Question 18677: Pairwise Zero Intersections Do Not Prove a Three-Way Direct Sum

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0036-18677 · q-20261014-advance-20261105-r01-direct-sum-0052

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, 2]]]. 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, 2]]]. 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

Step-by-step derivation

S001 · implies

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]

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 dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

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

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 dependencies

Rule: finite_linear_source_matrix_not_semantic_reordering · Uses: G001, S001

S003 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

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 dependencies

Rule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S003

S005 · implies

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

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 dependencies

Rule: finite_linear_complete_rank_certificate_not_samples · Uses: G001, S004

S006 · implies

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

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 dependencies

Rule: directsum_declared_basis_complete_real_span · Uses: G001, S005

S007 · implies

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

Conditions: none

Step identifiers and dependencies

Rule: finite_linear_source_matrix_not_semantic_reordering · Uses: G001, S006

S008 · implies

Before

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 dependencies

Rule: finite_linear_invertible_row_swap · Uses: G001, S007

S009 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

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 dependencies

Rule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S009

S011 · implies

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

Conditions: none

Step identifiers and dependencies

Rule: finite_linear_complete_rank_certificate_not_samples · Uses: G001, S010

S012 · implies

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

Conditions: none

Step identifiers and dependencies

Rule: directsum_declared_basis_complete_real_span · Uses: G001, S011

S013 · implies

Before

directsum subspace2:complete ordered matrix=[[1], [2]];rows=2;columns=1

After

use exact invertible ROW operations only;original COLUMN roles/order stay fixed

Conditions: none

Step identifiers and dependencies

Rule: finite_linear_source_matrix_not_semantic_reordering · Uses: G001, S012

S014 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · implies

Before

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 dependencies

Rule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S014

S016 · calculation

Before

directsum subspace2 eliminate row1 pivot0 entry0 product := mul(2, 1)

After

directsum subspace2 eliminate row1 pivot0 entry0 product = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

directsum subspace2 eliminate row1 pivot0 entry0 difference := sub(2, 2)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · implies

Before

directsum subspace2:target row=1;old=[2];factor=2;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 dependencies

Rule: finite_linear_invertible_all_entry_elimination · Uses: G001, S017

S019 · implies

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

Conditions: none

Step identifiers and dependencies

Rule: finite_linear_complete_rank_certificate_not_samples · Uses: G001, S018

S020 · implies

Before

directsum subspace2:complete ORIGINAL independent columns=[[1, 2]]

After

declared subspace dimension=1;basis of its full real span;empty basis means zero subspace

Conditions: none

Step identifiers and dependencies

Rule: directsum_declared_basis_complete_real_span · Uses: G001, S019

S021 · implies

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

Conditions: none

Step identifiers and dependencies

Rule: finite_linear_source_matrix_not_semantic_reordering · Uses: G001, S020

S022 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · implies

Before

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 dependencies

Rule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S023

S025 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · implies

Before

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 dependencies

Rule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S026

S028 · implies

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

Conditions: none

Step identifiers and dependencies

Rule: finite_linear_complete_rank_certificate_not_samples · Uses: G001, S027

S029 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · implies

Before

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 dependencies

Rule: directsum_pair_intersection_rank_nullity_not_global_directness · Uses: G001, S030

S032 · implies

Before

directsum pair0,2:complete ordered matrix=[[1, 1], [0, 2]];rows=2;columns=2

After

use exact invertible ROW operations only;original COLUMN roles/order stay fixed

Conditions: none

Step identifiers and dependencies

Rule: finite_linear_source_matrix_not_semantic_reordering · Uses: G001, S031

S033 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · implies

Before

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 dependencies

Rule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S034

S036 · calculation

Before

directsum pair0,2 pivot1 column1 normalized entry0 := div(0, 2)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

Before

directsum pair0,2 pivot1 column1 normalized entry1 := div(2, 2)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · implies

Before

directsum pair0,2:pivot row=1;column=1;nonzero pivot=2;old row=[0, 2]

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 dependencies

Rule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S037

S039 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · implies

Before

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 dependencies

Rule: finite_linear_invertible_all_entry_elimination · Uses: G001, S042

S044 · implies

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

Conditions: none

Step identifiers and dependencies

Rule: finite_linear_complete_rank_certificate_not_samples · Uses: G001, S043

S045 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S045

S047 · implies

Before

pair original subspaces=[0, 2];individual independent basis counts=[1, 1];combined rank=2

After

dim intersection=0;pair intersection zero=Yes

Conditions: none

Step identifiers and dependencies

Rule: directsum_pair_intersection_rank_nullity_not_global_directness · Uses: G001, S046

S048 · implies

Before

directsum pair1,2:complete ordered matrix=[[0, 1], [1, 2]];rows=2;columns=2

After

use exact invertible ROW operations only;original COLUMN roles/order stay fixed

Conditions: none

Step identifiers and dependencies

Rule: finite_linear_source_matrix_not_semantic_reordering · Uses: G001, S047

S049 · implies

Before

directsum pair1,2:matrix=[[0, 1], [1, 2]];selected row=1

After

swap rows0,1;matrix=[[1, 2], [0, 1]]

Conditions: none

Swap entire rows invertibly while retaining original column/coefficient labels.

Step identifiers and dependencies

Rule: finite_linear_invertible_row_swap · Uses: G001, S048

S050 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S049

S051 · calculation

Before

directsum pair1,2 pivot0 column0 normalized entry1 := div(2, 1)

After

directsum pair1,2 pivot0 column0 normalized entry1 = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S050

S052 · implies

Before

directsum pair1,2:pivot row=0;column=0;nonzero pivot=1;old row=[1, 2]

After

normalized full row=[1, 2]

Conditions: none

Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs.

Step identifiers and dependencies

Rule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S051

S053 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S052

S054 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S053

S055 · implies

Before

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 dependencies

Rule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S054

S056 · calculation

Before

directsum pair1,2 eliminate row0 pivot1 entry0 product := mul(2, 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S055

S057 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S056

S058 · calculation

Before

directsum pair1,2 eliminate row0 pivot1 entry1 product := mul(2, 1)

After

directsum pair1,2 eliminate row0 pivot1 entry1 product = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S057

S059 · calculation

Before

directsum pair1,2 eliminate row0 pivot1 entry1 difference := sub(2, 2)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S058

S060 · implies

Before

directsum pair1,2:target row=0;old=[1, 2];factor=2;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 dependencies

Rule: finite_linear_invertible_all_entry_elimination · Uses: G001, S059

S061 · implies

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

Conditions: none

Step identifiers and dependencies

Rule: finite_linear_complete_rank_certificate_not_samples · Uses: G001, S060

S062 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S061

S063 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S062

S064 · implies

Before

pair original subspaces=[1, 2];individual independent basis counts=[1, 1];combined rank=2

After

dim intersection=0;pair intersection zero=Yes

Conditions: none

Step identifiers and dependencies

Rule: directsum_pair_intersection_rank_nullity_not_global_directness · Uses: G001, S063

S065 · implies

Before

directsum ALL subspace columns:complete ordered matrix=[[1, 0, 1], [0, 1, 2]];rows=2;columns=3

After

use exact invertible ROW operations only;original COLUMN roles/order stay fixed

Conditions: none

Step identifiers and dependencies

Rule: finite_linear_source_matrix_not_semantic_reordering · Uses: G001, S064

S066 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S065

S067 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S066

S068 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S067

S069 · implies

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]

Conditions: none

Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs.

Step identifiers and dependencies

Rule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S068

S070 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S069

S071 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S070

S072 · calculation

Before

directsum ALL subspace columns pivot1 column1 normalized entry2 := div(2, 1)

After

directsum ALL subspace columns pivot1 column1 normalized entry2 = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S071

S073 · implies

Before

directsum ALL subspace columns:pivot row=1;column=1;nonzero pivot=1;old row=[0, 1, 2]

After

normalized full row=[0, 1, 2]

Conditions: none

Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs.

Step identifiers and dependencies

Rule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S072

S074 · implies

Before

directsum ALL subspace columns:complete exact reduced matrix=[[1, 0, 1], [0, 1, 2]];pivot columns=[0, 1]

After

rank=2;pivot columns independent and all original column relations preserved;zero rows retained

Conditions: none

Step identifiers and dependencies

Rule: finite_linear_complete_rank_certificate_not_samples · Uses: G001, S073

S075 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S074

S076 · calculation

Before

directsum ALL subspaces free2 pivot1 relation coefficient := mul(-1, 2)

After

directsum ALL subspaces free2 pivot1 relation coefficient = -2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S075

S077 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S076

S078 · calculation

Before

directsum ALL subspaces relation free2 original coordinate0 coordinate1 product := mul(0, -2)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S077

S079 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S078

S080 · calculation

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

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S079

S081 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S080

S082 · calculation

Before

directsum ALL subspaces relation free2 original coordinate1 coordinate1 product := mul(1, -2)

After

directsum ALL subspaces relation free2 original coordinate1 coordinate1 product = -2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S081

S083 · calculation

Before

directsum ALL subspaces relation free2 original coordinate1 coordinate2 product := mul(2, 1)

After

directsum ALL subspaces relation free2 original coordinate1 coordinate2 product = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S082

S084 · calculation

Before

directsum ALL subspaces relation free2 original coordinate1 complete sum := sum(0, -2, 2)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S083

S085 · implies

Before

directsum ALL subspaces:free column=2;original ordered columns=[[1, 0], [0, 1], [1, 2]];exact relation coefficients=[-1, -2, 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 dependencies

Rule: directsum_actual_global_relation_not_pairwise_guess · Uses: G001, S084

S086 · implies

Before

directsum ALL subspaces:pivot columns=[0, 1];ALL free columns=[2];full relation family=[[-1, -2, 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 dependencies

Rule: directsum_full_relation_space_both_directions · Uses: G001, S085

S087 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S086

S088 · implies

Before

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, -2, 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 dependencies

Rule: 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: 3be2b2928fdd3cb80b07ab4219b680a2635a188710e66035f0f793a87467c722
candidate record SHA-256: ec10838b01a53e9df3bc5251d396d7f8c16eadadc204107d63ef022428b0fa2c
semantic review receipt SHA-256: efee5462991619d6ed9f64f11ddb085e5614ef33f9f8fcd8cf29b9329f8fd301
question-first witness SHA-256: acf91fa1622f01d2cf04b2c513294254323665351f660f2628aaafdb869c4ff2
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON