Question 20082: Multiplication by a Unit Permutes the Complete Reduced Residues

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

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R0039-20082 · q-20261014-advance-20261107-r06-euler-theorem-number-theory-0032

Original question and givens

Use integer modulus n=11≥2 and original integer base a=8. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

G001 — Use integer modulus n=11≥2 and original integer base a=8. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (units, sorted_mapped_units, permutation, certificate).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

mko-euler-theorem-number-theory

Step-by-step derivation

S001 · implies

Before

source operation=eulerthm-unit-permutation;original snapshot and mathematics retained

After

explicit ordered source output fields=[units, sorted_mapped_units, permutation, certificate]

Conditions: none

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

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · calculation

Before

Euler ORIGINAL signed base normalization exact quotient := div(8, 11)

After

Euler ORIGINAL signed base normalization exact quotient = 8/11

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

Euler ORIGINAL signed base normalization floor quotient multiple := mul(0, 11)

After

Euler ORIGINAL signed base normalization floor quotient multiple = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Euler ORIGINAL signed base normalization normalized remainder := sub(8, 0)

After

Euler ORIGINAL signed base normalization normalized remainder = 8

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

Euler ORIGINAL signed base normalization:ORIGINAL signed integer=8;modulus=11;exact ratio=8/11;floor quotient=0

After

value=q*n+remainder;normalized residue=8 in[0,11);same congruence class,NOT absolute-value replacement

Conditions: none

Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.

Step identifiers and dependencies

Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S004

S006 · implies

Before

Euler ORIGINAL base applicability gcd:ORIGINAL gcd inputs=[8, 11];nonnegative initial pair=[8, 11]

After

ALL exact Euclidean rows[left,right,q,r]=[[8, 11, 0, 8], [11, 8, 1, 3], [8, 3, 2, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

Step identifiers and dependencies

Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S005

S007 · implies

Before

Euler residue representative0:ORIGINAL gcd inputs=[0, 11];nonnegative initial pair=[0, 11]

After

ALL exact Euclidean rows[left,right,q,r]=[[0, 11, 0, 0]];terminal gcd=11

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

Step identifiers and dependencies

Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S006

S008 · implies

Before

Euler residue representative1:ORIGINAL gcd inputs=[1, 11];nonnegative initial pair=[1, 11]

After

ALL exact Euclidean rows[left,right,q,r]=[[1, 11, 0, 1], [11, 1, 11, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

Step identifiers and dependencies

Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S007

S009 · implies

Before

Euler residue representative2:ORIGINAL gcd inputs=[2, 11];nonnegative initial pair=[2, 11]

After

ALL exact Euclidean rows[left,right,q,r]=[[2, 11, 0, 2], [11, 2, 5, 1], [2, 1, 2, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

Step identifiers and dependencies

Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S008

S010 · implies

Before

Euler residue representative3:ORIGINAL gcd inputs=[3, 11];nonnegative initial pair=[3, 11]

After

ALL exact Euclidean rows[left,right,q,r]=[[3, 11, 0, 3], [11, 3, 3, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

Step identifiers and dependencies

Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S009

S011 · implies

Before

Euler residue representative4:ORIGINAL gcd inputs=[4, 11];nonnegative initial pair=[4, 11]

After

ALL exact Euclidean rows[left,right,q,r]=[[4, 11, 0, 4], [11, 4, 2, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

Step identifiers and dependencies

Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S010

S012 · implies

Before

Euler residue representative5:ORIGINAL gcd inputs=[5, 11];nonnegative initial pair=[5, 11]

After

ALL exact Euclidean rows[left,right,q,r]=[[5, 11, 0, 5], [11, 5, 2, 1], [5, 1, 5, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

Step identifiers and dependencies

Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S011

S013 · implies

Before

Euler residue representative6:ORIGINAL gcd inputs=[6, 11];nonnegative initial pair=[6, 11]

After

ALL exact Euclidean rows[left,right,q,r]=[[6, 11, 0, 6], [11, 6, 1, 5], [6, 5, 1, 1], [5, 1, 5, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

Step identifiers and dependencies

Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S012

S014 · implies

Before

Euler residue representative7:ORIGINAL gcd inputs=[7, 11];nonnegative initial pair=[7, 11]

After

ALL exact Euclidean rows[left,right,q,r]=[[7, 11, 0, 7], [11, 7, 1, 4], [7, 4, 1, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

Step identifiers and dependencies

Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S013

S015 · implies

Before

Euler residue representative8:ORIGINAL gcd inputs=[8, 11];nonnegative initial pair=[8, 11]

After

ALL exact Euclidean rows[left,right,q,r]=[[8, 11, 0, 8], [11, 8, 1, 3], [8, 3, 2, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

Step identifiers and dependencies

Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S014

S016 · implies

Before

Euler residue representative9:ORIGINAL gcd inputs=[9, 11];nonnegative initial pair=[9, 11]

After

ALL exact Euclidean rows[left,right,q,r]=[[9, 11, 0, 9], [11, 9, 1, 2], [9, 2, 4, 1], [2, 1, 2, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

Step identifiers and dependencies

Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S015

S017 · implies

Before

Euler residue representative10:ORIGINAL gcd inputs=[10, 11];nonnegative initial pair=[10, 11]

After

ALL exact Euclidean rows[left,right,q,r]=[[10, 11, 0, 10], [11, 10, 1, 1], [10, 1, 10, 0]];terminal gcd=1

Conditions: none

Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.

Step identifiers and dependencies

Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S016

S018 · calculation

Before

Euler complete reduced residue count phi := sum(1, 1, 1, 1, 1, 1, 1, 1, 1, 1)

After

Euler complete reduced residue count phi = 10

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · implies

Before

COMPLETE original residue carrier0..10;ALL [representative,gcd,unit] rows=[[0, 11, No], [1, 1, Yes], [2, 1, Yes], [3, 1, Yes], [4, 1, Yes], [5, 1, Yes], [6, 1, Yes], [7, 1, Yes], [8, 1, Yes], [9, 1, Yes], [10, 1, Yes]]

After

increasing COMPLETE U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10];phi(n)=10;source count over1..n agrees because0 and n are BOTH nonunits for n>=2

Conditions: none

Test gcd1 for every representative to obtain the complete unit set,which contains1.The source totient count uses1..n while the unit list uses0..n−1.For n≥2 both swapped endpoints0 and n are nonunits,so the counts agree without assuming a prime modulus.

Step identifiers and dependencies

Rule: euler_full_units_totient_count_convention_not_prime_only · Uses: G001, S018

S020 · calculation

Before

Euler ACTUAL base-unit inverse product := mul(8, 7)

After

Euler ACTUAL base-unit inverse product = 56

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

Euler ACTUAL inverse check exact quotient := div(56, 11)

After

Euler ACTUAL inverse check exact quotient = 56/11

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

Euler ACTUAL inverse check floor quotient multiple := mul(5, 11)

After

Euler ACTUAL inverse check floor quotient multiple = 55

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

Euler ACTUAL inverse check normalized remainder := sub(56, 55)

After

Euler ACTUAL inverse check normalized remainder = 1

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · implies

Before

Euler ACTUAL inverse check:ORIGINAL signed integer=56;modulus=11;exact ratio=56/11;floor quotient=5

After

value=q*n+remainder;normalized residue=1 in[0,11);same congruence class,NOT absolute-value replacement

Conditions: none

Step identifiers and dependencies

Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S023

S025 · implies

Before

ACTUAL modulus n=11;gcd(original a=8,n)=1;normalized base=8;ACTUAL inverse=7;COMPLETE U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

After

for ANY r in U(n),a*r remains a unit;actual inverse makes multiplication injective;finite self-injection on U(n) is surjective,so a*U(n)=U(n)

Conditions: none

The reverse product of inverses inverts a product of units,so multiplication by a stays in U(n).Its actual inverse proves injectivity and a finite self-injection is surjective.The modulus may be composite;not every nonzero residue must be a unit.

