# Question 18683: Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum [Return to catalogue page 94](https://ocme.evemisslab.com/en/questions/page/94/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261105-r01-direct-sum-foundations/#q-20261014-advance-20261105-r01-direct-sum-0058) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0036-18683 · q-20261014-advance-20261105-r01-direct-sum-0058 ## Original question and givens Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 8]]]. 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). - G001 — Use R^2 and at least three subspaces, with complete independent column-vector bases subspace_bases=[[[1, 0]], [[0, 1]], [[1, 8]]]. 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). ## Definition and scope 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. - [mko-direct-sum](https://ocme.evemisslab.com/en/mko/mko-direct-sum/) ## Step-by-step derivation S001 · impliesBefore 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] Conditions: none 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. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore directsum subspace0:complete ordered matrix=[[1], [0]];rows=2;columns=1 After use exact invertible ROW operations only;original COLUMN roles/order stay fixed Conditions: none 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. Step identifiers and dependenciesRule: finite_linear_source_matrix_not_semantic_reordering · Uses: G001, S001 S003 · calculationBefore directsum subspace0 pivot0 column0 normalized entry0 := div(1, 1) After directsum subspace0 pivot0 column0 normalized entry0 = 1 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore directsum subspace0:pivot row=0;column=0;nonzero pivot=1;old row=[1] After normalized full row=[1] Conditions: none Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs. Step identifiers and dependenciesRule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S003 S005 · impliesBefore 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 Conditions: none 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. Step identifiers and dependenciesRule: finite_linear_complete_rank_certificate_not_samples · Uses: G001, S004 S006 · impliesBefore directsum subspace0:complete ORIGINAL independent columns=[[1, 0]] After declared subspace dimension=1;basis of its full real span;empty basis means zero subspace Conditions: none 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. Step identifiers and dependenciesRule: directsum_declared_basis_complete_real_span · Uses: G001, S005 S007 · impliesBefore directsum subspace1:complete ordered matrix=[[0], [1]];rows=2;columns=1 After use exact invertible ROW operations only;original COLUMN roles/order stay fixed Conditions: none 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. Step identifiers and dependenciesRule: finite_linear_source_matrix_not_semantic_reordering · Uses: G001, S006 S008 · impliesBefore directsum subspace1:matrix=[[0], [1]];selected row=1 After swap rows0,1;matrix=[[1], [0]] Conditions: none Swap entire rows invertibly while retaining original column/coefficient labels. Step identifiers and dependenciesRule: finite_linear_invertible_row_swap · Uses: G001, S007 S009 · calculationBefore directsum subspace1 pivot0 column0 normalized entry0 := div(1, 1) After directsum subspace1 pivot0 column0 normalized entry0 = 1 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore directsum subspace1:pivot row=0;column=0;nonzero pivot=1;old row=[1] After normalized full row=[1] Conditions: none Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs. Step identifiers and dependenciesRule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S009 S011 · impliesBefore 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 Conditions: none 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. Step identifiers and dependenciesRule: finite_linear_complete_rank_certificate_not_samples · Uses: G001, S010 S012 · impliesBefore directsum subspace1:complete ORIGINAL independent columns=[[0, 1]] After declared subspace dimension=1;basis of its full real span;empty basis means zero subspace Conditions: none 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. Step identifiers and dependenciesRule: directsum_declared_basis_complete_real_span · Uses: G001, S011 S013 · impliesBefore directsum subspace2:complete ordered matrix=[[1], [8]];rows=2;columns=1 After use exact invertible ROW operations only;original COLUMN roles/order stay fixed Conditions: none 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. Step identifiers and dependenciesRule: finite_linear_source_matrix_not_semantic_reordering · Uses: G001, S012 S014 · calculationBefore directsum subspace2 pivot0 column0 normalized entry0 := div(1, 1) After directsum subspace2 pivot0 column0 normalized entry0 = 1 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · impliesBefore directsum subspace2:pivot row=0;column=0;nonzero pivot=1;old row=[1] After normalized full row=[1] Conditions: none Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs. Step identifiers and dependenciesRule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S014 S016 · calculationBefore directsum subspace2 eliminate row1 pivot0 entry0 product := mul(8, 1) After