{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261105-r02-dual-vector-space-0083", "pilot_number": "R0036-18708", "number": 18708, "family_id": "qf-mod36-dual-coordinate-change-boundary", "title": { "zh-Hant": "第 18708 題：對偶向量空間", "en": "Question 18708: Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261105-r02-dual-vector-space-0083", "batch_id": "batch-20261014-advance-20261105-r02-dual-vector-space-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-dual-vector-space" ], "title_zh": "對偶向量空間", "stem_zh": "有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,8],[0,1]],\"primal_coordinates\":[9,3],\"covector_coordinates\":[2,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([-15, 3], [2, 24], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)", "explanation_zh": "Wave5第五層核對=([-15, 3], [2, 24], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)；依完整明示資料獨立重算；dual元素是線性泛函而不是primal vector；dual basis須滿足phi_i(v_j)=delta_ij；無額外內積不得宣稱V與V*典範識別；線性須由symbolic row certificate而非有限取樣。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "對偶向量空間" ] }, "variant": { "family_id": "qf-mod36-dual-coordinate-change-boundary", "seed": 202611052083, "parameters": { "change_of_basis": [ [ 1, 8 ], [ 0, 1 ] ], "primal_coordinates": [ 9, 3 ], "covector_coordinates": [ 2, 8 ], "domain": "Finite_dimensional_real_dual_coordinates", "operation": "dual-coordinate-change-boundary", "semantic_template": "qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,8],[0,1]],\"covector_coordinates\":[2,8],\"primal_coordinates\":[9,3]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或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-20261105-r02-dual-vector-space-0083", "batch_id": "batch-20261014-advance-20261105-r02-dual-vector-space-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "66c7043da6a1686aa50b8f48d6eee3d86eafb4a640d2597d2f27ef475bd107e7", "family_id": "qf-mod36-dual-coordinate-change-boundary", "seed": 202611052083, "semantic_template": "qf-mod36-dual-coordinate-change-boundary/{\"change_of_basis\":[[1,8],[0,1]],\"covector_coordinates\":[2,8],\"primal_coordinates\":[9,3]}" }, "fields": { "title": "Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws", "stem": "Use invertible change matrix P=[[1, 8], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[9, 3] and old covector row=[2, 8]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).", "answer": { "kind": "expression", "value": "([-15, 3], [2, 24], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)", "explanation": "Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-15, 3], [2, 24], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)." }, "skills": [ "Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws" ], "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": "有限維R^n中的線性泛函與明示基底座標。操作=dual-coordinate-change-boundary；資料={\"change_of_basis\":[[1,8],[0,1]],\"primal_coordinates\":[9,3],\"covector_coordinates\":[2,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use invertible change matrix P=[[1, 8], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[9, 3] and old covector row=[2, 8]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=dual-coordinate-change-boundary;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "原題歷史介面標記仍在快照；這裡直接給原始欄位順序，主文不需JavaScript或外部表。描述欄位不自動證成任何數學聲稱。", "en": "Keep the historical interface marker in the original snapshot while listing the original ordered fields here.No JavaScript or external table is needed for the main text,and field labels are not mathematical evidence." }, "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": "dual change augmented P with identity:complete ordered matrix=[[1, 8, 1, 0], [0, 1, 0, 1]];rows=2;columns=4", "after": "use exact invertible ROW operations only;original COLUMN roles/order stay fixed", "relation": "implies", "rule_id": "finite_linear_source_matrix_not_semantic_reordering", "conditions": [], "explanation": { "zh-Hant": "基向量放欄且保原順序；semantic用的排序不能拿來重排係數或候選泛函。列運算是左乘可逆映射，保核、欄相依及rank，不宣稱實際欄空間字面不變。", "en": "Keep original basis vectors in ordered columns.Semantic sorting must not reorder coefficients or candidate rows.Row operations multiply by an invertible map,retaining kernel,column relations and rank,not the literal original column space." }, "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": "dual change augmented P with identity pivot0 column0 normalized entry0 := div(1, 1)", "after": "dual change augmented P with identity pivot0 column0 normalized entry0 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual change augmented P with identity pivot0 column0 normalized entry0，不以小數近似替代。", "en": "Recompute dual change augmented P with identity pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "dual change augmented P with identity pivot0 