# Question 18070: Every Congruence Must Pass in the Same CRT System

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0035-18070 · q-20261014-advance-20261104-r05-chinese-remainder-theorem-0045



## Original question and givens

Use the complete integer congruence system with positive distinct moduli=[3, 5, 43] and aligned original residues=[19, 39, 59] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).


- G001 — Use the complete integer congruence system with positive distinct moduli=[3, 5, 43] and aligned original residues=[19, 39, 59] in supplied order. Normalize each residue to [0,modulus). The main CRT operations require pairwise coprimality and give a least nonnegative solution unique modulo the product, not a unique integer. A dedicated noncoprime audit instead tests all pairwise gcd compatibility conditions and uses the lcm period. Modulus 1 is a trivial constraint; modulus 0 is rejected. All arithmetic is exact. Require all moduli pairwise coprime. Give (moduli, normalized_residues, solution, product, congruence_checks, solved).





## 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-chinese-remainder-theorem](https://ocme.evemisslab.com/en/mko/mko-chinese-remainder-theorem/)





## Step-by-step derivation

S001 · impliesBefore

source operation=crt-solve-system;original snapshot and mathematics retained

After

explicit ordered source output fields=[moduli, normalized_residues, solution, product, congruence_checks, solved]

Conditions: none

Keep the historical interface marker in the original snapshot while giving ordered fields here,without JavaScript or an external table.Labels are not mathematical evidence,and each No marker retains the operation's claim scope.

Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · calculationBefore

original residue slot0 exact quotient := div(19, 3)

After

original residue slot0 exact quotient = 19/3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

original residue slot0 floor quotient multiple := mul(6, 3)

After

original residue slot0 floor quotient multiple = 18

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

original residue slot0 normalized remainder := sub(19, 18)

After

original residue slot0 normalized remainder = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · impliesBefore

original residue slot0:integer value=19;positive modulus=3;exact ratio=19/3;floor quotient=6

After

value=q*modulus+remainder;normalized residue=1 in[0,3)

Conditions: none

Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0.

Step identifiers and dependenciesRule: crt_exact_floor_modulo_signed_input · Uses: G001, S004





S006 · calculationBefore

original residue slot1 exact quotient := div(39, 5)

After

original residue slot1 exact quotient = 39/5

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · calculationBefore

original residue slot1 floor quotient multiple := mul(7, 5)

After

original residue slot1 floor quotient multiple = 35

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · calculationBefore

original residue slot1 normalized remainder := sub(39, 35)

After

original residue slot1 normalized remainder = 4

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · impliesBefore

original residue slot1:integer value=39;positive modulus=5;exact ratio=39/5;floor quotient=7

After

value=q*modulus+remainder;normalized residue=4 in[0,5)

Conditions: none

Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0.

Step identifiers and dependenciesRule: crt_exact_floor_modulo_signed_input · Uses: G001, S008





S010 · calculationBefore

original residue slot2 exact quotient := div(59, 43)

After

original residue slot2 exact quotient = 59/43

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009





S011 · calculationBefore

original residue slot2 floor quotient multiple := mul(1, 43)

After

original residue slot2 floor quotient multiple = 43

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010





S012 · calculationBefore

original residue slot2 normalized remainder := sub(59, 43)

After

original residue slot2 normalized remainder = 16

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011





S013 · impliesBefore

original residue slot2:integer value=59;positive modulus=43;exact ratio=59/43;floor quotient=1

After

value=q*modulus+remainder;normalized residue=16 in[0,43)

Conditions: none

Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0.

Step identifiers and dependenciesRule: crt_exact_floor_modulo_signed_input · Uses: G001, S012





S014 · calculationBefore

CRT modulus product prefix0 := mul(1, 3)

After

CRT modulus product prefix0 = 3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013





S015 · impliesBefore

LCM prefix0 gcd:positive original gcd inputs=[1, 3]

After

Euclidean rows[a,b,q,r] with a=q*b+r=[[1, 3, 0, 1], [3, 1, 3, 0]];positive gcd=1

Conditions: none

Each remainder replacement preserves common divisors;the last nonzero value is the gcd.List every quotient/remainder row for source-bound checking.

Step identifiers and dependenciesRule: crt_pair_gcd_exact_euclidean_witness · Uses: G001, S014





S016 · calculationBefore

CRT LCM prefix0 quotient := div(1, 1)

After

CRT LCM prefix0 quotient = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015





S017 · calculationBefore

CRT LCM prefix0 value := mul(1, 3)

After

CRT LCM prefix0 value = 3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016





S018 · calculationBefore

CRT modulus product prefix1 := mul(3, 5)

After

CRT modulus product prefix1 = 15

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017





S019 · impliesBefore

LCM prefix1 gcd:positive original gcd inputs=[3, 5]

After

Euclidean rows[a,b,q,r] with a=q*b+r=[[3, 5, 0, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];positive gcd=1

Conditions: none

Each remainder replacement preserves common divisors;the last nonzero value is the gcd.List every quotient/remainder row for source-bound checking.

Step identifiers and dependenciesRule: crt_pair_gcd_exact_euclidean_witness · Uses: G001, S018





S020 · calculationBefore

CRT LCM prefix1 quotient := div(3, 1)

After

CRT LCM prefix1 quotient = 3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019





S021 · calculationBefore

CRT LCM prefix1 value := mul(3, 5)

After

CRT LCM prefix1 value = 15

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020





S022 · calculationBefore

CRT modulus product prefix2 := mul(15, 43)

After

CRT modulus product prefix2 = 645

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021





S023 · impliesBefore

LCM prefix2 gcd:positive original gcd inputs=[15, 43]

After

Euclidean rows[a,b,q,r] with a=q*b+r=[[15, 43, 0, 15], [43, 15, 2, 13], [15, 13, 1, 2], [13, 2, 6, 1], [2, 1, 2, 0]];positive gcd=1

Conditions: none

Each remainder replacement preserves common divisors;the last nonzero value is the gcd.List every quotient/remainder row for source-bound checking.

Step identifiers and dependenciesRule: crt_pair_gcd_exact_euclidean_witness · Uses: G001, S022





S024 · calculationBefore

CRT LCM prefix2 quotient := div(15, 1)

After

CRT LCM prefix2 quotient = 15

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023





S025 · calculationBefore

CRT LCM prefix2 value := mul(15, 43)

After

CRT LCM prefix2 value = 645

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024





S026 · impliesBefore

original modulus pair0,1:positive original gcd inputs=[3, 5]

After

Euclidean rows[a,b,q,r] with a=q*b+r=[[3, 5, 0, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];positive gcd=1

Conditions: none

Each remainder replacement preserves common divisors;the last nonzero value is the gcd.List every quotient/remainder row for source-bound checking.

Step identifiers and dependenciesRule: crt_pair_gcd_exact_euclidean_witness · Uses: G001, S025





S027 · calculationBefore

CRT original pair0,1 residue difference := sub(1, 4)

After

CRT original pair0,1 residue difference = -3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026





S028 · calculationBefore

CRT original pair0,1 difference divided by gcd := div(-3, 1)

After

CRT original pair0,1 difference divided by gcd = -3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027





S029 · impliesBefore

original constraint slots=[0, 1];moduli=[3, 5];gcd=1;normalized residue difference=-3

After

difference/gcd=-3;pair coprime=Yes;pair compatible=Yes

Conditions: none

A common solution forces each residue difference to be divisible by the corresponding gcd.Check every pair;noncoprimality is not automatic inconsistency.

Step identifiers and dependenciesRule: crt_every_pair_compatibility_not_automatic_inconsistency · Uses: G001, S028





S030 · impliesBefore

original modulus pair0,2:positive original gcd inputs=[3, 43]

After

Euclidean rows[a,b,q,r] with a=q*b+r=[[3, 43, 0, 3], [43, 3, 14, 1], [3, 1, 3, 0]];positive gcd=1

Conditions: none

Each remainder replacement preserves common divisors;the last nonzero value is the gcd.List every quotient/remainder row for source-bound checking.

Step identifiers and dependenciesRule: crt_pair_gcd_exact_euclidean_witness · Uses: G001, S029





S031 · calculationBefore

CRT original pair0,2 residue difference := sub(1, 16)

After

CRT original pair0,2 residue difference = -15

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030





S032 · calculationBefore

CRT original pair0,2 difference divided by gcd := div(-15, 1)

After

CRT original pair0,2 difference divided by gcd = -15

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031





S033 · impliesBefore

original constraint slots=[0, 2];moduli=[3, 43];gcd=1;normalized residue difference=-15

After

difference/gcd=-15;pair coprime=Yes;pair compatible=Yes

Conditions: none

A common solution forces each residue difference to be divisible by the corresponding gcd.Check every pair;noncoprimality is not automatic inconsistency.

