{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261027-r04-real-bilinear-form-0056", "pilot_number": "R0024-12756", "number": 12756, "family_id": "qf-mod27-bilinear-second-slot-identity", "title": { "zh-Hant": "第 12756 題：實雙線性形式", "en": "Question 12756: Linearity in the Second Slot with the First Slot Fixed" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261027-r04-real-bilinear-form-0056", "batch_id": "batch-20261014-advance-20261027-r04-real-bilinear-form-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-real-bilinear-form" ], "title_zh": "實雙線性形式", "stem_zh": "雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[-7,2],[-2,1]],\"x\":[-2,2],\"u\":[1,-2],\"v\":[2,1],\"alpha\":-2,\"beta\":2}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。", "answer": { "kind": "expression", "value": "(-2, 2, [2, 6], 8, 8, 是)", "explanation_zh": "Wave4根節點核對=(-2, 2, [2, 6], 8, 8, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；矩陣公式提供全稱座標證書；不推出對稱正定非退化或內積。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "實雙線性形式" ] }, "variant": { "family_id": "qf-mod27-bilinear-second-slot-identity", "seed": 202610274056, "parameters": { "matrix": [ [ -7, 2 ], [ -2, 1 ] ], "x": [ -2, 2 ], "u": [ 1, -2 ], "v": [ 2, 1 ], "alpha": -2, "beta": 2, "domain": "Exact_matrix_real_bilinear_form", "operation": "bilinear-second-slot-identity", "semantic_template": "qf-mod27-bilinear-second-slot-identity/{\"alpha\":-2,\"beta\":2,\"matrix\":[[-7,2],[-2,1]],\"u\":[1,-2],\"v\":[2,1],\"x\":[-2,2]}" } }, "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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261027-r04-real-bilinear-form-0056", "batch_id": "batch-20261014-advance-20261027-r04-real-bilinear-form-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "9798a009a8d8f5b026498944893883faa5fe2404274ae61297e69d5b4bdd24f5", "family_id": "qf-mod27-bilinear-second-slot-identity", "seed": 202610274056, "semantic_template": "qf-mod27-bilinear-second-slot-identity/{\"alpha\":-2,\"beta\":2,\"matrix\":[[-7,2],[-2,1]],\"u\":[1,-2],\"v\":[2,1],\"x\":[-2,2]}" }, "fields": { "title": "Linearity in the Second Slot with the First Slot Fixed", "stem": "Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-7, 2], [-2, 1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[-2, 2], u=[1, -2], v=[2, 1], alpha=-2, beta=2. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).", "answer": { "kind": "expression", "value": "(-2, 2, [2, 6], 8, 8, Yes)", "explanation": "Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, 2, [2, 6], 8, 8, Yes)." }, "skills": [ "Linearity in the Second Slot with the First Slot Fixed" ], "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": "雙線性必須分別核第一槽與第二槽。操作=bilinear-second-slot-identity；資料={\"matrix\":[[-7,2],[-2,1]],\"x\":[-2,2],\"u\":[1,-2],\"v\":[2,1],\"alpha\":-2,\"beta\":2}。請依 [alpha,beta,combined,lhs,rhs,equal] 欄序回答。", "en": "Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-7, 2], [-2, 1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[-2, 2], u=[1, -2], v=[2, 1], alpha=-2, beta=2. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "fixed real matrix=[[-7, 2], [-2, 1]];B(x,y)=sum_i sum_j x_i*A_ij*y_j", "after": "distributivity proves real linearity in each slot separately for all vectors and scalars", "relation": "implies", "rule_id": "matrix_formula_bilinearity_universal_not_samples", "conditions": [], "explanation": { "zh-Hant": "有限雙重和中實數分配律可把任意第一槽或第二槽線性組合展開，得到各自的全稱線性證書；整數實例只是此固定公式的求值，不是黑盒抽樣證明。", "en": "Distributivity expands any real linear combination in either slot of the finite double sum,giving separate universal linearity certificates.Integer instances evaluate this fixed formula,not prove a black box by sampling." }, "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": "combined coordinate0 alpha term := mul(-2, 1)", "after": "combined coordinate0 alpha term = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 combined coordinate0 alpha term，不以小數近似替代。", "en": "Recompute combined coordinate0 alpha term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "1" ], "output": "-2", "label": "combined coordinate0 alpha term" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "combined coordinate0 beta term := mul(2, 2)", "after": "combined coordinate0 beta term = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 combined coordinate0 beta term，不以小數近似替代。", "en": "Recompute combined coordinate0 beta term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "2" ], "output": "4", "label": "combined coordinate0 beta term" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "combined coordinate0 sum := sum(-2, 4)", "after": "combined coordinate0 sum = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 combined coordinate0 sum，不以小數近似替代。", "en": "Recompute combined coordinate0 sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2", "4" ], "output": "2", "label": "combined coordinate0 sum" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "combined coordinate1 alpha term := mul(-2, -2)", "after": "combined coordinate1 alpha term = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 combined coordinate1 alpha term，不以小數近似替代。", "en": "Recompute combined coordinate1 alpha term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "-2" ], "output": "4", "label": "combined coordinate1 alpha term" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "combined coordinate1 beta term := mul(2, 1)", "after": "combined coordinate1 beta term = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 combined coordinate1 beta term，不以小數近似替代。", "en": "Recompute combined coordinate1 beta term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "1" ], "output": "2", "label": "combined coordinate1 beta term" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "combined coordinate1 sum := sum(4, 2)", "after": "combined coordinate1 sum = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 combined coordinate1 sum，不以小數近似替代。", "en": "Recompute combined coordinate1 sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "4", "2" ], "output": "6", "label": "combined coordinate1 sum" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "expanding second slot;alpha=-2;beta=2;u=[1, -2];v=[2, 1];fixed other input=[-2, 2]", "after": "combined=[2, 6];other slot held fixed", "relation": "implies", "rule_id": "bilinear_selected_slot_not_symmetry", "conditions": [], "explanation": { "zh-Hant": "只在指定槽代入alpha*u+beta*v，另一槽固定；這不是利用對稱性把兩槽互換。", "en": "Substitute alpha*u+beta*v only in the selected slot,holding the other input fixed.This does not exchange slots by symmetry." }, "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": "S009", "uses": [ "G001", "S008" ], "before": "combination lhs:matrix=[[-7, 2], [-2, 1]];first slot=[-2, 2];second slot=[2, 6]", "after": "B(first,second)=first^T*(matrix*second);slot order fixed", "relation": "implies", "rule_id": "bilinear_ordered_matrix_slots", "conditions": [], "explanation": { "zh-Hant": "先算A乘第二槽，再與第一槽內積；一般A不對稱，不可交換兩槽。", "en": "First multiply A by the second slot,then dot with the first.A general matrix is not symmetric,so the two slots cannot be exchanged." }, "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": "S010", "uses": [ "G001", "S009" ], "before": "combination lhs matrix row0 column0 term := mul(-7, 2)", "after": "combination lhs matrix row0 column0 term = -14", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 combination lhs matrix row0 column0 term，不以小數近似替代。", "en": "Recompute combination lhs matrix row0 column0 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-7", "2" ], "output": "-14", "label": "combination lhs matrix row0 column0 term" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "combination lhs matrix row0 column1 term := mul(2, 6)", "after": "combination lhs matrix row0 column1 term = 12", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 combination lhs matrix row0 column1 term，不以小數近似替代。", "en": "Recompute combination lhs matrix row0 column1 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "6" ], "output": "12", "label": "combination lhs matrix row0 column1 term" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "combination lhs matrix-times-second row0 := sum(-14, 12)", "after": "combination lhs matrix-times-second row0 = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 combination lhs matrix-times-second row0，不以小數近似替代。", "en": "Recompute combination lhs matrix-times-second row0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-14", "12" ], "output": "-2", "label": "combination lhs matrix-times-second row0" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "combination lhs matrix row1 column0 term := mul(-2, 2)", "after": "combination lhs matrix row1 column0 term = -4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 combination lhs matrix row1 column0 term，不以小數近似替代。", "en": "Recompute combination lhs matrix row1 column0 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "2" ], "output": "-4", "label": "combination lhs matrix row1 column0 term" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "combination lhs matrix row1 column1 term := mul(1, 6)", "after": "combination lhs matrix row1 column1 term = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 combination lhs matrix row1 column1 term，不以小數近似替代。", "en": "Recompute combination lhs matrix row1 column1 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "6" ], "output": "6", "label": "combination lhs matrix row1 column1 term" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "combination lhs matrix-times-second row1 := sum(-4, 6)", "after": "combination lhs matrix-times-second row1 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 combination lhs matrix-times-second row1，不以小數近似替代。", "en": "Recompute combination lhs matrix-times-second row1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-4", "6" ], "output": "2", "label": "combination lhs matrix-times-second row1" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "combination lhs first-slot coordinate0 term := mul(-2, -2)", "after": "combination lhs first-slot coordinate0 term = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 combination lhs first-slot coordinate0 term，不以小數近似替代。", "en": "Recompute combination lhs first-slot coordinate0 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "-2" ], "output": "4", "label": "combination lhs first-slot coordinate0 term" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "combination lhs first-slot coordinate1 term := mul(2, 2)", "after": "combination lhs first-slot coordinate1 term = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 combination lhs first-slot coordinate1 term，不以小數近似替代。", "en": "Recompute combination lhs first-slot coordinate1 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "2" ], "output": "4", "label": "combination lhs first-slot coordinate1 term" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "combination lhs final first-slot dot sum := sum(4, 4)", "after": "combination lhs final first-slot dot sum = 8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 combination lhs final first-slot dot sum，不以小數近似替代。", "en": "Recompute combination lhs final first-slot dot sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "4", "4" ], "output": "8", "label": "combination lhs final first-slot dot sum" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "u form:matrix=[[-7, 2], [-2, 1]];first slot=[-2, 2];second slot=[1, -2]", "after": "B(first,second)=first^T*(matrix*second);slot order fixed", "relation": "implies", "rule_id": "bilinear_ordered_matrix_slots", "conditions": [], "explanation": { "zh-Hant": "先算A乘第二槽，再與第一槽內積；一般A不對稱，不可交換兩槽。", "en": "First multiply A by the second slot,then dot with the first.A general matrix is not symmetric,so the two slots cannot be exchanged." }, "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": "S020", "uses": [ "G001", "S019" ], "before": "u form matrix row0 column0 term := mul(-7, 1)", "after": "u form matrix row0 column0 term = -7", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 u form matrix row0 column0 term，不以小數近似替代。", "en": "Recompute u form matrix row0 column0 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-7", "1" ], "output": "-7", "label": "u form matrix row0 column0 term" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "u form matrix row0 column1 term := mul(2, -2)", "after": "u form matrix row0 column1 term = -4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 u form matrix row0 column1 term，不以小數近似替代。", "en": "Recompute u form matrix row0 column1 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "-2" ], "output": "-4", "label": "u form matrix row0 column1 term" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "u form matrix-times-second row0 := sum(-7, -4)", "after": "u form matrix-times-second row0 = -11", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 u form matrix-times-second row0，不以小數近似替代。", "en": "Recompute u form matrix-times-second row0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-7", "-4" ], "output": "-11", "label": "u form matrix-times-second row0" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "u form matrix row1 column0 term := mul(-2, 1)", "after": "u form matrix row1 column0 term = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 u form matrix row1 column0 term，不以小數近似替代。", "en": "Recompute u form matrix row1 column0 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "1" ], "output": "-2", "label": "u form matrix row1 column0 term" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "u form matrix row1 column1 term := mul(1, -2)", "after": "u form matrix row1 column1 term = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 u form matrix row1 column1 term，不以小數近似替代。", "en": "Recompute u form matrix row1 column1 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "-2" ], "output": "-2", "label": "u form matrix row1 column1 term" } }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "u form matrix-times-second row1 := sum(-2, -2)", "after": "u form matrix-times-second row1 = -4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 u form matrix-times-second row1，不以小數近似替代。", "en": "Recompute u form matrix-times-second row1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2", "-2" ], "output": "-4", "label": "u form matrix-times-second row1" } }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "u form first-slot coordinate0 term := mul(-2, -11)", "after": "u form first-slot coordinate0 term = 22", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 u form first-slot coordinate0 term，不以小數近似替代。", "en": "Recompute u form first-slot coordinate0 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "-11" ], "output": "22", "label": "u form first-slot coordinate0 term" } }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "u form first-slot coordinate1 term := mul(2, -4)", "after": "u form first-slot coordinate1 term = -8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 u form first-slot coordinate1 term，不以小數近似替代。", "en": "Recompute u form first-slot coordinate1 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "-4" ], "output": "-8", "label": "u form first-slot coordinate1 term" } }, { "id": "S028", "uses": [ "G001", "S027" ], "before": "u form final first-slot dot sum := sum(22, -8)", "after": "u form final first-slot dot sum = 14", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 u form final first-slot dot sum，不以小數近似替代。", "en": "Recompute u form final first-slot dot sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "22", "-8" ], "output": "14", "label": "u form final first-slot dot sum" } }, { "id": "S029", "uses": [ "G001", "S028" ], "before": "v form:matrix=[[-7, 2], [-2, 1]];first slot=[-2, 2];second slot=[2, 1]", "after": "B(first,second)=first^T*(matrix*second);slot order fixed", "relation": "implies", "rule_id": "bilinear_ordered_matrix_slots", "conditions": [], "explanation": { "zh-Hant": "先算A乘第二槽，再與第一槽內積；一般A不對稱，不可交換兩槽。", "en": "First multiply A by the second slot,then dot with the first.A general matrix is not symmetric,so the two slots cannot be exchanged." }, "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": "S030", "uses": [ "G001", "S029" ], "before": "v form matrix row0 column0 term := mul(-7, 2)", "after": "v form matrix row0 column0 term = -14", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 v form matrix row0 column0 term，不以小數近似替代。", "en": "Recompute v form matrix row0 column0 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-7", "2" ], "output": "-14", "label": "v form matrix row0 column0 term" } }, { "id": "S031", "uses": [ "G001", "S030" ], "before": "v form matrix row0 column1 term := mul(2, 1)", "after": "v form matrix row0 column1 term = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 v form matrix row0 column1 term，不以小數近似替代。", "en": "Recompute v form matrix row0 column1 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "1" ], "output": "2", "label": "v form matrix row0 column1 term" } }, { "id": "S032", "uses": [ "G001", "S031" ], "before": "v form matrix-times-second row0 := sum(-14, 2)", "after": "v form matrix-times-second row0 = -12", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 v form matrix-times-second row0，不以小數近似替代。", "en": "Recompute v form matrix-times-second row0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-14", "2" ], "output": "-12", "label": "v form matrix-times-second row0" } }, { "id": "S033", "uses": [ "G001", "S032" ], "before": "v form matrix row1 column0 term := mul(-2, 2)", "after": "v form matrix row1 column0 term = -4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 v form matrix row1 column0 term，不以小數近似替代。", "en": "Recompute v form matrix row1 column0 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "2" ], "output": "-4", "label": "v form matrix row1 column0 term" } }, { "id": "S034", "uses": [ "G001", "S033" ], "before": "v form matrix row1 column1 term := mul(1, 1)", "after": "v form matrix row1 column1 term = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 v form matrix row1 column1 term，不以小數近似替代。", "en": "Recompute v form matrix row1 column1 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "v form matrix row1 column1 term" } }, { "id": "S035", "uses": [ "G001", "S034" ], "before": "v form matrix-times-second row1 := sum(-4, 1)", "after": "v form matrix-times-second row1 = -3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 v form matrix-times-second row1，不以小數近似替代。", "en": "Recompute v form matrix-times-second row1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-4", "1" ], "output": "-3", "label": "v form matrix-times-second row1" } }, { "id": "S036", "uses": [ "G001", "S035" ], "before": "v form first-slot coordinate0 term := mul(-2, -12)", "after": "v form first-slot coordinate0 term = 24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 v form first-slot coordinate0 term，不以小數近似替代。", "en": "Recompute v form first-slot coordinate0 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "-12" ], "output": "24", "label": "v form first-slot