column0 normalized entry0" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "dual change augmented P with identity pivot0 column0 normalized entry1 := div(8, 1)", "after": "dual change augmented P with identity pivot0 column0 normalized entry1 = 8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual change augmented P with identity pivot0 column0 normalized entry1，不以小數近似替代。", "en": "Recompute dual change augmented P with identity pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "8", "1" ], "output": "8", "label": "dual change augmented P with identity pivot0 column0 normalized entry1" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "dual change augmented P with identity pivot0 column0 normalized entry2 := div(1, 1)", "after": "dual change augmented P with identity pivot0 column0 normalized entry2 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual change augmented P with identity pivot0 column0 normalized entry2，不以小數近似替代。", "en": "Recompute dual change augmented P with identity pivot0 column0 normalized entry2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "dual change augmented P with identity pivot0 column0 normalized entry2" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "dual change augmented P with identity pivot0 column0 normalized entry3 := div(0, 1)", "after": "dual change augmented P with identity pivot0 column0 normalized entry3 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual change augmented P with identity pivot0 column0 normalized entry3，不以小數近似替代。", "en": "Recompute dual change augmented P with identity pivot0 column0 normalized entry3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "0", "1" ], "output": "0", "label": "dual change augmented P with identity pivot0 column0 normalized entry3" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "dual change augmented P with identity:pivot row=0;column=0;nonzero pivot=1;old row=[1, 8, 1, 0]", "after": "normalized full row=[1, 8, 1, 0]", "relation": "implies", "rule_id": "finite_linear_nonzero_pivot_complete_row_scaling", "conditions": [], "explanation": { "zh-Hant": "整列每格同除非零pivot，精確有理數保零值與符號。", "en": "Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs." }, "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": "dual change augmented P with identity pivot1 column1 normalized entry0 := div(0, 1)", "after": "dual change augmented P with identity pivot1 column1 normalized entry0 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual change augmented P with identity pivot1 column1 normalized entry0，不以小數近似替代。", "en": "Recompute dual change augmented P with identity pivot1 column1 normalized entry0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "0", "1" ], "output": "0", "label": "dual change augmented P with identity pivot1 column1 normalized entry0" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "dual change augmented P with identity pivot1 column1 normalized entry1 := div(1, 1)", "after": "dual change augmented P with identity pivot1 column1 normalized entry1 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual change augmented P with identity pivot1 column1 normalized entry1，不以小數近似替代。", "en": "Recompute dual change augmented P with identity pivot1 column1 normalized entry1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "dual change augmented P with identity pivot1 column1 normalized entry1" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "dual change augmented P with identity pivot1 column1 normalized entry2 := div(0, 1)", "after": "dual change augmented P with identity pivot1 column1 normalized entry2 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual change augmented P with identity pivot1 column1 normalized entry2，不以小數近似替代。", "en": "Recompute dual change augmented P with identity pivot1 column1 normalized entry2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "0", "1" ], "output": "0", "label": "dual change augmented P with identity pivot1 column1 normalized entry2" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "dual change augmented P with identity pivot1 column1 normalized entry3 := div(1, 1)", "after": "dual change augmented P with identity pivot1 column1 normalized entry3 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual change augmented P with identity pivot1 column1 normalized entry3，不以小數近似替代。", "en": "Recompute dual change augmented P with identity pivot1 column1 normalized entry3 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "dual change augmented P with identity pivot1 column1 normalized entry3" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "dual change augmented P with identity:pivot row=1;column=1;nonzero pivot=1;old row=[0, 1, 0, 1]", "after": "normalized full row=[0, 1, 0, 1]", "relation": "implies", "rule_id": "finite_linear_nonzero_pivot_complete_row_scaling", "conditions": [], "explanation": { "zh-Hant": "整列每格同除非零pivot，精確有理數保零值與符號。", "en": "Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs." }, "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": "S013", "uses": [ "G001", "S012" ], "before": "dual change augmented P with identity eliminate row0 pivot1 entry0 product := mul(8, 0)", "after": "dual change augmented P with identity eliminate row0 pivot1 entry0 product = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual change augmented P with identity eliminate row0 pivot1 entry0 product，不以小數近似替代。", "en": "Recompute dual change augmented P with identity eliminate row0 pivot1 entry0 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "8", "0" ], "output": "0", "label": "dual change augmented P with identity eliminate row0 pivot1 entry0 product" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "dual change augmented P with identity eliminate row0 pivot1 entry0 difference := sub(1, 0)", "after": "dual change augmented P with identity eliminate row0 pivot1 entry0 difference = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual change augmented P with identity eliminate row0 pivot1 entry0 difference，不以小數近似替代。", "en": "Recompute dual change augmented P with identity eliminate row0 pivot1 entry0 difference with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "0" ], "output": "1", "label": "dual change augmented P with identity eliminate row0 pivot1 entry0 difference" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "dual change augmented P with identity eliminate row0 pivot1 entry1 product := mul(8, 1)", "after": "dual change augmented P with identity eliminate row0 pivot1 entry1 product = 8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual change augmented P with identity eliminate row0 pivot1 entry1 product，不以小數近似替代。", "en": "Recompute dual change augmented P with identity eliminate row0 pivot1 entry1 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "8", "1" ], "output": "8", "label": "dual change augmented P with identity eliminate row0 pivot1 entry1 product" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "dual change augmented P with identity eliminate row0 pivot1 entry1 difference := sub(8, 8)", "after": "dual change augmented P with identity eliminate row0 pivot1 entry1 difference = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual change augmented P with identity eliminate row0 pivot1 entry1 difference，不以小數近似替代。", "en": "Recompute dual change augmented P with identity eliminate row0 pivot1 entry1 difference with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "8", "8" ], "output": "0", "label": "dual change augmented P with identity eliminate row0 pivot1 entry1 difference" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "dual change augmented P with identity eliminate row0 pivot1 entry2 product := mul(8, 0)", "after": "dual change augmented P with identity eliminate row0 pivot1 entry2 product = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual change augmented P with identity eliminate row0 pivot1 entry2 product，不以小數近似替代。", "en": "Recompute dual change augmented P with identity eliminate row0 pivot1 entry2 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "8", "0" ], "output": "0", "label": "dual change augmented P with identity eliminate row0 pivot1 entry2 product" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "dual change augmented P with identity eliminate row0 pivot1 entry2 difference := sub(1, 0)", "after": "dual change augmented P with identity eliminate row0 pivot1 entry2 difference = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual change augmented P with identity eliminate row0 pivot1 entry2 difference，不以小數近似替代。", "en": "Recompute dual change augmented P with identity eliminate row0 pivot1 entry2 difference with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "0" ], "output": "1", "label": "dual change augmented P with identity eliminate row0 pivot1 entry2 difference" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "dual change augmented P with identity eliminate row0 pivot1 entry3 product := mul(8, 1)", "after": "dual change augmented P with identity eliminate row0 pivot1 entry3 product = 8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual change augmented P with identity eliminate row0 pivot1 entry3 product，不以小數近似替代。", "en": "Recompute dual change augmented P with identity eliminate row0 pivot1 entry3 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "8", "1" ], "output": "8", "label": "dual change augmented P with identity eliminate row0 pivot1 entry3 product" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "dual change augmented P with identity eliminate row0 pivot1 entry3 difference := sub(0, 8)", "after": "dual change augmented P with identity eliminate row0 pivot1 entry3 difference = -8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual change augmented P with identity eliminate row0 pivot1 entry3 difference，不以小數近似替代。", "en": "Recompute dual change augmented P with identity eliminate row0 pivot1 entry3 difference with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "8" ], "output": "-8", "label": "dual change augmented P with identity eliminate row0 pivot1 entry3 difference" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "dual change augmented P with identity:target row=0;old=[1, 8, 1, 0];factor=8;pivot row=[0, 1, 0, 1]", "after": "new full row=[1, 0, 1, -8]", "relation": "implies", "rule_id": "finite_linear_invertible_all_entry_elimination", "conditions": [], "explanation": { "zh-Hant": "同一列的所有格都減同一倍pivot列；反向加回即逆運算。", "en": "Subtract the same pivot-row multiple in every entry;adding it back is the inverse." }, "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": "S022", "uses": [ "G001", "S021" ], "before": "dual change augmented P with identity:complete exact reduced matrix=[[1, 0, 1, -8], [0, 1, 0, 1]];pivot columns=[0, 1]", "after": "rank=2;pivot columns independent and all original column relations preserved;zero rows retained", "relation": "implies", "rule_id": "finite_linear_complete_rank_certificate_not_samples", "conditions": [], "explanation": { "zh-Hant": "主元欄有不同的首1座標而獨立，其餘欄由主元列座標生成；可逆列運算把依賴關係帶回原矩陣。零列不刪除，不從有限向量測試猜rank。", "en": "Distinct pivot identity coordinates prove independence;other columns are generated by their pivot-row coordinates.Invertible operations transport column relations back to the original matrix.Keep zero rows;do not infer rank from sampled vectors." }, "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": "S023", "uses": [ "G001", "S022" ], "before": "dual P times inverse row0 column0 coordinate0 product := mul(1, 1)", "after": "dual P times inverse row0 column0 coordinate0 product = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual P times inverse row0 column0 coordinate0 product，不以小數近似替代。", "en": "Recompute dual P times inverse row0 column0 coordinate0 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "dual P times inverse row0 column0 coordinate0 product" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "dual P times inverse row0 column0 coordinate1 product := mul(8, 0)", "after": "dual P times inverse row0 column0 coordinate1 product = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual P times inverse row0 column0 coordinate1 product，不以小數近似替代。", "en": "Recompute dual P times inverse row0 column0 coordinate1 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "8", "0" ], "output": "0", "label": "dual P times inverse row0 column0 coordinate1 product" } }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "dual P times inverse row0 column0 complete sum := sum(1, 0)", "after": "dual P times inverse row0 column0 complete sum = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual P times inverse row0 column0 complete sum，不以小數近似替代。", "en": "Recompute dual P times inverse row0 column0 complete sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "0" ], "output": "1", "label": "dual P times inverse row0 column0 complete sum" } }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "dual P times inverse row0 column1 coordinate0 product := mul(1, -8)", "after": "dual P times inverse row0 column1 coordinate0 product = -8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual P times inverse row0 column1 coordinate0 product，不以小數近似替代。", "en": "Recompute dual P times inverse row0 column1 coordinate0 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "-8" ], "output": "-8", "label": "dual P times inverse row0 column1 coordinate0 product" } }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "dual P times inverse row0 column1 coordinate1 product := mul(8, 1)", "after": "dual P times inverse row0 column1 coordinate1 product = 8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual P times inverse row0 column1 coordinate1 product，不以小數近似替代。", "en": "Recompute dual P times inverse row0 column1 coordinate1 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "8", "1" ], "output": "8", "label": "dual P times inverse row0 column1 coordinate1 product" } }, { "id": "S028", "uses": [ "G001", "S027" ], "before": "dual P times inverse row0 column1 complete sum := sum(-8, 8)", "after": "dual P times inverse row0 column1 complete sum = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual P times inverse row0 column1 complete sum，不以小數近似替代。", "en": "Recompute dual P times inverse row0 column1 complete sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-8", "8" ], "output": "0", "label": "dual P times inverse row0 column1 