Step identifiers and dependenciesRule: crt_every_pair_compatibility_not_automatic_inconsistency · Uses: G001, S032





S034 · impliesBefore

original modulus pair1,2:positive original gcd inputs=[5, 43]

After

Euclidean rows[a,b,q,r] with a=q*b+r=[[5, 43, 0, 5], [43, 5, 8, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]];positive gcd=1

Conditions: none

Each remainder replacement preserves common divisors;the last nonzero value is the gcd.List every quotient/remainder row for source-bound checking.

Step identifiers and dependenciesRule: crt_pair_gcd_exact_euclidean_witness · Uses: G001, S033





S035 · calculationBefore

CRT original pair1,2 residue difference := sub(4, 16)

After

CRT original pair1,2 residue difference = -12

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034





S036 · calculationBefore

CRT original pair1,2 difference divided by gcd := div(-12, 1)

After

CRT original pair1,2 difference divided by gcd = -12

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035





S037 · impliesBefore

original constraint slots=[1, 2];moduli=[5, 43];gcd=1;normalized residue difference=-12

After

difference/gcd=-12;pair coprime=Yes;pair compatible=Yes

Conditions: none

A common solution forces each residue difference to be divisible by the corresponding gcd.Check every pair;noncoprimality is not automatic inconsistency.

Step identifiers and dependenciesRule: crt_every_pair_compatibility_not_automatic_inconsistency · Uses: G001, S036





S038 · impliesBefore

original aligned moduli=[3, 5, 43];original signed residues=[19, 39, 59];normalized residues=[1, 4, 16];all pair audits=[[0, 1, 1, Yes], [0, 2, 1, Yes], [1, 2, 1, Yes]]

After

pairwise coprime=Yes;gcd compatible=Yes;product=645;LCM period=645

Conditions: none

Retain original aligned constraint order,not separate semantic sorting.Main coprime operations use the product;the noncoprime boundary retains the lcm.Modulus1 adds no restriction.

Step identifiers and dependenciesRule: crt_original_constraint_order_product_vs_lcm · Uses: G001, S037





S039 · calculationBefore

merge original slot0 Euclidean step0 quotient fraction := div(1, 3)

After

merge original slot0 Euclidean step0 quotient fraction = 1/3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038





S040 · calculationBefore

merge original slot0 Euclidean step0 divisor multiple := mul(0, 3)

After

merge original slot0 Euclidean step0 divisor multiple = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039





S041 · calculationBefore

merge original slot0 Euclidean step0 remainder := sub(1, 0)

After

merge original slot0 Euclidean step0 remainder = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040





S042 · calculationBefore

merge original slot0 Euclidean step0 first coefficient multiple := mul(0, 0)

After

merge original slot0 Euclidean step0 first coefficient multiple = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041





S043 · calculationBefore

merge original slot0 Euclidean step0 first coefficient difference := sub(1, 0)

After

merge original slot0 Euclidean step0 first coefficient difference = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042





S044 · calculationBefore

merge original slot0 Euclidean step0 second coefficient multiple := mul(0, 1)

After

merge original slot0 Euclidean step0 second coefficient multiple = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043





S045 · calculationBefore

merge original slot0 Euclidean step0 second coefficient difference := sub(0, 0)

After

merge original slot0 Euclidean step0 second coefficient difference = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044





S046 · impliesBefore

merge original slot0:step=0;current remainders=[1, 3];coefficient pairs=[[1, 0], [0, 1]];exact ratio=1/3;floor q=0

After

next remainder=1;next signed coefficient pair=[1, 0];a*x+b*y equals next remainder

Conditions: none

Use the same quotient for remainders and both signed coefficient recurrences,preserving a*x+b*y=remainder.Do not compute only a gcd and guess an inverse.

Step identifiers and dependenciesRule: crt_extended_euclid_invariant_signed_coefficients · Uses: G001, S045





S047 · calculationBefore

merge original slot0 Euclidean step1 quotient fraction := div(3, 1)

After

merge original slot0 Euclidean step1 quotient fraction = 3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046





S048 · calculationBefore

merge original slot0 Euclidean step1 divisor multiple := mul(3, 1)

After

merge original slot0 Euclidean step1 divisor multiple = 3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S047





S049 · calculationBefore

merge original slot0 Euclidean step1 remainder := sub(3, 3)

After

merge original slot0 Euclidean step1 remainder = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S048





S050 · calculationBefore

merge original slot0 Euclidean step1 first coefficient multiple := mul(3, 1)

After

merge original slot0 Euclidean step1 first coefficient multiple = 3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S049





S051 · calculationBefore

merge original slot0 Euclidean step1 first coefficient difference := sub(0, 3)

After

merge original slot0 Euclidean step1 first coefficient difference = -3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S050





S052 · calculationBefore

merge original slot0 Euclidean step1 second coefficient multiple := mul(3, 0)

After

merge original slot0 Euclidean step1 second coefficient multiple = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S051





S053 · calculationBefore

merge original slot0 Euclidean step1 second coefficient difference := sub(1, 0)

After

merge original slot0 Euclidean step1 second coefficient difference = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S052





S054 · impliesBefore

merge original slot0:step=1;current remainders=[3, 1];coefficient pairs=[[0, 1], [1, 0]];exact ratio=3;floor q=3

After

next remainder=0;next signed coefficient pair=[-3, 1];a*x+b*y equals next remainder

Conditions: none

Use the same quotient for remainders and both signed coefficient recurrences,preserving a*x+b*y=remainder.Do not compute only a gcd and guess an inverse.

Step identifiers and dependenciesRule: crt_extended_euclid_invariant_signed_coefficients · Uses: G001, S053





S055 · calculationBefore

merge original slot0 final first Bezout term := mul(1, 1)

After

merge original slot0 final first Bezout term = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S054





S056 · calculationBefore

merge original slot0 final second Bezout term := mul(3, 0)

After

merge original slot0 final second Bezout term = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S055





S057 · calculationBefore

merge original slot0 final Bezout identity := sum(1, 0)

After

merge original slot0 final Bezout identity = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S056





S058 · calculationBefore

merge slot0 new residue minus old class := sub(1, 0)

After

merge slot0 new residue minus old class = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S057





S059 · calculationBefore

merge slot0 difference divided by gcd := div(1, 1)

After

merge slot0 difference divided by gcd = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S058





S060 · calculationBefore

merge slot0 reduced old modulus := div(1, 1)

After

merge slot0 reduced old modulus = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S059





S061 · calculationBefore

merge slot0 reduced new modulus := div(3, 1)

After

merge slot0 reduced new modulus = 3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S060





S062 · calculationBefore

merge slot0 new LCM period := mul(1, 3)

After

merge slot0 new LCM period = 3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S061





S063 · calculationBefore

merge slot0 reduced-unit inverse exact quotient := div(1, 3)

After

merge slot0 reduced-unit inverse exact quotient = 1/3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S062





S064 · calculationBefore

merge slot0 reduced-unit inverse floor quotient multiple := mul(0, 3)

After

merge slot0 reduced-unit inverse floor quotient multiple = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S063





S065 · calculationBefore

merge slot0 reduced-unit inverse normalized remainder := sub(1, 0)

After

merge slot0 reduced-unit inverse normalized remainder = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S064





S066 · impliesBefore

merge slot0 reduced-unit inverse:integer value=1;positive modulus=3;exact ratio=1/3;floor quotient=0

After

value=q*modulus+remainder;normalized residue=1 in[0,3)

Conditions: none

Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0.

Step identifiers and dependenciesRule: crt_exact_floor_modulo_signed_input · Uses: G001, S065





S067 · calculationBefore

merge slot0 parameter congruence product := mul(1, 1)

After

merge slot0 parameter congruence product = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S066





S068 · calculationBefore

merge slot0 normalized parameter exact quotient := div(1, 3)

After

merge slot0 normalized parameter exact quotient = 1/3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S067





S069 · calculationBefore

merge slot0 normalized parameter floor quotient multiple := mul(0, 3)

After

merge slot0 normalized parameter floor quotient multiple = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S068





S070 · calculationBefore

merge slot0 normalized parameter normalized remainder := sub(1, 0)

After

merge slot0 normalized parameter normalized remainder = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S069





S071 · impliesBefore

merge slot0 normalized parameter:integer value=1;positive modulus=3;exact ratio=1/3;floor quotient=0

After

value=q*modulus+remainder;normalized residue=1 in[0,3)

Conditions: none

Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0.

