Question 18819: Restore Input Signs in the Extended Euclidean Witness

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

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R0036-18819 · q-20261014-advance-20261105-r03-euclidean-algorithm-0069

Original question and givens

Use original integers a=815, b=125. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, certificate).

G001 — Use original integers a=815, b=125. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (gcd, x, y, linear_combination, equal, 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-euclidean-algorithm

Step-by-step derivation

S001 · implies

Before

source operation=euclid-bezout-certificate;original snapshot and mathematics retained

After

explicit ordered source output fields=[gcd, x, y, linear_combination, equal, certificate]

Conditions: none

Keep the historical interface marker in the original snapshot while listing the original ordered fields here.No JavaScript or external table is needed for the main text,and field labels are not mathematical evidence.

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

original signed inputs a=815,b=125

After

initial nonnegative chain values=[815, 125];gcd(a,b)=gcd(abs(a),abs(b));never divide by a zero right input

Conditions: none

Divisibility is unaffected by input signs,and the gcd is nonnegative.Ordinary chain/invariant families require b≠0;the zero boundary separately retains its(0,0)convention.

Step identifiers and dependencies

Rule: euclid_signed_inputs_absolute_chain_scope · Uses: G001, S001

S003 · calculation

Before

Euclid step0 exact ratio := div(815, 125)

After

Euclid step0 exact ratio = 163/25

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Euclid step0 quotient times divisor := mul(6, 125)

After

Euclid step0 quotient times divisor = 750

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Euclid step0 remainder := sub(815, 750)

After

Euclid step0 remainder = 65

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

step=0;left=815;positive right=125;exact ratio=163/25;floor quotient=6

After

left=right*q+r with r=65 in[0,125);gcd pair=[5, 5]

Conditions: none

An integer d divides both left and right iff it divides right and left−q*right,so the gcd is unchanged.The next nonnegative right is the strictly smaller remainder,so infinite descent is impossible;termination is separate from invariance.

Step identifiers and dependencies

Rule: euclid_each_division_common_divisors_and_strict_descent · Uses: G001, S005

S007 · calculation

Before

Euclid step0 first coefficient multiple := mul(6, 0)

After

Euclid step0 first coefficient multiple = 0

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

Euclid step0 first coefficient difference := sub(1, 0)

After

Euclid step0 first coefficient difference = 1

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

Euclid step0 second coefficient multiple := mul(6, 1)

After

Euclid step0 second coefficient multiple = 6

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

Euclid step0 second coefficient difference := sub(0, 6)

After

Euclid step0 second coefficient difference = -6

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · implies

Before

step=0;two original-absolute-input coefficient pairs=[[1, 0], [0, 1]];same quotient=6

After

next remainder coefficient pair=[1, -6];abs(a)*s_next+abs(b)*t_next=65

Conditions: none

Update both coefficient sequences with the same quotient as the remainder.Inductively every row remains a combination of the original absolute inputs;do not guess coefficients after finding the gcd.

Step identifiers and dependencies

Rule: euclid_extended_coefficients_same_quotient_recurrence · Uses: G001, S010

S012 · calculation

Before

Euclid step1 exact ratio := div(125, 65)

After

Euclid step1 exact ratio = 25/13

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

Euclid step1 quotient times divisor := mul(1, 65)

After

Euclid step1 quotient times divisor = 65

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

Euclid step1 remainder := sub(125, 65)

After

Euclid step1 remainder = 60

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · implies

Before

step=1;left=125;positive right=65;exact ratio=25/13;floor quotient=1

After

left=right*q+r with r=60 in[0,65);gcd pair=[5, 5]

Conditions: none

Step identifiers and dependencies

Rule: euclid_each_division_common_divisors_and_strict_descent · Uses: G001, S014

S016 · calculation

Before

Euclid step1 first coefficient multiple := mul(1, 1)

After

Euclid step1 first coefficient multiple = 1

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

Euclid step1 first coefficient difference := sub(0, 1)

After

Euclid step1 first coefficient difference = -1

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

Euclid step1 second coefficient multiple := mul(1, -6)

After

Euclid step1 second coefficient multiple = -6

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

Euclid step1 second coefficient difference := sub(1, -6)

After

Euclid step1 second coefficient difference = 7

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · implies

Before

step=1;two original-absolute-input coefficient pairs=[[0, 1], [1, -6]];same quotient=1

After

next remainder coefficient pair=[-1, 7];abs(a)*s_next+abs(b)*t_next=60

Conditions: none

Update both coefficient sequences with the same quotient as the remainder.Inductively every row remains a combination of the original absolute inputs;do not guess coefficients after finding the gcd.

Step identifiers and dependencies

Rule: euclid_extended_coefficients_same_quotient_recurrence · Uses: G001, S019

S021 · calculation

Before

Euclid step2 exact ratio := div(65, 60)

After

Euclid step2 exact ratio = 13/12

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

Euclid step2 quotient times divisor := mul(1, 60)

After

Euclid step2 quotient times divisor = 60

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

Euclid step2 remainder := sub(65, 60)

After

Euclid step2 remainder = 5

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · implies

Before

step=2;left=65;positive right=60;exact ratio=13/12;floor quotient=1

After

left=right*q+r with r=5 in[0,60);gcd pair=[5, 5]

Conditions: none

Step identifiers and dependencies

Rule: euclid_each_division_common_divisors_and_strict_descent · Uses: G001, S023

S025 · calculation

Before

Euclid step2 first coefficient multiple := mul(1, -1)

After

Euclid step2 first coefficient multiple = -1

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

Euclid step2 first coefficient difference := sub(1, -1)

After

Euclid step2 first coefficient difference = 2

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

Euclid step2 second coefficient multiple := mul(1, 7)

After

Euclid step2 second coefficient multiple = 7

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · calculation

Before

Euclid step2 second coefficient difference := sub(-6, 7)

After

Euclid step2 second coefficient difference = -13

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · implies

Before

step=2;two original-absolute-input coefficient pairs=[[1, -6], [-1, 7]];same quotient=1

After

next remainder coefficient pair=[2, -13];abs(a)*s_next+abs(b)*t_next=5

Conditions: none

Update both coefficient sequences with the same quotient as the remainder.Inductively every row remains a combination of the original absolute inputs;do not guess coefficients after finding the gcd.

Step identifiers and dependencies

Rule: euclid_extended_coefficients_same_quotient_recurrence · Uses: G001, S028

S030 · calculation

Before

Euclid step3 exact ratio := div(60, 5)

After

Euclid step3 exact ratio = 12

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

Euclid step3 quotient times divisor := mul(12, 5)

After

Euclid step3 quotient times divisor = 60

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

Before

Euclid step3 remainder := sub(60, 60)

After

Euclid step3 remainder = 0

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · implies

Before

step=3;left=60;positive right=5;exact ratio=12;floor quotient=12

After

left=right*q+r with r=0 in[0,5);gcd pair=[5, 5]

Conditions: none

Step identifiers and dependencies

Rule: euclid_each_division_common_divisors_and_strict_descent · Uses: G001, S032

S034 · calculation

Before

Euclid step3 first coefficient multiple := mul(12, 2)

After

Euclid step3 first coefficient multiple = 24

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

Euclid step3 first coefficient difference := sub(-1, 24)

After

Euclid step3 first coefficient difference = -25

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · calculation

Before

Euclid step3 second coefficient multiple := mul(12, -13)

After

Euclid step3 second coefficient multiple = -156

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

Before

Euclid step3 second coefficient difference := sub(7, -156)

After

Euclid step3 second coefficient difference = 163

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · implies

Before

step=3;two original-absolute-input coefficient pairs=[[-1, 7], [2, -13]];same quotient=12

After

next remainder coefficient pair=[-25, 163];abs(a)*s_next+abs(b)*t_next=0

Conditions: none

Update both coefficient sequences with the same quotient as the remainder.Inductively every row remains a combination of the original absolute inputs;do not guess coefficients after finding the gcd.

Step identifiers and dependencies

Rule: euclid_extended_coefficients_same_quotient_recurrence · Uses: G001, S037

S039 · implies

Before

complete division rows=[[815, 125, 6, 65], [125, 65, 1, 60], [65, 60, 1, 5], [60, 5, 12, 0]];terminal pair=[5, 0]

After

gcd(a,b)=gcd(5,0)=5;all divisions used positive divisors;nonnegative gcd convention

Conditions: none

At termination common divisors are those of the terminal left value,giving its nonnegative gcd.The(0,0)value0 is an explicit separate convention.Do not append a fictitious division by zero.

Step identifiers and dependencies

Rule: euclid_terminal_gcd_not_extra_zero_division · Uses: G001, S038

S040 · calculation

Before

Euclid restore original a sign in coefficient := mul(1, 2)

After

Euclid restore original a sign in coefficient = 2

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · calculation

Before

Euclid restore original b sign in coefficient := mul(1, -13)

After

Euclid restore original b sign in coefficient = -13

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · calculation

Before

Euclid ORIGINAL signed a times x := mul(815, 2)

After

Euclid ORIGINAL signed a times x = 1630

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

Euclid ORIGINAL signed b times y := mul(125, -13)

After

Euclid ORIGINAL signed b times y = -1625

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · calculation

Before

Euclid final signed Bezout linear combination := sum(1630, -1625)

After

Euclid final signed Bezout linear combination = 5

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · implies

Before

original a=815,b=125;deterministic absolute-chain coefficients=[2, -13]

After

sign-restored source coefficients x=2,y=-13;a*x+b*y=5=nonnegative gcd

Conditions: none

Retain the deterministic coefficients of this source extended chain even though other Bezout pairs may be valid.Use sign+1 for zero;(0,0)keeps(1,0)and0=0,without a uniqueness claim.

Step identifiers and dependencies

Rule: euclid_deterministic_signed_bezout_not_just_any_coefficients · Uses: G001, S044

Reverse checks

S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S044 · premise_and_result_audit — Recompute Euclid final signed Bezout linear combination with exact rational arithmetic, not rounded decimals.
S043 · premise_and_result_audit — Recompute Euclid ORIGINAL signed b times y with exact rational arithmetic, not rounded decimals.
S042 · premise_and_result_audit — Recompute Euclid ORIGINAL signed a times x with exact rational arithmetic, not rounded decimals.
S041 · premise_and_result_audit — Recompute Euclid restore original b sign in coefficient with exact rational arithmetic, not rounded decimals.
S040 · premise_and_result_audit — Recompute Euclid restore original a sign in coefficient with exact rational arithmetic, not rounded decimals.
S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S037 · premise_and_result_audit — Recompute Euclid step3 second coefficient difference with exact rational arithmetic, not rounded decimals.
S036 · premise_and_result_audit — Recompute Euclid step3 second coefficient multiple with exact rational arithmetic, not rounded decimals.
S035 · premise_and_result_audit — Recompute Euclid step3 first coefficient difference with exact rational arithmetic, not rounded decimals.
S034 · premise_and_result_audit — Recompute Euclid step3 first coefficient multiple with exact rational arithmetic, not rounded decimals.
S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S032 · premise_and_result_audit — Recompute Euclid step3 remainder with exact rational arithmetic, not rounded decimals.
S031 · premise_and_result_audit — Recompute Euclid step3 quotient times divisor with exact rational arithmetic, not rounded decimals.
S030 · premise_and_result_audit — Recompute Euclid step3 exact ratio with exact rational arithmetic, not rounded decimals.
S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S028 · premise_and_result_audit — Recompute Euclid step2 second coefficient difference with exact rational arithmetic, not rounded decimals.
S027 · premise_and_result_audit — Recompute Euclid step2 second coefficient multiple with exact rational arithmetic, not rounded decimals.
S026 · premise_and_result_audit — Recompute Euclid step2 first coefficient difference with exact rational arithmetic, not rounded decimals.
S025 · premise_and_result_audit — Recompute Euclid step2 first coefficient multiple 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 Euclid step2 remainder with exact rational arithmetic, not rounded decimals.
S022 · premise_and_result_audit — Recompute Euclid step2 quotient times divisor with exact rational arithmetic, not rounded decimals.
S021 · premise_and_result_audit — Recompute Euclid step2 exact ratio with exact rational arithmetic, not rounded decimals.
S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S019 · premise_and_result_audit — Recompute Euclid step1 second coefficient difference with exact rational arithmetic, not rounded decimals.
S018 · premise_and_result_audit — Recompute Euclid step1 second coefficient multiple with exact rational arithmetic, not rounded decimals.
S017 · premise_and_result_audit — Recompute Euclid step1 first coefficient difference with exact rational arithmetic, not rounded decimals.
S016 · premise_and_result_audit — Recompute Euclid step1 first coefficient multiple with exact rational arithmetic, not rounded decimals.
S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S014 · premise_and_result_audit — Recompute Euclid step1 remainder with exact rational arithmetic, not rounded decimals.
S013 · premise_and_result_audit — Recompute Euclid step1 quotient times divisor with exact rational arithmetic, not rounded decimals.
S012 · premise_and_result_audit — Recompute Euclid step1 exact ratio with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S010 · premise_and_result_audit — Recompute Euclid step0 second coefficient difference with exact rational arithmetic, not rounded decimals.
S009 · premise_and_result_audit — Recompute Euclid step0 second coefficient multiple with exact rational arithmetic, not rounded decimals.
S008 · premise_and_result_audit — Recompute Euclid step0 first coefficient difference with exact rational arithmetic, not rounded decimals.
S007 · premise_and_result_audit — Recompute Euclid step0 first coefficient multiple with exact rational arithmetic, not rounded decimals.
S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S005 · premise_and_result_audit — Recompute Euclid step0 remainder with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Recompute Euclid step0 quotient times divisor with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Recompute Euclid step0 exact ratio 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

(5, 2, -13, 5, Yes, extended_euclidean_back_substitution)

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: 86f7e8129bf3198d333900016ab261a5796f2834d61600b2421c1c643c7dbf85
candidate record SHA-256: cb5868b505cf9d1bb97e2b190ed92a30d30e332911ebd7c57b09c5ec6eb8bccc
semantic review receipt SHA-256: efee5462991619d6ed9f64f11ddb085e5614ef33f9f8fcd8cf29b9329f8fd301
question-first witness SHA-256: acf91fa1622f01d2cf04b2c513294254323665351f660f2628aaafdb869c4ff2
Lean: NOT_RUN · human review: NOT_PERFORMED

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