complete sum" } }, { "id": "S029", "uses": [ "G001", "S028" ], "before": "dual P times inverse row1 column0 coordinate0 product := mul(0, 1)", "after": "dual P times inverse row1 column0 coordinate0 product = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual P times inverse row1 column0 coordinate0 product，不以小數近似替代。", "en": "Recompute dual P times inverse row1 column0 coordinate0 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "1" ], "output": "0", "label": "dual P times inverse row1 column0 coordinate0 product" } }, { "id": "S030", "uses": [ "G001", "S029" ], "before": "dual P times inverse row1 column0 coordinate1 product := mul(1, 0)", "after": "dual P times inverse row1 column0 coordinate1 product = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual P times inverse row1 column0 coordinate1 product，不以小數近似替代。", "en": "Recompute dual P times inverse row1 column0 coordinate1 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "0" ], "output": "0", "label": "dual P times inverse row1 column0 coordinate1 product" } }, { "id": "S031", "uses": [ "G001", "S030" ], "before": "dual P times inverse row1 column0 complete sum := sum(0, 0)", "after": "dual P times inverse row1 column0 complete sum = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual P times inverse row1 column0 complete sum，不以小數近似替代。", "en": "Recompute dual P times inverse row1 column0 complete sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "0" ], "output": "0", "label": "dual P times inverse row1 column0 complete sum" } }, { "id": "S032", "uses": [ "G001", "S031" ], "before": "dual P times inverse row1 column1 coordinate0 product := mul(0, -8)", "after": "dual P times inverse row1 column1 coordinate0 product = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual P times inverse row1 column1 coordinate0 product，不以小數近似替代。", "en": "Recompute dual P times inverse row1 column1 coordinate0 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "-8" ], "output": "0", "label": "dual P times inverse row1 column1 coordinate0 product" } }, { "id": "S033", "uses": [ "G001", "S032" ], "before": "dual P times inverse row1 column1 coordinate1 product := mul(1, 1)", "after": "dual P times inverse row1 column1 coordinate1 product = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual P times inverse row1 column1 coordinate1 product，不以小數近似替代。", "en": "Recompute dual P times inverse row1 column1 coordinate1 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "dual P times inverse row1 column1 coordinate1 product" } }, { "id": "S034", "uses": [ "G001", "S033" ], "before": "dual P times inverse row1 column1 complete sum := sum(0, 1)", "after": "dual P times inverse row1 column1 complete sum = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual P times inverse row1 column1 complete sum，不以小數近似替代。", "en": "Recompute dual P times inverse row1 column1 complete sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "1" ], "output": "1", "label": "dual P times inverse row1 column1 complete sum" } }, { "id": "S035", "uses": [ "G001", "S034" ], "before": "original P=[[1, 8], [0, 1]];exact row reduction [P|I] to [I|inverse];inverse=[[1, -8], [0, 1]]", "after": "complete P*inverse=identity;given old_primal=P*new_primal therefore new_primal=inverse*old_primal", "relation": "implies", "rule_id": "dual_invertible_change_not_same_primal_covector_law", "conditions": [], "explanation": { "zh-Hant": "對整個增廣矩陣做同一可逆列運算，左半為I才接受P可逆，並逐格核P乘inverse。依來源old=P·new的方向解new，不反套P。", "en": "Apply the same invertible row operations to the complete augmented matrix.Accept invertibility only when the left block is I,and check every entry of P times the inverse.Solve new from the source direction old=P*new,not by multiplying the primal by P." }, "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": "S036", "uses": [ "G001", "S035" ], "before": "dual new primal coordinate0 coordinate0 product := mul(1, 9)", "after": "dual new primal coordinate0 coordinate0 product = 9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual new primal coordinate0 coordinate0 product，不以小數近似替代。", "en": "Recompute dual new primal coordinate0 coordinate0 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "9" ], "output": "9", "label": "dual new primal coordinate0 coordinate0 product" } }, { "id": "S037", "uses": [ "G001", "S036" ], "before": "dual new primal coordinate0 coordinate1 product := mul(-8, 3)", "after": "dual new primal coordinate0 coordinate1 product = -24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual new primal coordinate0 coordinate1 product，不以小數近似替代。", "en": "Recompute dual new primal coordinate0 coordinate1 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-8", "3" ], "output": "-24", "label": "dual new primal coordinate0 coordinate1 