Step identifiers and dependenciesRule: crt_exact_floor_modulo_signed_input · Uses: G001, S070





S072 · calculationBefore

merge slot0 old modulus times parameter := mul(1, 1)

After

merge slot0 old modulus times parameter = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S071





S073 · calculationBefore

merge slot0 unnormalized common solution := sum(0, 1)

After

merge slot0 unnormalized common solution = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S072





S074 · calculationBefore

merge slot0 least class representative exact quotient := div(1, 3)

After

merge slot0 least class representative exact quotient = 1/3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S073





S075 · calculationBefore

merge slot0 least class representative floor quotient multiple := mul(0, 3)

After

merge slot0 least class representative floor quotient multiple = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S074





S076 · calculationBefore

merge slot0 least class representative normalized remainder := sub(1, 0)

After

merge slot0 least class representative normalized remainder = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S075





S077 · impliesBefore

merge slot0 least class representative:integer value=1;positive modulus=3;exact ratio=1/3;floor quotient=0

After

value=q*modulus+remainder;normalized residue=1 in[0,3)

Conditions: none

Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0.

Step identifiers and dependenciesRule: crt_exact_floor_modulo_signed_input · Uses: G001, S076





S078 · impliesBefore

old FULL solution class=0 mod1;new original constraint=1 mod3;gcd=1;Bezout coefficients=[1, 0];reduced coprime moduli=[1, 3]

After

write old x=a+M*t;reduced equation gives t=1 mod3;ALL merged solutions x=1 mod3

Conditions: none

Substitute into the old solution class and divide by the gcd.Bezout divided by that gcd supplies the coprime inverse,so t lies in exactly one class modulo m/g.Multiplying back by M gives the lcm period and all solutions,not merely one checked point.

Step identifiers and dependenciesRule: crt_constructive_merge_all_solutions_not_one_witness · Uses: G001, S077





S079 · calculationBefore

merge original slot1 Euclidean step0 quotient fraction := div(3, 5)

After

merge original slot1 Euclidean step0 quotient fraction = 3/5

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S078





S080 · calculationBefore

merge original slot1 Euclidean step0 divisor multiple := mul(0, 5)

After

merge original slot1 Euclidean step0 divisor multiple = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S079





S081 · calculationBefore

merge original slot1 Euclidean step0 remainder := sub(3, 0)

After

merge original slot1 Euclidean step0 remainder = 3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S080





S082 · calculationBefore

merge original slot1 Euclidean step0 first coefficient multiple := mul(0, 0)

After

merge original slot1 Euclidean step0 first coefficient multiple = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S081





S083 · calculationBefore

merge original slot1 Euclidean step0 first coefficient difference := sub(1, 0)

After

merge original slot1 Euclidean step0 first coefficient difference = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S082





S084 · calculationBefore

merge original slot1 Euclidean step0 second coefficient multiple := mul(0, 1)

After

merge original slot1 Euclidean step0 second coefficient multiple = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S083





S085 · calculationBefore

merge original slot1 Euclidean step0 second coefficient difference := sub(0, 0)

After

merge original slot1 Euclidean step0 second coefficient difference = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S084





S086 · impliesBefore

merge original slot1:step=0;current remainders=[3, 5];coefficient pairs=[[1, 0], [0, 1]];exact ratio=3/5;floor q=0

After

next remainder=3;next signed coefficient pair=[1, 0];a*x+b*y equals next remainder

Conditions: none

Use the same quotient for remainders and both signed coefficient recurrences,preserving a*x+b*y=remainder.Do not compute only a gcd and guess an inverse.

Step identifiers and dependenciesRule: crt_extended_euclid_invariant_signed_coefficients · Uses: G001, S085





S087 · calculationBefore

merge original slot1 Euclidean step1 quotient fraction := div(5, 3)

After

merge original slot1 Euclidean step1 quotient fraction = 5/3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S086





S088 · calculationBefore

merge original slot1 Euclidean step1 divisor multiple := mul(1, 3)

After

merge original slot1 Euclidean step1 divisor multiple = 3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S087





S089 · calculationBefore

merge original slot1 Euclidean step1 remainder := sub(5, 3)

After

merge original slot1 Euclidean step1 remainder = 2

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S088





S090 · calculationBefore

merge original slot1 Euclidean step1 first coefficient multiple := mul(1, 1)

After

merge original slot1 Euclidean step1 first coefficient multiple = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S089





S091 · calculationBefore

merge original slot1 Euclidean step1 first coefficient difference := sub(0, 1)

After

merge original slot1 Euclidean step1 first coefficient difference = -1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S090





S092 · calculationBefore

merge original slot1 Euclidean step1 second coefficient multiple := mul(1, 0)

After

merge original slot1 Euclidean step1 second coefficient multiple = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S091





S093 · calculationBefore

merge original slot1 Euclidean step1 second coefficient difference := sub(1, 0)

After

merge original slot1 Euclidean step1 second coefficient difference = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S092





S094 · impliesBefore

merge original slot1:step=1;current remainders=[5, 3];coefficient pairs=[[0, 1], [1, 0]];exact ratio=5/3;floor q=1

After

next remainder=2;next signed coefficient pair=[-1, 1];a*x+b*y equals next remainder

Conditions: none

Use the same quotient for remainders and both signed coefficient recurrences,preserving a*x+b*y=remainder.Do not compute only a gcd and guess an inverse.

Step identifiers and dependenciesRule: crt_extended_euclid_invariant_signed_coefficients · Uses: G001, S093





S095 · calculationBefore

merge original slot1 Euclidean step2 quotient fraction := div(3, 2)

After

merge original slot1 Euclidean step2 quotient fraction = 3/2

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S094





S096 · calculationBefore

merge original slot1 Euclidean step2 divisor multiple := mul(1, 2)

After

merge original slot1 Euclidean step2 divisor multiple = 2

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S095





S097 · calculationBefore

merge original slot1 Euclidean step2 remainder := sub(3, 2)

After

merge original slot1 Euclidean step2 remainder = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S096





S098 · calculationBefore

merge original slot1 Euclidean step2 first coefficient multiple := mul(1, -1)

After

merge original slot1 Euclidean step2 first coefficient multiple = -1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S097





S099 · calculationBefore

merge original slot1 Euclidean step2 first coefficient difference := sub(1, -1)

After

merge original slot1 Euclidean step2 first coefficient difference = 2

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S098





S100 · calculationBefore

merge original slot1 Euclidean step2 second coefficient multiple := mul(1, 1)

After

merge original slot1 Euclidean step2 second coefficient multiple = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S099





S101 · calculationBefore

merge original slot1 Euclidean step2 second coefficient difference := sub(0, 1)

After

merge original slot1 Euclidean step2 second coefficient difference = -1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S100





S102 · impliesBefore

merge original slot1:step=2;current remainders=[3, 2];coefficient pairs=[[1, 0], [-1, 1]];exact ratio=3/2;floor q=1

After

next remainder=1;next signed coefficient pair=[2, -1];a*x+b*y equals next remainder

Conditions: none

Use the same quotient for remainders and both signed coefficient recurrences,preserving a*x+b*y=remainder.Do not compute only a gcd and guess an inverse.

Step identifiers and dependenciesRule: crt_extended_euclid_invariant_signed_coefficients · Uses: G001, S101





S103 · calculationBefore

merge original slot1 Euclidean step3 quotient fraction := div(2, 1)

After

merge original slot1 Euclidean step3 quotient fraction = 2

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S102





S104 · calculationBefore

merge original slot1 Euclidean step3 divisor multiple := mul(2, 1)

After

merge original slot1 Euclidean step3 divisor multiple = 2

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S103





S105 · calculationBefore

merge original slot1 Euclidean step3 remainder := sub(2, 2)

After

merge original slot1 Euclidean step3 remainder = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S104





S106 · calculationBefore

merge original slot1 Euclidean step3 first coefficient multiple := mul(2, 2)

After

merge original slot1 Euclidean step3 first coefficient multiple = 4

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S105





S107 · calculationBefore

merge original slot1 Euclidean step3 first coefficient difference := sub(-1, 4)

After

merge original slot1 Euclidean step3 first coefficient difference = -5

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S106





S108 · calculationBefore

merge original slot1 Euclidean step3 second coefficient multiple := mul(2, -1)

After

merge original slot1 Euclidean step3 second coefficient multiple = -2

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S107





S109 · calculationBefore

merge original slot1 Euclidean step3 second coefficient difference := sub(1, -2)

After

merge original slot1 Euclidean step3 second coefficient difference = 3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S108





S110 · impliesBefore

merge original slot1:step=3;current remainders=[2, 1];coefficient pairs=[[-1, 1], [2, -1]];exact ratio=2;floor q=2

After

next remainder=0;next signed coefficient pair=[-5, 3];a*x+b*y equals next remainder

Conditions: none

Use the same quotient for remainders and both signed coefficient recurrences,preserving a*x+b*y=remainder.Do not compute only a gcd and guess an inverse.

Step identifiers and dependenciesRule: crt_extended_euclid_invariant_signed_coefficients · Uses: G001, S109





S111 · calculationBefore

merge original slot1 final first Bezout term := mul(3, 2)

After

merge original slot1 final first Bezout term = 6

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S110





S112 · calculationBefore

merge original slot1 final second Bezout term := mul(5, -1)

After

merge original slot1 final second Bezout term = -5

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S111





S113 · calculationBefore

merge original slot1 final Bezout identity := sum(6, -5)

After

merge original slot1 final Bezout identity = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S112





S114 · calculationBefore

merge slot1 new residue minus old class := sub(4, 1)

After

merge slot1 new residue minus old class = 3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S113





S115 · calculationBefore

merge slot1 difference divided by gcd := div(3, 1)

After

merge slot1 difference divided by gcd = 3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S114





S116 · calculationBefore

merge slot1 reduced old modulus := div(3, 1)

After

merge slot1 reduced old modulus = 3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S115





S117 · calculationBefore

merge slot1 reduced new modulus := div(5, 1)

After

merge slot1 reduced new modulus = 5

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S116





S118 · calculationBefore

merge slot1 new LCM period := mul(3, 5)

After

merge slot1 new LCM period = 15

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S117





S119 · calculationBefore

merge slot1 reduced-unit inverse exact quotient := div(2, 5)

After

merge slot1 reduced-unit inverse exact quotient = 2/5

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S118





S120 · calculationBefore

merge slot1 reduced-unit inverse floor quotient multiple := mul(0, 5)

After

merge slot1 reduced-unit inverse floor quotient multiple = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S119





S121 · calculationBefore

merge slot1 reduced-unit inverse normalized remainder := sub(2, 0)

After

merge slot1 reduced-unit inverse normalized remainder = 2

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S120





S122 · impliesBefore

merge slot1 reduced-unit inverse:integer value=2;positive modulus=5;exact ratio=2/5;floor quotient=0

After

value=q*modulus+remainder;normalized residue=2 in[0,5)

Conditions: none

Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0.

Step identifiers and dependenciesRule: crt_exact_floor_modulo_signed_input · Uses: G001, S121





S123 · calculationBefore

merge slot1 parameter congruence product := mul(3, 2)

After

merge slot1 parameter congruence product = 6

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S122





S124 · calculationBefore

merge slot1 normalized parameter exact quotient := div(6, 5)

After

merge slot1 normalized parameter exact quotient = 6/5

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S123





S125 · calculationBefore

merge slot1 normalized parameter floor quotient multiple := mul(1, 5)

After

merge slot1 normalized parameter floor quotient multiple = 5

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S124





S126 · calculationBefore

merge slot1 normalized parameter normalized remainder := sub(6, 5)

After

merge slot1 normalized parameter normalized remainder = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S125





S127 · impliesBefore

merge slot1 normalized parameter:integer value=6;positive modulus=5;exact ratio=6/5;floor quotient=1

After

value=q*modulus+remainder;normalized residue=1 in[0,5)

Conditions: none

Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0.

Step identifiers and dependenciesRule: crt_exact_floor_modulo_signed_input · Uses: G001, S126





S128 · calculationBefore

merge slot1 old modulus times parameter := mul(3, 1)

After

merge slot1 old modulus times parameter = 3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S127





S129 · calculationBefore

merge slot1 unnormalized common solution := sum(1, 3)

After

merge slot1 unnormalized common solution = 4

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S128





S130 · calculationBefore

merge slot1 least class representative exact quotient := div(4, 15)

After

merge slot1 least class representative exact quotient = 4/15

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S129





S131 · calculationBefore

merge slot1 least class representative floor quotient multiple := mul(0, 15)

After

merge slot1 least class representative floor quotient multiple = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S130





S132 · calculationBefore

merge slot1 least class representative normalized remainder := sub(4, 0)

After

merge slot1 least class representative normalized remainder = 4

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S131





S133 · impliesBefore

merge slot1 least class representative:integer value=4;positive modulus=15;exact ratio=4/15;floor quotient=0

After

value=q*modulus+remainder;normalized residue=4 in[0,15)

Conditions: none

Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0.

Step identifiers and dependenciesRule: crt_exact_floor_modulo_signed_input · Uses: G001, S132





S134 · impliesBefore

old FULL solution class=1 mod3;new original constraint=4 mod5;gcd=1;Bezout coefficients=[2, -1];reduced coprime moduli=[3, 5]

After

write old x=a+M*t;reduced equation gives t=1 mod5;ALL merged solutions x=4 mod15

Conditions: none

Substitute into the old solution class and divide by the gcd.Bezout divided by that gcd supplies the coprime inverse,so t lies in exactly one class modulo m/g.Multiplying back by M gives the lcm period and all solutions,not merely one checked point.

Step identifiers and dependenciesRule: crt_constructive_merge_all_solutions_not_one_witness · Uses: G001, S133





S135 · calculationBefore

merge original slot2 Euclidean step0 quotient fraction := div(15, 43)

After

merge original slot2 Euclidean step0 quotient fraction = 15/43

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S134





S136 · calculationBefore

merge original slot2 Euclidean step0 divisor multiple := mul(0, 43)

After

merge original slot2 Euclidean step0 divisor multiple = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S135





S137 · calculationBefore

merge original slot2 Euclidean step0 remainder := sub(15, 0)

After

merge original slot2 Euclidean step0 remainder = 15

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S136





S138 · calculationBefore

merge original slot2 Euclidean step0 first coefficient multiple := mul(0, 0)

After

merge original slot2 Euclidean step0 first coefficient multiple = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S137





S139 · calculationBefore

merge original slot2 Euclidean step0 first coefficient difference := sub(1, 0)

After

merge original slot2 Euclidean step0 first coefficient difference = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S138





S140 · calculationBefore

merge original slot2 Euclidean step0 second coefficient multiple := mul(0, 1)

After

merge original slot2 Euclidean step0 second coefficient multiple = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S139





S141 · calculationBefore

merge original slot2 Euclidean step0 second coefficient difference := sub(0, 0)

After

merge original slot2 Euclidean step0 second coefficient difference = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S140





S142 · impliesBefore

merge original slot2:step=0;current remainders=[15, 43];coefficient pairs=[[1, 0], [0, 1]];exact ratio=15/43;floor q=0

After

next remainder=15;next signed coefficient pair=[1, 0];a*x+b*y equals next remainder

Conditions: none

Use the same quotient for remainders and both signed coefficient recurrences,preserving a*x+b*y=remainder.Do not compute only a gcd and guess an inverse.

Step identifiers and dependenciesRule: crt_extended_euclid_invariant_signed_coefficients · Uses: G001, S141





S143 · calculationBefore

merge original slot2 Euclidean step1 quotient fraction := div(43, 15)

After

merge original slot2 Euclidean step1 quotient fraction = 43/15

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S142





S144 · calculationBefore

merge original slot2 Euclidean step1 divisor multiple := mul(2, 15)

After

merge original slot2 Euclidean step1 divisor multiple = 30

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S143





S145 · calculationBefore

merge original slot2 Euclidean step1 remainder := sub(43, 30)

After

merge original slot2 Euclidean step1 remainder = 13

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S144





S146 · calculationBefore

merge original slot2 Euclidean step1 first coefficient multiple := mul(2, 1)

After

merge original slot2 Euclidean step1 first coefficient multiple = 2

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S145





S147 · calculationBefore

merge original slot2 Euclidean step1 first coefficient difference := sub(0, 2)

After

merge original slot2 Euclidean step1 first coefficient difference = -2

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S146





S148 · calculationBefore

merge original slot2 Euclidean step1 second coefficient multiple := mul(2, 0)

After

merge original slot2 Euclidean step1 second coefficient multiple = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S147





S149 · calculationBefore

merge original slot2 Euclidean step1 second coefficient difference := sub(1, 0)

After

merge original slot2 Euclidean step1 second coefficient difference = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S148





S150 · impliesBefore

merge original slot2:step=1;current remainders=[43, 15];coefficient pairs=[[0, 1], [1, 0]];exact ratio=43/15;floor q=2

After

next remainder=13;next signed coefficient pair=[-2, 1];a*x+b*y equals next remainder

Conditions: none

Use the same quotient for remainders and both signed coefficient recurrences,preserving a*x+b*y=remainder.Do not compute only a gcd and guess an inverse.

Step identifiers and dependenciesRule: crt_extended_euclid_invariant_signed_coefficients · Uses: G001, S149





S151 · calculationBefore

merge original slot2 Euclidean step2 quotient fraction := div(15, 13)

After

merge original slot2 Euclidean step2 quotient fraction = 15/13

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S150





S152 · calculationBefore

merge original slot2 Euclidean step2 divisor multiple := mul(1, 13)

After

merge original slot2 Euclidean step2 divisor multiple = 13

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S151





S153 · calculationBefore

merge original slot2 Euclidean step2 remainder := sub(15, 13)

After

merge original slot2 Euclidean step2 remainder = 2

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S152





S154 · calculationBefore

merge original slot2 Euclidean step2 first coefficient multiple := mul(1, -2)

After

merge original slot2 Euclidean step2 first coefficient multiple = -2

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S153





S155 · calculationBefore

merge original slot2 Euclidean step2 first coefficient difference := sub(1, -2)

After

merge original slot2 Euclidean step2 first coefficient difference = 3

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S154





S156 · calculationBefore

merge original slot2 Euclidean step2 second coefficient multiple := mul(1, 1)

After

merge original slot2 Euclidean step2 second coefficient multiple = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S155





S157 · calculationBefore

merge original slot2 Euclidean step2 second coefficient difference := sub(0, 1)

After

merge original slot2 Euclidean step2 second coefficient difference = -1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S156





S158 · impliesBefore

merge original slot2:step=2;current remainders=[15, 13];coefficient pairs=[[1, 0], [-2, 1]];exact ratio=15/13;floor q=1

After

next remainder=2;next signed coefficient pair=[3, -1];a*x+b*y equals next remainder

Conditions: none

Use the same quotient for remainders and both signed coefficient recurrences,preserving a*x+b*y=remainder.Do not compute only a gcd and guess an inverse.

Step identifiers and dependenciesRule: crt_extended_euclid_invariant_signed_coefficients · Uses: G001, S157





S159 · calculationBefore

merge original slot2 Euclidean step3 quotient fraction := div(13, 2)

After

merge original slot2 Euclidean step3 quotient fraction = 13/2

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S158





S160 · calculationBefore

merge original slot2 Euclidean step3 divisor multiple := mul(6, 2)

After

merge original slot2 Euclidean step3 divisor multiple = 12

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S159





S161 · calculationBefore

merge original slot2 Euclidean step3 remainder := sub(13, 12)

After

merge original slot2 Euclidean step3 remainder = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S160





S162 · calculationBefore

merge original slot2 Euclidean step3 first coefficient multiple := mul(6, 3)

After

merge original slot2 Euclidean step3 first coefficient multiple = 18

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S161





S163 · calculationBefore

merge original slot2 Euclidean step3 first coefficient difference := sub(-2, 18)

After

merge original slot2 Euclidean step3 first coefficient difference = -20

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S162





S164 · calculationBefore

merge original slot2 Euclidean step3 second coefficient multiple := mul(6, -1)

After

merge original slot2 Euclidean step3 second coefficient multiple = -6

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S163





S165 · calculationBefore

merge original slot2 Euclidean step3 second coefficient difference := sub(1, -6)

After

merge original slot2 Euclidean step3 second coefficient difference = 7

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S164





S166 · impliesBefore

merge original slot2:step=3;current remainders=[13, 2];coefficient pairs=[[-2, 1], [3, -1]];exact ratio=13/2;floor q=6

After

next remainder=1;next signed coefficient pair=[-20, 7];a*x+b*y equals next remainder

Conditions: none

Use the same quotient for remainders and both signed coefficient recurrences,preserving a*x+b*y=remainder.Do not compute only a gcd and guess an inverse.

Step identifiers and dependenciesRule: crt_extended_euclid_invariant_signed_coefficients · Uses: G001, S165





S167 · calculationBefore

merge original slot2 Euclidean step4 quotient fraction := div(2, 1)

After

merge original slot2 Euclidean step4 quotient fraction = 2

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S166





S168 · calculationBefore

merge original slot2 Euclidean step4 divisor multiple := mul(2, 1)

After

merge original slot2 Euclidean step4 divisor multiple = 2

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S167





S169 · calculationBefore

merge original slot2 Euclidean step4 remainder := sub(2, 2)

After

merge original slot2 Euclidean step4 remainder = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S168





S170 · calculationBefore

merge original slot2 Euclidean step4 first coefficient multiple := mul(2, -20)

After

merge original slot2 Euclidean step4 first coefficient multiple = -40

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S169





S171 · calculationBefore

merge original slot2 Euclidean step4 first coefficient difference := sub(3, -40)

After

merge original slot2 Euclidean step4 first coefficient difference = 43

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S170





S172 · calculationBefore

merge original slot2 Euclidean step4 second coefficient multiple := mul(2, 7)

After

merge original slot2 Euclidean step4 second coefficient multiple = 14

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S171





S173 · calculationBefore

merge original slot2 Euclidean step4 second coefficient difference := sub(-1, 14)

After

merge original slot2 Euclidean step4 second coefficient difference = -15

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S172





S174 · impliesBefore

merge original slot2:step=4;current remainders=[2, 1];coefficient pairs=[[3, -1], [-20, 7]];exact ratio=2;floor q=2

After

next remainder=0;next signed coefficient pair=[43, -15];a*x+b*y equals next remainder

Conditions: none

Use the same quotient for remainders and both signed coefficient recurrences,preserving a*x+b*y=remainder.Do not compute only a gcd and guess an inverse.

Step identifiers and dependenciesRule: crt_extended_euclid_invariant_signed_coefficients · Uses: G001, S173





S175 · calculationBefore

merge original slot2 final first Bezout term := mul(15, -20)

After

merge original slot2 final first Bezout term = -300

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S174





S176 · calculationBefore

merge original slot2 final second Bezout term := mul(43, 7)

After

merge original slot2 final second Bezout term = 301

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S175





S177 · calculationBefore

merge original slot2 final Bezout identity := sum(-300, 301)

After

merge original slot2 final Bezout identity = 1

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S176





S178 · calculationBefore

merge slot2 new residue minus old class := sub(16, 4)

After

merge slot2 new residue minus old class = 12

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S177





S179 · calculationBefore

merge slot2 difference divided by gcd := div(12, 1)

After

merge slot2 difference divided by gcd = 12

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S178





S180 · calculationBefore

merge slot2 reduced old modulus := div(15, 1)

After

merge slot2 reduced old modulus = 15

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S179





S181 · calculationBefore

merge slot2 reduced new modulus := div(43, 1)

After

merge slot2 reduced new modulus = 43

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S180





S182 · calculationBefore

merge slot2 new LCM period := mul(15, 43)

After

merge slot2 new LCM period = 645

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S181





S183 · calculationBefore

merge slot2 reduced-unit inverse exact quotient := div(-20, 43)

After

merge slot2 reduced-unit inverse exact quotient = -20/43

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S182





S184 · calculationBefore

merge slot2 reduced-unit inverse floor quotient multiple := mul(-1, 43)

After

merge slot2 reduced-unit inverse floor quotient multiple = -43

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S183





S185 · calculationBefore

merge slot2 reduced-unit inverse normalized remainder := sub(-20, -43)

After

merge slot2 reduced-unit inverse normalized remainder = 23

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S184





S186 · impliesBefore

merge slot2 reduced-unit inverse:integer value=-20;positive modulus=43;exact ratio=-20/43;floor quotient=-1

After

value=q*modulus+remainder;normalized residue=23 in[0,43)

Conditions: none

Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0.

Step identifiers and dependenciesRule: crt_exact_floor_modulo_signed_input · Uses: G001, S185





S187 · calculationBefore

merge slot2 parameter congruence product := mul(12, 23)

After

merge slot2 parameter congruence product = 276

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S186





S188 · calculationBefore

merge slot2 normalized parameter exact quotient := div(276, 43)

After

merge slot2 normalized parameter exact quotient = 276/43

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S187





S189 · calculationBefore

merge slot2 normalized parameter floor quotient multiple := mul(6, 43)

After

merge slot2 normalized parameter floor quotient multiple = 258

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S188





S190 · calculationBefore

merge slot2 normalized parameter normalized remainder := sub(276, 258)

After

merge slot2 normalized parameter normalized remainder = 18

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S189





S191 · impliesBefore

merge slot2 normalized parameter:integer value=276;positive modulus=43;exact ratio=276/43;floor quotient=6

After

value=q*modulus+remainder;normalized residue=18 in[0,43)

Conditions: none

Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0.

Step identifiers and dependenciesRule: crt_exact_floor_modulo_signed_input · Uses: G001, S190





S192 · calculationBefore

merge slot2 old modulus times parameter := mul(15, 18)

After

merge slot2 old modulus times parameter = 270

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S191





S193 · calculationBefore

merge slot2 unnormalized common solution := sum(4, 270)

After

merge slot2 unnormalized common solution = 274

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S192





S194 · calculationBefore

merge slot2 least class representative exact quotient := div(274, 645)

After

merge slot2 least class representative exact quotient = 274/645

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S193





S195 · calculationBefore

merge slot2 least class representative floor quotient multiple := mul(0, 645)

After

merge slot2 least class representative floor quotient multiple = 0

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S194





S196 · calculationBefore

merge slot2 least class representative normalized remainder := sub(274, 0)

After

merge slot2 least class representative normalized remainder = 274

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S195





S197 · impliesBefore

merge slot2 least class representative:integer value=274;positive modulus=645;exact ratio=274/645;floor quotient=0

After

value=q*modulus+remainder;normalized residue=274 in[0,645)

Conditions: none

Use the floor quotient for a positive modulus,including signed inputs,to obtain the representative from0 through modulus−1.This is not truncation toward zero or a change of congruence class.Modulus1 is a valid trivial constraint with residue0.

Step identifiers and dependenciesRule: crt_exact_floor_modulo_signed_input · Uses: G001, S196





S198 · impliesBefore

old FULL solution class=4 mod15;new original constraint=16 mod43;gcd=1;Bezout coefficients=[-20, 7];reduced coprime moduli=[15, 43]

After

write old x=a+M*t;reduced equation gives t=18 mod43;ALL merged solutions x=274 mod645

Conditions: none

Substitute into the old solution class and divide by the gcd.Bezout divided by that gcd supplies the coprime inverse,so t lies in exactly one class modulo m/g.Multiplying back by M gives the lcm period and all solutions,not merely one checked point.

Step identifiers and dependenciesRule: crt_constructive_merge_all_solutions_not_one_witness · Uses: G001, S197









## Reverse checks


- S198 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S197 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S196 · premise_and_result_audit — Recompute merge slot2 least class representative normalized remainder with exact rational arithmetic, not rounded decimals.
- S195 · premise_and_result_audit — Recompute merge slot2 least class representative floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S194 · premise_and_result_audit — Recompute merge slot2 least class representative exact quotient with exact rational arithmetic, not rounded decimals.
- S193 · premise_and_result_audit — Recompute merge slot2 unnormalized common solution with exact rational arithmetic, not rounded decimals.
- S192 · premise_and_result_audit — Recompute merge slot2 old modulus times parameter with exact rational arithmetic, not rounded decimals.
- S191 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S190 · premise_and_result_audit — Recompute merge slot2 normalized parameter normalized remainder with exact rational arithmetic, not rounded decimals.
- S189 · premise_and_result_audit — Recompute merge slot2 normalized parameter floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S188 · premise_and_result_audit — Recompute merge slot2 normalized parameter exact quotient with exact rational arithmetic, not rounded decimals.
- S187 · premise_and_result_audit — Recompute merge slot2 parameter congruence product with exact rational arithmetic, not rounded decimals.
- S186 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S185 · premise_and_result_audit — Recompute merge slot2 reduced-unit inverse normalized remainder with exact rational arithmetic, not rounded decimals.
- S184 · premise_and_result_audit — Recompute merge slot2 reduced-unit inverse floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S183 · premise_and_result_audit — Recompute merge slot2 reduced-unit inverse exact quotient with exact rational arithmetic, not rounded decimals.
- S182 · premise_and_result_audit — Recompute merge slot2 new LCM period with exact rational arithmetic, not rounded decimals.
- S181 · premise_and_result_audit — Recompute merge slot2 reduced new modulus with exact rational arithmetic, not rounded decimals.
- S180 · premise_and_result_audit — Recompute merge slot2 reduced old modulus with exact rational arithmetic, not rounded decimals.
- S179 · premise_and_result_audit — Recompute merge slot2 difference divided by gcd with exact rational arithmetic, not rounded decimals.
- S178 · premise_and_result_audit — Recompute merge slot2 new residue minus old class with exact rational arithmetic, not rounded decimals.
- S177 · premise_and_result_audit — Recompute merge original slot2 final Bezout identity with exact rational arithmetic, not rounded decimals.
- S176 · premise_and_result_audit — Recompute merge original slot2 final second Bezout term with exact rational arithmetic, not rounded decimals.
- S175 · premise_and_result_audit — Recompute merge original slot2 final first Bezout term with exact rational arithmetic, not rounded decimals.
- S174 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S173 · premise_and_result_audit — Recompute merge original slot2 Euclidean step4 second coefficient difference with exact rational arithmetic, not rounded decimals.
- S172 · premise_and_result_audit — Recompute merge original slot2 Euclidean step4 second coefficient multiple with exact rational arithmetic, not rounded decimals.
- S171 · premise_and_result_audit — Recompute merge original slot2 Euclidean step4 first coefficient difference with exact rational arithmetic, not rounded decimals.
- S170 · premise_and_result_audit — Recompute merge original slot2 Euclidean step4 first coefficient multiple with exact rational arithmetic, not rounded decimals.
- S169 · premise_and_result_audit — Recompute merge original slot2 Euclidean step4 remainder with exact rational arithmetic, not rounded decimals.
- S168 · premise_and_result_audit — Recompute merge original slot2 Euclidean step4 divisor multiple with exact rational arithmetic, not rounded decimals.
- S167 · premise_and_result_audit — Recompute merge original slot2 Euclidean step4 quotient fraction with exact rational arithmetic, not rounded decimals.
- S166 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S165 · premise_and_result_audit — Recompute merge original slot2 Euclidean step3 second coefficient difference with exact rational arithmetic, not rounded decimals.
- S164 · premise_and_result_audit — Recompute merge original slot2 Euclidean step3 second coefficient multiple with exact rational arithmetic, not rounded decimals.
- S163 · premise_and_result_audit — Recompute merge original slot2 Euclidean step3 first coefficient difference with exact rational arithmetic, not rounded decimals.
- S162 · premise_and_result_audit — Recompute merge original slot2 Euclidean step3 first coefficient multiple with exact rational arithmetic, not rounded decimals.
- S161 · premise_and_result_audit — Recompute merge original slot2 Euclidean step3 remainder with exact rational arithmetic, not rounded decimals.
- S160 · premise_and_result_audit — Recompute merge original slot2 Euclidean step3 divisor multiple with exact rational arithmetic, not rounded decimals.
- S159 · premise_and_result_audit — Recompute merge original slot2 Euclidean step3 quotient fraction with exact rational arithmetic, not rounded decimals.
- S158 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S157 · premise_and_result_audit — Recompute merge original slot2 Euclidean step2 second coefficient difference with exact rational arithmetic, not rounded decimals.
- S156 · premise_and_result_audit — Recompute merge original slot2 Euclidean step2 second coefficient multiple with exact rational arithmetic, not rounded decimals.
- S155 · premise_and_result_audit — Recompute merge original slot2 Euclidean step2 first coefficient difference with exact rational arithmetic, not rounded decimals.
- S154 · premise_and_result_audit — Recompute merge original slot2 Euclidean step2 first coefficient multiple with exact rational arithmetic, not rounded decimals.
- S153 · premise_and_result_audit — Recompute merge original slot2 Euclidean step2 remainder with exact rational arithmetic, not rounded decimals.
- S152 · premise_and_result_audit — Recompute merge original slot2 Euclidean step2 divisor multiple with exact rational arithmetic, not rounded decimals.
- S151 · premise_and_result_audit — Recompute merge original slot2 Euclidean step2 quotient fraction with exact rational arithmetic, not rounded decimals.
- S150 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S149 · premise_and_result_audit — Recompute merge original slot2 Euclidean step1 second coefficient difference with exact rational arithmetic, not rounded decimals.
- S148 · premise_and_result_audit — Recompute merge original slot2 Euclidean step1 second coefficient multiple with exact rational arithmetic, not rounded decimals.
- S147 · premise_and_result_audit — Recompute merge original slot2 Euclidean step1 first coefficient difference with exact rational arithmetic, not rounded decimals.
- S146 · premise_and_result_audit — Recompute merge original slot2 Euclidean step1 first coefficient multiple with exact rational arithmetic, not rounded decimals.
- S145 · premise_and_result_audit — Recompute merge original slot2 Euclidean step1 remainder with exact rational arithmetic, not rounded decimals.
- S144 · premise_and_result_audit — Recompute merge original slot2 Euclidean step1 divisor multiple with exact rational arithmetic, not rounded decimals.
- S143 · premise_and_result_audit — Recompute merge original slot2 Euclidean step1 quotient fraction with exact rational arithmetic, not rounded decimals.
- S142 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S141 · premise_and_result_audit — Recompute merge original slot2 Euclidean step0 second coefficient difference with exact rational arithmetic, not rounded decimals.
- S140 · premise_and_result_audit — Recompute merge original slot2 Euclidean step0 second coefficient multiple with exact rational arithmetic, not rounded decimals.
- S139 · premise_and_result_audit — Recompute merge original slot2 Euclidean step0 first coefficient difference with exact rational arithmetic, not rounded decimals.
- S138 · premise_and_result_audit — Recompute merge original slot2 Euclidean step0 first coefficient multiple with exact rational arithmetic, not rounded decimals.
- S137 · premise_and_result_audit — Recompute merge original slot2 Euclidean step0 remainder with exact rational arithmetic, not rounded decimals.
- S136 · premise_and_result_audit — Recompute merge original slot2 Euclidean step0 divisor multiple with exact rational arithmetic, not rounded decimals.
- S135 · premise_and_result_audit — Recompute merge original slot2 Euclidean step0 quotient fraction with exact rational arithmetic, not rounded decimals.
- S134 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S133 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S132 · premise_and_result_audit — Recompute merge slot1 least class representative normalized remainder with exact rational arithmetic, not rounded decimals.
- S131 · premise_and_result_audit — Recompute merge slot1 least class representative floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S130 · premise_and_result_audit — Recompute merge slot1 least class representative exact quotient with exact rational arithmetic, not rounded decimals.
- S129 · premise_and_result_audit — Recompute merge slot1 unnormalized common solution with exact rational arithmetic, not rounded decimals.
- S128 · premise_and_result_audit — Recompute merge slot1 old modulus times parameter with exact rational arithmetic, not rounded decimals.
- S127 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S126 · premise_and_result_audit — Recompute merge slot1 normalized parameter normalized remainder with exact rational arithmetic, not rounded decimals.
- S125 · premise_and_result_audit — Recompute merge slot1 normalized parameter floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S124 · premise_and_result_audit — Recompute merge slot1 normalized parameter exact quotient with exact rational arithmetic, not rounded decimals.
- S123 · premise_and_result_audit — Recompute merge slot1 parameter congruence product with exact rational arithmetic, not rounded decimals.
- S122 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S121 · premise_and_result_audit — Recompute merge slot1 reduced-unit inverse normalized remainder with exact rational arithmetic, not rounded decimals.
- S120 · premise_and_result_audit — Recompute merge slot1 reduced-unit inverse floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S119 · premise_and_result_audit — Recompute merge slot1 reduced-unit inverse exact quotient with exact rational arithmetic, not rounded decimals.
- S118 · premise_and_result_audit — Recompute merge slot1 new LCM period with exact rational arithmetic, not rounded decimals.
- S117 · premise_and_result_audit — Recompute merge slot1 reduced new modulus with exact rational arithmetic, not rounded decimals.
- S116 · premise_and_result_audit — Recompute merge slot1 reduced old modulus with exact rational arithmetic, not rounded decimals.
- S115 · premise_and_result_audit — Recompute merge slot1 difference divided by gcd with exact rational arithmetic, not rounded decimals.
- S114 · premise_and_result_audit — Recompute merge slot1 new residue minus old class with exact rational arithmetic, not rounded decimals.
- S113 · premise_and_result_audit — Recompute merge original slot1 final Bezout identity with exact rational arithmetic, not rounded decimals.
- S112 · premise_and_result_audit — Recompute merge original slot1 final second Bezout term with exact rational arithmetic, not rounded decimals.
- S111 · premise_and_result_audit — Recompute merge original slot1 final first Bezout term with exact rational arithmetic, not rounded decimals.
- S110 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S109 · premise_and_result_audit — Recompute merge original slot1 Euclidean step3 second coefficient difference with exact rational arithmetic, not rounded decimals.
- S108 · premise_and_result_audit — Recompute merge original slot1 Euclidean step3 second coefficient multiple with exact rational arithmetic, not rounded decimals.
- S107 · premise_and_result_audit — Recompute merge original slot1 Euclidean step3 first coefficient difference with exact rational arithmetic, not rounded decimals.
- S106 · premise_and_result_audit — Recompute merge original slot1 Euclidean step3 first coefficient multiple with exact rational arithmetic, not rounded decimals.
- S105 · premise_and_result_audit — Recompute merge original slot1 Euclidean step3 remainder with exact rational arithmetic, not rounded decimals.
- S104 · premise_and_result_audit — Recompute merge original slot1 Euclidean step3 divisor multiple with exact rational arithmetic, not rounded decimals.