directsum subspace2 eliminate row1 pivot0 entry0 product = 8 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore directsum subspace2 eliminate row1 pivot0 entry0 difference := sub(8, 8) After directsum subspace2 eliminate row1 pivot0 entry0 difference = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · impliesBefore directsum subspace2:target row=1;old=[8];factor=8;pivot row=[1] After new full row=[0] Conditions: none Subtract the same pivot-row multiple in every entry;adding it back is the inverse. Step identifiers and dependenciesRule: finite_linear_invertible_all_entry_elimination · Uses: G001, S017 S019 · impliesBefore 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 Conditions: none 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. Step identifiers and dependenciesRule: finite_linear_complete_rank_certificate_not_samples · Uses: G001, S018 S020 · impliesBefore directsum subspace2:complete ORIGINAL independent columns=[[1, 8]] After declared subspace dimension=1;basis of its full real span;empty basis means zero subspace Conditions: none 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. Step identifiers and dependenciesRule: directsum_declared_basis_complete_real_span · Uses: G001, S019 S021 · impliesBefore 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 Conditions: none 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. Step identifiers and dependenciesRule: finite_linear_source_matrix_not_semantic_reordering · Uses: G001, S020 S022 · calculationBefore directsum pair0,1 pivot0 column0 normalized entry0 := div(1, 1) After directsum pair0,1 pivot0 column0 normalized entry0 = 1 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore directsum pair0,1 pivot0 column0 normalized entry1 := div(0, 1) After directsum pair0,1 pivot0 column0 normalized entry1 = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · impliesBefore directsum pair0,1:pivot row=0;column=0;nonzero pivot=1;old row=[1, 0] After normalized full row=[1, 0] Conditions: none Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs. Step identifiers and dependenciesRule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S023 S025 · calculationBefore directsum pair0,1 pivot1 column1 normalized entry0 := div(0, 1) After directsum pair0,1 pivot1 column1 normalized entry0 = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore directsum pair0,1 pivot1 column1 normalized entry1 := div(1, 1) After directsum pair0,1 pivot1 column1 normalized entry1 = 1 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · impliesBefore directsum pair0,1:pivot row=1;column=1;nonzero pivot=1;old row=[0, 1] After normalized full row=[0, 1] Conditions: none Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs. Step identifiers and dependenciesRule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S026 S028 · impliesBefore 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 Conditions: none 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. Step identifiers and dependenciesRule: finite_linear_complete_rank_certificate_not_samples · Uses: G001, S027 S029 · calculationBefore directsum pair0,1 basis counts := sum(1, 1) After directsum pair0,1 basis counts = 2 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore directsum pair0,1 intersection dimension := sub(2, 2) After directsum pair0,1 intersection dimension = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · impliesBefore pair original subspaces=[0, 1];individual independent basis counts=[1, 1];combined rank=2 After dim intersection=0;pair intersection zero=Yes Conditions: none 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. Step identifiers and dependenciesRule: directsum_pair_intersection_rank_nullity_not_global_directness · Uses: G001, S030 S032 · impliesBefore directsum pair0,2:complete ordered matrix=[[1, 1], [0, 8]];rows=2;columns=2 After use exact invertible ROW operations only;original COLUMN roles/order stay fixed Conditions: none 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. Step identifiers and dependenciesRule: finite_linear_source_matrix_not_semantic_reordering · Uses: G001, S031 S033 · calculationBefore directsum pair0,2 pivot0 column0 normalized entry0 := div(1, 1) After directsum pair0,2 pivot0 column0 normalized entry0 = 1 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore directsum pair0,2 pivot0 column0 normalized entry1 := div(1, 1) After directsum pair0,2 pivot0 column0 normalized entry1 = 1 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · impliesBefore directsum pair0,2:pivot row=0;column=0;nonzero pivot=1;old row=[1, 1] After normalized full row=[1, 1] Conditions: none Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs. Step identifiers and dependenciesRule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S034 S036 · calculationBefore directsum pair0,2 pivot1 column1 normalized entry0 := div(0, 8) After directsum pair0,2 pivot1 column1 normalized entry0 = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore directsum pair0,2 pivot1 column1 normalized entry1 := div(8, 8) After directsum pair0,2 pivot1 column1 normalized entry1 = 1 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · impliesBefore directsum pair0,2:pivot row=1;column=1;nonzero pivot=8;old row=[0, 8] After normalized full row=[0, 1] Conditions: none Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs. Step identifiers and dependenciesRule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S037 S039 · calculationBefore directsum pair0,2 eliminate row0 pivot1 entry0 product := mul(1, 0) After directsum pair0,2 eliminate row0 pivot1 entry0 product = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore directsum pair0,2 eliminate row0 pivot1 entry0 difference := sub(1, 0) After directsum pair0,2 eliminate row0 pivot1 entry0 difference = 1 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore directsum pair0,2 eliminate row0 pivot1 entry1 product := mul(1, 1) After directsum pair0,2 eliminate row0 pivot1 entry1 product = 1 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore directsum pair0,2 eliminate row0 pivot1 entry1 difference := sub(1, 1) After directsum pair0,2 eliminate row0 pivot1 entry1 difference = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · impliesBefore directsum pair0,2:target row=0;old=[1, 1];factor=1;pivot row=[0, 1] After new full row=[1, 0] Conditions: none Subtract the same pivot-row multiple in every entry;adding it back is the inverse. Step identifiers and dependenciesRule: finite_linear_invertible_all_entry_elimination · Uses: G001, S042 S044 · impliesBefore 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 Conditions: none 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. Step identifiers and dependenciesRule: finite_linear_complete_rank_certificate_not_samples · Uses: G001, S043 S045 · calculationBefore directsum pair0,2 basis counts := sum(1, 1) After directsum pair0,2 basis counts = 2 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · calculationBefore directsum pair0,2 intersection dimension := sub(2, 2) After directsum pair0,2 intersection dimension = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · impliesBefore pair original subspaces=[0, 2];individual independent basis counts=[1, 1];combined rank=2 After dim intersection=0;pair intersection zero=Yes Conditions: none 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. Step identifiers and dependenciesRule: directsum_pair_intersection_rank_nullity_not_global_directness · Uses: G001, S046 S048 · impliesBefore directsum pair1,2:complete ordered matrix=[[0, 1], [1, 8]];rows=2;columns=2 After use exact invertible ROW operations only;original COLUMN roles/order stay fixed Conditions: none 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. Step identifiers and dependenciesRule: finite_linear_source_matrix_not_semantic_reordering · Uses: G001, S047 S049 · impliesBefore directsum pair1,2:matrix=[[0, 1], [1, 8]];selected row=1 After swap rows0,1;matrix=[[1, 8], [0, 1]] Conditions: none Swap entire rows invertibly while retaining original column/coefficient labels. Step identifiers and dependenciesRule: finite_linear_invertible_row_swap · Uses: G001, S048 S050 · calculationBefore directsum pair1,2 pivot0 column0 normalized entry0 := div(1, 1) After directsum pair1,2 pivot0 column0 normalized entry0 = 1 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S049 S051 · calculationBefore directsum pair1,2 pivot0 column0 normalized entry1 := div(8, 1) After directsum pair1,2 pivot0 column0 normalized entry1 = 8 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S050 S052 · impliesBefore directsum pair1,2:pivot row=0;column=0;nonzero pivot=1;old row=[1, 8] After normalized full row=[1, 8] Conditions: none Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs. Step identifiers and dependenciesRule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S051 S053 · calculationBefore directsum pair1,2 pivot1 column1 normalized entry0 := div(0, 1) After directsum pair1,2 pivot1 column1 normalized entry0 = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S052 S054 · calculationBefore directsum pair1,2 pivot1 column1 normalized entry1 := div(1, 1) After directsum pair1,2 pivot1 column1 normalized entry1 = 1 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S053 S055 · impliesBefore directsum pair1,2:pivot row=1;column=1;nonzero pivot=1;old row=[0, 1] After normalized full row=[0, 1] Conditions: none Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs. Step identifiers and dependenciesRule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S054 S056 · calculationBefore directsum pair1,2 eliminate row0 pivot1 entry0 product := mul(8, 0) After directsum pair1,2 eliminate row0 pivot1 entry0 product = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S055 S057 · calculationBefore directsum pair1,2 eliminate row0 pivot1 entry0 difference := sub(1, 0) After directsum pair1,2 eliminate row0 pivot1 entry0 difference = 1 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S056 S058 · calculationBefore directsum pair1,2 eliminate row0 pivot1 entry1 product := mul(8, 1) After directsum pair1,2 eliminate row0 pivot1 entry1 product = 8 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S057 S059 · calculationBefore directsum pair1,2 eliminate row0 pivot1 entry1 difference := sub(8, 8) After directsum pair1,2 eliminate row0 pivot1 entry1 difference = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S058 S060 · impliesBefore directsum pair1,2:target row=0;old=[1, 