product" } }, { "id": "S038", "uses": [ "G001", "S037" ], "before": "dual new primal coordinate0 complete sum := sum(9, -24)", "after": "dual new primal coordinate0 complete sum = -15", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual new primal coordinate0 complete sum，不以小數近似替代。", "en": "Recompute dual new primal coordinate0 complete sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "9", "-24" ], "output": "-15", "label": "dual new primal coordinate0 complete sum" } }, { "id": "S039", "uses": [ "G001", "S038" ], "before": "dual new primal coordinate1 coordinate0 product := mul(0, 9)", "after": "dual new primal coordinate1 coordinate0 product = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual new primal coordinate1 coordinate0 product，不以小數近似替代。", "en": "Recompute dual new primal coordinate1 coordinate0 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "9" ], "output": "0", "label": "dual new primal coordinate1 coordinate0 product" } }, { "id": "S040", "uses": [ "G001", "S039" ], "before": "dual new primal coordinate1 coordinate1 product := mul(1, 3)", "after": "dual new primal coordinate1 coordinate1 product = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual new primal coordinate1 coordinate1 product，不以小數近似替代。", "en": "Recompute dual new primal coordinate1 coordinate1 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "3" ], "output": "3", "label": "dual new primal coordinate1 coordinate1 product" } }, { "id": "S041", "uses": [ "G001", "S040" ], "before": "dual new primal coordinate1 complete sum := sum(0, 3)", "after": "dual new primal coordinate1 complete sum = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual new primal coordinate1 complete sum，不以小數近似替代。", "en": "Recompute dual new primal coordinate1 complete sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "3" ], "output": "3", "label": "dual new primal coordinate1 complete sum" } }, { "id": "S042", "uses": [ "G001", "S041" ], "before": "dual new covector coordinate0 coordinate0 product := mul(2, 1)", "after": "dual new covector coordinate0 coordinate0 product = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual new covector coordinate0 coordinate0 product，不以小數近似替代。", "en": "Recompute dual new covector coordinate0 coordinate0 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "1" ], "output": "2", "label": "dual new covector coordinate0 coordinate0 product" } }, { "id": "S043", "uses": [ "G001", "S042" ], "before": "dual new covector coordinate0 coordinate1 product := mul(8, 0)", "after": "dual new covector coordinate0 coordinate1 product = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual new covector coordinate0 coordinate1 product，不以小數近似替代。", "en": "Recompute dual new covector coordinate0 coordinate1 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "8", "0" ], "output": "0", "label": "dual new covector coordinate0 coordinate1 product" } }, { "id": "S044", "uses": [ "G001", "S043" ], "before": "dual new covector coordinate0 complete sum := sum(2, 0)", "after": "dual new covector coordinate0 complete sum = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual new covector coordinate0 complete sum，不以小數近似替代。", "en": "Recompute dual new covector coordinate0 complete sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "2", "0" ], "output": "2", "label": "dual new covector coordinate0 complete sum" } }, { "id": "S045", "uses": [ "G001", "S044" ], "before": "dual new covector coordinate1 coordinate0 product := mul(2, 8)", "after": "dual new covector coordinate1 coordinate0 product = 16", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual new covector coordinate1 coordinate0 product，不以小數近似替代。", "en": "Recompute dual new covector coordinate1 coordinate0 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "8" ], "output": "16", "label": "dual new covector coordinate1 coordinate0 product" } }, { "id": "S046", "uses": [ "G001", "S045" ], "before": "dual new covector coordinate1 coordinate1 product := mul(8, 1)", "after": "dual new covector coordinate1 coordinate1 product = 8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual new covector coordinate1 coordinate1 product，不以小數近似替代。", "en": "Recompute dual new covector coordinate1 coordinate1 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "8", "1" ], "output": "8", "label": "dual new covector coordinate1 coordinate1 product" } }, { "id": "S047", "uses": [ "G001", "S046" ], "before": "dual new covector coordinate1 complete sum := sum(16, 8)", "after": "dual new covector coordinate1 complete sum = 24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual new covector coordinate1 complete sum，不以小數近似替代。", "en": "Recompute dual new covector coordinate1 complete sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "16", "8" ], "output": "24", "label": "dual new covector coordinate1 complete sum" } }, { "id": "S048", "uses": [ "G001", "S047" ], "before": "dual recover old primal coordinate0 coordinate0 product := mul(1, -15)", "after": "dual recover old primal coordinate0 coordinate0 product = -15", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual recover old primal coordinate0 coordinate0 product，不以小數近似替代。", "en": "Recompute dual recover old primal coordinate0 coordinate0 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "-15" ], "output": "-15", "label": "dual recover old primal coordinate0 coordinate0 product" } }, { "id": "S049", "uses": [ "G001", "S048" ], "before": "dual recover old primal coordinate0 coordinate1 product := mul(8, 3)", "after": "dual recover old primal coordinate0 coordinate1 product = 24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual recover old primal coordinate0 coordinate1 product，不以小數近似替代。", "en": "Recompute dual recover old primal coordinate0 coordinate1 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "8", "3" ], "output": "24", "label": "dual recover old primal coordinate0 coordinate1 product" } }, { "id": "S050", "uses": [ "G001", "S049" ], "before": "dual recover old primal coordinate0 complete sum := sum(-15, 24)", "after": "dual recover old primal coordinate0 complete sum = 9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual recover old primal coordinate0 complete sum，不以小數近似替代。", "en": "Recompute dual recover old primal coordinate0 complete sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-15", "24" ], "output": "9", "label": "dual recover old primal coordinate0 complete sum" } }, { "id": "S051", "uses": [ "G001", "S050" ], "before": "dual recover old primal coordinate1 coordinate0 product := mul(0, -15)", "after": "dual recover old primal coordinate1 coordinate0 product = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual recover old primal coordinate1 coordinate0 product，不以小數近似替代。", "en": "Recompute dual recover old primal coordinate1 coordinate0 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "-15" ], "output": "0", "label": "dual recover old primal coordinate1 coordinate0 product" } }, { "id": "S052", "uses": [ "G001", "S051" ], "before": "dual recover old primal coordinate1 coordinate1 product := mul(1, 3)", "after": "dual recover old primal coordinate1 coordinate1 product = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual recover old primal coordinate1 coordinate1 product，不以小數近似替代。", "en": "Recompute dual recover old primal coordinate1 coordinate1 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "3" ], "output": "3", "label": "dual recover old primal coordinate1 coordinate1 product" } }, { "id": "S053", "uses": [ "G001", "S052" ], "before": "dual recover old primal coordinate1 complete sum := sum(0, 3)", "after": "dual recover old primal coordinate1 complete sum = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual recover old primal coordinate1 complete sum，不以小數近似替代。", "en": "Recompute dual recover old primal coordinate1 complete sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "3" ], "output": "3", "label": "dual recover old primal coordinate1 complete sum" } }, { "id": "S054", "uses": [ "G001", "S053" ], "before": "dual old covector primal pairing coordinate0 product := mul(2, 9)", "after": "dual old covector primal pairing coordinate0 product = 18", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual old covector primal pairing coordinate0 product，不以小數近似替代。", "en": "Recompute dual old covector primal pairing coordinate0 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "9" ], "output": "18", "label": "dual old covector primal pairing coordinate0 product" } }, { "id": "S055", "uses": [ "G001", "S054" ], "before": "dual old covector primal pairing coordinate1 product := mul(8, 3)", "after": "dual old covector primal pairing coordinate1 product = 24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual old covector primal pairing coordinate1 product，不以小數近似替代。", "en": "Recompute dual old covector primal pairing coordinate1 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "8", "3" ], "output": "24", "label": "dual old covector primal pairing coordinate1 product" } }, { "id": "S056", "uses": [ "G001", "S055" ], "before": "dual old covector primal pairing complete sum := sum(18, 24)", "after": "dual old covector primal pairing complete sum = 42", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual old covector primal pairing complete sum，不以小數近似替代。", "en": "Recompute dual old covector primal pairing complete sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "18", "24" ], "output": "42", "label": "dual old covector primal pairing complete sum" } }, { "id": "S057", "uses": [ "G001", "S056" ], "before": "dual new covector primal pairing coordinate0 product := mul(2, -15)", "after": "dual new covector primal pairing coordinate0 product = -30", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual new covector primal pairing coordinate0 product，不以小數近似替代。", "en": "Recompute dual new covector primal pairing coordinate0 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "-15" ], "output": "-30", "label": "dual new covector primal pairing coordinate0 product" } }, { "id": "S058", "uses": [ "G001", "S057" ], "before": "dual new covector primal pairing coordinate1 product := mul(24, 3)", "after": "dual new covector primal pairing coordinate1 product = 72", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual new covector primal pairing coordinate1 product，不以小數近似替代。", "en": "Recompute dual new covector primal pairing coordinate1 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "24", "3" ], "output": "72", "label": "dual new covector primal pairing coordinate1 product" } }, { "id": "S059", "uses": [ "G001", "S058" ], "before": "dual new covector primal pairing complete sum := sum(-30, 72)", "after": "dual new covector primal pairing complete sum = 42", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。", "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 dual new covector primal pairing complete sum，不以小數近似替代。", "en": "Recompute dual new covector primal pairing complete sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-30", "72" ], "output": "42", "label": "dual new covector primal pairing complete sum" } }, { "id": "S060", "uses": [ "G001", "S059" ], "before": "original old primal=[9, 3];old covector ROW=[2, 8];given old_primal=P*new_primal", "after": "new primal=[-15, 3];new covector=old_covector*P=[2, 24];pairing preserved=42=42;canonical identification CLAIM=否", "relation": "implies", "rule_id": "dual_contragredient_pairing_preserved_no_canonical_identification", "conditions": [], "explanation": { "zh-Hant": "因對每個new vector都需c_old·P·new=(c_old P)·new，所以covector在右乘P，與primal乘inverse方向相反。No只保留無額外結構不作典範識別，不否定另選basis或inner product可建同構；這也不宣稱無限維連續對偶。", "en": "For every new-coordinate vector,c_old*P*new=(c_old P)*new,so the covector row multiplies P on the right,opposite the inverse law for the primal.The No withholds a canonical identification without extra structure,not identifications chosen via a basis or inner product,and makes no infinite-dimensional continuous-dual claim." }, "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": [ [ "-15", "3" ], [ "2", "24" ], "contragredient_coordinate_laws", false, "no_canonical_V_to_dual_identification_without_extra_structure" ] }, "conclusion": { "canonical_value": "([-15, 3], [2, 24], contragredient_coordinate_laws, 否, no_canonical_V_to_dual_identification_without_extra_structure)", "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-20261105-r02-dual-vector-space-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261105-r02-dual-vector-space-foundations/questions.jsonl", "batch_sha256": "710a303eff4a61aad547a2ca9501595d14cef5d84a5e44e51d08864c41eb9f14", "original_line": 83, "original_line_sha256": "da729c8f29f29ed77b3829994c0841bb533c255d9ba255ab6b4c84daeeff1667", "question_compact_sha256": "66c7043da6a1686aa50b8f48d6eee3d86eafb4a640d2597d2f27ef475bd107e7", "english_row_compact_sha256": "671d14247bf648b411c1752528de47a81bdb54ab6a89fb3f507aeea542679b45", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261105-r02-dual-vector-space-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261105-r02-dual-vector-space-foundations.en.jsonl" }, "mko_bindings": { "mko-dual-vector-space": { "sha256": "1994bbbcc73dff239e5c7cfdaba60f5dc5aadb1a0a4603f613e4b1371d59234a", "english_sha256": "1e23ee2fa2f740284ff7f2e81139957b6196a81326cb161bae3a56d54fe07153", "original_path": "public/data/mko/source13-20261018-dual-vector-space.json", "english_path": "public/data/i18n/editions/mko-dual-vector-space.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "49e5d7674cefa01298c2386543f777d2bdfc99446b4027156b80142f10031994", "original_candidate_sha256": "a02782ff1b1ba9dd196115524cf050d6b34e96e07e30f1f9765b0bbec78213e3", "original_candidate_line_sha256": "a02782ff1b1ba9dd196115524cf050d6b34e96e07e30f1f9765b0bbec78213e3" }, "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-20261026-008-p37", "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": "efee5462991619d6ed9f64f11ddb085e5614ef33f9f8fcd8cf29b9329f8fd301", "question_first_witness_sha256": "acf91fa1622f01d2cf04b2c513294254323665351f660f2628aaafdb869c4ff2" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 60, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "c62dbdea8a1ae3ceea1b954bfdcb8f244382f8be24f0a79b97d61b92c5335e76", "arithmetic_steps": 52 }, "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." } }