- S103 · premise_and_result_audit — Recompute merge original slot1 Euclidean step3 quotient fraction with exact rational arithmetic, not rounded decimals.
- 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 merge original slot1 Euclidean step2 second coefficient difference with exact rational arithmetic, not rounded decimals.
- S100 · premise_and_result_audit — Recompute merge original slot1 Euclidean step2 second coefficient multiple with exact rational arithmetic, not rounded decimals.
- S099 · premise_and_result_audit — Recompute merge original slot1 Euclidean step2 first coefficient difference with exact rational arithmetic, not rounded decimals.
- S098 · premise_and_result_audit — Recompute merge original slot1 Euclidean step2 first coefficient multiple with exact rational arithmetic, not rounded decimals.
- S097 · premise_and_result_audit — Recompute merge original slot1 Euclidean step2 remainder with exact rational arithmetic, not rounded decimals.
- S096 · premise_and_result_audit — Recompute merge original slot1 Euclidean step2 divisor multiple with exact rational arithmetic, not rounded decimals.
- S095 · premise_and_result_audit — Recompute merge original slot1 Euclidean step2 quotient fraction with exact rational arithmetic, not rounded decimals.
- 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 — Recompute merge original slot1 Euclidean step1 second coefficient difference with exact rational arithmetic, not rounded decimals.
- S092 · premise_and_result_audit — Recompute merge original slot1 Euclidean step1 second coefficient multiple with exact rational arithmetic, not rounded decimals.
- S091 · premise_and_result_audit — Recompute merge original slot1 Euclidean step1 first coefficient difference with exact rational arithmetic, not rounded decimals.
- S090 · premise_and_result_audit — Recompute merge original slot1 Euclidean step1 first coefficient multiple with exact rational arithmetic, not rounded decimals.
- S089 · premise_and_result_audit — Recompute merge original slot1 Euclidean step1 remainder with exact rational arithmetic, not rounded decimals.
- S088 · premise_and_result_audit — Recompute merge original slot1 Euclidean step1 divisor multiple with exact rational arithmetic, not rounded decimals.
- S087 · premise_and_result_audit — Recompute merge original slot1 Euclidean step1 quotient fraction with exact rational arithmetic, not rounded decimals.
- S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S085 · premise_and_result_audit — Recompute merge original slot1 Euclidean step0 second coefficient difference with exact rational arithmetic, not rounded decimals.
- S084 · premise_and_result_audit — Recompute merge original slot1 Euclidean step0 second coefficient multiple with exact rational arithmetic, not rounded decimals.
- S083 · premise_and_result_audit — Recompute merge original slot1 Euclidean step0 first coefficient difference with exact rational arithmetic, not rounded decimals.
- S082 · premise_and_result_audit — Recompute merge original slot1 Euclidean step0 first coefficient multiple with exact rational arithmetic, not rounded decimals.
- S081 · premise_and_result_audit — Recompute merge original slot1 Euclidean step0 remainder with exact rational arithmetic, not rounded decimals.
- S080 · premise_and_result_audit — Recompute merge original slot1 Euclidean step0 divisor multiple with exact rational arithmetic, not rounded decimals.
- S079 · premise_and_result_audit — Recompute merge original slot1 Euclidean step0 quotient fraction with exact rational arithmetic, not rounded decimals.
- S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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 — Recompute merge slot0 least class representative normalized remainder with exact rational arithmetic, not rounded decimals.
- S075 · premise_and_result_audit — Recompute merge slot0 least class representative floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S074 · premise_and_result_audit — Recompute merge slot0 least class representative exact quotient with exact rational arithmetic, not rounded decimals.
- S073 · premise_and_result_audit — Recompute merge slot0 unnormalized common solution with exact rational arithmetic, not rounded decimals.
- S072 · premise_and_result_audit — Recompute merge slot0 old modulus times parameter with exact rational arithmetic, not rounded decimals.
- S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S070 · premise_and_result_audit — Recompute merge slot0 normalized parameter normalized remainder with exact rational arithmetic, not rounded decimals.
- S069 · premise_and_result_audit — Recompute merge slot0 normalized parameter floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S068 · premise_and_result_audit — Recompute merge slot0 normalized parameter exact quotient with exact rational arithmetic, not rounded decimals.
- S067 · premise_and_result_audit — Recompute merge slot0 parameter congruence product with exact rational arithmetic, not rounded decimals.
- 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 merge slot0 reduced-unit inverse normalized remainder with exact rational arithmetic, not rounded decimals.
- S064 · premise_and_result_audit — Recompute merge slot0 reduced-unit inverse floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S063 · premise_and_result_audit — Recompute merge slot0 reduced-unit inverse exact quotient with exact rational arithmetic, not rounded decimals.
- S062 · premise_and_result_audit — Recompute merge slot0 new LCM period with exact rational arithmetic, not rounded decimals.
- S061 · premise_and_result_audit — Recompute merge slot0 reduced new modulus with exact rational arithmetic, not rounded decimals.
- S060 · premise_and_result_audit — Recompute merge slot0 reduced old modulus with exact rational arithmetic, not rounded decimals.
- S059 · premise_and_result_audit — Recompute merge slot0 difference divided by gcd with exact rational arithmetic, not rounded decimals.
- S058 · premise_and_result_audit — Recompute merge slot0 new residue minus old class with exact rational arithmetic, not rounded decimals.
- S057 · premise_and_result_audit — Recompute merge original slot0 final Bezout identity with exact rational arithmetic, not rounded decimals.
- S056 · premise_and_result_audit — Recompute merge original slot0 final second Bezout term with exact rational arithmetic, not rounded decimals.
- S055 · premise_and_result_audit — Recompute merge original slot0 final first Bezout term with exact rational arithmetic, not rounded decimals.
- S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S053 · premise_and_result_audit — Recompute merge original slot0 Euclidean step1 second coefficient difference with exact rational arithmetic, not rounded decimals.
- S052 · premise_and_result_audit — Recompute merge original slot0 Euclidean step1 second coefficient multiple with exact rational arithmetic, not rounded decimals.
- S051 · premise_and_result_audit — Recompute merge original slot0 Euclidean step1 first coefficient difference with exact rational arithmetic, not rounded decimals.
- S050 · premise_and_result_audit — Recompute merge original slot0 Euclidean step1 first coefficient multiple with exact rational arithmetic, not rounded decimals.
- S049 · premise_and_result_audit — Recompute merge original slot0 Euclidean step1 remainder with exact rational arithmetic, not rounded decimals.
- S048 · premise_and_result_audit — Recompute merge original slot0 Euclidean step1 divisor multiple with exact rational arithmetic, not rounded decimals.
- S047 · premise_and_result_audit — Recompute merge original slot0 Euclidean step1 quotient fraction with exact rational arithmetic, not rounded decimals.
- S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S045 · premise_and_result_audit — Recompute merge original slot0 Euclidean step0 second coefficient difference with exact rational arithmetic, not rounded decimals.
- S044 · premise_and_result_audit — Recompute merge original slot0 Euclidean step0 second coefficient multiple with exact rational arithmetic, not rounded decimals.
- S043 · premise_and_result_audit — Recompute merge original slot0 Euclidean step0 first coefficient difference with exact rational arithmetic, not rounded decimals.
- S042 · premise_and_result_audit — Recompute merge original slot0 Euclidean step0 first coefficient multiple with exact rational arithmetic, not rounded decimals.
- S041 · premise_and_result_audit — Recompute merge original slot0 Euclidean step0 remainder with exact rational arithmetic, not rounded decimals.
- S040 · premise_and_result_audit — Recompute merge original slot0 Euclidean step0 divisor multiple with exact rational arithmetic, not rounded decimals.
- S039 · premise_and_result_audit — Recompute merge original slot0 Euclidean step0 quotient fraction with exact rational arithmetic, not rounded decimals.
- S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S036 · premise_and_result_audit — Recompute CRT original pair1,2 difference divided by gcd with exact rational arithmetic, not rounded decimals.
- S035 · premise_and_result_audit — Recompute CRT original pair1,2 residue difference with exact rational arithmetic, not rounded decimals.
- S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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 CRT original pair0,2 difference divided by gcd with exact rational arithmetic, not rounded decimals.
- S031 · premise_and_result_audit — Recompute CRT original pair0,2 residue difference with exact rational arithmetic, not rounded decimals.
- S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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 CRT original pair0,1 difference divided by gcd with exact rational arithmetic, not rounded decimals.
- S027 · premise_and_result_audit — Recompute CRT original pair0,1 residue difference with exact rational arithmetic, not rounded decimals.
- S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S025 · premise_and_result_audit — Recompute CRT LCM prefix2 value with exact rational arithmetic, not rounded decimals.
- S024 · premise_and_result_audit — Recompute CRT LCM prefix2 quotient with exact rational arithmetic, not rounded decimals.
- S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S022 · premise_and_result_audit — Recompute CRT modulus product prefix2 with exact rational arithmetic, not rounded decimals.
- S021 · premise_and_result_audit — Recompute CRT LCM prefix1 value with exact rational arithmetic, not rounded decimals.
- S020 · premise_and_result_audit — Recompute CRT LCM prefix1 quotient with exact rational arithmetic, not rounded decimals.
- S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S018 · premise_and_result_audit — Recompute CRT modulus product prefix1 with exact rational arithmetic, not rounded decimals.
- S017 · premise_and_result_audit — Recompute CRT LCM prefix0 value with exact rational arithmetic, not rounded decimals.
- S016 · premise_and_result_audit — Recompute CRT LCM prefix0 quotient 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 CRT modulus product prefix0 with exact rational arithmetic, not rounded decimals.
- S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S012 · premise_and_result_audit — Recompute original residue slot2 normalized remainder with exact rational arithmetic, not rounded decimals.
- S011 · premise_and_result_audit — Recompute original residue slot2 floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S010 · premise_and_result_audit — Recompute original residue slot2 exact quotient with exact rational arithmetic, not rounded decimals.
- 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 — Recompute original residue slot1 normalized remainder with exact rational arithmetic, not rounded decimals.
- S007 · premise_and_result_audit — Recompute original residue slot1 floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Recompute original residue slot1 exact quotient with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S004 · premise_and_result_audit — Recompute original residue slot0 normalized remainder with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute original residue slot0 floor quotient multiple with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute original residue slot0 exact quotient with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

([3, 5, 43], [1, 4, 16], 274, 645, [Yes, Yes, Yes], Yes)

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: 5adb243d9e835e4e5c9bbce9eab5ab24241719b40a7dc07c88ad246b7b48c38b
- candidate record SHA-256: 4bdd841edf4859949174efbd9d03274ade01da7fc3d5f470ec8c6d115b7c42e4
- semantic review receipt SHA-256: 713db6309b22bf8b31f0253d4aaa7aca9011ab908587ae0e9d4a365edce1f9bc
- question-first witness SHA-256: 0338a3db344ab237113fc63be29153c3684da96e388109e9dcd913a0a693aed2
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261104-r05-chinese-remainder-theorem-0045.json)