8];factor=8;pivot row=[0, 1] After new full row=[1, 0] Conditions: none Subtract the same pivot-row multiple in every entry;adding it back is the inverse. Step identifiers and dependenciesRule: finite_linear_invertible_all_entry_elimination · Uses: G001, S059 S061 · impliesBefore 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 Conditions: none 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. Step identifiers and dependenciesRule: finite_linear_complete_rank_certificate_not_samples · Uses: G001, S060 S062 · calculationBefore directsum pair1,2 basis counts := sum(1, 1) After directsum pair1,2 basis counts = 2 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S061 S063 · calculationBefore directsum pair1,2 intersection dimension := sub(2, 2) After directsum pair1,2 intersection dimension = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S062 S064 · impliesBefore pair original subspaces=[1, 2];individual independent basis counts=[1, 1];combined rank=2 After dim intersection=0;pair intersection zero=Yes Conditions: none 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. Step identifiers and dependenciesRule: directsum_pair_intersection_rank_nullity_not_global_directness · Uses: G001, S063 S065 · impliesBefore directsum ALL subspace columns:complete ordered matrix=[[1, 0, 1], [0, 1, 8]];rows=2;columns=3 After use exact invertible ROW operations only;original COLUMN roles/order stay fixed Conditions: none 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. Step identifiers and dependenciesRule: finite_linear_source_matrix_not_semantic_reordering · Uses: G001, S064 S066 · calculationBefore directsum ALL subspace columns pivot0 column0 normalized entry0 := div(1, 1) After directsum ALL subspace columns pivot0 column0 normalized entry0 = 1 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S065 S067 · calculationBefore directsum ALL subspace columns pivot0 column0 normalized entry1 := div(0, 1) After directsum ALL subspace columns pivot0 column0 normalized entry1 = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S066 S068 · calculationBefore directsum ALL subspace columns pivot0 column0 normalized entry2 := div(1, 1) After directsum ALL subspace columns pivot0 column0 normalized entry2 = 1 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S067 S069 · impliesBefore directsum ALL subspace columns:pivot row=0;column=0;nonzero pivot=1;old row=[1, 0, 1] After normalized full row=[1, 0, 1] Conditions: none Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs. Step identifiers and dependenciesRule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S068 S070 · calculationBefore directsum ALL subspace columns pivot1 column1 normalized entry0 := div(0, 1) After directsum ALL subspace columns pivot1 column1 normalized entry0 = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S069 S071 · calculationBefore directsum ALL subspace columns pivot1 column1 normalized entry1 := div(1, 1) After directsum ALL subspace columns pivot1 column1 normalized entry1 = 1 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S070 S072 · calculationBefore directsum ALL subspace columns pivot1 column1 normalized entry2 := div(8, 1) After directsum ALL subspace columns pivot1 column1 normalized entry2 = 8 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S071 S073 · impliesBefore directsum ALL subspace columns:pivot row=1;column=1;nonzero pivot=1;old row=[0, 1, 8] After normalized full row=[0, 1, 8] Conditions: none Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs. Step identifiers and dependenciesRule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S072 S074 · impliesBefore directsum ALL subspace columns:complete exact reduced matrix=[[1, 0, 1], [0, 1, 8]];pivot columns=[0, 1] After rank=2;pivot columns independent and all original column relations preserved;zero rows retained Conditions: none 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. Step identifiers and dependenciesRule: finite_linear_complete_rank_certificate_not_samples · Uses: G001, S073 S075 · calculationBefore directsum ALL subspaces free2 pivot0 relation coefficient := mul(-1, 1) After directsum ALL subspaces free2 pivot0 relation coefficient = -1 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S074 S076 · calculationBefore directsum ALL subspaces free2 pivot1 relation coefficient := mul(-1, 8) After directsum ALL subspaces free2 pivot1 relation coefficient = -8 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S075 S077 · calculationBefore directsum ALL subspaces relation free2 original coordinate0 coordinate0 product := mul(1, -1) After directsum ALL subspaces relation free2 original coordinate0 coordinate0 product = -1 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S076 S078 · calculationBefore directsum ALL subspaces relation free2 original coordinate0 coordinate1 product := mul(0, -8) After directsum ALL subspaces relation free2 original coordinate0 coordinate1 product = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S077 S079 · calculationBefore directsum ALL subspaces relation free2 original coordinate0 coordinate2 product := mul(1, 1) After directsum ALL subspaces relation free2 original coordinate0 coordinate2 product = 1 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S078 