Question 19990: Changing Representatives by W Must Preserve Both Operations

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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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R0038-19990 · q-20261014-advance-20261106-r07-quotient-vector-space-0065

Original question and givens

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 15, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[16, 2, 3], u=[3, 15, 4], shift_v_coefficients=[16], shift_u_coefficients=[-15] in supplied basis order, scalar=15. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

G001 — Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 15, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[16, 2, 3], u=[3, 15, 4], shift_v_coefficients=[16], shift_u_coefficients=[-15] in supplied basis order, scalar=15. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

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-quotient-vector-space

Step-by-step derivation

S001 · implies

Before

source operation=quotient-representative-independence;original snapshot and mathematics retained

After

explicit ordered source output fields=[shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate]

Conditions: none

Keep the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not mathematical evidence,and No retains the operation's claim scope.

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

quotient ORIGINAL ordered W basis:COMPLETE exact original ordered matrix=[[1], [15], [0]];rows=3;columns=1

After

perform invertible ROW operations on EVERY entry;original COLUMN and right-side roles remain fixed

Conditions: none

Entrywise invertible row operations preserve the whole real solution set,without bounding coefficients to a finite integer box.Keep original basis-column order and right-side roles rather than semantic sorting.

Step identifiers and dependencies

Rule: quotient_full_matrix_real_linear_equations_not_grid · Uses: G001, S001

S003 · calculation

Before

quotient ORIGINAL ordered W basis pivot0 column0 normalized entry0 := div(1, 1)

After

quotient ORIGINAL ordered W basis pivot0 column0 normalized entry0 = 1

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

quotient ORIGINAL ordered W basis:row=0;column=0;nonzero exact pivot=1;old FULL row=[1]

After

normalized FULL row=[1]

Conditions: none

Divide every entry by the same nonzero pivot,not only the left block or one entry.

Step identifiers and dependencies

Rule: quotient_nonzero_pivot_all_entry_scaling · Uses: G001, S003

S005 · calculation

Before

quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product := mul(15, 1)

After

quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product = 15

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 difference := sub(15, 15)

After

quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 difference = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · implies

Before

quotient ORIGINAL ordered W basis:target row=1;old row=[15];subtract factor=15 times FULL pivot row=[1]

After

new exact FULL row=[0]

Conditions: none

Subtract the same multiple in every entry.The inverse adds it back,preserving every equation rather than replacing the full system with sampled solutions.

Step identifiers and dependencies

Rule: quotient_invertible_complete_row_elimination · Uses: G001, S006

S008 · implies

Before

quotient ORIGINAL ordered W basis:complete exact reduced matrix=[[1], [0], [0]];ORIGINAL pivot-column indices=[0]

After

rank=1;pivot coordinates independent;ALL original column relations and full real system solution set preserved

Conditions: none

Pivot identity coordinates prove independence and express other columns by their pivot-row coordinates,so pivot count gives rank.Invertible operations carry these relations back to the original matrix.

Step identifiers and dependencies

Rule: quotient_full_rank_and_solution_set_certificate · Uses: G001, S007

S009 · implies

Before

ACTUAL ambient real space R^3;complete ORIGINAL independent basis columns of W=[[1, 15, 0]]

After

W is the FULL real span of these columns;dim W=1;zero coset is the WHOLE subspace W,not the vector0 alone

Conditions: none

Exact rank verifies independence of the declared complete basis,whose count is dim W.The list is not the set of all W elements.An empty basis gives W={0};the zero coset is still the set W,not a representative vector.

Step identifiers and dependencies

Rule: quotient_declared_subspace_full_real_span_zero_coset · Uses: G001, S008

S010 · implies

Before

ACTUAL subspace W full real span in R^3;original basis=[[1, 15, 0]]

After

for ANY representatives v'=v+w,u'=u+z with w,z in W,and ANY real lambda,sum difference=w+z in W and scalar difference=lambda*w in W;thus quotient addition/scaling independent of representatives

Conditions: none

Subspace closure under addition,negation and every real scalar keeps all representative-change differences in W.This proves well-definedness for arbitrary changes,not just supplied shifts.Quotient objects are cosets:representatives are generally nonunique,with uniqueness possible when W={0},without equating the coset object type with vectors.

Step identifiers and dependencies

Rule: quotient_arbitrary_representatives_well_defined_both_operations · Uses: G001, S009

S011 · calculation

Before

quotient original v shift original coordinate0 coefficient0 product := mul(1, 16)

After

quotient original v shift original coordinate0 coefficient0 product = 16

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

quotient original v shift original coordinate0 complete sum := sum(16)

After

quotient original v shift original coordinate0 complete sum = 16

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

quotient original v shift original coordinate1 coefficient0 product := mul(15, 16)

After

quotient original v shift original coordinate1 coefficient0 product = 240

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

quotient original v shift original coordinate1 complete sum := sum(240)

After

quotient original v shift original coordinate1 complete sum = 240

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

quotient original v shift original coordinate2 coefficient0 product := mul(0, 16)

After

quotient original v shift original coordinate2 coefficient0 product = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

quotient original v shift original coordinate2 complete sum := sum(0)

After

quotient original v shift original coordinate2 complete sum = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · implies

Before

quotient original v shift:ORIGINAL W basis order=[[1, 15, 0]];aligned coefficient block=[16]

After

computed whole vector=[16, 240, 0];a real linear combination in W by its declared span

Conditions: none

Align coefficients with the original basis columns,without semantic reordering.The complete linear combination itself witnesses membership in W.

