{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261023-r02-linear-combination-span-0009", "pilot_number": "R0020-10859", "number": 10859, "family_id": "qf-mod23-span-weighted-sum", "title": { "zh-Hant": "第 10859 題：有限線性組合", "en": "Question 10859: An Ordered Generator-Coefficient Weighted Sum" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261023-r02-linear-combination-span-0009", "batch_id": "batch-20261013-advance-20261023-r02-linear-combination-span-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-linear-combination-span" ], "title_zh": "有限線性組合", "stem_zh": "固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-4,-2,1]}。請依 [vectors,coefficients,sum] 欄序回答。", "answer": { "kind": "expression", "value": "([[1, 0], [0, 1], [1, 1]], [-4, -2, 1], [-3, -1])", "explanation_zh": "現代橋梁核對=([[1, 0], [0, 1], [1, 1]], [-4, -2, 1], [-3, -1])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "有限線性組合" ] }, "variant": { "family_id": "qf-mod23-span-weighted-sum", "seed": 202610232009, "parameters": { "vectors": [ [ 1, 0 ], [ 0, 1 ], [ 1, 1 ] ], "coefficients": [ -4, -2, 1 ], "domain": "Finite_coordinate_linear_combinations", "operation": "span-weighted-sum", "semantic_template": "qf-mod23-span-weighted-sum/{\"coefficients\":[-4,-2,1],\"vectors\":[[0,1],[1,0],[1,1]]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、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-20261023-r02-linear-combination-span-0009", "batch_id": "batch-20261013-advance-20261023-r02-linear-combination-span-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "d42b5cc93415bfeea2bd5ab67e22a1a2b87534c839d923adbc4ff3ac3cb6c281", "family_id": "qf-mod23-span-weighted-sum", "seed": 202610232009, "semantic_template": "qf-mod23-span-weighted-sum/{\"coefficients\":[-4,-2,1],\"vectors\":[[0,1],[1,0],[1,1]]}" }, "fields": { "title": "An Ordered Generator-Coefficient Weighted Sum", "stem": "Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-4, -2, 1]. Give (original vectors,original coefficients,the coordinate sum).", "answer": { "kind": "expression", "value": "([[1, 0], [0, 1], [1, 1]], [-4, -2, 1], [-3, -1])", "explanation": "Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-4, -2, 1], [-3, -1])." }, "skills": [ "An Ordered Generator-Coefficient Weighted Sum" ], "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": "固定維度與明示有限生成族。操作=span-weighted-sum；資料={\"vectors\":[[1,0],[0,1],[1,1]],\"coefficients\":[-4,-2,1]}。請依 [vectors,coefficients,sum] 欄序回答。", "en": "Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-4, -2, 1]. Give (original vectors,original coefficients,the coordinate sum)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "fixed real coordinate space;span uses all finite real linear combinations", "after": "given coefficient witness and existence of some coefficients are separate", "relation": "implies", "rule_id": "span_quantifier_and_witness_scope", "conditions": [], "explanation": { "zh-Hant": "span量化所有有限實係數組合；本題的整數／有理資料是明示切片，一個指定見證失敗不等於所有係數都失敗。", "en": "Span quantifies all finite real linear combinations.These integer/rational instances are declared slices;a failed designated witness does not show every coefficient choice fails." }, "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": "original ordered generators=[[1, 0], [0, 1], [1, 1]];aligned coefficients=[-4, -2, 1]", "after": "fixed coordinate dimension=2;generator duplicates/order retained", "relation": "implies", "rule_id": "span_generator_coefficient_pairing", "conditions": [], "explanation": { "zh-Hant": "每個係數只配自己的生成元，保留重複與順序；不要依semantic身份排序或去重，亦不把有限組合換成無窮級數。", "en": "Pair each coefficient with its own generator,retaining duplicates and order.Do not sort/deduplicate by semantic identity or replace a finite combination by an infinite series." }, "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": "generator0 coordinate0 contribution := mul(-4, 1)", "after": "generator0 coordinate0 contribution = -4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。", "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 generator0 coordinate0 contribution，不以小數近似替代。", "en": "Recompute generator0 coordinate0 contribution with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-4", "1" ], "output": "-4", "label": "generator0 coordinate0 contribution" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "generator1 coordinate0 contribution := mul(-2, 0)", "after": "generator1 coordinate0 contribution = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。", "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 generator1 coordinate0 contribution，不以小數近似替代。", "en": "Recompute generator1 coordinate0 contribution with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "0" ], "output": "0", "label": "generator1 coordinate0 contribution" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "generator2 coordinate0 contribution := mul(1, 1)", "after": "generator2 coordinate0 contribution = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。", "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 generator2 coordinate0 contribution，不以小數近似替代。", "en": "Recompute generator2 coordinate0 contribution with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "generator2 