S080 · calculationBefore directsum ALL subspaces relation free2 original coordinate0 complete sum := sum(-1, 0, 1) After directsum ALL subspaces relation free2 original coordinate0 complete sum = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S079 S081 · calculationBefore directsum ALL subspaces relation free2 original coordinate1 coordinate0 product := mul(0, -1) After directsum ALL subspaces relation free2 original coordinate1 coordinate0 product = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S080 S082 · calculationBefore directsum ALL subspaces relation free2 original coordinate1 coordinate1 product := mul(1, -8) After directsum ALL subspaces relation free2 original coordinate1 coordinate1 product = -8 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S081 S083 · calculationBefore directsum ALL subspaces relation free2 original coordinate1 coordinate2 product := mul(8, 1) After directsum ALL subspaces relation free2 original coordinate1 coordinate2 product = 8 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S082 S084 · calculationBefore directsum ALL subspaces relation free2 original coordinate1 complete sum := sum(0, -8, 8) After directsum ALL subspaces relation free2 original coordinate1 complete sum = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S083 S085 · impliesBefore directsum ALL subspaces:free column=2;original ordered columns=[[1, 0], [0, 1], [1, 8]];exact relation coefficients=[-1, -8, 1] After nonzero coefficient at free position;full original combination=[0, 0] Conditions: none 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. Step identifiers and dependenciesRule: directsum_actual_global_relation_not_pairwise_guess · Uses: G001, S084 S086 · impliesBefore directsum ALL subspaces:pivot columns=[0, 1];ALL free columns=[2];full relation family=[[-1, -8, 1]] After EVERY zero combination is uniquely determined by its free coordinates;these free-coordinate unit relations form the full kernel;dimension=1 Conditions: none 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. Step identifiers and dependenciesRule: directsum_full_relation_space_both_directions · Uses: G001, S085 S087 · calculationBefore directsum total ordered basis count := sum(1, 1, 1) After directsum total ordered basis count = 3 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S086 S088 · impliesBefore ALL pair intersection dimensions=[[0, 1, 0], [0, 2, 0], [1, 2, 0]];pairwise zero=Yes;GLOBAL original ordered basis count=3;global rank=2;full relations=[[-1, -8, 1]] After global directness=No from injectivity of the FULL sum map;ambient coverage=Yes separately;pairwise_zero_not_sufficient_for_three_or_more Conditions: none 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. Step identifiers and dependenciesRule: directsum_threeplus_global_kernel_not_pairwise_test · Uses: G001, S087 ## Reverse checks - S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S087 · premise_and_result_audit — Recompute directsum total ordered basis count with exact rational arithmetic, not rounded decimals. - S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S085 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S084 · premise_and_result_audit — Recompute directsum ALL subspaces relation free2 original coordinate1 complete sum with exact rational arithmetic, not rounded decimals. - S083 · premise_and_result_audit — Recompute directsum ALL subspaces relation free2 original coordinate1 coordinate2 product with exact rational arithmetic, not rounded decimals. - S082 · premise_and_result_audit — Recompute directsum ALL subspaces relation free2 original coordinate1 coordinate1 product with exact rational arithmetic, not rounded decimals. - S081 · premise_and_result_audit — Recompute directsum ALL subspaces relation free2 original coordinate1 coordinate0 product with exact rational arithmetic, not rounded decimals. - S080 · premise_and_result_audit — Recompute directsum ALL subspaces relation free2 original coordinate0 complete sum with exact rational arithmetic, not rounded decimals. - S079 · premise_and_result_audit — Recompute directsum ALL subspaces relation free2 original coordinate0 coordinate2 product with exact rational arithmetic, not rounded decimals. - S078 · premise_and_result_audit — Recompute directsum ALL subspaces relation free2 original coordinate0 coordinate1 product with exact rational arithmetic, not rounded decimals. - S077 · premise_and_result_audit — Recompute directsum ALL subspaces relation free2 original coordinate0 coordinate0 product with exact rational arithmetic, not rounded decimals. - S076 · premise_and_result_audit — Recompute directsum ALL subspaces free2 pivot1 relation coefficient with exact rational arithmetic, not rounded decimals. - S075 · premise_and_result_audit — Recompute directsum ALL subspaces free2 pivot0 relation coefficient with exact rational arithmetic, not rounded decimals. - S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S072 · premise_and_result_audit — Recompute directsum ALL subspace columns pivot1 column1 normalized entry2 with exact rational arithmetic, not rounded decimals. - S071 · premise_and_result_audit — Recompute directsum ALL subspace columns pivot1 column1 normalized entry1 with exact rational arithmetic, not rounded decimals. - S070 · premise_and_result_audit — Recompute directsum ALL subspace columns pivot1 column1 normalized entry0 with exact rational arithmetic, not rounded decimals. - S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S068 · premise_and_result_audit — Recompute directsum ALL subspace columns pivot0 column0 normalized entry2 with exact rational arithmetic, not rounded decimals. - S067 · premise_and_result_audit — Recompute directsum ALL subspace columns pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals. - S066 · premise_and_result_audit — Recompute directsum ALL subspace columns pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals. - S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S063 · premise_and_result_audit — Recompute directsum pair1,2 intersection dimension with exact rational arithmetic, not rounded decimals. - S062 · premise_and_result_audit — Recompute directsum pair1,2 basis counts with exact rational arithmetic, not rounded decimals. - S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S059 · premise_and_result_audit — Recompute directsum pair1,2 eliminate row0 pivot1 entry1 difference with exact rational arithmetic, not rounded decimals. - S058 · premise_and_result_audit — Recompute directsum pair1,2 eliminate row0 pivot1 entry1 product with exact rational arithmetic, not rounded decimals. - S057 · premise_and_result_audit — Recompute directsum pair1,2 eliminate row0 pivot1 entry0 difference with exact rational arithmetic, not rounded decimals. - S056 · premise_and_result_audit — Recompute directsum pair1,2 eliminate row0 pivot1 entry0 product with exact rational arithmetic, not rounded decimals. - S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S054 · premise_and_result_audit — Recompute directsum pair1,2 pivot1 column1 normalized entry1 with exact rational arithmetic, not rounded decimals. - S053 · premise_and_result_audit — Recompute directsum pair1,2 pivot1 column1 normalized entry0 with exact rational arithmetic, not rounded decimals. - S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S051 · premise_and_result_audit — Recompute directsum pair1,2 pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute directsum pair1,2 pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S046 · premise_and_result_audit — Recompute directsum pair0,2 intersection dimension with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute directsum pair0,2 basis counts with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S042 · premise_and_result_audit — Recompute directsum pair0,2 eliminate row0 pivot1 entry1 difference with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute directsum pair0,2 eliminate row0 pivot1 entry1 product with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute directsum pair0,2 eliminate row0 pivot1 entry0 difference with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute directsum pair0,2 eliminate row0 pivot1 entry0 product with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S037 · premise_and_result_audit — Recompute directsum pair0,2 pivot1 column1 normalized entry1 with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute directsum pair0,2 pivot1 column1 normalized entry0 with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S034 · premise_and_result_audit — Recompute directsum pair0,2 pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute directsum pair0,2 pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S030 · premise_and_result_audit — Recompute directsum pair0,1 intersection dimension with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute directsum pair0,1 basis counts with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S026 · premise_and_result_audit — Recompute directsum pair0,1 pivot1 column1 normalized entry1 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute directsum pair0,1 pivot1 column1 normalized entry0 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S023 · premise_and_result_audit — Recompute directsum pair0,1 pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute directsum pair0,1 pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Recompute directsum subspace2 eliminate row1 pivot0 entry0 difference with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute directsum subspace2 eliminate row1 pivot0 entry0 product with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Recompute directsum subspace2 pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Recompute directsum subspace1 pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Recompute directsum subspace0 pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (Yes, 3, 2, No, Yes, pairwise_zero_not_sufficient_for_three_or_more) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 3d20adc0c8952a4aa1509b62979799d5dec9882d0aa0e6471d6761fa4c01d181 - candidate record SHA-256: c96ad7cd45de4633d428e399c85024f6424de6c6a9c002d846d453d191b9f267 - semantic review receipt SHA-256: efee5462991619d6ed9f64f11ddb085e5614ef33f9f8fcd8cf29b9329f8fd301 - question-first witness SHA-256: acf91fa1622f01d2cf04b2c513294254323665351f660f2628aaafdb869c4ff2 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261105-r01-direct-sum-0058.json)