Question 19995: Changing Representatives by W Must Preserve Both Operations

Original question and givens

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 20, 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=[21, 2, 3], u=[3, 20, 4], shift_v_coefficients=[21], shift_u_coefficients=[-20] in supplied basis order, scalar=20. 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, 20, 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=[21, 2, 3], u=[3, 20, 4], shift_v_coefficients=[21], shift_u_coefficients=[-20] in supplied basis order, scalar=20. 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

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

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]

quotient ORIGINAL ordered W basis:COMPLETE exact original ordered matrix=[[1], [20], [0]];rows=3;columns=1

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

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

quotient ORIGINAL ordered W basis pivot0 column0 normalized entry0 = 1

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

normalized FULL row=[1]

quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product := mul(20, 1)

quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product = 20

quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 difference := sub(20, 20)

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

quotient ORIGINAL ordered W basis:target row=1;old row=[20];subtract factor=20 times FULL pivot row=[1]

new exact FULL row=[0]

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

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

ACTUAL ambient real space R^3;complete ORIGINAL independent basis columns of W=[[1, 20, 0]]

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

ACTUAL subspace W full real span in R^3;original basis=[[1, 20, 0]]

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

quotient original v shift original coordinate0 coefficient0 product := mul(1, 21)

quotient original v shift original coordinate0 coefficient0 product = 21

quotient original v shift original coordinate0 complete sum := sum(21)

quotient original v shift original coordinate0 complete sum = 21

quotient original v shift original coordinate1 coefficient0 product := mul(20, 21)

quotient original v shift original coordinate1 coefficient0 product = 420

quotient original v shift original coordinate1 complete sum := sum(420)

quotient original v shift original coordinate1 complete sum = 420

quotient original v shift original coordinate2 coefficient0 product := mul(0, 21)

quotient original v shift original coordinate2 coefficient0 product = 0

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

quotient original v shift original coordinate2 complete sum = 0

quotient original v shift:ORIGINAL W basis order=[[1, 20, 0]];aligned coefficient block=[21]

computed whole vector=[21, 420, 0];a real linear combination in W by its declared span

quotient original u shift original coordinate0 coefficient0 product := mul(1, -20)

quotient original u shift original coordinate0 coefficient0 product = -20

quotient original u shift original coordinate0 complete sum := sum(-20)

quotient original u shift original coordinate0 complete sum = -20

quotient original u shift original coordinate1 coefficient0 product := mul(20, -20)

quotient original u shift original coordinate1 coefficient0 product = -400

quotient original u shift original coordinate1 complete sum := sum(-400)

quotient original u shift original coordinate1 complete sum = -400

quotient original u shift original coordinate2 coefficient0 product := mul(0, -20)

quotient original u shift original coordinate2 coefficient0 product = 0

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

quotient original u shift original coordinate2 complete sum = 0

quotient original u shift:ORIGINAL W basis order=[[1, 20, 0]];aligned coefficient block=[-20]

computed whole vector=[-20, -400, 0];a real linear combination in W by its declared span

quotient changed v coordinate0 := sum(21, 21)

quotient changed v coordinate0 = 42

quotient changed v coordinate1 := sum(2, 420)

quotient changed v coordinate1 = 422

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

quotient changed v coordinate2 = 3

quotient changed u coordinate0 := sum(3, -20)

quotient changed u coordinate0 = -17

quotient changed u coordinate1 := sum(20, -400)

quotient changed u coordinate1 = -380

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

quotient changed u coordinate2 = 4

quotient new sum coordinate0 := sum(42, -17)

quotient new sum coordinate0 = 25

quotient old sum coordinate0 := sum(21, 3)

quotient old sum coordinate0 = 24

quotient changed sum difference coordinate0 := sub(25, 24)

quotient changed sum difference coordinate0 = 1

quotient new scalar representative coordinate0 := mul(20, 42)

quotient new scalar representative coordinate0 = 840

quotient old scalar representative coordinate0 := mul(20, 21)

quotient old scalar representative coordinate0 = 420

quotient changed scalar difference coordinate0 := sub(840, 420)

quotient changed scalar difference coordinate0 = 420

quotient new sum coordinate1 := sum(422, -380)

quotient new sum coordinate1 = 42

quotient old sum coordinate1 := sum(2, 20)

quotient old sum coordinate1 = 22

quotient changed sum difference coordinate1 := sub(42, 22)

quotient changed sum difference coordinate1 = 20

quotient new scalar representative coordinate1 := mul(20, 422)

quotient new scalar representative coordinate1 = 8440

quotient old scalar representative coordinate1 := mul(20, 2)

quotient old scalar representative coordinate1 = 40

quotient changed scalar difference coordinate1 := sub(8440, 40)

quotient changed scalar difference coordinate1 = 8400

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

quotient new sum coordinate2 = 7

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

quotient old sum coordinate2 = 7

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

quotient changed sum difference coordinate2 = 0

quotient new scalar representative coordinate2 := mul(20, 3)

quotient new scalar representative coordinate2 = 60

quotient old scalar representative coordinate2 := mul(20, 3)

quotient old scalar representative coordinate2 = 60

quotient changed scalar difference coordinate2 := sub(60, 60)

quotient changed scalar difference coordinate2 = 0

quotient supplied v shift membership:COMPLETE exact original ordered matrix=[[1, 21], [20, 420], [0, 0]];rows=3;columns=2

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

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

quotient supplied v shift membership pivot0 column0 normalized entry0 = 1

quotient supplied v shift membership pivot0 column0 normalized entry1 := div(21, 1)

quotient supplied v shift membership pivot0 column0 normalized entry1 = 21

quotient supplied v shift membership:row=0;column=0;nonzero exact pivot=1;old FULL row=[1, 21]

normalized FULL row=[1, 21]

quotient supplied v shift membership eliminate row1 pivot0 entry0 product := mul(20, 1)

quotient supplied v shift membership eliminate row1 pivot0 entry0 product = 20

quotient supplied v shift membership eliminate row1 pivot0 entry0 difference := sub(20, 20)

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

quotient supplied v shift membership eliminate row1 pivot0 entry1 product := mul(20, 21)

quotient supplied v shift membership eliminate row1 pivot0 entry1 product = 420

quotient supplied v shift membership eliminate row1 pivot0 entry1 difference := sub(420, 420)

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

quotient supplied v shift membership:target row=1;old row=[20, 420];subtract factor=20 times FULL pivot row=[1, 21]

new exact FULL row=[0, 0]

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

no pivot added;current rank=1

quotient supplied v shift membership:complete exact reduced matrix=[[1, 21], [0, 0], [0, 0]];ORIGINAL pivot-column indices=[0]

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

quotient supplied v shift membership original coordinate0 reconstruction coefficient0 product := mul(1, 21)

quotient supplied v shift membership original coordinate0 reconstruction coefficient0 product = 21

quotient supplied v shift membership original coordinate0 reconstruction complete sum := sum(21)

quotient supplied v shift membership original coordinate0 reconstruction complete sum = 21

quotient supplied v shift membership original coordinate1 reconstruction coefficient0 product := mul(20, 21)

quotient supplied v shift membership original coordinate1 reconstruction coefficient0 product = 420

quotient supplied v shift membership original coordinate1 reconstruction complete sum := sum(420)

quotient supplied v shift membership original coordinate1 reconstruction complete sum = 420

quotient supplied v shift membership original coordinate2 reconstruction coefficient0 product := mul(0, 21)

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

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

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

quotient supplied v shift membership:original proposed vector=[21, 420, 0];original ordered independent columns=[[1, 20, 0]];complete reduced augmented system=[[1, 21], [0, 0], [0, 0]]

exact real-span coefficient witness=[21];FULL original reconstruction=[21, 420, 0];membership=Yes

quotient supplied u shift membership:COMPLETE exact original ordered matrix=[[1, -20], [20, -400], [0, 0]];rows=3;columns=2

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

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

quotient supplied u shift membership pivot0 column0 normalized entry0 = 1

quotient supplied u shift membership pivot0 column0 normalized entry1 := div(-20, 1)

quotient supplied u shift membership pivot0 column0 normalized entry1 = -20

quotient supplied u shift membership:row=0;column=0;nonzero exact pivot=1;old FULL row=[1, -20]

normalized FULL row=[1, -20]

quotient supplied u shift membership eliminate row1 pivot0 entry0 product := mul(20, 1)

quotient supplied u shift membership eliminate row1 pivot0 entry0 product = 20

quotient supplied u shift membership eliminate row1 pivot0 entry0 difference := sub(20, 20)

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

quotient supplied u shift membership eliminate row1 pivot0 entry1 product := mul(20, -20)

quotient supplied u shift membership eliminate row1 pivot0 entry1 product = -400

quotient supplied u shift membership eliminate row1 pivot0 entry1 difference := sub(-400, -400)

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

quotient supplied u shift membership:target row=1;old row=[20, -400];subtract factor=20 times FULL pivot row=[1, -20]

new exact FULL row=[0, 0]

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

no pivot added;current rank=1

quotient supplied u shift membership:complete exact reduced matrix=[[1, -20], [0, 0], [0, 0]];ORIGINAL pivot-column indices=[0]

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

quotient supplied u shift membership original coordinate0 reconstruction coefficient0 product := mul(1, -20)

quotient supplied u shift membership original coordinate0 reconstruction coefficient0 product = -20

quotient supplied u shift membership original coordinate0 reconstruction complete sum := sum(-20)

quotient supplied u shift membership original coordinate0 reconstruction complete sum = -20

quotient supplied u shift membership original coordinate1 reconstruction coefficient0 product := mul(20, -20)

quotient supplied u shift membership original coordinate1 reconstruction coefficient0 product = -400

quotient supplied u shift membership original coordinate1 reconstruction complete sum := sum(-400)

quotient supplied u shift membership original coordinate1 reconstruction complete sum = -400

quotient supplied u shift membership original coordinate2 reconstruction coefficient0 product := mul(0, -20)

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

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

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

quotient supplied u shift membership:original proposed vector=[-20, -400, 0];original ordered independent columns=[[1, 20, 0]];complete reduced augmented system=[[1, -20], [0, 0], [0, 0]]

exact real-span coefficient witness=[-20];FULL original reconstruction=[-20, -400, 0];membership=Yes

quotient supplied sum difference membership:COMPLETE exact original ordered matrix=[[1, 1], [20, 20], [0, 0]];rows=3;columns=2

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

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

quotient supplied sum difference membership pivot0 column0 normalized entry0 = 1

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

quotient supplied sum difference membership pivot0 column0 normalized entry1 = 1

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

normalized FULL row=[1, 1]

quotient supplied sum difference membership eliminate row1 pivot0 entry0 product := mul(20, 1)

quotient supplied sum difference membership eliminate row1 pivot0 entry0 product = 20

quotient supplied sum difference membership eliminate row1 pivot0 entry0 difference := sub(20, 20)

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

quotient supplied sum difference membership eliminate row1 pivot0 entry1 product := mul(20, 1)

quotient supplied sum difference membership eliminate row1 pivot0 entry1 product = 20

quotient supplied sum difference membership eliminate row1 pivot0 entry1 difference := sub(20, 20)

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

quotient supplied sum difference membership:target row=1;old row=[20, 20];subtract factor=20 times FULL pivot row=[1, 1]

new exact FULL row=[0, 0]

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

no pivot added;current rank=1

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

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

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

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

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

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

quotient supplied sum difference membership original coordinate1 reconstruction coefficient0 product := mul(20, 1)

quotient supplied sum difference membership original coordinate1 reconstruction coefficient0 product = 20

quotient supplied sum difference membership original coordinate1 reconstruction complete sum := sum(20)

quotient supplied sum difference membership original coordinate1 reconstruction complete sum = 20

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

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

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

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

quotient supplied sum difference membership:original proposed vector=[1, 20, 0];original ordered independent columns=[[1, 20, 0]];complete reduced augmented system=[[1, 1], [0, 0], [0, 0]]

exact real-span coefficient witness=[1];FULL original reconstruction=[1, 20, 0];membership=Yes

quotient supplied scalar difference membership:COMPLETE exact original ordered matrix=[[1, 420], [20, 8400], [0, 0]];rows=3;columns=2

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

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

quotient supplied scalar difference membership pivot0 column0 normalized entry0 = 1

quotient supplied scalar difference membership pivot0 column0 normalized entry1 := div(420, 1)

quotient supplied scalar difference membership pivot0 column0 normalized entry1 = 420

quotient supplied scalar difference membership:row=0;column=0;nonzero exact pivot=1;old FULL row=[1, 420]

normalized FULL row=[1, 420]

quotient supplied scalar difference membership eliminate row1 pivot0 entry0 product := mul(20, 1)

quotient supplied scalar difference membership eliminate row1 pivot0 entry0 product = 20

quotient supplied scalar difference membership eliminate row1 pivot0 entry0 difference := sub(20, 20)

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

quotient supplied scalar difference membership eliminate row1 pivot0 entry1 product := mul(20, 420)

quotient supplied scalar difference membership eliminate row1 pivot0 entry1 product = 8400

quotient supplied scalar difference membership eliminate row1 pivot0 entry1 difference := sub(8400, 8400)

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

quotient supplied scalar difference membership:target row=1;old row=[20, 8400];subtract factor=20 times FULL pivot row=[1, 420]

new exact FULL row=[0, 0]

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

no pivot added;current rank=1

quotient supplied scalar difference membership:complete exact reduced matrix=[[1, 420], [0, 0], [0, 0]];ORIGINAL pivot-column indices=[0]

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

quotient supplied scalar difference membership original coordinate0 reconstruction coefficient0 product := mul(1, 420)

quotient supplied scalar difference membership original coordinate0 reconstruction coefficient0 product = 420

quotient supplied scalar difference membership original coordinate0 reconstruction complete sum := sum(420)

quotient supplied scalar difference membership original coordinate0 reconstruction complete sum = 420

quotient supplied scalar difference membership original coordinate1 reconstruction coefficient0 product := mul(20, 420)

quotient supplied scalar difference membership original coordinate1 reconstruction coefficient0 product = 8400

quotient supplied scalar difference membership original coordinate1 reconstruction complete sum := sum(8400)

quotient supplied scalar difference membership original coordinate1 reconstruction complete sum = 8400

quotient supplied scalar difference membership original coordinate2 reconstruction coefficient0 product := mul(0, 420)

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

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

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

quotient supplied scalar difference membership:original proposed vector=[420, 8400, 0];original ordered independent columns=[[1, 20, 0]];complete reduced augmented system=[[1, 420], [0, 0], [0, 0]]

exact real-span coefficient witness=[420];FULL original reconstruction=[420, 8400, 0];membership=Yes

ACTUAL original W shifts=[[21, 420, 0], [-20, -400, 0]];changed representatives=[[42, 422, 3], [-17, -380, 4]];actual sum/scalar differences=[[1, 20, 0], [420, 8400, 0]]

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

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.