Step identifiers and dependencies

Rule: euler_all_unit_residue_permutation_composite_allowed · Uses: G001, S024

S026 · calculation

Before

Euler ORIGINAL a times unit representative1 := mul(8, 1)

After

Euler ORIGINAL a times unit representative1 = 8

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

Euler unit representative1 raw image exact quotient := div(8, 11)

After

Euler unit representative1 raw image exact quotient = 8/11

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · calculation

Before

Euler unit representative1 raw image floor quotient multiple := mul(0, 11)

After

Euler unit representative1 raw image floor quotient multiple = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · calculation

Before

Euler unit representative1 raw image normalized remainder := sub(8, 0)

After

Euler unit representative1 raw image normalized remainder = 8

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · implies

Before

Euler unit representative1 raw image:ORIGINAL signed integer=8;modulus=11;exact ratio=8/11;floor quotient=0

After

value=q*n+remainder;normalized residue=8 in[0,11);same congruence class,NOT absolute-value replacement

Conditions: none

Step identifiers and dependencies

Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S029

S031 · calculation

Before

Euler ORIGINAL a times unit representative2 := mul(8, 2)

After

Euler ORIGINAL a times unit representative2 = 16

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

Before

Euler unit representative2 raw image exact quotient := div(16, 11)

After

Euler unit representative2 raw image exact quotient = 16/11

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

Euler unit representative2 raw image floor quotient multiple := mul(1, 11)

After

Euler unit representative2 raw image floor quotient multiple = 11

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

Euler unit representative2 raw image normalized remainder := sub(16, 11)

After

Euler unit representative2 raw image normalized remainder = 5

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · implies

Before

Euler unit representative2 raw image:ORIGINAL signed integer=16;modulus=11;exact ratio=16/11;floor quotient=1

After

value=q*n+remainder;normalized residue=5 in[0,11);same congruence class,NOT absolute-value replacement

Conditions: none

Step identifiers and dependencies

Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S034

S036 · calculation

Before

Euler ORIGINAL a times unit representative3 := mul(8, 3)

After

Euler ORIGINAL a times unit representative3 = 24

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

Before

Euler unit representative3 raw image exact quotient := div(24, 11)

After

Euler unit representative3 raw image exact quotient = 24/11

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · calculation

Before

Euler unit representative3 raw image floor quotient multiple := mul(2, 11)

After

Euler unit representative3 raw image floor quotient multiple = 22

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · calculation

Before

Euler unit representative3 raw image normalized remainder := sub(24, 22)

After

Euler unit representative3 raw image normalized remainder = 2

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · implies

Before

Euler unit representative3 raw image:ORIGINAL signed integer=24;modulus=11;exact ratio=24/11;floor quotient=2

After

value=q*n+remainder;normalized residue=2 in[0,11);same congruence class,NOT absolute-value replacement

Conditions: none

Step identifiers and dependencies

Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S039

S041 · calculation

Before

Euler ORIGINAL a times unit representative4 := mul(8, 4)

After

Euler ORIGINAL a times unit representative4 = 32

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · calculation

Before

Euler unit representative4 raw image exact quotient := div(32, 11)

After

Euler unit representative4 raw image exact quotient = 32/11

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

Euler unit representative4 raw image floor quotient multiple := mul(2, 11)

After

Euler unit representative4 raw image floor quotient multiple = 22

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · calculation

Before

Euler unit representative4 raw image normalized remainder := sub(32, 22)

After

Euler unit representative4 raw image normalized remainder = 10

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · implies

Before

Euler unit representative4 raw image:ORIGINAL signed integer=32;modulus=11;exact ratio=32/11;floor quotient=2

After

value=q*n+remainder;normalized residue=10 in[0,11);same congruence class,NOT absolute-value replacement

Conditions: none

Step identifiers and dependencies

Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S044

S046 · calculation

Before

Euler ORIGINAL a times unit representative5 := mul(8, 5)

After

Euler ORIGINAL a times unit representative5 = 40

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S045

S047 · calculation

Before

Euler unit representative5 raw image exact quotient := div(40, 11)

After

Euler unit representative5 raw image exact quotient = 40/11

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S046

S048 · calculation

Before

Euler unit representative5 raw image floor quotient multiple := mul(3, 11)

After

Euler unit representative5 raw image floor quotient multiple = 33

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S047

S049 · calculation

Before

Euler unit representative5 raw image normalized remainder := sub(40, 33)

After

Euler unit representative5 raw image normalized remainder = 7

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S048

S050 · implies

Before

Euler unit representative5 raw image:ORIGINAL signed integer=40;modulus=11;exact ratio=40/11;floor quotient=3

After

value=q*n+remainder;normalized residue=7 in[0,11);same congruence class,NOT absolute-value replacement

Conditions: none

Step identifiers and dependencies

Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S049

S051 · calculation

Before

Euler ORIGINAL a times unit representative6 := mul(8, 6)

After

Euler ORIGINAL a times unit representative6 = 48

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S050

S052 · calculation

Before

Euler unit representative6 raw image exact quotient := div(48, 11)

After

Euler unit representative6 raw image exact quotient = 48/11

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S051

S053 · calculation

Before

Euler unit representative6 raw image floor quotient multiple := mul(4, 11)

After

Euler unit representative6 raw image floor quotient multiple = 44

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S052

S054 · calculation

Before

Euler unit representative6 raw image normalized remainder := sub(48, 44)

After

Euler unit representative6 raw image normalized remainder = 4

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S053

S055 · implies

Before

Euler unit representative6 raw image:ORIGINAL signed integer=48;modulus=11;exact ratio=48/11;floor quotient=4

After

value=q*n+remainder;normalized residue=4 in[0,11);same congruence class,NOT absolute-value replacement

Conditions: none

Step identifiers and dependencies

Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S054

S056 · calculation

Before

Euler ORIGINAL a times unit representative7 := mul(8, 7)

After

Euler ORIGINAL a times unit representative7 = 56

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S055

S057 · calculation

Before

Euler unit representative7 raw image exact quotient := div(56, 11)

After

Euler unit representative7 raw image exact quotient = 56/11

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S056

S058 · calculation

Before

Euler unit representative7 raw image floor quotient multiple := mul(5, 11)

After

Euler unit representative7 raw image floor quotient multiple = 55

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S057

S059 · calculation

Before

Euler unit representative7 raw image normalized remainder := sub(56, 55)

After

Euler unit representative7 raw image normalized remainder = 1

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S058

S060 · implies

Before

Euler unit representative7 raw image:ORIGINAL signed integer=56;modulus=11;exact ratio=56/11;floor quotient=5

After

value=q*n+remainder;normalized residue=1 in[0,11);same congruence class,NOT absolute-value replacement

Conditions: none

Step identifiers and dependencies

Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S059

S061 · calculation

Before

Euler ORIGINAL a times unit representative8 := mul(8, 8)

After

Euler ORIGINAL a times unit representative8 = 64

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S060

S062 · calculation

Before

Euler unit representative8 raw image exact quotient := div(64, 11)

After

Euler unit representative8 raw image exact quotient = 64/11

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S061

S063 · calculation

Before

Euler unit representative8 raw image floor quotient multiple := mul(5, 11)

After

Euler unit representative8 raw image floor quotient multiple = 55

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S062

S064 · calculation

Before

Euler unit representative8 raw image normalized remainder := sub(64, 55)

After

Euler unit representative8 raw image normalized remainder = 9

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S063

S065 · implies

Before

Euler unit representative8 raw image:ORIGINAL signed integer=64;modulus=11;exact ratio=64/11;floor quotient=5

After

value=q*n+remainder;normalized residue=9 in[0,11);same congruence class,NOT absolute-value replacement

Conditions: none

Step identifiers and dependencies

Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S064

S066 · calculation

Before

Euler ORIGINAL a times unit representative9 := mul(8, 9)

After

Euler ORIGINAL a times unit representative9 = 72

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S065

S067 · calculation

Before

Euler unit representative9 raw image exact quotient := div(72, 11)

After

Euler unit representative9 raw image exact quotient = 72/11

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S066

S068 · calculation

Before

Euler unit representative9 raw image floor quotient multiple := mul(6, 11)

After

Euler unit representative9 raw image floor quotient multiple = 66

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S067

S069 · calculation

Before

Euler unit representative9 raw image normalized remainder := sub(72, 66)

After

Euler unit representative9 raw image normalized remainder = 6

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S068

S070 · implies

Before

Euler unit representative9 raw image:ORIGINAL signed integer=72;modulus=11;exact ratio=72/11;floor quotient=6

After

value=q*n+remainder;normalized residue=6 in[0,11);same congruence class,NOT absolute-value replacement

Conditions: none

Step identifiers and dependencies

Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S069

S071 · calculation

Before

Euler ORIGINAL a times unit representative10 := mul(8, 10)

After

Euler ORIGINAL a times unit representative10 = 80

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S070

S072 · calculation

Before

Euler unit representative10 raw image exact quotient := div(80, 11)

After

Euler unit representative10 raw image exact quotient = 80/11

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S071

S073 · calculation

Before

Euler unit representative10 raw image floor quotient multiple := mul(7, 11)

After

Euler unit representative10 raw image floor quotient multiple = 77

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S072

S074 · calculation

Before

Euler unit representative10 raw image normalized remainder := sub(80, 77)

After

Euler unit representative10 raw image normalized remainder = 3

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S073

S075 · implies

Before

Euler unit representative10 raw image:ORIGINAL signed integer=80;modulus=11;exact ratio=80/11;floor quotient=7

After

value=q*n+remainder;normalized residue=3 in[0,11);same congruence class,NOT absolute-value replacement

Conditions: none

Step identifiers and dependencies

Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S074

S076 · implies

Before

COMPLETE original increasing U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10];actual images in ORIGINAL unit order=[8, 5, 2, 10, 7, 4, 1, 9, 6, 3]

After

SOURCE output sorts the mapped list numerically=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10];full-list equality=Yes;not a claim each point is fixed

Conditions: none

The source's second field is the sorted complete image list,not images in original unit order.Equal lists certify a permutation,not that multiplication by a fixes every point.The raw images are listed separately here.

Step identifiers and dependencies

Rule: euler_sorted_mapped_units_not_original_image_order_or_identity · Uses: G001, S075

Reverse checks

S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S074 · premise_and_result_audit — Recompute Euler unit representative10 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
S073 · premise_and_result_audit — Recompute Euler unit representative10 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
S072 · premise_and_result_audit — Recompute Euler unit representative10 raw image exact quotient with exact rational arithmetic, not rounded decimals.
S071 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative10 with exact rational arithmetic, not rounded decimals.
S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S069 · premise_and_result_audit — Recompute Euler unit representative9 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
S068 · premise_and_result_audit — Recompute Euler unit representative9 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
S067 · premise_and_result_audit — Recompute Euler unit representative9 raw image exact quotient with exact rational arithmetic, not rounded decimals.
S066 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative9 with exact rational arithmetic, not rounded decimals.
S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S064 · premise_and_result_audit — Recompute Euler unit representative8 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
S063 · premise_and_result_audit — Recompute Euler unit representative8 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
S062 · premise_and_result_audit — Recompute Euler unit representative8 raw image exact quotient with exact rational arithmetic, not rounded decimals.
S061 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative8 with exact rational arithmetic, not rounded decimals.
S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S059 · premise_and_result_audit — Recompute Euler unit representative7 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
S058 · premise_and_result_audit — Recompute Euler unit representative7 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
S057 · premise_and_result_audit — Recompute Euler unit representative7 raw image exact quotient with exact rational arithmetic, not rounded decimals.
S056 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative7 with exact rational arithmetic, not rounded decimals.
S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S054 · premise_and_result_audit — Recompute Euler unit representative6 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
S053 · premise_and_result_audit — Recompute Euler unit representative6 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
S052 · premise_and_result_audit — Recompute Euler unit representative6 raw image exact quotient with exact rational arithmetic, not rounded decimals.
S051 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative6 with exact rational arithmetic, not rounded decimals.
S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S049 · premise_and_result_audit — Recompute Euler unit representative5 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
S048 · premise_and_result_audit — Recompute Euler unit representative5 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
S047 · premise_and_result_audit — Recompute Euler unit representative5 raw image exact quotient with exact rational arithmetic, not rounded decimals.
S046 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative5 with exact rational arithmetic, not rounded decimals.
S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S044 · premise_and_result_audit — Recompute Euler unit representative4 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
S043 · premise_and_result_audit — Recompute Euler unit representative4 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
S042 · premise_and_result_audit — Recompute Euler unit representative4 raw image exact quotient with exact rational arithmetic, not rounded decimals.
S041 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative4 with exact rational arithmetic, not rounded decimals.
S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S039 · premise_and_result_audit — Recompute Euler unit representative3 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
S038 · premise_and_result_audit — Recompute Euler unit representative3 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
S037 · premise_and_result_audit — Recompute Euler unit representative3 raw image exact quotient with exact rational arithmetic, not rounded decimals.
S036 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative3 with exact rational arithmetic, not rounded decimals.
S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S034 · premise_and_result_audit — Recompute Euler unit representative2 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
S033 · premise_and_result_audit — Recompute Euler unit representative2 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
S032 · premise_and_result_audit — Recompute Euler unit representative2 raw image exact quotient with exact rational arithmetic, not rounded decimals.
S031 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative2 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 — Recompute Euler unit representative1 raw image normalized remainder with exact rational arithmetic, not rounded decimals.
S028 · premise_and_result_audit — Recompute Euler unit representative1 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals.
S027 · premise_and_result_audit — Recompute Euler unit representative1 raw image exact quotient with exact rational arithmetic, not rounded decimals.
S026 · premise_and_result_audit — Recompute Euler ORIGINAL a times unit representative1 with exact rational arithmetic, not rounded decimals.
S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S023 · premise_and_result_audit — Recompute Euler ACTUAL inverse check normalized remainder with exact rational arithmetic, not rounded decimals.
S022 · premise_and_result_audit — Recompute Euler ACTUAL inverse check floor quotient multiple with exact rational arithmetic, not rounded decimals.
S021 · premise_and_result_audit — Recompute Euler ACTUAL inverse check exact quotient with exact rational arithmetic, not rounded decimals.
S020 · premise_and_result_audit — Recompute Euler ACTUAL base-unit inverse product 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 Euler complete reduced residue count phi with exact rational arithmetic, not rounded decimals.
S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S004 · premise_and_result_audit — Recompute Euler ORIGINAL signed base normalization normalized remainder with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Recompute Euler ORIGINAL signed base normalization floor quotient multiple with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Recompute Euler ORIGINAL signed base normalization 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

([1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], Yes, multiplication_by_a_permutes_reduced_residues)

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: c2969b35a4d7e8265064a12d50f1d1979d23b8e69b08ddca578fe1c4c3ea76b0
candidate record SHA-256: 3b62f66b295b59645bf5bda3d8f9710f89585b9a20d582f017e73ef7201bed8f
semantic review receipt SHA-256: 6e11e35fafe32f16900ce9b522d01acb7b2ee6b463ab179392076cfbe1af8034
question-first witness SHA-256: cc0fa361fc25151729e70264340d2eae94f179f6f2a2edfe80ba5704fbbd0682
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON