{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261105-r01-direct-sum-0053", "pilot_number": "R0036-18678", "number": 18678, "family_id": "qf-mod36-directsum-multi-subspace-global", "title": { "zh-Hant": "第 18678 題：子空間直和", "en": "Question 18678: Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261105-r01-direct-sum-0053", "batch_id": "batch-20261014-advance-20261105-r01-direct-sum-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-direct-sum" ], "title_zh": "子空間直和", "stem_zh": "有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,3]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)", "explanation_zh": "Wave5第五層核對=(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)；依完整明示資料獨立重算；內直和唯一性與覆蓋母空間是兩項不同責任；二項交集為零只證和空間內唯一分解；三項以上兩兩交集為零不足；外直和載體不得與內直和混同。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "子空間直和" ] }, "variant": { "family_id": "qf-mod36-directsum-multi-subspace-global", "seed": 202611051053, "parameters": { "ambient_dimension": 2, "subspace_bases": [ [ [ 1, 0 ] ], [ [ 0, 1 ] ], [ [ 1, 3 ] ] ], "domain": "Exact_finite_dimensional_real_subspaces", "operation": "directsum-multi-subspace-global", "semantic_template": "qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,3]]]}" } }, "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-r01-direct-sum-0053", "batch_id": "batch-20261014-advance-20261105-r01-direct-sum-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "d55e0a26336613eda5704e00afd0d2b448cecf0607dbc7b29e6de20958b19037", "family_id": "qf-mod36-directsum-multi-subspace-global", "seed": 202611051053, "semantic_template": "qf-mod36-directsum-multi-subspace-global/{\"ambient_dimension\":2,\"subspace_bases\":[[[0,1]],[[1,0]],[[1,3]]]}" }, "fields": { "title": "Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum", "stem": "Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 3]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary).", "answer": { "kind": "expression", "value": "(Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)", "explanation": "Check pairwise intersections separately, then compare the rank of all basis columns together with their total count. Global independence gives directness of the sum; comparison of the total rank with ambient dimension gives the separate coverage flag. The exact result, retaining the requested order, is (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more)." }, "skills": [ "Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={\"ambient_dimension\":2,\"subspace_bases\":[[[1,0]],[[0,1]],[[1,3]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 3]]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=directsum-multi-subspace-global;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, 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": "directsum subspace0:complete ordered matrix=[[1], [0]];rows=2;columns=1", "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": "directsum subspace0 pivot0 column0 normalized entry0 := div(1, 1)", "after": "directsum subspace0 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 directsum subspace0 pivot0 column0 normalized entry0，不以小數近似替代。", "en": "Recompute directsum subspace0 pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "directsum subspace0 pivot0 column0 normalized entry0" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "directsum subspace0:pivot row=0;column=0;nonzero pivot=1;old row=[1]", "after": "normalized full row=[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": "S005", "uses": [ "G001", "S004" ], "before": "directsum subspace0:complete exact reduced matrix=[[1], [0]];pivot columns=[0]", "after": "rank=1;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": "S006", "uses": [ "G001", "S005" ], "before": "directsum subspace0:complete ORIGINAL independent columns=[[1, 0]]", "after": "declared subspace dimension=1;basis of its full real span;empty basis means zero subspace", "relation": "implies", "rule_id": "directsum_declared_basis_complete_real_span", "conditions": [], "explanation": { "zh-Hant": "原資料宣告完整basis，精確rank再核獨立，故數量才是此子空間維度；允許空basis代表零子空間，不把有限列表當子空間所有元素。", "en": "The source declares a full basis and exact rank checks independence,so its count is the subspace dimension.An empty basis spans the zero subspace;the finite list is not the set of all subspace elements." }, "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": "S007", "uses": [ "G001", "S006" ], "before": "directsum subspace1:complete ordered matrix=[[0], [1]];rows=2;columns=1", "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": "S008", "uses": [ "G001", "S007" ], "before": "directsum subspace1:matrix=[[0], [1]];selected row=1", "after": "swap rows0,1;matrix=[[1], [0]]", "relation": "implies", "rule_id": "finite_linear_invertible_row_swap", "conditions": [], "explanation": { "zh-Hant": "整列交換可逆，原欄與其係數標籤不變。", "en": "Swap entire rows invertibly while retaining original column/coefficient labels." }, "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": "directsum subspace1 pivot0 column0 normalized entry0 := div(1, 1)", "after": "directsum subspace1 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 directsum subspace1 pivot0 column0 normalized entry0，不以小數近似替代。", "en": "Recompute directsum subspace1 pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "directsum subspace1 pivot0 column0 normalized entry0" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "directsum subspace1:pivot row=0;column=0;nonzero pivot=1;old row=[1]", "after": "normalized full row=[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": "S011", "uses": [ "G001", "S010" ], "before": "directsum subspace1:complete exact reduced matrix=[[1], [0]];pivot columns=[0]", "after": "rank=1;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": "S012", "uses": [ "G001", "S011" ], "before": "directsum subspace1:complete ORIGINAL independent columns=[[0, 1]]", "after": "declared subspace dimension=1;basis of its full real span;empty basis means zero subspace", "relation": "implies", "rule_id": "directsum_declared_basis_complete_real_span", "conditions": [], "explanation": { "zh-Hant": "原資料宣告完整basis，精確rank再核獨立，故數量才是此子空間維度；允許空basis代表零子空間，不把有限列表當子空間所有元素。", "en": "The source declares a full basis and exact rank checks independence,so its count is the subspace dimension.An empty basis spans the zero subspace;the finite list is not the set of all subspace elements." }, "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": "directsum subspace2:complete ordered matrix=[[1], [3]];rows=2;columns=1", "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": "S014", "uses": [ "G001", "S013" ], "before": "directsum subspace2 pivot0 column0 normalized entry0 := div(1, 1)", "after": "directsum subspace2 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 directsum subspace2 pivot0 column0 normalized entry0，不以小數近似替代。", "en": "Recompute directsum subspace2 pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "directsum subspace2 pivot0 column0 normalized entry0" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "directsum subspace2:pivot row=0;column=0;nonzero pivot=1;old row=[1]", "after": "normalized full row=[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": "S016", "uses": [ "G001", "S015" ], "before": "directsum subspace2 eliminate row1 pivot0 entry0 product := mul(3, 1)", "after": "directsum subspace2 eliminate row1 pivot0 entry0 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 directsum subspace2 eliminate row1 pivot0 entry0 product，不以小數近似替代。", "en": "Recompute directsum subspace2 eliminate row1 pivot0 entry0 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "1" ], "output": "3", "label": "directsum subspace2 eliminate row1 pivot0 entry0 product" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "directsum subspace2 eliminate row1 pivot0 entry0 difference := sub(3, 3)", "after": "directsum subspace2 eliminate row1 pivot0 entry0 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 directsum subspace2 eliminate row1 pivot0 entry0 difference，不以小數近似替代。", "en": "Recompute directsum subspace2 eliminate row1 pivot0 entry0 difference with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "3", "3" ], "output": "0", "label": "directsum subspace2 eliminate row1 pivot0 entry0 difference" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "directsum subspace2:target row=1;old=[3];factor=3;pivot row=[1]", "after": "new full row=[0]", "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": "S019", "uses": [ "G001", "S018" ], "before": "directsum subspace2:complete exact reduced matrix=[[1], [0]];pivot columns=[0]", "after": "rank=1;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": "S020", "uses": [ "G001", "S019" ], "before": "directsum subspace2:complete ORIGINAL independent columns=[[1, 3]]", "after": "declared subspace dimension=1;basis of its full real span;empty basis means zero subspace", "relation": "implies", "rule_id": "directsum_declared_basis_complete_real_span", "conditions": [], "explanation": { "zh-Hant": "原資料宣告完整basis，精確rank再核獨立，故數量才是此子空間維度；允許空basis代表零子空間，不把有限列表當子空間所有元素。", "en": "The source declares a full basis and exact rank checks independence,so its count is the subspace dimension.An empty basis spans the zero subspace;the finite list is not the set of all subspace elements." }, "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": "S021", "uses": [ "G001", "S020" ], "before": "directsum pair0,1:complete ordered matrix=[[1, 0], [0, 1]];rows=2;columns=2", "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": "S022", "uses": [ "G001", "S021" ], "before": "directsum pair0,1 pivot0 column0 normalized entry0 := div(1, 1)", "after": "directsum pair0,1 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 directsum pair0,1 pivot0 column0 normalized entry0，不以小數近似替代。", "en": "Recompute directsum pair0,1 pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "directsum pair0,1 pivot0 column0 normalized entry0" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "directsum pair0,1 pivot0 column0 normalized entry1 := div(0, 1)", "after": "directsum pair0,1 pivot0 column0 normalized entry1 = 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 directsum pair0,1 pivot0 column0 normalized entry1，不以小數近似替代。", "en": "Recompute directsum pair0,1 pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "0", "1" ], "output": "0", "label": "directsum pair0,1 pivot0 column0 normalized entry1" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "directsum pair0,1:pivot row=0;column=0;nonzero pivot=1;old row=[1, 0]", "after": "normalized full row=[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": "S025", "uses": [ "G001", "S024" ], "before": "directsum pair0,1 pivot1 column1 normalized entry0 := div(0, 1)", "after": "directsum pair0,1 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 directsum pair0,1 pivot1 column1 normalized entry0，不以小數近似替代。", "en": "Recompute directsum pair0,1 pivot1 column1 normalized entry0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "0", "1" ], "output": "0", "label": "directsum pair0,1 pivot1 column1 normalized entry0" } }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "directsum pair0,1 pivot1 column1 normalized entry1 := div(1, 1)", "after": "directsum pair0,1 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 directsum pair0,1 pivot1 column1 normalized entry1，不以小數近似替代。", "en": "Recompute directsum pair0,1 pivot1 column1 normalized entry1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "directsum pair0,1 pivot1 column1 normalized entry1" } }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "directsum pair0,1:pivot row=1;column=1;nonzero pivot=1;old row=[0, 1]", "after": "normalized full row=[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": "S028", "uses": [ "G001", "S027" ], "before": "directsum pair0,1:complete exact reduced matrix=[[1, 0], [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": "S029", "uses": [ "G001", "S028" ], "before": "directsum pair0,1 basis counts := sum(1, 1)", "after": "directsum pair0,1 basis counts = 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 directsum pair0,1 basis counts，不以小數近似替代。", "en": "Recompute directsum pair0,1 basis counts with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "1" ], "output": "2", "label": "directsum pair0,1 basis counts" } }, { "id": "S030", "uses": [ "G001", "S029" ], "before": "directsum pair0,1 intersection dimension := sub(2, 2)", "after": "directsum pair0,1 intersection dimension = 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 directsum pair0,1 intersection dimension，不以小數近似替代。", "en": "Recompute directsum pair0,1 intersection dimension with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "2", "2" ], "output": "0", "label": "directsum pair0,1 intersection dimension" } }, { "id": "S031", "uses": [ "G001", "S030" ], "before": "pair original subspaces=[0, 1];individual independent basis counts=[1, 1];combined rank=2", "after": "dim intersection=0;pair intersection zero=是", "relation": "implies", "rule_id": "directsum_pair_intersection_rank_nullity_not_global_directness", "conditions": [], "explanation": { "zh-Hant": "映射(u,w)到u+w滿射到和空間，核是(v,−v)的交集副本，所以兩項維度公式成立。這個兩項結果不能直接升格三項以上全域直和。", "en": "The map(u,w)↦u+w surjects onto the sum and has kernel consisting of(v,−v)from the intersection,giving the two-term dimension formula.This pairwise fact does not establish global directness of three or more subspaces." }, "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": "S032", "uses": [ "G001", "S031" ], "before": "directsum pair0,2:complete ordered matrix=[[1, 1], [0, 3]];rows=2;columns=2", "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": "S033", "uses": [ "G001", "S032" ], "before": "directsum pair0,2 pivot0 column0 normalized entry0 := div(1, 1)", "after": "directsum pair0,2 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 directsum pair0,2 pivot0 column0 normalized entry0，不以小數近似替代。", "en": "Recompute directsum pair0,2 pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "directsum pair0,2 pivot0 column0 normalized entry0" } }, { "id": "S034", "uses": [ "G001", "S033" ], "before": "directsum pair0,2 pivot0 column0 normalized entry1 := div(1, 1)", "after": "directsum pair0,2 pivot0 column0 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 directsum pair0,2 pivot0 column0 normalized entry1，不以小數近似替代。", "en": "Recompute directsum pair0,2 pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "directsum pair0,2 pivot0 column0 normalized entry1" } }, { "id": "S035", "uses": [ "G001", "S034" ], "before": "directsum pair0,2:pivot row=0;column=0;nonzero pivot=1;old row=[1, 1]", "after": "normalized full row=[1, 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": "S036", "uses": [ "G001", "S035" ], "before": "directsum pair0,2 pivot1 column1 normalized entry0 := div(0, 3)", "after": "directsum pair0,2 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 directsum pair0,2 pivot1 column1 normalized entry0，不以小數近似替代。", "en": "Recompute directsum pair0,2 pivot1 column1 normalized entry0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "0", "3" ], "output": "0", "label": "directsum pair0,2 pivot1 column1 normalized entry0" } }, { "id": "S037", "uses": [ "G001", "S036" ], "before": "directsum pair0,2 pivot1 column1 normalized entry1 := div(3, 3)", "after": "directsum pair0,2 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 directsum pair0,2 pivot1 column1 normalized entry1，不以小數近似替代。", "en": "Recompute directsum pair0,2 pivot1 column1 normalized entry1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "3", "3" ], "output": "1", "label": "directsum pair0,2 pivot1 column1 normalized entry1" } }, { "id": "S038", "uses": [ "G001", "S037" ], "before": "directsum pair0,2:pivot row=1;column=1;nonzero pivot=3;old row=[0, 3]", "after": "normalized full row=[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": "S039", "uses": [ "G001", "S038" ], "before": "directsum pair0,2 eliminate row0 pivot1 entry0 product := mul(1, 0)", "after": "directsum pair0,2 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 directsum pair0,2 eliminate row0 pivot1 entry0 product，不以小數近似替代。", "en": "Recompute directsum pair0,2 eliminate row0 pivot1 entry0 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "0" ], "output": "0", "label": "directsum pair0,2 eliminate row0 pivot1 entry0 product" } }, { "id": "S040", "uses": [ "G001", "S039" ], "before": "directsum pair0,2 eliminate row0 pivot1 entry0 difference := sub(1, 0)", "after": "directsum pair0,2 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 directsum pair0,2 eliminate row0 pivot1 entry0 difference，不以小數近似替代。", "en": "Recompute directsum pair0,2 eliminate row0 pivot1 entry0 difference with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "0" ], "output": "1", "label": "directsum pair0,2 eliminate row0 pivot1 entry0 difference" } }, { "id": "S041", "uses": [ "G001", "S040" ], "before": "directsum pair0,2 eliminate row0 pivot1 entry1 product := mul(1, 1)", "after": "directsum pair0,2 eliminate row0 pivot1 entry1 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 directsum pair0,2 eliminate row0 pivot1 entry1 product，不以小數近似替代。", "en": "Recompute directsum pair0,2 eliminate row0 pivot1 entry1 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "directsum pair0,2 eliminate row0 pivot1 entry1 product" } }, { "id": "S042", "uses": [ "G001", "S041" ], "before": "directsum pair0,2 eliminate row0 pivot1 entry1 difference := sub(1, 1)", "after": "directsum pair0,2 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 directsum pair0,2 eliminate row0 pivot1 entry1 difference，不以小數近似替代。", "en": "Recompute directsum pair0,2 eliminate row0 pivot1 entry1 difference with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "1" ], "output": "0", "label": "directsum pair0,2 eliminate row0 pivot1 entry1 difference" } }, { "id": "S043", "uses": [ "G001", "S042" ], "before": "directsum pair0,2:target row=0;old=[1, 1];factor=1;pivot row=[0, 1]", "after": "new full row=[1, 0]", "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": "S044", "uses": [ "G001", "S043" ], "before": "directsum pair0,2:complete exact reduced matrix=[[1, 0], [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": "S045", "uses": [ "G001", "S044" ], "before": "directsum pair0,2 basis counts := sum(1, 1)", "after": "directsum pair0,2 basis counts = 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 directsum pair0,2 basis counts，不以小數近似替代。", "en": "Recompute directsum pair0,2 basis counts with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "1" ], "output": "2", "label": "directsum pair0,2 basis counts" } }, { "id": "S046", "uses": [ "G001", "S045" ], "before": "directsum pair0,2 intersection dimension := sub(2, 2)", "after": "directsum pair0,2 intersection dimension = 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 directsum pair0,2 intersection dimension，不以小數近似替代。", "en": "Recompute directsum pair0,2 intersection dimension with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "2", "2" ], "output": "0", "label": "directsum pair0,2 intersection dimension" } }, { "id": "S047", "uses": [ "G001", "S046" ], "before": "pair original subspaces=[0, 2];individual independent basis counts=[1, 1];combined rank=2", "after": "dim intersection=0;pair intersection zero=是", "relation": "implies", "rule_id": "directsum_pair_intersection_rank_nullity_not_global_directness", "conditions": [], "explanation": { "zh-Hant": "映射(u,w)到u+w滿射到和空間，核是(v,−v)的交集副本，所以兩項維度公式成立。這個兩項結果不能直接升格三項以上全域直和。", "en": "The map(u,w)↦u+w surjects onto the sum and has kernel consisting of(v,−v)from the intersection,giving the two-term dimension formula.This pairwise fact does not establish global directness of three or more subspaces." }, "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": "S048", "uses": [ "G001", "S047" ], "before": "directsum pair1,2:complete ordered matrix=[[0, 1], [1, 3]];rows=2;columns=2", "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": "S049", "uses": [ "G001", "S048" ], "before": "directsum pair1,2:matrix=[[0, 1], [1, 3]];selected row=1", "after": "swap rows0,1;matrix=[[1, 3], [0, 1]]", "relation": "implies", "rule_id": "finite_linear_invertible_row_swap", "conditions": [], "explanation": { "zh-Hant": "整列交換可逆，原欄與其係數標籤不變。", "en": "Swap entire rows invertibly while retaining original column/coefficient labels." }, "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": "S050", "uses": [ "G001", "S049" ], "before": "directsum pair1,2 pivot0 column0 normalized entry0 := div(1, 1)", "after": "directsum pair1,2 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 directsum pair1,2 pivot0 column0 normalized entry0，不以小數近似替代。", "en": "Recompute directsum pair1,2 pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "directsum pair1,2 pivot0 column0 normalized entry0" } }, { "id": "S051", "uses": [ "G001", "S050" ], "before": "directsum pair1,2 pivot0 column0 normalized entry1 := div(3, 1)", "after": "directsum pair1,2 pivot0 column0 normalized entry1 = 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 directsum pair1,2 pivot0 column0 normalized entry1，不以小數近似替代。", "en": "Recompute directsum pair1,2 pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "3", "1" ], "output": "3", "label": "directsum pair1,2 pivot0 column0 normalized entry1" } }, { "id": "S052", "uses": [ "G001", "S051" ], "before": "directsum pair1,2:pivot row=0;column=0;nonzero pivot=1;old row=[1, 3]", "after": "normalized full row=[1, 3]", "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": "S053", "uses": [ "G001", "S052" ], "before": "directsum pair1,2 pivot1 column1 normalized entry0 := div(0, 1)", "after": "directsum pair1,2 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 directsum pair1,2 pivot1 column1 normalized entry0，不以小數近似替代。", "en": "Recompute directsum pair1,2 pivot1 column1 normalized entry0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "0", "1" ], "output": "0", "label": "directsum pair1,2 pivot1 column1 normalized entry0" } }, { "id": "S054", "uses": [ "G001", "S053" ], "before": "directsum pair1,2 pivot1 column1 normalized entry1 := div(1, 1)", "after": "directsum pair1,2 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 directsum pair1,2 pivot1 column1 normalized entry1，不以小數近似替代。", "en": "Recompute directsum pair1,2 pivot1 column1 normalized entry1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "directsum pair1,2 pivot1 column1 normalized entry1" } }, { "id": "S055", "uses": [ "G001", "S054" ], "before": "directsum pair1,2:pivot row=1;column=1;nonzero pivot=1;old row=[0, 1]", "after": "normalized full row=[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": "S056", "uses": [ "G001", "S055" ], "before": "directsum pair1,2 eliminate row0 pivot1 entry0 product := mul(3, 0)", "after": "directsum pair1,2 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 directsum pair1,2 eliminate row0 pivot1 entry0 product，不以小數近似替代。", "en": "Recompute directsum pair1,2 eliminate row0 pivot1 entry0 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "0" ], "output": "0", "label": "directsum pair1,2 eliminate row0 pivot1 entry0 product" } }, { "id": "S057", "uses": [ "G001", "S056" ], "before": "directsum pair1,2 eliminate row0 pivot1 entry0 difference := sub(1, 0)", "after": "directsum pair1,2 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 directsum pair1,2 eliminate row0 pivot1 entry0 difference，不以小數近似替代。", "en": "Recompute directsum pair1,2 eliminate row0 pivot1 entry0 difference with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "0" ], "output": "1", "label": "directsum pair1,2 eliminate row0 pivot1 entry0 difference" } }, { "id": "S058", "uses": [ "G001", "S057" ], "before": "directsum pair1,2 eliminate row0 pivot1 entry1 product := mul(3, 1)", "after": "directsum pair1,2 eliminate row0 pivot1 entry1 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 directsum pair1,2 eliminate row0 pivot1 entry1 product，不以小數近似替代。", "en": "Recompute directsum pair1,2 eliminate row0 pivot1 entry1 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "1" ], "output": "3", "label": "directsum pair1,2 eliminate row0 pivot1 entry1 product" } }, { "id": "S059", "uses": [ "G001", "S058" ], "before": "directsum pair1,2 eliminate row0 pivot1 entry1 difference := sub(3, 3)", "after": "directsum pair1,2 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 directsum pair1,2 eliminate row0 pivot1 entry1 difference，不以小數近似替代。", "en": "Recompute directsum pair1,2 eliminate row0 pivot1 entry1 difference with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "3", "3" ], "output": "0", "label": "directsum pair1,2 eliminate row0 pivot1 entry1 difference" } }, { "id": "S060", "uses": [ "G001", "S059" ], "before": "directsum pair1,2:target row=0;old=[1, 3];factor=3;pivot row=[0, 1]", "after": "new full row=[1, 0]", "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": "S061", "uses": [ "G001", "S060" ], "before": "directsum pair1,2:complete exact reduced matrix=[[1, 0], [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": "S062", "uses": [ "G001", "S061" ], "before": "directsum pair1,2 basis counts := sum(1, 1)", "after": "directsum pair1,2 basis counts = 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 directsum pair1,2 basis counts，不以小數近似替代。", "en": "Recompute directsum pair1,2 basis counts with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "1" ], "output": "2", "label": "directsum pair1,2 basis counts" } }, { "id": "S063", "uses": [ "G001", "S062" ], "before": "directsum pair1,2 intersection dimension := sub(2, 2)", "after": "directsum pair1,2 intersection dimension = 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 directsum pair1,2 intersection dimension，不以小數近似替代。", "en": "Recompute directsum pair1,2 intersection dimension with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "2", "2" ], "output": "0", "label": "directsum pair1,2 intersection dimension" } }, { "id": "S064", "uses": [ "G001", "S063" ], "before": "pair original subspaces=[1, 2];individual independent basis counts=[1, 1];combined rank=2", "after": "dim intersection=0;pair intersection zero=是", "relation": "implies", "rule_id": "directsum_pair_intersection_rank_nullity_not_global_directness", "conditions": [], "explanation": { "zh-Hant": "映射(u,w)到u+w滿射到和空間，核是(v,−v)的交集副本，所以兩項維度公式成立。這個兩項結果不能直接升格三項以上全域直和。", "en": "The map(u,w)↦u+w surjects onto the sum and has kernel consisting of(v,−v)from the intersection,giving the two-term dimension formula.This pairwise fact does not establish global directness of three or more subspaces." }, "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": "S065", "uses": [ "G001", "S064" ], "before": "directsum ALL subspace columns:complete ordered matrix=[[1, 0, 1], [0, 1, 3]];rows=2;columns=3", "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": "S066", "uses": [ "G001", "S065" ], "before": "directsum ALL subspace columns pivot0 column0 normalized entry0 := div(1, 1)", "after": "directsum ALL subspace columns 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 directsum ALL subspace columns pivot0 column0 normalized entry0，不以小數近似替代。", "en": "Recompute directsum ALL subspace columns pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "directsum ALL subspace columns pivot0 column0 normalized entry0" } }, { "id": "S067", "uses": [ "G001", "S066" ], "before": "directsum ALL subspace columns pivot0 column0 normalized entry1 := div(0, 1)", "after": "directsum ALL subspace columns pivot0 column0 normalized entry1 = 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 directsum ALL subspace columns pivot0 column0 normalized entry1，不以小數近似替代。", "en": "Recompute directsum ALL subspace columns pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "0", "1" ], "output": "0", "label": "directsum ALL subspace columns pivot0 column0 normalized entry1" } }, { "id": "S068", "uses": [ "G001", "S067" ], "before": "directsum ALL subspace columns pivot0 column0 normalized entry2 := div(1, 1)", "after": "directsum ALL subspace columns 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 directsum ALL subspace columns pivot0 column0 normalized entry2，不以小數近似替代。", "en": "Recompute directsum ALL subspace columns pivot0 column0 normalized entry2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "directsum ALL subspace columns pivot0 column0 normalized entry2" } }, { "id": "S069", "uses": [ "G001", "S068" ], "before": "directsum ALL subspace columns:pivot row=0;column=0;nonzero pivot=1;old row=[1, 0, 1]", "after": "normalized full row=[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": "S070", "uses": [ "G001", "S069" ], "before": "directsum ALL subspace columns pivot1 column1 normalized entry0 := div(0, 1)", "after": "directsum ALL subspace columns 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 directsum ALL subspace columns pivot1 column1 normalized entry0，不以小數近似替代。", "en": "Recompute directsum ALL subspace columns pivot1 column1 normalized entry0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "0", "1" ], "output": "0", "label": "directsum ALL subspace columns pivot1 column1 normalized entry0" } }, { "id": "S071", "uses": [ "G001", "S070" ], "before": "directsum ALL subspace columns pivot1 column1 normalized entry1 := div(1, 1)", "after": "directsum ALL subspace columns 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 directsum ALL subspace columns pivot1 column1 normalized entry1，不以小數近似替代。", "en": "Recompute directsum ALL subspace columns pivot1 column1 normalized entry1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "directsum ALL subspace columns pivot1 column1 normalized entry1" } }, { "id": "S072", "uses": [ "G001", "S071" ], "before": "directsum ALL subspace columns pivot1 column1 normalized entry2 := div(3, 1)", "after": "directsum ALL subspace columns pivot1 column1 normalized entry2 = 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 directsum ALL subspace columns pivot1 column1 normalized entry2，不以小數近似替代。", "en": "Recompute directsum ALL subspace columns pivot1 column1 normalized entry2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "3", "1" ], "output": "3", "label": "directsum ALL subspace columns pivot1 column1 normalized entry2" } }, { "id": "S073", "uses": [ "G001", "S072" ], "before": "directsum ALL subspace columns:pivot row=1;column=1;nonzero pivot=1;old row=[0, 1, 3]", "after": "normalized full row=[0, 1, 3]", "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": "S074", "uses": [ "G001", "S073" ], "before": "directsum ALL subspace columns:complete exact reduced matrix=[[1, 0, 1], [0, 1, 3]];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": "S075", "uses": [ "G001", "S074" ], "before": "directsum ALL subspaces free2 pivot0 relation coefficient := mul(-1, 1)", "after": "directsum ALL subspaces free2 pivot0 relation coefficient = -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 directsum ALL subspaces free2 pivot0 relation coefficient，不以小數近似替代。", "en": "Recompute directsum ALL subspaces free2 pivot0 relation coefficient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "1" ], "output": "-1", "label": "directsum ALL subspaces free2 pivot0 relation coefficient" } }, { "id": "S076", "uses": [ "G001", "S075" ], "before": "directsum ALL subspaces free2 pivot1 relation coefficient := mul(-1, 3)", "after": "directsum ALL subspaces free2 pivot1 relation coefficient = -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 directsum ALL subspaces free2 pivot1 relation coefficient，不以小數近似替代。", "en": "Recompute directsum ALL subspaces free2 pivot1 relation coefficient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "3" ], "output": "-3", "label": "directsum ALL subspaces free2 pivot1 relation coefficient" } }, { "id": "S077", "uses": [ "G001", "S076" ], "before": "directsum ALL subspaces relation free2 original coordinate0 coordinate0 product := mul(1, -1)", "after": "directsum ALL subspaces relation free2 original coordinate0 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 directsum ALL subspaces relation free2 original coordinate0 coordinate0 product，不以小數近似替代。", "en": "Recompute directsum ALL subspaces relation free2 original coordinate0 coordinate0 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "-1" ], "output": "-1", "label": "directsum ALL subspaces relation free2 original coordinate0 coordinate0 product" } }, { "id": "S078", "uses": [ "G001", "S077" ], "before": "directsum ALL subspaces relation free2 original coordinate0 coordinate1 product := mul(0, -3)", "after": "directsum ALL subspaces relation free2 original 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 directsum ALL subspaces relation free2 original coordinate0 coordinate1 product，不以小數近似替代。", "en": "Recompute directsum ALL subspaces relation free2 original coordinate0 coordinate1 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "-3" ], "output": "0", "label": "directsum ALL subspaces relation free2 original coordinate0 coordinate1 product" } }, { "id": "S079", "uses": [ "G001", "S078" ], "before": "directsum ALL subspaces relation free2 original coordinate0 coordinate2 product := mul(1, 1)", "after": "directsum ALL subspaces relation free2 original coordinate0 coordinate2 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 directsum ALL subspaces relation free2 original coordinate0 coordinate2 product，不以小數近似替代。", "en": "Recompute directsum ALL subspaces relation free2 original coordinate0 coordinate2 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "1" ], "output": "1", "label": "directsum ALL subspaces relation free2 original coordinate0 coordinate2 product" } }, { "id": "S080", "uses": [ "G001", "S079" ], "before": "directsum ALL subspaces relation free2 original coordinate0 complete sum := sum(-1, 0, 1)", "after": "directsum ALL subspaces relation free2 original coordinate0 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 directsum ALL subspaces relation free2 original coordinate0 complete sum，不以小數近似替代。", "en": "Recompute directsum ALL subspaces relation free2 original coordinate0 complete sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-1", "0", "1" ], "output": "0", "label": "directsum ALL subspaces relation free2 original coordinate0 complete sum" } }, { "id": "S081", "uses": [ "G001", "S080" ], "before": "directsum ALL subspaces relation free2 original coordinate1 coordinate0 product := mul(0, -1)", "after": "directsum ALL subspaces relation free2 original 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 directsum ALL subspaces relation free2 original coordinate1 coordinate0 product，不以小數近似替代。", "en": "Recompute directsum ALL subspaces relation free2 original coordinate1 coordinate0 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "-1" ], "output": "0", "label": "directsum ALL subspaces relation free2 original coordinate1 coordinate0 product" } }, { "id": "S082", "uses": [ "G001", "S081" ], "before": "directsum ALL subspaces relation free2 original coordinate1 coordinate1 product := mul(1, -3)", "after": "directsum ALL subspaces relation free2 original 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 directsum ALL subspaces relation free2 original coordinate1 coordinate1 product，不以小數近似替代。", "en": "Recompute directsum ALL subspaces relation free2 original coordinate1 coordinate1 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "-3" ], "output": "-3", "label": "directsum ALL subspaces relation free2 original coordinate1 coordinate1 product" } }, { "id": "S083", "uses": [ "G001", "S082" ], "before": "directsum ALL subspaces relation free2 original coordinate1 coordinate2 product := mul(3, 1)", "after": "directsum ALL subspaces relation free2 original coordinate1 coordinate2 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 directsum ALL subspaces relation free2 original coordinate1 coordinate2 product，不以小數近似替代。", "en": "Recompute directsum ALL subspaces relation free2 original coordinate1 coordinate2 product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "1" ], "output": "3", "label": "directsum ALL subspaces relation free2 original coordinate1 coordinate2 product" } }, { "id": "S084", "uses": [ "G001", "S083" ], "before": "directsum ALL subspaces relation free2 original coordinate1 complete sum := sum(0, -3, 3)", "after": "directsum ALL subspaces relation free2 original coordinate1 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 directsum ALL subspaces relation free2 original coordinate1 complete sum，不以小數近似替代。", "en": "Recompute directsum ALL subspaces relation free2 original coordinate1 complete sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "-3", "3" ], "output": "0", "label": "directsum ALL subspaces relation free2 original coordinate1 complete sum" } }, { "id": "S085", "uses": [ "G001", "S084" ], "before": "directsum ALL subspaces:free column=2;original ordered columns=[[1, 0], [0, 1], [1, 3]];exact relation coefficients=[-1, -3, 1]", "after": "nonzero coefficient at free position;full original combination=[0, 0]", "relation": "implies", "rule_id": "directsum_actual_global_relation_not_pairwise_guess", "conditions": [], "explanation": { "zh-Hant": "自由座標設1後解所有主元座標，並對原欄完整代回。關係是原實向量空間的非零係數關係，不是列簡化後換了一組基向量。", "en": "Set one free coordinate to1,solve all pivot coordinates and substitute into the original columns.This is a nonzero coefficient relation in the original real space,not a changed family of reduced columns." }, "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": "S086", "uses": [ "G001", "S085" ], "before": "directsum ALL subspaces:pivot columns=[0, 1];ALL free columns=[2];full relation family=[[-1, -3, 1]]", "after": "EVERY zero combination is uniquely determined by its free coordinates;these free-coordinate unit relations form the full kernel;dimension=1", "relation": "implies", "rule_id": "directsum_full_relation_space_both_directions", "conditions": [], "explanation": { "zh-Hant": "可逆列運算保原矩陣核，任意零組合的自由座標就是此族的組合係數，主元方程定其餘座標；反向代回皆零。自由座標identity亦證獨立，故不只列幾個樣本關係。", "en": "Invertible row operations preserve the original kernel.Every zero combination's free coordinates are its unique coefficients in this family,and pivot equations determine the rest.Conversely all these combinations map to zero.The free-coordinate identity proves independence,not merely sampled relations." }, "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": "S087", "uses": [ "G001", "S086" ], "before": "directsum total ordered basis count := sum(1, 1, 1)", "after": "directsum total ordered basis count = 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 directsum total ordered basis count，不以小數近似替代。", "en": "Recompute directsum total ordered basis count with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "1", "1" ], "output": "3", "label": "directsum total ordered basis count" } }, { "id": "S088", "uses": [ "G001", "S087" ], "before": "ALL pair intersection dimensions=[[0, 1, 0], [0, 2, 0], [1, 2, 0]];pairwise zero=是;GLOBAL original ordered basis count=3;global rank=2;full relations=[[-1, -3, 1]]", "after": "global directness=否 from injectivity of the FULL sum map;ambient coverage=是 separately;pairwise_zero_not_sufficient_for_three_or_more", "relation": "implies", "rule_id": "directsum_threeplus_global_kernel_not_pairwise_test", "conditions": [], "explanation": { "zh-Hant": "全域直和要任意全部分量和為0時每個分量都0。每個子空間內basis獨立，故這等價合併全部欄獨立，而非僅每對交集為0。另一方面，rank等ambient dimension才給覆蓋；兩項flag不混同。", "en": "Global directness requires every component to vanish whenever their complete sum is zero.Independence within each subspace makes this equivalent to independence of all combined columns,not merely pairwise zero intersections.Coverage separately requires global rank equal to ambient dimension." }, "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": [ true, 3, 2, false, true, "pairwise_zero_not_sufficient_for_three_or_more" ] }, "conclusion": { "canonical_value": "(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)", "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-r01-direct-sum-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261105-r01-direct-sum-foundations/questions.jsonl", "batch_sha256": "7e0295c5944e58b0de639be2d989049e9f65b8f21905ff6597dbf939a5f3be6d", "original_line": 53, "original_line_sha256": "56544a77aa835a686282bdf9b79ffd3bd6538b7ca4a1b2ab034474c4d320c119", "question_compact_sha256": "d55e0a26336613eda5704e00afd0d2b448cecf0607dbc7b29e6de20958b19037", "english_row_compact_sha256": "0e120a660bdb8d05324077ac98bc0a9e09987ebb57cfc6d9b241046dd61549b1", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261105-r01-direct-sum-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261105-r01-direct-sum-foundations.en.jsonl" }, "mko_bindings": { "mko-direct-sum": { "sha256": "3eedbed2a646d34aa3bf6b0c5eb4e3482fcd8c816d40ca7c5652a6bf7224b9e8", "english_sha256": "1fe3402da619d87b341bcf2c97aad834c0b1ea87391581c56651a5dc46bc3779", "original_path": "public/data/mko/direct-sum.json", "english_path": "public/data/i18n/editions/mko-direct-sum.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "49e5d7674cefa01298c2386543f777d2bdfc99446b4027156b80142f10031994", "original_candidate_sha256": "3cb97bcc4eea4cf8c4914ab9a2f91ab12c6c9307bd102232b6396d9d7ecd92cf", "original_candidate_line_sha256": "3cb97bcc4eea4cf8c4914ab9a2f91ab12c6c9307bd102232b6396d9d7ecd92cf" }, "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": 88, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "a7e3349df2431789c7f2a66eed9ab0a83107fe71e9909f157ec4effbf754d035", "arithmetic_steps": 48 }, "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." } }