([2, -1, 0], 2, [4, -2, 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；量綱一致是必要而非充分條件；不證物理模型正確。" }, "difficulty": { "band": "upper_secondary", "level": 1, "skills": [ "量綱指數向量" ] }, "variant": { "family_id": "qf-app20-dimension-power", "seed": 202610205056, "parameters": { "base": [ 2, -1, 0 ], "exponent": 2, "domain": "Dimension_exponent_vectors", "operation": "dimension-power", "semantic_template": "qf-app20-dimension-power/{\"base\":[2,-1,0],\"exponent\":2}" } }, "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-r05-dimensional-analysis-0056", "batch_id": "batch-20261013-advance-20261020-r05-dimensional-analysis-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "b1023da3526b9cc935674cff039cd207a4e04cdeb52205de6da4abfc5034299b", "family_id": "qf-app20-dimension-power", "seed": 202610205056, "semantic_template": "qf-app20-dimension-power/{\"base\":[2,-1,0],\"exponent\":2}" }, "fields": { "title": "Integer Power of a Dimension Vector", "stem": "Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[2, -1, 0], exponent=2. Give (base vector, exponent, exponent-scaled vector).", "answer": { "kind": "expression", "value": "([2, -1, 0], 2, [4, -2, 0])", "explanation": "Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, -1, 0], 2, [4, -2, 0])." }, "skills": [ "Integer Power of a Dimension Vector" ], "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": "基本量綱指數向量[L,M,T]。操作=dimension-power；資料={\"base\":[2,-1,0],\"exponent\":2}。請依 [base,exponent,result] 欄序回答。", "en": "Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[2, -1, 0], exponent=2. Give (base vector, exponent, exponent-scaled vector)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "dimension coordinate order=[L,M,T]", "after": "left/base=[2, -1, 0]; right=null", "relation": "implies", "rule_id": "dimension_order", "conditions": [], "explanation": { "zh-Hant": "這些是量綱的指數，不是三個量值；順序固定為長度、質量、時間。", "en": "These are dimension exponents,not measured magnitudes;the order is length,mass,time." }, "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": "formal units L^a M^b T^c", "after": "product adds;quotient subtracts;power scales exponents", "relation": "implies", "rule_id": "dimension_exponent_laws", "conditions": [], "explanation": { "zh-Hant": "這是指數法則在固定基本量綱上的逐分量運算。", "en": "Apply exponent laws componentwise in the fixed base dimensions." }, "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": "result exponent L := mul(2, 2)", "after": "result exponent L = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依同一基本量綱的位置運算，不交叉相乘不同座標。", "en": "Operate at the same base-dimension position,not across different coordinates." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 result exponent L，不以小數近似替代。", "en": "Recompute result exponent L with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "2" ], "output": "4", "label": "result exponent L" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "result exponent M := mul(-1, 2)", "after": "result exponent M = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依同一基本量綱的位置運算，不交叉相乘不同座標。", "en": "Operate at the same base-dimension position,not across different coordinates." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 result exponent M，不以小數近似替代。", "en": "Recompute result exponent M with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "2" ], "output": "-2", "label": "result exponent M" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "result exponent T := mul(0, 2)", "after": "result exponent T = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依同一基本量綱的位置運算，不交叉相乘不同座標。", "en": "Operate at the same base-dimension position,not across different coordinates." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 result exponent T，不以小數近似替代。", "en": "Recompute result exponent T with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "2" ], "output": "0", "label": "result exponent T" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "[4, -2, 0]", "after": "formal dimensional result only", "relation": "implies", "rule_id": "dimension_not_magnitude", "conditions": [], "explanation": { "zh-Hant": "負指數表示形式倒數量綱；實際商或負冪仍須量值運算有定義，本題不由量綱證明分母非零。", "en": "Negative exponents denote reciprocal dimensions;an actual quotient or negative power still needs defined magnitude arithmetic,and dimensions alone do not prove a nonzero denominator." }, "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": { "left": [ 2, -1, 0 ], "right": null, "exponent": 2, "result": [ 4, -2, 0 ] }, "conclusion": { "canonical_value": "([2, -1, 0], 2, [4, -2, 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-r05-dimensional-analysis-foundations", "batch_path": "public/data/questions/batches/batch-20261013-advance-20261020-r05-dimensional-analysis-foundations/questions.jsonl", "batch_sha256": "1c6c10d0b9ac12f3099d904d0e58111c0223d555684da4bbff2448eaafa6b3c0", "original_line": 56, "original_line_sha256": "8d13044c4753ec8441adfc346b7a6fc8c77c3f8c16990f0e3ba678a684f68ba4", "question_compact_sha256": "b1023da3526b9cc935674cff039cd207a4e04cdeb52205de6da4abfc5034299b", "english_row_compact_sha256": "1d1735ddd52ac26a8dc1e01bb46bba673d179e04d99c3156b2a434bc97b39edd", "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261020-r05-dimensional-analysis-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261020-r05-dimensional-analysis-foundations.en.jsonl" }, "mko_bindings": { "mko-dimensional-analysis": { "sha256": "05ee29db7e8f4b787e36b2e536eb249e1b92597007252752dd6347b2dffab5eb", "english_sha256": "cc3b3348730aa124197861366975fb0d94ed08956269687fd9bcb2c71ac58e7d", "original_path": "public/data/mko/dimensional-analysis.json", "english_path": "public/data/i18n/editions/mko-dimensional-analysis.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "b9fe70a7a264c90c20349be0449b2bd1a1b4d0e0a97092bc9fbcb7ca36ce0694", "original_candidate_sha256": "01e17b98783d8ce8f9e26f0a7e4bf2fb5a7ae7e3219e693ccf89f776421098cc", "original_candidate_line_sha256": "01e17b98783d8ce8f9e26f0a7e4bf2fb5a7ae7e3219e693ccf89f776421098cc" }, "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-p2", "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": "c3b44ce8ed60ada6902fb206627537d5f90ba2a867eaba43a36f903280c496cc", "question_first_witness_sha256": "3e91602adc7f81a2671a2a3ffaf6a447028b56abc5853e8271673d76f30cd7ad" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 6, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "591612cfc033297cca8a6ac0b6581ec822cbb19e02245346b6efb16b3fa47edf", "arithmetic_steps": 3 }, "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." } }