Step identifiers and dependencies

Rule: quotient_original_basis_coefficients_not_semantic_reordering · Uses: G001, S016

S018 · calculation

Before

quotient original u shift original coordinate0 coefficient0 product := mul(1, -15)

After

quotient original u shift original coordinate0 coefficient0 product = -15

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

quotient original u shift original coordinate0 complete sum := sum(-15)

After

quotient original u shift original coordinate0 complete sum = -15

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

quotient original u shift original coordinate1 coefficient0 product := mul(15, -15)

After

quotient original u shift original coordinate1 coefficient0 product = -225

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

quotient original u shift original coordinate1 complete sum := sum(-225)

After

quotient original u shift original coordinate1 complete sum = -225

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

quotient original u shift original coordinate2 coefficient0 product := mul(0, -15)

After

quotient original u shift original coordinate2 coefficient0 product = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

quotient original u shift original coordinate2 complete sum := sum(0)

After

quotient original u shift original coordinate2 complete sum = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · implies

Before

quotient original u shift:ORIGINAL W basis order=[[1, 15, 0]];aligned coefficient block=[-15]

After

computed whole vector=[-15, -225, 0];a real linear combination in W by its declared span

Conditions: none

Align coefficients with the original basis columns,without semantic reordering.The complete linear combination itself witnesses membership in W.

Step identifiers and dependencies

Rule: quotient_original_basis_coefficients_not_semantic_reordering · Uses: G001, S023

S025 · calculation

Before

quotient changed v coordinate0 := sum(16, 16)

After

quotient changed v coordinate0 = 32

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

quotient changed v coordinate1 := sum(2, 240)

After

quotient changed v coordinate1 = 242

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

quotient changed v coordinate2 := sum(3, 0)

After

quotient changed v coordinate2 = 3

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · calculation

Before

quotient changed u coordinate0 := sum(3, -15)

After

quotient changed u coordinate0 = -12

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · calculation

Before

quotient changed u coordinate1 := sum(15, -225)

After

quotient changed u coordinate1 = -210

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

quotient changed u coordinate2 := sum(4, 0)

After

quotient changed u coordinate2 = 4

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

quotient new sum coordinate0 := sum(32, -12)

After

quotient new sum coordinate0 = 20

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

Before

quotient old sum coordinate0 := sum(16, 3)

After

quotient old sum coordinate0 = 19

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

quotient changed sum difference coordinate0 := sub(20, 19)

After

quotient changed sum difference coordinate0 = 1

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

quotient new scalar representative coordinate0 := mul(15, 32)

After

quotient new scalar representative coordinate0 = 480

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

quotient old scalar representative coordinate0 := mul(15, 16)

After

quotient old scalar representative coordinate0 = 240

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · calculation

Before

quotient changed scalar difference coordinate0 := sub(480, 240)

After

quotient changed scalar difference coordinate0 = 240

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

Before

quotient new sum coordinate1 := sum(242, -210)

After

quotient new sum coordinate1 = 32

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · calculation

Before

quotient old sum coordinate1 := sum(2, 15)

After

quotient old sum coordinate1 = 17

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · calculation

Before

quotient changed sum difference coordinate1 := sub(32, 17)

After

quotient changed sum difference coordinate1 = 15

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · calculation

Before

quotient new scalar representative coordinate1 := mul(15, 242)

After

quotient new scalar representative coordinate1 = 3630

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · calculation

Before

quotient old scalar representative coordinate1 := mul(15, 2)

After

quotient old scalar representative coordinate1 = 30

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · calculation

Before

quotient changed scalar difference coordinate1 := sub(3630, 30)

After

quotient changed scalar difference coordinate1 = 3600

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

quotient new sum coordinate2 := sum(3, 4)

After

quotient new sum coordinate2 = 7

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · calculation

Before

quotient old sum coordinate2 := sum(3, 4)

After

quotient old sum coordinate2 = 7

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · calculation

Before

quotient changed sum difference coordinate2 := sub(7, 7)

After

quotient changed sum difference coordinate2 = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · calculation

Before

quotient new scalar representative coordinate2 := mul(15, 3)

After

quotient new scalar representative coordinate2 = 45

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S045

S047 · calculation

Before

quotient old scalar representative coordinate2 := mul(15, 3)

After

quotient old scalar representative coordinate2 = 45

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S046

S048 · calculation

Before

quotient changed scalar difference coordinate2 := sub(45, 45)

After

quotient changed scalar difference coordinate2 = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S047

S049 · implies

Before

quotient supplied v shift membership:COMPLETE exact original ordered matrix=[[1, 16], [15, 240], [0, 0]];rows=3;columns=2

After

perform invertible ROW operations on EVERY entry;original COLUMN and right-side roles remain fixed

Conditions: none

Step identifiers and dependencies

Rule: quotient_full_matrix_real_linear_equations_not_grid · Uses: G001, S048

S050 · calculation

Before

quotient supplied v shift membership pivot0 column0 normalized entry0 := div(1, 1)

After

quotient supplied v shift membership pivot0 column0 normalized entry0 = 1

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S049

S051 · calculation

Before

quotient supplied v shift membership pivot0 column0 normalized entry1 := div(16, 1)

After

quotient supplied v shift membership pivot0 column0 normalized entry1 = 16

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S050

S052 · implies

Before

quotient supplied v shift membership:row=0;column=0;nonzero exact pivot=1;old FULL row=[1, 16]

After

normalized FULL row=[1, 16]

Conditions: none

Divide every entry by the same nonzero pivot,not only the left block or one entry.

Step identifiers and dependencies

Rule: quotient_nonzero_pivot_all_entry_scaling · Uses: G001, S051

S053 · calculation

Before

quotient supplied v shift membership eliminate row1 pivot0 entry0 product := mul(15, 1)

After

quotient supplied v shift membership eliminate row1 pivot0 entry0 product = 15

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S052

S054 · calculation

Before

quotient supplied v shift membership eliminate row1 pivot0 entry0 difference := sub(15, 15)

After

quotient supplied v shift membership eliminate row1 pivot0 entry0 difference = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S053

S055 · calculation

Before

quotient supplied v shift membership eliminate row1 pivot0 entry1 product := mul(15, 16)

After

quotient supplied v shift membership eliminate row1 pivot0 entry1 product = 240

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S054

S056 · calculation

Before

quotient supplied v shift membership eliminate row1 pivot0 entry1 difference := sub(240, 240)

After

quotient supplied v shift membership eliminate row1 pivot0 entry1 difference = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S055

S057 · implies

Before

quotient supplied v shift membership:target row=1;old row=[15, 240];subtract factor=15 times FULL pivot row=[1, 16]

After

new exact FULL row=[0, 0]

Conditions: none

Subtract the same multiple in every entry.The inverse adds it back,preserving every equation rather than replacing the full system with sampled solutions.

Step identifiers and dependencies

Rule: quotient_invertible_complete_row_elimination · Uses: G001, S056

S058 · implies

Before

quotient supplied v shift membership:column=1;remaining exact entries=[0, 0]

After

no pivot added;current rank=1

Conditions: none

Do not add a pivot for an all-zero remaining column;retain its original role and index.

Step identifiers and dependencies

Rule: quotient_zero_remaining_column_no_fake_pivot · Uses: G001, S057

S059 · implies

Before

quotient supplied v shift membership:complete exact reduced matrix=[[1, 16], [0, 0], [0, 0]];ORIGINAL pivot-column indices=[0]

After

rank=1;pivot coordinates independent;ALL original column relations and full real system solution set preserved

Conditions: none

Step identifiers and dependencies

Rule: quotient_full_rank_and_solution_set_certificate · Uses: G001, S058

S060 · calculation

Before

quotient supplied v shift membership original coordinate0 reconstruction coefficient0 product := mul(1, 16)

After

quotient supplied v shift membership original coordinate0 reconstruction coefficient0 product = 16

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S059

S061 · calculation

Before

quotient supplied v shift membership original coordinate0 reconstruction complete sum := sum(16)

After

quotient supplied v shift membership original coordinate0 reconstruction complete sum = 16

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S060

S062 · calculation

Before

quotient supplied v shift membership original coordinate1 reconstruction coefficient0 product := mul(15, 16)

After

quotient supplied v shift membership original coordinate1 reconstruction coefficient0 product = 240

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S061

S063 · calculation

Before

quotient supplied v shift membership original coordinate1 reconstruction complete sum := sum(240)

After

quotient supplied v shift membership original coordinate1 reconstruction complete sum = 240

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S062

S064 · calculation

Before

quotient supplied v shift membership original coordinate2 reconstruction coefficient0 product := mul(0, 16)

After

quotient supplied v shift membership original coordinate2 reconstruction coefficient0 product = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S063

S065 · calculation

Before

quotient supplied v shift membership original coordinate2 reconstruction complete sum := sum(0)

After

quotient supplied v shift membership original coordinate2 reconstruction complete sum = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S064

S066 · implies

Before

quotient supplied v shift membership:original proposed vector=[16, 240, 0];original ordered independent columns=[[1, 15, 0]];complete reduced augmented system=[[1, 16], [0, 0], [0, 0]]

After

exact real-span coefficient witness=[16];FULL original reconstruction=[16, 240, 0];membership=Yes

Conditions: none

Read exact coefficients from the pivot equations and reconstruct every original coordinate.Empty coefficients reconstruct only the zero vector for an empty basis.Valid coefficients for nonempty bases may be fractional or outside original input bounds and must not be excluded.

Step identifiers and dependencies

Rule: quotient_full_span_coefficient_witness_not_sampled_membership · Uses: G001, S065

S067 · implies

Before

quotient supplied u shift membership:COMPLETE exact original ordered matrix=[[1, -15], [15, -225], [0, 0]];rows=3;columns=2

After

perform invertible ROW operations on EVERY entry;original COLUMN and right-side roles remain fixed

Conditions: none

Step identifiers and dependencies

Rule: quotient_full_matrix_real_linear_equations_not_grid · Uses: G001, S066

S068 · calculation

Before

quotient supplied u shift membership pivot0 column0 normalized entry0 := div(1, 1)

After

quotient supplied u shift membership pivot0 column0 normalized entry0 = 1

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S067

S069 · calculation

Before

quotient supplied u shift membership pivot0 column0 normalized entry1 := div(-15, 1)

After

quotient supplied u shift membership pivot0 column0 normalized entry1 = -15

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S068

S070 · implies

Before

quotient supplied u shift membership:row=0;column=0;nonzero exact pivot=1;old FULL row=[1, -15]

After

normalized FULL row=[1, -15]

Conditions: none

Divide every entry by the same nonzero pivot,not only the left block or one entry.

Step identifiers and dependencies

Rule: quotient_nonzero_pivot_all_entry_scaling · Uses: G001, S069

S071 · calculation

Before

quotient supplied u shift membership eliminate row1 pivot0 entry0 product := mul(15, 1)

After

quotient supplied u shift membership eliminate row1 pivot0 entry0 product = 15

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S070

S072 · calculation

Before

quotient supplied u shift membership eliminate row1 pivot0 entry0 difference := sub(15, 15)

After

quotient supplied u shift membership eliminate row1 pivot0 entry0 difference = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S071

S073 · calculation

Before

quotient supplied u shift membership eliminate row1 pivot0 entry1 product := mul(15, -15)

After

quotient supplied u shift membership eliminate row1 pivot0 entry1 product = -225

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S072

S074 · calculation

Before

quotient supplied u shift membership eliminate row1 pivot0 entry1 difference := sub(-225, -225)

After

quotient supplied u shift membership eliminate row1 pivot0 entry1 difference = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S073

S075 · implies

Before

quotient supplied u shift membership:target row=1;old row=[15, -225];subtract factor=15 times FULL pivot row=[1, -15]

After

new exact FULL row=[0, 0]

Conditions: none

Subtract the same multiple in every entry.The inverse adds it back,preserving every equation rather than replacing the full system with sampled solutions.

Step identifiers and dependencies

Rule: quotient_invertible_complete_row_elimination · Uses: G001, S074

S076 · implies

Before

quotient supplied u shift membership:column=1;remaining exact entries=[0, 0]

After

no pivot added;current rank=1

Conditions: none

Do not add a pivot for an all-zero remaining column;retain its original role and index.

Step identifiers and dependencies

Rule: quotient_zero_remaining_column_no_fake_pivot · Uses: G001, S075

S077 · implies

Before

quotient supplied u shift membership:complete exact reduced matrix=[[1, -15], [0, 0], [0, 0]];ORIGINAL pivot-column indices=[0]

After

rank=1;pivot coordinates independent;ALL original column relations and full real system solution set preserved

Conditions: none

Step identifiers and dependencies

Rule: quotient_full_rank_and_solution_set_certificate · Uses: G001, S076

S078 · calculation

Before

quotient supplied u shift membership original coordinate0 reconstruction coefficient0 product := mul(1, -15)

After

quotient supplied u shift membership original coordinate0 reconstruction coefficient0 product = -15

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S077

S079 · calculation

Before

quotient supplied u shift membership original coordinate0 reconstruction complete sum := sum(-15)

After

quotient supplied u shift membership original coordinate0 reconstruction complete sum = -15

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S078

S080 · calculation

Before

quotient supplied u shift membership original coordinate1 reconstruction coefficient0 product := mul(15, -15)

After

quotient supplied u shift membership original coordinate1 reconstruction coefficient0 product = -225

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S079

S081 · calculation

Before

quotient supplied u shift membership original coordinate1 reconstruction complete sum := sum(-225)

After

quotient supplied u shift membership original coordinate1 reconstruction complete sum = -225

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S080

S082 · calculation

Before

quotient supplied u shift membership original coordinate2 reconstruction coefficient0 product := mul(0, -15)

After

quotient supplied u shift membership original coordinate2 reconstruction coefficient0 product = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S081

S083 · calculation

Before

quotient supplied u shift membership original coordinate2 reconstruction complete sum := sum(0)

After

quotient supplied u shift membership original coordinate2 reconstruction complete sum = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S082

S084 · implies

Before

quotient supplied u shift membership:original proposed vector=[-15, -225, 0];original ordered independent columns=[[1, 15, 0]];complete reduced augmented system=[[1, -15], [0, 0], [0, 0]]

After

exact real-span coefficient witness=[-15];FULL original reconstruction=[-15, -225, 0];membership=Yes

Conditions: none

Step identifiers and dependencies

Rule: quotient_full_span_coefficient_witness_not_sampled_membership · Uses: G001, S083

S085 · implies

Before

quotient supplied sum difference membership:COMPLETE exact original ordered matrix=[[1, 1], [15, 15], [0, 0]];rows=3;columns=2

After

perform invertible ROW operations on EVERY entry;original COLUMN and right-side roles remain fixed

Conditions: none

Step identifiers and dependencies

Rule: quotient_full_matrix_real_linear_equations_not_grid · Uses: G001, S084

S086 · calculation

Before

quotient supplied sum difference membership pivot0 column0 normalized entry0 := div(1, 1)

After

quotient supplied sum difference membership pivot0 column0 normalized entry0 = 1

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S085

S087 · calculation

Before

quotient supplied sum difference membership pivot0 column0 normalized entry1 := div(1, 1)

After

quotient supplied sum difference membership pivot0 column0 normalized entry1 = 1

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S086

S088 · implies

Before

quotient supplied sum difference membership:row=0;column=0;nonzero exact pivot=1;old FULL row=[1, 1]

After

normalized FULL row=[1, 1]

Conditions: none

Divide every entry by the same nonzero pivot,not only the left block or one entry.

Step identifiers and dependencies

Rule: quotient_nonzero_pivot_all_entry_scaling · Uses: G001, S087

S089 · calculation

Before

quotient supplied sum difference membership eliminate row1 pivot0 entry0 product := mul(15, 1)

After

quotient supplied sum difference membership eliminate row1 pivot0 entry0 product = 15

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S088

S090 · calculation

Before

quotient supplied sum difference membership eliminate row1 pivot0 entry0 difference := sub(15, 15)

After

quotient supplied sum difference membership eliminate row1 pivot0 entry0 difference = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S089

S091 · calculation

Before

quotient supplied sum difference membership eliminate row1 pivot0 entry1 product := mul(15, 1)

After

quotient supplied sum difference membership eliminate row1 pivot0 entry1 product = 15

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S090

S092 · calculation

Before

quotient supplied sum difference membership eliminate row1 pivot0 entry1 difference := sub(15, 15)

After

quotient supplied sum difference membership eliminate row1 pivot0 entry1 difference = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S091

S093 · implies

Before

quotient supplied sum difference membership:target row=1;old row=[15, 15];subtract factor=15 times FULL pivot row=[1, 1]

After

new exact FULL row=[0, 0]

Conditions: none

Subtract the same multiple in every entry.The inverse adds it back,preserving every equation rather than replacing the full system with sampled solutions.

Step identifiers and dependencies

Rule: quotient_invertible_complete_row_elimination · Uses: G001, S092

S094 · implies

Before

quotient supplied sum difference membership:column=1;remaining exact entries=[0, 0]

After

no pivot added;current rank=1

Conditions: none

Do not add a pivot for an all-zero remaining column;retain its original role and index.

Step identifiers and dependencies

Rule: quotient_zero_remaining_column_no_fake_pivot · Uses: G001, S093

S095 · implies

Before

quotient supplied sum difference membership:complete exact reduced matrix=[[1, 1], [0, 0], [0, 0]];ORIGINAL pivot-column indices=[0]

After

rank=1;pivot coordinates independent;ALL original column relations and full real system solution set preserved

Conditions: none

Step identifiers and dependencies

Rule: quotient_full_rank_and_solution_set_certificate · Uses: G001, S094

S096 · calculation

Before

quotient supplied sum difference membership original coordinate0 reconstruction coefficient0 product := mul(1, 1)

After

quotient supplied sum difference membership original coordinate0 reconstruction coefficient0 product = 1

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S095

S097 · calculation

Before

quotient supplied sum difference membership original coordinate0 reconstruction complete sum := sum(1)

After

quotient supplied sum difference membership original coordinate0 reconstruction complete sum = 1

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S096

S098 · calculation

Before

quotient supplied sum difference membership original coordinate1 reconstruction coefficient0 product := mul(15, 1)

After

quotient supplied sum difference membership original coordinate1 reconstruction coefficient0 product = 15

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S097

S099 · calculation

Before

quotient supplied sum difference membership original coordinate1 reconstruction complete sum := sum(15)

After

quotient supplied sum difference membership original coordinate1 reconstruction complete sum = 15

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S098

S100 · calculation

Before

quotient supplied sum difference membership original coordinate2 reconstruction coefficient0 product := mul(0, 1)

After

quotient supplied sum difference membership original coordinate2 reconstruction coefficient0 product = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S099

S101 · calculation

Before

quotient supplied sum difference membership original coordinate2 reconstruction complete sum := sum(0)

After

quotient supplied sum difference membership original coordinate2 reconstruction complete sum = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S100

S102 · implies

Before

quotient supplied sum difference membership:original proposed vector=[1, 15, 0];original ordered independent columns=[[1, 15, 0]];complete reduced augmented system=[[1, 1], [0, 0], [0, 0]]

After

exact real-span coefficient witness=[1];FULL original reconstruction=[1, 15, 0];membership=Yes

Conditions: none

Step identifiers and dependencies

Rule: quotient_full_span_coefficient_witness_not_sampled_membership · Uses: G001, S101

S103 · implies

Before

quotient supplied scalar difference membership:COMPLETE exact original ordered matrix=[[1, 240], [15, 3600], [0, 0]];rows=3;columns=2

After

perform invertible ROW operations on EVERY entry;original COLUMN and right-side roles remain fixed

Conditions: none

Step identifiers and dependencies

Rule: quotient_full_matrix_real_linear_equations_not_grid · Uses: G001, S102

S104 · calculation

Before

quotient supplied scalar difference membership pivot0 column0 normalized entry0 := div(1, 1)

After

quotient supplied scalar difference membership pivot0 column0 normalized entry0 = 1

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S103

S105 · calculation

Before

quotient supplied scalar difference membership pivot0 column0 normalized entry1 := div(240, 1)

After

quotient supplied scalar difference membership pivot0 column0 normalized entry1 = 240

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S104

S106 · implies

Before

quotient supplied scalar difference membership:row=0;column=0;nonzero exact pivot=1;old FULL row=[1, 240]

After

normalized FULL row=[1, 240]

Conditions: none

Divide every entry by the same nonzero pivot,not only the left block or one entry.

Step identifiers and dependencies

Rule: quotient_nonzero_pivot_all_entry_scaling · Uses: G001, S105

S107 · calculation

Before

quotient supplied scalar difference membership eliminate row1 pivot0 entry0 product := mul(15, 1)

After

quotient supplied scalar difference membership eliminate row1 pivot0 entry0 product = 15

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S106

S108 · calculation

Before

quotient supplied scalar difference membership eliminate row1 pivot0 entry0 difference := sub(15, 15)

After

quotient supplied scalar difference membership eliminate row1 pivot0 entry0 difference = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S107

S109 · calculation

Before

quotient supplied scalar difference membership eliminate row1 pivot0 entry1 product := mul(15, 240)

After

quotient supplied scalar difference membership eliminate row1 pivot0 entry1 product = 3600

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S108

S110 · calculation

Before

quotient supplied scalar difference membership eliminate row1 pivot0 entry1 difference := sub(3600, 3600)

After

quotient supplied scalar difference membership eliminate row1 pivot0 entry1 difference = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S109

S111 · implies

Before

quotient supplied scalar difference membership:target row=1;old row=[15, 3600];subtract factor=15 times FULL pivot row=[1, 240]

After

new exact FULL row=[0, 0]

Conditions: none

Subtract the same multiple in every entry.The inverse adds it back,preserving every equation rather than replacing the full system with sampled solutions.

Step identifiers and dependencies

Rule: quotient_invertible_complete_row_elimination · Uses: G001, S110

S112 · implies

Before

quotient supplied scalar difference membership:column=1;remaining exact entries=[0, 0]

After

no pivot added;current rank=1

Conditions: none

Do not add a pivot for an all-zero remaining column;retain its original role and index.

Step identifiers and dependencies

Rule: quotient_zero_remaining_column_no_fake_pivot · Uses: G001, S111

S113 · implies

Before

quotient supplied scalar difference membership:complete exact reduced matrix=[[1, 240], [0, 0], [0, 0]];ORIGINAL pivot-column indices=[0]

After

rank=1;pivot coordinates independent;ALL original column relations and full real system solution set preserved

Conditions: none

Step identifiers and dependencies

Rule: quotient_full_rank_and_solution_set_certificate · Uses: G001, S112

S114 · calculation

Before

quotient supplied scalar difference membership original coordinate0 reconstruction coefficient0 product := mul(1, 240)

After

quotient supplied scalar difference membership original coordinate0 reconstruction coefficient0 product = 240

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S113

S115 · calculation

Before

quotient supplied scalar difference membership original coordinate0 reconstruction complete sum := sum(240)

After

quotient supplied scalar difference membership original coordinate0 reconstruction complete sum = 240

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S114

S116 · calculation

Before

quotient supplied scalar difference membership original coordinate1 reconstruction coefficient0 product := mul(15, 240)

After

quotient supplied scalar difference membership original coordinate1 reconstruction coefficient0 product = 3600

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S115

S117 · calculation

Before

quotient supplied scalar difference membership original coordinate1 reconstruction complete sum := sum(3600)

After

quotient supplied scalar difference membership original coordinate1 reconstruction complete sum = 3600

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S116

S118 · calculation

Before

quotient supplied scalar difference membership original coordinate2 reconstruction coefficient0 product := mul(0, 240)

After

quotient supplied scalar difference membership original coordinate2 reconstruction coefficient0 product = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S117

S119 · calculation

Before

quotient supplied scalar difference membership original coordinate2 reconstruction complete sum := sum(0)

After

quotient supplied scalar difference membership original coordinate2 reconstruction complete sum = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S118

S120 · implies

Before

quotient supplied scalar difference membership:original proposed vector=[240, 3600, 0];original ordered independent columns=[[1, 15, 0]];complete reduced augmented system=[[1, 240], [0, 0], [0, 0]]

After

exact real-span coefficient witness=[240];FULL original reconstruction=[240, 3600, 0];membership=Yes

Conditions: none

Step identifiers and dependencies

Rule: quotient_full_span_coefficient_witness_not_sampled_membership · Uses: G001, S119

S121 · implies

Before

ACTUAL original W shifts=[[16, 240, 0], [-15, -225, 0]];changed representatives=[[32, 242, 3], [-12, -210, 4]];actual sum/scalar differences=[[1, 15, 0], [240, 3600, 0]]

After

ALL four exact full-span memberships=[Yes, Yes, Yes, Yes];differences equal shift_v+shift_u and lambda*shift_v,matching the arbitrary-representative proof

Conditions: none

Check every supplied difference in the complete original W and connect it to the arbitrary-change algebra above.The four passing instances are not the sole universal argument.Keep legitimate lambda=0 and W=0 cases.

Step identifiers and dependencies

Rule: quotient_supplied_changes_match_general_well_definedness · Uses: G001, S120

Reverse checks

S121 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S120 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S119 · premise_and_result_audit — Recompute quotient supplied scalar difference membership original coordinate2 reconstruction complete sum with exact rational arithmetic, not rounded decimals.
S118 · premise_and_result_audit — Recompute quotient supplied scalar difference membership original coordinate2 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals.
S117 · premise_and_result_audit — Recompute quotient supplied scalar difference membership original coordinate1 reconstruction complete sum with exact rational arithmetic, not rounded decimals.
S116 · premise_and_result_audit — Recompute quotient supplied scalar difference membership original coordinate1 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals.
S115 · premise_and_result_audit — Recompute quotient supplied scalar difference membership original coordinate0 reconstruction complete sum with exact rational arithmetic, not rounded decimals.
S114 · premise_and_result_audit — Recompute quotient supplied scalar difference membership original coordinate0 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals.
S113 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S112 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S111 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S110 · premise_and_result_audit — Recompute quotient supplied scalar difference membership eliminate row1 pivot0 entry1 difference with exact rational arithmetic, not rounded decimals.
S109 · premise_and_result_audit — Recompute quotient supplied scalar difference membership eliminate row1 pivot0 entry1 product with exact rational arithmetic, not rounded decimals.
S108 · premise_and_result_audit — Recompute quotient supplied scalar difference membership eliminate row1 pivot0 entry0 difference with exact rational arithmetic, not rounded decimals.
S107 · premise_and_result_audit — Recompute quotient supplied scalar difference membership eliminate row1 pivot0 entry0 product with exact rational arithmetic, not rounded decimals.
S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S105 · premise_and_result_audit — Recompute quotient supplied scalar difference membership pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals.
S104 · premise_and_result_audit — Recompute quotient supplied scalar difference membership pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals.
S103 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S101 · premise_and_result_audit — Recompute quotient supplied sum difference membership original coordinate2 reconstruction complete sum with exact rational arithmetic, not rounded decimals.
S100 · premise_and_result_audit — Recompute quotient supplied sum difference membership original coordinate2 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals.
S099 · premise_and_result_audit — Recompute quotient supplied sum difference membership original coordinate1 reconstruction complete sum with exact rational arithmetic, not rounded decimals.
S098 · premise_and_result_audit — Recompute quotient supplied sum difference membership original coordinate1 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals.
S097 · premise_and_result_audit — Recompute quotient supplied sum difference membership original coordinate0 reconstruction complete sum with exact rational arithmetic, not rounded decimals.
S096 · premise_and_result_audit — Recompute quotient supplied sum difference membership original coordinate0 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals.
S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S094 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S092 · premise_and_result_audit — Recompute quotient supplied sum difference membership eliminate row1 pivot0 entry1 difference with exact rational arithmetic, not rounded decimals.
S091 · premise_and_result_audit — Recompute quotient supplied sum difference membership eliminate row1 pivot0 entry1 product with exact rational arithmetic, not rounded decimals.
S090 · premise_and_result_audit — Recompute quotient supplied sum difference membership eliminate row1 pivot0 entry0 difference with exact rational arithmetic, not rounded decimals.
S089 · premise_and_result_audit — Recompute quotient supplied sum difference membership eliminate row1 pivot0 entry0 product with exact rational arithmetic, not rounded decimals.
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 quotient supplied sum difference membership pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals.
S086 · premise_and_result_audit — Recompute quotient supplied sum difference membership pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals.
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S083 · premise_and_result_audit — Recompute quotient supplied u shift membership original coordinate2 reconstruction complete sum with exact rational arithmetic, not rounded decimals.
S082 · premise_and_result_audit — Recompute quotient supplied u shift membership original coordinate2 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals.
S081 · premise_and_result_audit — Recompute quotient supplied u shift membership original coordinate1 reconstruction complete sum with exact rational arithmetic, not rounded decimals.
S080 · premise_and_result_audit — Recompute quotient supplied u shift membership original coordinate1 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals.
S079 · premise_and_result_audit — Recompute quotient supplied u shift membership original coordinate0 reconstruction complete sum with exact rational arithmetic, not rounded decimals.
S078 · premise_and_result_audit — Recompute quotient supplied u shift membership original coordinate0 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals.
S077 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S074 · premise_and_result_audit — Recompute quotient supplied u shift membership eliminate row1 pivot0 entry1 difference with exact rational arithmetic, not rounded decimals.
S073 · premise_and_result_audit — Recompute quotient supplied u shift membership eliminate row1 pivot0 entry1 product with exact rational arithmetic, not rounded decimals.
S072 · premise_and_result_audit — Recompute quotient supplied u shift membership eliminate row1 pivot0 entry0 difference with exact rational arithmetic, not rounded decimals.
S071 · premise_and_result_audit — Recompute quotient supplied u shift membership eliminate row1 pivot0 entry0 product with exact rational arithmetic, not rounded decimals.
S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S069 · premise_and_result_audit — Recompute quotient supplied u shift membership pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals.
S068 · premise_and_result_audit — Recompute quotient supplied u shift membership pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals.
S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S065 · premise_and_result_audit — Recompute quotient supplied v shift membership original coordinate2 reconstruction complete sum with exact rational arithmetic, not rounded decimals.
S064 · premise_and_result_audit — Recompute quotient supplied v shift membership original coordinate2 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals.
S063 · premise_and_result_audit — Recompute quotient supplied v shift membership original coordinate1 reconstruction complete sum with exact rational arithmetic, not rounded decimals.
S062 · premise_and_result_audit — Recompute quotient supplied v shift membership original coordinate1 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals.
S061 · premise_and_result_audit — Recompute quotient supplied v shift membership original coordinate0 reconstruction complete sum with exact rational arithmetic, not rounded decimals.
S060 · premise_and_result_audit — Recompute quotient supplied v shift membership original coordinate0 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals.
S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S056 · premise_and_result_audit — Recompute quotient supplied v shift membership eliminate row1 pivot0 entry1 difference with exact rational arithmetic, not rounded decimals.
S055 · premise_and_result_audit — Recompute quotient supplied v shift membership eliminate row1 pivot0 entry1 product with exact rational arithmetic, not rounded decimals.
S054 · premise_and_result_audit — Recompute quotient supplied v shift membership eliminate row1 pivot0 entry0 difference with exact rational arithmetic, not rounded decimals.
S053 · premise_and_result_audit — Recompute quotient supplied v shift membership eliminate row1 pivot0 entry0 product 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 quotient supplied v shift membership pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals.
S050 · premise_and_result_audit — Recompute quotient supplied v shift membership 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 — Recompute quotient changed scalar difference coordinate2 with exact rational arithmetic, not rounded decimals.
S047 · premise_and_result_audit — Recompute quotient old scalar representative coordinate2 with exact rational arithmetic, not rounded decimals.
S046 · premise_and_result_audit — Recompute quotient new scalar representative coordinate2 with exact rational arithmetic, not rounded decimals.
S045 · premise_and_result_audit — Recompute quotient changed sum difference coordinate2 with exact rational arithmetic, not rounded decimals.
S044 · premise_and_result_audit — Recompute quotient old sum coordinate2 with exact rational arithmetic, not rounded decimals.
S043 · premise_and_result_audit — Recompute quotient new sum coordinate2 with exact rational arithmetic, not rounded decimals.
S042 · premise_and_result_audit — Recompute quotient changed scalar difference coordinate1 with exact rational arithmetic, not rounded decimals.
S041 · premise_and_result_audit — Recompute quotient old scalar representative coordinate1 with exact rational arithmetic, not rounded decimals.
S040 · premise_and_result_audit — Recompute quotient new scalar representative coordinate1 with exact rational arithmetic, not rounded decimals.
S039 · premise_and_result_audit — Recompute quotient changed sum difference coordinate1 with exact rational arithmetic, not rounded decimals.
S038 · premise_and_result_audit — Recompute quotient old sum coordinate1 with exact rational arithmetic, not rounded decimals.
S037 · premise_and_result_audit — Recompute quotient new sum coordinate1 with exact rational arithmetic, not rounded decimals.
S036 · premise_and_result_audit — Recompute quotient changed scalar difference coordinate0 with exact rational arithmetic, not rounded decimals.
S035 · premise_and_result_audit — Recompute quotient old scalar representative coordinate0 with exact rational arithmetic, not rounded decimals.
S034 · premise_and_result_audit — Recompute quotient new scalar representative coordinate0 with exact rational arithmetic, not rounded decimals.
S033 · premise_and_result_audit — Recompute quotient changed sum difference coordinate0 with exact rational arithmetic, not rounded decimals.
S032 · premise_and_result_audit — Recompute quotient old sum coordinate0 with exact rational arithmetic, not rounded decimals.
S031 · premise_and_result_audit — Recompute quotient new sum coordinate0 with exact rational arithmetic, not rounded decimals.
S030 · premise_and_result_audit — Recompute quotient changed u coordinate2 with exact rational arithmetic, not rounded decimals.
S029 · premise_and_result_audit — Recompute quotient changed u coordinate1 with exact rational arithmetic, not rounded decimals.
S028 · premise_and_result_audit — Recompute quotient changed u coordinate0 with exact rational arithmetic, not rounded decimals.
S027 · premise_and_result_audit — Recompute quotient changed v coordinate2 with exact rational arithmetic, not rounded decimals.
S026 · premise_and_result_audit — Recompute quotient changed v coordinate1 with exact rational arithmetic, not rounded decimals.
S025 · premise_and_result_audit — Recompute quotient changed v coordinate0 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 quotient original u shift original coordinate2 complete sum with exact rational arithmetic, not rounded decimals.
S022 · premise_and_result_audit — Recompute quotient original u shift original coordinate2 coefficient0 product with exact rational arithmetic, not rounded decimals.
S021 · premise_and_result_audit — Recompute quotient original u shift original coordinate1 complete sum with exact rational arithmetic, not rounded decimals.
S020 · premise_and_result_audit — Recompute quotient original u shift original coordinate1 coefficient0 product with exact rational arithmetic, not rounded decimals.
S019 · premise_and_result_audit — Recompute quotient original u shift original coordinate0 complete sum with exact rational arithmetic, not rounded decimals.
S018 · premise_and_result_audit — Recompute quotient original u shift original coordinate0 coefficient0 product with exact rational arithmetic, not rounded decimals.
S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S016 · premise_and_result_audit — Recompute quotient original v shift original coordinate2 complete sum with exact rational arithmetic, not rounded decimals.
S015 · premise_and_result_audit — Recompute quotient original v shift original coordinate2 coefficient0 product with exact rational arithmetic, not rounded decimals.
S014 · premise_and_result_audit — Recompute quotient original v shift original coordinate1 complete sum with exact rational arithmetic, not rounded decimals.
S013 · premise_and_result_audit — Recompute quotient original v shift original coordinate1 coefficient0 product with exact rational arithmetic, not rounded decimals.
S012 · premise_and_result_audit — Recompute quotient original v shift original coordinate0 complete sum with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Recompute quotient original v shift original coordinate0 coefficient0 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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 — Recompute quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 difference with exact rational arithmetic, not rounded decimals.
S005 · premise_and_result_audit — Recompute quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product with exact rational arithmetic, not rounded decimals.
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 quotient ORIGINAL ordered W basis 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, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)

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: a9827877d9ac84715e3de9ba8757e47c16c26d2ae417aa4a521edd9fd17a9e20
candidate record SHA-256: eb83266455dd5a0bdb41a844337929fd3bd246b48d61cb0f2285904a00b97c17
semantic review receipt SHA-256: d165cad30849a3aba67390a59c572c0d965269d4b405e43df36da19e28d85612
question-first witness SHA-256: c222ca47a07939c019865a1e4d9ce633205cdee68db92bf96a96f5683a62ebc7
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON