{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261003-advance-20261004-r05-vector-0056", "pilot_number": "R0070-35931", "number": 35931, "family_id": "qf-vector-coordinate-difference", "title": { "zh-Hant": "第 35931 題：固定基底向量的有方向減法", "en": "Question 35931: Ordered Vector Subtraction in a Fixed Basis" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261003-advance-20261004-r05-vector-0056", "batch_id": "batch-20261003-advance-20261004-r05-vector-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-vector" ], "title_zh": "固定基底向量的有方向減法", "stem_zh": "在固定基底下的實數平面向量 R²，u=(-7, -2)，v=(2, 7)。求 u−v（不可交換順序） 的精確分量。", "answer": { "kind": "expression", "value": "(-9, -9)", "explanation_zh": "分量結果=(-9, -9)。逐分量做有方向減法，交換得相反向量；此處是向量，不把它與幾何點或長度混同。" }, "difficulty": { "band": "upper_secondary", "level": 1, "skills": [ "固定基底向量的有方向減法" ] }, "variant": { "family_id": "qf-vector-coordinate-difference", "seed": 202610045056, "parameters": { "u": [ -7, -2 ], "domain": "reals", "operation": "vector_subtract", "v": [ 2, 7 ], "semantic_template": "qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[-7,-2],\"v\":[2,7]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_integer_rational_arithmetic", "producer_finite_enumeration", "producer_source_snapshot_check", "producer_substantive_template_deduplication" ], "warning_zh": "僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "Codex independent peer (inherited model; exact runtime model not exposed)", "generated_at": "2026-10-03", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261003-advance-20261004-r05-vector-0056", "batch_id": "batch-20261003-advance-20261004-r05-vector-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "c51afaf0fa34d354456f2eb7a625a35eda873b551d742fe14b0e4758ed33777e", "family_id": "qf-vector-coordinate-difference", "seed": 202610045056, "semantic_template": "qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[-7,-2],\"v\":[2,7]}" }, "fields": { "title": "Ordered Vector Subtraction in a Fixed Basis", "stem": "In the fixed basis of the real plane R², u=(-7, -2) and v=(2, 7). Compute the exact components of u−v, preserving the subtraction order.", "answer": { "kind": "expression", "value": "(-9, -9)", "explanation": "Apply ordered subtraction componentwise to obtain (-9, -9). These are vector components, not a geometric point or a length." }, "skills": [ "Ordered Vector Subtraction in a Fixed Basis" ], "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": "在固定基底下的實數平面向量 R²，u=(-7, -2)，v=(2, 7)。求 u−v（不可交換順序） 的精確分量。", "en": "In the fixed basis of the real plane R², u=(-7, -2) and v=(2, 7). Compute the exact components of u−v, preserving the subtraction order." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-vector-coordinate-difference, [domain, operation, u, v]]", "after": "derived = [vector, vector_subtract, reals]", "relation": "implies", "rule_id": "R70_SOURCE_bank_specific_interface", "conditions": [], "explanation": { "zh-Hant": "只使用此bank真正明示的欄位、operation/domain與型別；BASE容許保留額外metadata，不把LDL SAFE/集合上限或其他bank的去重規則搬來。", "en": "Use THIS bank’s actual fields,operation/domain and types.BASE retains permissible extra metadata.Do not import LDL SAFE/set limits or a different bank’s deduplication rules." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反查本題原始操作數、角色與必要條件；重建下一步，不把單向推論倒當等價。數值/有限表核對不是任意命題的形式證明。原題、中英來源、MKO、AST、狀態與歷史均保留。", "en": "Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [[-7, -2], [2, 7], null]", "after": "derived = [two real coordinates in ONE ordered basis]", "relation": "implies", "rule_id": "Vector_GIVEN_common_ordered_basis_RAW_two_SAFE_components", "conditions": [], "explanation": { "zh-Hant": "兩向量座標在同一有序基底，每個raw分量與k驗SAFE；此題向量不是幾何點、長度或內積，不能只比較分量和。", "en": "The vectors use ONE ordered basis.Type ALL TWO RAW SAFE components and k.These are vector coordinates,not points,lengths or a dot product.Component sums alone do not determine equality." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反查本題原始操作數、角色與必要條件；重建下一步，不把單向推論倒當等價。數值/有限表核對不是任意命題的形式證明。原題、中英來源、MKO、AST、狀態與歷史均保留。", "en": "Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Vector ORIGINAL coordinate i=0 := sub(-7, 2)", "after": "Vector ORIGINAL coordinate i=0 = -9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。", "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Vector ORIGINAL coordinate i=0，不以小數近似替代。", "en": "Recompute Vector ORIGINAL coordinate i=0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-7", "2" ], "output": "-9", "label": "Vector ORIGINAL coordinate i=0" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Vector ORIGINAL coordinate i=1 := sub(-2, 7)", "after": "Vector ORIGINAL coordinate i=1 = -9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。", "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Vector ORIGINAL coordinate i=1，不以小數近似替代。", "en": "Recompute Vector ORIGINAL coordinate i=1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-2", "7" ], "output": "-9", "label": "Vector ORIGINAL coordinate i=1" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "source roles = [[-7, -2], [2, 7], null, vector_subtract]", "after": "derived = [[-9, -9], (-9, -9)]", "relation": "implies", "rule_id": "Vector_ORDERED_componentwise_result", "conditions": [], "explanation": { "zh-Hant": "依原有順序逐分量運算，u−v不可倒轉，零純量得零向量。結果是有序tuple，不把它換成集合或長度；衍生分量可超raw SAFE。", "en": "Operate componentwise in ORIGINAL order.u−v must not reverse;zero scaling yields the zero vector.The result is an ORDERED tuple,not a set or length.Derived components may exceed RAW SAFE." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反查本題原始操作數、角色與必要條件；重建下一步，不把單向推論倒當等價。數值/有限表核對不是任意命題的形式證明。原題、中英來源、MKO、AST、狀態與歷史均保留。", "en": "Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved." } }, "arithmetic": null } ], "math_model": { "result": "(-9, -9)" }, "conclusion": { "canonical_value": "(-9, -9)", "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-20261003-advance-20261004-r05-vector-foundations", "batch_path": "public/data/questions/batches/batch-20261003-advance-20261004-r05-vector-foundations/questions.jsonl", "batch_sha256": "dcec7d66d887707f0c830babd46721978583fb3c50e70c97f0049d83f1576701", "original_line": 56, "original_line_sha256": "fbaa87c7816640a409497ca251ad72e8691113b0ce15728c741b611195191aa1", "question_compact_sha256": "c51afaf0fa34d354456f2eb7a625a35eda873b551d742fe14b0e4758ed33777e", "english_row_compact_sha256": "72c3ee81673ee630ea744481d7618c756bfa225eb80db6b6ca1bef13280eccab", "english_manifest_path": "public/data/i18n/editions/batch-20261003-advance-20261004-r05-vector-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261003-advance-20261004-r05-vector-foundations.en.jsonl" }, "mko_bindings": { "mko-vector": { "sha256": "0d8db0f5e7511aaa8401ebe39e808af23991063892ca349e742b0551c91cba3d", "english_sha256": "8b23c6e9698bb13ffb12ef863fc1ba12d88c96f880c720e6504681e634fcd99d", "original_path": "public/data/mko/vector.json", "english_path": "public/data/i18n/editions/mko-vector.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "20c37a7e64ee5969e7606c5cc3a00e575002ebebdfe82873521cc3d7c608caff", "original_candidate_sha256": "0ac072dcb51c0286194bc2e14f477302fe35cd2f6e2d2dadb00eb3ab438a2872", "original_candidate_line_sha256": "0ac072dcb51c0286194bc2e14f477302fe35cd2f6e2d2dadb00eb3ab438a2872" }, "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-20261026-012-p71", "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": "9820ac6529139bd53d516c8bfdfdd41184b0cd52dad69fa433344010db37af80", "question_first_witness_sha256": "c9c939762bbd4d3ea3403084c8e95541dc2c7e56eb82fc74eaba8c80a4517dab" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 5, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "e2ec810e065fa3f7783a3eeffa73485ee464fd75d54b57703d63fe31e0db8d96", "arithmetic_steps": 2 }, "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." } }