coordinate0 contribution" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "weighted coordinate0 complete sum := sum(-4, 0, 1)", "after": "weighted coordinate0 complete sum = -3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。", "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 weighted coordinate0 complete sum，不以小數近似替代。", "en": "Recompute weighted coordinate0 complete sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-4", "0", "1" ], "output": "-3", "label": "weighted coordinate0 complete sum" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "generator0 coordinate1 contribution := mul(-4, 0)", "after": "generator0 coordinate1 contribution = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。", "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 generator0 coordinate1 contribution，不以小數近似替代。", "en": "Recompute generator0 coordinate1 contribution with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-4", "0" ], "output": "0", "label": "generator0 coordinate1 contribution" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "generator1 coordinate1 contribution := mul(-2, 1)", "after": "generator1 coordinate1 contribution = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。", "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 generator1 coordinate1 contribution，不以小數近似替代。", "en": "Recompute generator1 coordinate1 contribution with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "1" ], "output": "-2", "label": "generator1 coordinate1 contribution" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "generator2 coordinate1 contribution := mul(1, 1)", "after": "generator2 coordinate1 contribution = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。", "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 generator2 coordinate1 contribution，不以小數近似替代。", "en": "Recompute generator2 coordinate1 contribution with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "generator2 coordinate1 contribution" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "weighted coordinate1 complete sum := sum(0, -2, 1)", "after": "weighted coordinate1 complete sum = -1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。", "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 weighted coordinate1 complete sum，不以小數近似替代。", "en": "Recompute weighted coordinate1 complete sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "-2", "1" ], "output": "-1", "label": "weighted coordinate1 complete sum" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "all ordered weighted coordinates=[-3, -1]", "after": "finite combination=[-3, -1]", "relation": "implies", "rule_id": "span_complete_componentwise_reconstruction", "conditions": [], "explanation": { "zh-Hant": "同一座標內把全部生成元的貢獻加完，不取內積或長度代替向量。", "en": "Add every generator contribution at the same coordinate;an inner product or norm is not the requested vector." }, "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": [ [ [ 1, 0 ], [ 0, 1 ], [ 1, 1 ] ], [ -4, -2, 1 ], [ -3, -1 ] ] }, "conclusion": { "canonical_value": "([[1, 0], [0, 1], [1, 1]], [-4, -2, 1], [-3, -1])", "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-20261023-r02-linear-combination-span-foundations", "batch_path": "public/data/questions/batches/batch-20261013-advance-20261023-r02-linear-combination-span-foundations/questions.jsonl", "batch_sha256": "da062918e67f92a171ca3b576c4e8fd07cc820916a8047a29a22a0da6e0df6d4", "original_line": 9, "original_line_sha256": "8e573e1fc63c6b9465cc463d6693853bfdea0f7baab196832d75b7b57b730b3d", "question_compact_sha256": "d42b5cc93415bfeea2bd5ab67e22a1a2b87534c839d923adbc4ff3ac3cb6c281", "english_row_compact_sha256": "97c64424961fd888f592936c5662891d85d827a76a24ef44d965aad12f8d927c", "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261023-r02-linear-combination-span-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261023-r02-linear-combination-span-foundations.en.jsonl" }, "mko_bindings": { "mko-linear-combination-span": { "sha256": "6f4f8ff504708ab9492740d70255cb2d85c406d8eaa891eef387935b3d46dad3", "english_sha256": "3d16797b83376915e08d16c68eee0e87ae7d9b28ca5d5ba76a9e2466ce8195b3", "original_path": "public/data/mko/linear-combination-span.json", "english_path": "public/data/i18n/editions/mko-linear-combination-span.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "bd67681fffd9e29972c322d497cd945a668a794dd25e0c8cd26f1a6ce22e4dfb", "original_candidate_sha256": "773c273ce8f07cef88ff69cfd848c2f7549bc50c5aa8d2cab1932f5d427092ba", "original_candidate_line_sha256": "773c273ce8f07cef88ff69cfd848c2f7549bc50c5aa8d2cab1932f5d427092ba" }, "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-20261026-006-p21", "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": "b4c47938fd38085d44967226cfa8337d61334455472e7307c60d3a488cef598f", "question_first_witness_sha256": "3c23baa97a5933bc07b44f2257017ef4e845bf1429bd6f5381410681c43b97e4" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 11, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "cf0e51f74d105c86dc6b78aeb4bf2160c4d0cd9c68927766e1fb94efd90365a1", "arithmetic_steps": 8 }, "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." } }