coordinate0 term" } }, { "id": "S037", "uses": [ "G001", "S036" ], "before": "v form first-slot coordinate1 term := mul(2, -3)", "after": "v form first-slot coordinate1 term = -6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 v form first-slot coordinate1 term，不以小數近似替代。", "en": "Recompute v form first-slot coordinate1 term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "-3" ], "output": "-6", "label": "v form first-slot coordinate1 term" } }, { "id": "S038", "uses": [ "G001", "S037" ], "before": "v form final first-slot dot sum := sum(24, -6)", "after": "v form final first-slot dot sum = 18", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 v form final first-slot dot sum，不以小數近似替代。", "en": "Recompute v form final first-slot dot sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "24", "-6" ], "output": "18", "label": "v form final first-slot dot sum" } }, { "id": "S039", "uses": [ "G001", "S038" ], "before": "rhs alpha form value := mul(-2, 14)", "after": "rhs alpha form value = -28", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 rhs alpha form value，不以小數近似替代。", "en": "Recompute rhs alpha form value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "14" ], "output": "-28", "label": "rhs alpha form value" } }, { "id": "S040", "uses": [ "G001", "S039" ], "before": "rhs beta form value := mul(2, 18)", "after": "rhs beta form value = 36", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 rhs beta form value，不以小數近似替代。", "en": "Recompute rhs beta form value with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "18" ], "output": "36", "label": "rhs beta form value" } }, { "id": "S041", "uses": [ "G001", "S040" ], "before": "combination rhs sum := sum(-28, 36)", "after": "combination rhs sum = 8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。", "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 combination rhs sum，不以小數近似替代。", "en": "Recompute combination rhs sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-28", "36" ], "output": "8", "label": "combination rhs sum" } }, { "id": "S042", "uses": [ "G001", "S041" ], "before": "fixed matrix formula;second slot expanded;lhs=8;rhs=8", "after": "equal=是;universal justification is matrix distributivity,not one passing sample", "relation": "implies", "rule_id": "bilinear_slot_identity_and_formula_certificate", "conditions": [], "explanation": { "zh-Hant": "本題數值兩側一致回核代值；對所有實數端點與純量的結論來自前述矩陣分配律，不由這一次相等推廣。", "en": "The equal numerical sides check this substitution.The all-real-vector-and-scalar result follows from matrix distributivity above,not extrapolation from one passing instance." }, "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": [ -2, 2, [ 2, 6 ], 8, 8, true ] }, "conclusion": { "canonical_value": "(-2, 2, [2, 6], 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-20261014-advance-20261027-r04-real-bilinear-form-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261027-r04-real-bilinear-form-foundations/questions.jsonl", "batch_sha256": "79ffb57ba6a4960f78c0a25d3bd6db5c8a35f5fb9b48e8379516b459afa44c19", "original_line": 56, "original_line_sha256": "119ce5b5875a478ebf89a9ba6fdc4732d7929e70a820682ca38048eb4a4e5a81", "question_compact_sha256": "9798a009a8d8f5b026498944893883faa5fe2404274ae61297e69d5b4bdd24f5", "english_row_compact_sha256": "ec3083ed28181f6ded94a7d073e68a16873cb19ca42ad476761c27f905f580c9", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261027-r04-real-bilinear-form-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261027-r04-real-bilinear-form-foundations.en.jsonl" }, "mko_bindings": { "mko-real-bilinear-form": { "sha256": "fa2ccad4a26e9cf94b590f02594c9e1c0db0ba86fc50c60ee410e2b934d4c7d5", "english_sha256": "f3af9343056ca2070bf7df334b6c362b6abb13d0b8c5cb69981beec9201b8e72", "original_path": "public/data/mko/real-bilinear-form.json", "english_path": "public/data/i18n/editions/mko-real-bilinear-form.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "17a80c8482f64498e5a88bd4368d0a11b92d45c259784a0a536a29fbb98662d5", "original_candidate_sha256": "2997de12877fb9e756673542a51db56f81bc9fac724ee4fce9e4d663fc529a99", "original_candidate_line_sha256": "2997de12877fb9e756673542a51db56f81bc9fac724ee4fce9e4d663fc529a99" }, "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-p25", "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": "1c8f6c2c6c75852cb93c08235684ce9f32ebe86065c58c693a04257e772c2d35", "question_first_witness_sha256": "05fd764b5f9f7fff6185003efe576d87632f6daf40606ac3839a7feafe4f80d8" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 42, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "366f6d06773b54e54707a472b02fd08e8cfc49a34f608297002761eede303afd", "arithmetic_steps": 36 }, "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." } }