Question 18619: Check This Decomposition Separately from a Whole-Space Direct Sum

On this page

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

← Question derivation index · Original question batch

R0036-18619 · q-20261014-advance-20261105-r01-direct-sum-0044

Original question and givens

Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 19]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[19, 2, 38], coefficients_u=[19], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_sum, covers_ambient, boundary).

G001 — Use the finite real ambient space R^3, complete independent column-vector bases basis_u=[[1, 0, 0]] and basis_w=[[0, 1, 19]] in supplied order. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Uniqueness in U+W and coverage of the entire ambient space are separate obligations. Target vector=[19, 2, 38], coefficients_u=[19], coefficients_w=[2]. Give (u_part, w_part, reconstructed_vector, matches_target, unique_in_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-ambient-decomposition;original snapshot and mathematics retained

After

explicit ordered source output fields=[u_part, w_part, reconstructed_vector, matches_target, unique_in_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 U:complete ordered matrix=[[1], [0], [0]];rows=3;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 U pivot0 column0 normalized entry0 := div(1, 1)

After

directsum U 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 U: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 U:complete exact reduced matrix=[[1], [0], [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 U:complete ORIGINAL independent columns=[[1, 0, 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 W:complete ordered matrix=[[0], [1], [19]];rows=3;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 W:matrix=[[0], [1], [19]];selected row=1

After

swap rows0,1;matrix=[[1], [0], [19]]

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 W pivot0 column0 normalized entry0 := div(1, 1)

After

directsum W 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 W: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 · calculation

Before

directsum W eliminate row2 pivot0 entry0 product := mul(19, 1)

After

directsum W eliminate row2 pivot0 entry0 product = 19

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, S010

S012 · calculation

Before

directsum W eliminate row2 pivot0 entry0 difference := sub(19, 19)

After

directsum W eliminate row2 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, S011

S013 · implies

Before

directsum W:target row=2;old=[19];factor=19;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, S012

S014 · implies

Before

directsum W:complete exact reduced matrix=[[1], [0], [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, S013

S015 · implies

Before

directsum W:complete ORIGINAL independent columns=[[0, 1, 19]]

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, S014

S016 · implies

Before

directsum ordered U then W:complete ordered matrix=[[1, 0], [0, 1], [0, 19]];rows=3;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, S015

S017 · calculation

Before

directsum ordered U then W pivot0 column0 normalized entry0 := div(1, 1)

After

directsum ordered U then W 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, S016

S018 · calculation

Before

directsum ordered U then W pivot0 column0 normalized entry1 := div(0, 1)

After

directsum ordered U then W 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, S017

S019 · implies

Before

directsum ordered U then W: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, S018

S020 · calculation

Before

directsum ordered U then W pivot1 column1 normalized entry0 := div(0, 1)

After

directsum ordered U then W 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, S019

S021 · calculation

Before

directsum ordered U then W pivot1 column1 normalized entry1 := div(1, 1)

After

directsum ordered U then W 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, S020

S022 · implies

Before

directsum ordered U then W: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, S021

S023 · calculation

Before

directsum ordered U then W eliminate row2 pivot1 entry0 product := mul(19, 0)

After

directsum ordered U then W eliminate row2 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, S022

S024 · calculation

Before

directsum ordered U then W eliminate row2 pivot1 entry0 difference := sub(0, 0)

After

directsum ordered U then W eliminate row2 pivot1 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, S023

S025 · calculation

Before

directsum ordered U then W eliminate row2 pivot1 entry1 product := mul(19, 1)

After

directsum ordered U then W eliminate row2 pivot1 entry1 product = 19

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 ordered U then W eliminate row2 pivot1 entry1 difference := sub(19, 19)

After

directsum ordered U then W eliminate row2 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, S025

S027 · implies

Before

directsum ordered U then W:target row=2;old=[0, 19];factor=19;pivot row=[0, 1]

After

new full row=[0, 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, S026

S028 · implies

Before

directsum ordered U then W:complete exact reduced matrix=[[1, 0], [0, 1], [0, 0]];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 · implies

Before

directsum U plus W:pivot columns=[0, 1];ALL free columns=[];full relation family=[]

After

EVERY zero combination is uniquely determined by its free coordinates;these free-coordinate unit relations form the full kernel;dimension=0

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, S028

S030 · calculation

Before

directsum U and W basis count sum := sum(1, 1)

After

directsum U and W basis count sum = 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, S029

S031 · calculation

Before

directsum actual intersection dimension := sub(2, 2)

After

directsum actual 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, S030

S032 · implies

Before

ORIGINAL U basis=[[1, 0, 0]];W basis=[[0, 1, 19]];combined rank=2;full zero-combination relations=[];common vectors=[]

After

kernel of sum map consists of(u,-u) for u in U intersection W;intersection dimension=0;unique decomposition INSIDE U+W=Yes;coverage of R^3=No

Conditions: none

The sum map surjects onto U+W,and its kernel is linearly isomorphic to U∩W via v↦(v,−v).The complete kernel basis gives the intersection dimension.Decomposition differences lie in this kernel,so zero intersection gives uniqueness within the sum.Full ambient directness additionally requires rank equal to ambient dimension,not one successfully reconstructed target.

Step identifiers and dependencies

Rule: directsum_unique_in_sum_and_ambient_coverage_separate · Uses: G001, S031

S033 · calculation

Before

directsum target U coordinate0 coordinate0 product := mul(1, 19)

After

directsum target U coordinate0 coordinate0 product = 19

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 target U coordinate0 complete sum := sum(19)

After

directsum target U coordinate0 complete sum = 19

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 · calculation

Before

directsum target U coordinate1 coordinate0 product := mul(0, 19)

After

directsum target U 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, S034

S036 · calculation

Before

directsum target U coordinate1 complete sum := sum(0)

After

directsum target U 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, S035

S037 · calculation

Before

directsum target U coordinate2 coordinate0 product := mul(0, 19)

After

directsum target U coordinate2 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, S036

S038 · calculation

Before

directsum target U coordinate2 complete sum := sum(0)

After

directsum target U coordinate2 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, S037

S039 · calculation

Before

directsum target W coordinate0 coordinate0 product := mul(0, 2)

After

directsum target W coordinate0 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, S038

S040 · calculation

Before

directsum target W coordinate0 complete sum := sum(0)

After

directsum target W 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, S039

S041 · calculation

Before

directsum target W coordinate1 coordinate0 product := mul(1, 2)

After

directsum target W coordinate1 coordinate0 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, S040

S042 · calculation

Before

directsum target W coordinate1 complete sum := sum(2)

After

directsum target W coordinate1 complete sum = 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, S041

S043 · calculation

Before

directsum target W coordinate2 coordinate0 product := mul(19, 2)

After

directsum target W coordinate2 coordinate0 product = 38

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, S042

S044 · calculation

Before

directsum target W coordinate2 complete sum := sum(38)

After

directsum target W coordinate2 complete sum = 38

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, S043

S045 · calculation

Before

directsum target reconstructed coordinate0 := sum(19, 0)

After

directsum target reconstructed coordinate0 = 19

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 target reconstructed coordinate1 := sum(0, 2)

After

directsum target reconstructed coordinate1 = 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, S045

S047 · calculation

Before

directsum target reconstructed coordinate2 := sum(0, 38)

After

directsum target reconstructed coordinate2 = 38

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, S046

S048 · implies

Before

original coefficient blocks U=[19],W=[2] in ORIGINAL basis order;target=[19, 2, 38]

After

computed parts=[[19, 0, 0], [0, 2, 38]];reconstructed=[19, 2, 38];matches THIS target=Yes;unique-in-sum=Yes;ambient coverage=No

Conditions: none

Use the original basis order for both linear combinations and compare every coordinate with the original target.Do not reorder coefficients by semantic sorting or turn one equality into coverage of every vector.

Step identifiers and dependencies

Rule: directsum_one_target_reconstruction_not_all_ambient_coverage · Uses: G001, S047

Reverse checks

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 — Recompute directsum target reconstructed coordinate2 with exact rational arithmetic, not rounded decimals.
S046 · premise_and_result_audit — Recompute directsum target reconstructed coordinate1 with exact rational arithmetic, not rounded decimals.
S045 · premise_and_result_audit — Recompute directsum target reconstructed coordinate0 with exact rational arithmetic, not rounded decimals.
S044 · premise_and_result_audit — Recompute directsum target W coordinate2 complete sum with exact rational arithmetic, not rounded decimals.
S043 · premise_and_result_audit — Recompute directsum target W coordinate2 coordinate0 product with exact rational arithmetic, not rounded decimals.
S042 · premise_and_result_audit — Recompute directsum target W coordinate1 complete sum with exact rational arithmetic, not rounded decimals.
S041 · premise_and_result_audit — Recompute directsum target W coordinate1 coordinate0 product with exact rational arithmetic, not rounded decimals.
S040 · premise_and_result_audit — Recompute directsum target W coordinate0 complete sum with exact rational arithmetic, not rounded decimals.
S039 · premise_and_result_audit — Recompute directsum target W coordinate0 coordinate0 product with exact rational arithmetic, not rounded decimals.
S038 · premise_and_result_audit — Recompute directsum target U coordinate2 complete sum with exact rational arithmetic, not rounded decimals.
S037 · premise_and_result_audit — Recompute directsum target U coordinate2 coordinate0 product with exact rational arithmetic, not rounded decimals.
S036 · premise_and_result_audit — Recompute directsum target U coordinate1 complete sum with exact rational arithmetic, not rounded decimals.
S035 · premise_and_result_audit — Recompute directsum target U coordinate1 coordinate0 product with exact rational arithmetic, not rounded decimals.
S034 · premise_and_result_audit — Recompute directsum target U coordinate0 complete sum with exact rational arithmetic, not rounded decimals.
S033 · premise_and_result_audit — Recompute directsum target U coordinate0 coordinate0 product 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 — Recompute directsum actual intersection dimension with exact rational arithmetic, not rounded decimals.
S030 · premise_and_result_audit — Recompute directsum U and W basis count sum with exact rational arithmetic, not rounded decimals.
S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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 ordered U then W eliminate row2 pivot1 entry1 difference with exact rational arithmetic, not rounded decimals.
S025 · premise_and_result_audit — Recompute directsum ordered U then W eliminate row2 pivot1 entry1 product with exact rational arithmetic, not rounded decimals.
S024 · premise_and_result_audit — Recompute directsum ordered U then W eliminate row2 pivot1 entry0 difference with exact rational arithmetic, not rounded decimals.
S023 · premise_and_result_audit — Recompute directsum ordered U then W eliminate row2 pivot1 entry0 product with exact rational arithmetic, not rounded decimals.
S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S021 · premise_and_result_audit — Recompute directsum ordered U then W pivot1 column1 normalized entry1 with exact rational arithmetic, not rounded decimals.
S020 · premise_and_result_audit — Recompute directsum ordered U then W pivot1 column1 normalized entry0 with exact rational arithmetic, not rounded decimals.
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 — Recompute directsum ordered U then W pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals.
S017 · premise_and_result_audit — Recompute directsum ordered U then W pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals.
S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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 — Recompute directsum W eliminate row2 pivot0 entry0 difference with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Recompute directsum W eliminate row2 pivot0 entry0 product with exact rational arithmetic, not rounded decimals.
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 W 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 U 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

([19, 0, 0], [0, 2, 38], [19, 2, 38], Yes, Yes, No, whole_ambient_direct_sum_requires_uniqueness_and_coverage)

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: 1537b1cefa3244daafaea88cae6096b6be8f33d08858f92edb6dd3a6ae9b71fe
candidate record SHA-256: 7df7b96b5193678a9d374f2bfb6b1a9413d93f52a147e55fa97b63f29eb1a8e0
semantic review receipt SHA-256: efee5462991619d6ed9f64f11ddb085e5614ef33f9f8fcd8cf29b9329f8fd301
question-first witness SHA-256: acf91fa1622f01d2cf04b2c513294254323665351f660f2628aaafdb869c4ff2
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON