# Question 16000: Changing Both Representatives Must Preserve Both Operations

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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.

R0030-16000 · q-20261014-advance-20261031-r04-finite-quotient-ring-0100



## Original question and givens

Use the parent ring Z_36 with residues 0 through 35 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=0, b=1, a_prime=6, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).


- G001 — Use the parent ring Z_36 with residues 0 through 35 and the proper two-sided ideal I of multiples of d=6, where 2≤d≤n and d divides n. Its cosets are residue classes modulo d, ordered by representatives 0 through d−1 with increasing members. The d=n zero ideal is allowed; d=1 gives I=R and is excluded by the project's 1≠0 convention. Retain parent arithmetic modulo n separately from the quotient class label modulo d. No field conclusion follows. Representatives a=0, b=1, a_prime=6, b_prime=7 must denote the same respective input cosets. Give ([a,b,a_prime,b_prime],addition_classes,multiplication_classes,addition_well_defined?,multiplication_well_defined?,quotient_operations_well_defined?).





## 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-finite-quotient-ring](https://ocme.evemisslab.com/en/mko/mko-finite-quotient-ring/)





## Step-by-step derivation

S001 · calculationBefore

quotient ring parent modulus divided by ideal generator := div(36, 6)

After

quotient ring parent modulus divided by ideal generator = 6

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001





S002 · impliesBefore

same declared parent Z_36;generator d=6;n/d=6;ideal multiples=[0, 6, 12, 18, 24, 30]

After

I=kernel of residue map Z_36 to Z_6;proper two-sided ideal=Yes;zero ideal allowed when d=n

Conditions: none

Since d divides n,changing an integer representative by n does not change mod d.The map π:Z_n→Z_d is well-defined and preserves addition and multiplication.Its kernel consists of multiples of d,closed under differences and multiplication on either side by parent elements.With d≥2 it excludes1 and is proper;d=n gives the allowed zero ideal.

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





S003 · impliesBefore

complete parent residues=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35];same ideal I=[0, 6, 12, 18, 24, 30]

After

all additive cosets in representative order0..d-1=[[0, 6, 12, 18, 24, 30], [1, 7, 13, 19, 25, 31], [2, 8, 14, 20, 26, 32], [3, 9, 15, 21, 27, 33], [4, 10, 16, 22, 28, 34], [5, 11, 17, 23, 29, 35]];quotient size=6;no field conclusion

Conditions: none

List every parent element congruent mod d in each additive coset,ordered by representatives0 through d−1.They are disjoint and cover Z_n.Quotient elements are whole cosets;class labels are indices only,and a finite quotient ring is not automatically a field.

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





S004 · calculationBefore

representative slot0 input class exact quotient fraction := div(0, 6)

After

representative slot0 input class exact quotient fraction = 0

Conditions: none

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

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





S005 · impliesBefore

representative slot0 input class:integer value=0;positive modulus=6;exact ratio=0

After

floor quotient=0;quotient<=ratio<quotient+1

Conditions: none

Take the exact rational quotient and its floor q,not the rational division result as a residue.With positive modulus,q≤value/m<q+1,so subtraction of qm yields a residue from0 through m−1.

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





S006 · calculationBefore

representative slot0 input class quotient multiple := mul(0, 6)

After

representative slot0 input class quotient multiple = 0

Conditions: none

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

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





S007 · calculationBefore

representative slot0 input class canonical nonnegative residue := sub(0, 0)

After

representative slot0 input class canonical nonnegative residue = 0

Conditions: none

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

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





S008 · calculationBefore

representative slot1 input class exact quotient fraction := div(1, 6)

After

representative slot1 input class exact quotient fraction = 1/6

Conditions: none

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

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





S009 · impliesBefore

representative slot1 input class:integer value=1;positive modulus=6;exact ratio=1/6

After

floor quotient=0;quotient<=ratio<quotient+1

Conditions: none

Take the exact rational quotient and its floor q,not the rational division result as a residue.With positive modulus,q≤value/m<q+1,so subtraction of qm yields a residue from0 through m−1.

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





S010 · calculationBefore

representative slot1 input class quotient multiple := mul(0, 6)

After

representative slot1 input class quotient multiple = 0

Conditions: none

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

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





S011 · calculationBefore

representative slot1 input class canonical nonnegative residue := sub(1, 0)

After

representative slot1 input class canonical nonnegative residue = 1

Conditions: none

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

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





S012 · calculationBefore

representative slot2 input class exact quotient fraction := div(6, 6)

After

representative slot2 input class exact quotient fraction = 1

Conditions: none

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

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





S013 · impliesBefore

representative slot2 input class:integer value=6;positive modulus=6;exact ratio=1

After

floor quotient=1;quotient<=ratio<quotient+1

Conditions: none

Take the exact rational quotient and its floor q,not the rational division result as a residue.With positive modulus,q≤value/m<q+1,so subtraction of qm yields a residue from0 through m−1.

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





S014 · calculationBefore

representative slot2 input class quotient multiple := mul(1, 6)

After

representative slot2 input class quotient multiple = 6

Conditions: none

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

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





S015 · calculationBefore

representative slot2 input class canonical nonnegative residue := sub(6, 6)

After

representative slot2 input class canonical nonnegative residue = 0

Conditions: none

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

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





S016 · calculationBefore

representative slot3 input class exact quotient fraction := div(7, 6)

After

representative slot3 input class exact quotient fraction = 7/6

Conditions: none

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

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





S017 · impliesBefore

representative slot3 input class:integer value=7;positive modulus=6;exact ratio=7/6

After

floor quotient=1;quotient<=ratio<quotient+1

Conditions: none

Take the exact rational quotient and its floor q,not the rational division result as a residue.With positive modulus,q≤value/m<q+1,so subtraction of qm yields a residue from0 through m−1.

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





S018 · calculationBefore

representative slot3 input class quotient multiple := mul(1, 6)

After

representative slot3 input class quotient multiple = 6

Conditions: none

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

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





S019 · calculationBefore

representative slot3 input class canonical nonnegative residue := sub(7, 6)

After

representative slot3 input class canonical nonnegative residue = 1

Conditions: none

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

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





S020 · calculationBefore

first representative change := sub(0, 6)

After

first representative change = -6

Conditions: none

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

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





S021 · calculationBefore

second representative change := sub(1, 7)

After

second representative change = -6

Conditions: none

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

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





S022 · calculationBefore

first representative change divided by d := div(-6, 6)

After

first representative change divided by d = -1

Conditions: none

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

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





S023 · calculationBefore

second representative change divided by d := div(-6, 6)

After

second representative change divided by d = -1

Conditions: none

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

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





S024 · impliesBefore

original representative choices=[0, 1, 6, 7];class labels in slot order=[0, 1, 0, 1];changes a-a_prime=-6;b-b_prime=-6

After

integer ideal multiples=[-1, -1];corresponding INPUT cosets agree

Conditions: none

First require the corresponding representatives to denote the same two input cosets.A mismatch is a failed premise,not a counterexample to well-definedness.Integer differences divided by d explicitly witness membership in the same ideal.

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





S025 · calculationBefore

representative choice0 addition integer addition := sum(0, 1)

After

representative choice0 addition integer addition = 1

Conditions: none

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

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





S026 · calculationBefore

representative choice0 addition parent mod n exact quotient fraction := div(1, 36)

After

representative choice0 addition parent mod n exact quotient fraction = 1/36

Conditions: none

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

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





S027 · impliesBefore

representative choice0 addition parent mod n:integer value=1;positive modulus=36;exact ratio=1/36

After

floor quotient=0;quotient<=ratio<quotient+1

Conditions: none

Take the exact rational quotient and its floor q,not the rational division result as a residue.With positive modulus,q≤value/m<q+1,so subtraction of qm yields a residue from0 through m−1.

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





S028 · calculationBefore

representative choice0 addition parent mod n quotient multiple := mul(0, 36)

After

representative choice0 addition parent mod n quotient multiple = 0

Conditions: none

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

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





S029 · calculationBefore

representative choice0 addition parent mod n canonical nonnegative residue := sub(1, 0)

After

representative choice0 addition parent mod n canonical nonnegative residue = 1

Conditions: none

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

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





S030 · calculationBefore

representative choice0 addition quotient class mod d exact quotient fraction := div(1, 6)

After

representative choice0 addition quotient class mod d exact quotient fraction = 1/6

Conditions: none

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

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





S031 · impliesBefore

representative choice0 addition quotient class mod d:integer value=1;positive modulus=6;exact ratio=1/6

After

floor quotient=0;quotient<=ratio<quotient+1

Conditions: none

Take the exact rational quotient and its floor q,not the rational division result as a residue.With positive modulus,q≤value/m<q+1,so subtraction of qm yields a residue from0 through m−1.

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





S032 · calculationBefore

representative choice0 addition quotient class mod d quotient multiple := mul(0, 6)

After

representative choice0 addition quotient class mod d quotient multiple = 0

Conditions: none

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

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





S033 · calculationBefore

representative choice0 addition quotient class mod d canonical nonnegative residue := sub(1, 0)

After

representative choice0 addition quotient class mod d canonical nonnegative residue = 1

Conditions: none

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

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





S034 · impliesBefore

representative choice0 addition:original parent representatives=[0, 1];integer result=1;parent modulus=36

After

parent result=1;quotient class label=1;class modulus=6

Conditions: none

Operate in the specified parent modulo n,then take the quotient class modulo d.These fields can differ;do not conflate their residues or confuse a class label with its full coset.

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





S035 · calculationBefore

representative choice0 multiplication integer multiplication := mul(0, 1)

After

representative choice0 multiplication integer multiplication = 0

Conditions: none

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

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





S036 · calculationBefore

representative choice0 multiplication parent mod n exact quotient fraction := div(0, 36)

After

representative choice0 multiplication parent mod n exact quotient fraction = 0

Conditions: none

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

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





S037 · impliesBefore

representative choice0 multiplication parent mod n:integer value=0;positive modulus=36;exact ratio=0

After

floor quotient=0;quotient<=ratio<quotient+1

Conditions: none

Take the exact rational quotient and its floor q,not the rational division result as a residue.With positive modulus,q≤value/m<q+1,so subtraction of qm yields a residue from0 through m−1.

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





S038 · calculationBefore

representative choice0 multiplication parent mod n quotient multiple := mul(0, 36)

After

representative choice0 multiplication parent mod n quotient multiple = 0

Conditions: none

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

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





S039 · calculationBefore

representative choice0 multiplication parent mod n canonical nonnegative residue := sub(0, 0)

After

representative choice0 multiplication parent mod n canonical nonnegative residue = 0

Conditions: none

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

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





S040 · calculationBefore

representative choice0 multiplication quotient class mod d exact quotient fraction := div(0, 6)

After

representative choice0 multiplication quotient class mod d exact quotient fraction = 0

Conditions: none

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

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





S041 · impliesBefore

representative choice0 multiplication quotient class mod d:integer value=0;positive modulus=6;exact ratio=0

After

floor quotient=0;quotient<=ratio<quotient+1

Conditions: none

Take the exact rational quotient and its floor q,not the rational division result as a residue.With positive modulus,q≤value/m<q+1,so subtraction of qm yields a residue from0 through m−1.

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





S042 · calculationBefore

representative choice0 multiplication quotient class mod d quotient multiple := mul(0, 6)

After

representative choice0 multiplication quotient class mod d quotient multiple = 0

Conditions: none

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

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





S043 · calculationBefore

representative choice0 multiplication quotient class mod d canonical nonnegative residue := sub(0, 0)

After

representative choice0 multiplication quotient class mod d canonical nonnegative residue = 0

Conditions: none

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

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





S044 · impliesBefore

representative choice0 multiplication:original parent representatives=[0, 1];integer result=0;parent modulus=36

After

parent result=0;quotient class label=0;class modulus=6

Conditions: none

Operate in the specified parent modulo n,then take the quotient class modulo d.These fields can differ;do not conflate their residues or confuse a class label with its full coset.

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





S045 · calculationBefore

representative choice1 addition integer addition := sum(6, 7)

After

representative choice1 addition integer addition = 13

Conditions: none

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

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





S046 · calculationBefore

representative choice1 addition parent mod n exact quotient fraction := div(13, 36)

After

representative choice1 addition parent mod n exact quotient fraction = 13/36

Conditions: none

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

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





S047 · impliesBefore

representative choice1 addition parent mod n:integer value=13;positive modulus=36;exact ratio=13/36

After

floor quotient=0;quotient<=ratio<quotient+1

Conditions: none

Take the exact rational quotient and its floor q,not the rational division result as a residue.With positive modulus,q≤value/m<q+1,so subtraction of qm yields a residue from0 through m−1.

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





S048 · calculationBefore

representative choice1 addition parent mod n quotient multiple := mul(0, 36)

After

representative choice1 addition parent mod n quotient multiple = 0

Conditions: none

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

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





S049 · calculationBefore

representative choice1 addition parent mod n canonical nonnegative residue := sub(13, 0)

After

representative choice1 addition parent mod n canonical nonnegative residue = 13

Conditions: none

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

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





S050 · calculationBefore

representative choice1 addition quotient class mod d exact quotient fraction := div(13, 6)

After

representative choice1 addition quotient class mod d exact quotient fraction = 13/6

Conditions: none

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

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





S051 · impliesBefore

representative choice1 addition quotient class mod d:integer value=13;positive modulus=6;exact ratio=13/6

After

floor quotient=2;quotient<=ratio<quotient+1

Conditions: none

Take the exact rational quotient and its floor q,not the rational division result as a residue.With positive modulus,q≤value/m<q+1,so subtraction of qm yields a residue from0 through m−1.

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





S052 · calculationBefore

representative choice1 addition quotient class mod d quotient multiple := mul(2, 6)

After

representative choice1 addition quotient class mod d quotient multiple = 12

Conditions: none

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

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





S053 · calculationBefore

representative choice1 addition quotient class mod d canonical nonnegative residue := sub(13, 12)

After

representative choice1 addition quotient class mod d canonical nonnegative residue = 1

Conditions: none

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

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





S054 · impliesBefore

representative choice1 addition:original parent representatives=[6, 7];integer result=13;parent modulus=36

After

parent result=13;quotient class label=1;class modulus=6

Conditions: none

Operate in the specified parent modulo n,then take the quotient class modulo d.These fields can differ;do not conflate their residues or confuse a class label with its full coset.

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





S055 · calculationBefore

representative choice1 multiplication integer multiplication := mul(6, 7)

After

representative choice1 multiplication integer multiplication = 42

Conditions: none

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

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





S056 · calculationBefore

representative choice1 multiplication parent mod n exact quotient fraction := div(42, 36)

After

representative choice1 multiplication parent mod n exact quotient fraction = 7/6

Conditions: none

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

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





S057 · impliesBefore

representative choice1 multiplication parent mod n:integer value=42;positive modulus=36;exact ratio=7/6

After

floor quotient=1;quotient<=ratio<quotient+1

Conditions: none

Take the exact rational quotient and its floor q,not the rational division result as a residue.With positive modulus,q≤value/m<q+1,so subtraction of qm yields a residue from0 through m−1.

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





S058 · calculationBefore

representative choice1 multiplication parent mod n quotient multiple := mul(1, 36)

After

representative choice1 multiplication parent mod n quotient multiple = 36

Conditions: none

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

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





S059 · calculationBefore

representative choice1 multiplication parent mod n canonical nonnegative residue := sub(42, 36)

After

representative choice1 multiplication parent mod n canonical nonnegative residue = 6

Conditions: none

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

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





S060 · calculationBefore

representative choice1 multiplication quotient class mod d exact quotient fraction := div(6, 6)

After

representative choice1 multiplication quotient class mod d exact quotient fraction = 1

Conditions: none

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

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





S061 · impliesBefore

representative choice1 multiplication quotient class mod d:integer value=6;positive modulus=6;exact ratio=1

After

floor quotient=1;quotient<=ratio<quotient+1

Conditions: none

Take the exact rational quotient and its floor q,not the rational division result as a residue.With positive modulus,q≤value/m<q+1,so subtraction of qm yields a residue from0 through m−1.

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





S062 · calculationBefore

representative choice1 multiplication quotient class mod d quotient multiple := mul(1, 6)

After

representative choice1 multiplication quotient class mod d quotient multiple = 6

Conditions: none

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

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





S063 · calculationBefore

representative choice1 multiplication quotient class mod d canonical nonnegative residue := sub(6, 6)

After

representative choice1 multiplication quotient class mod d canonical nonnegative residue = 0

Conditions: none

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

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





S064 · impliesBefore

representative choice1 multiplication:original parent representatives=[6, 7];integer result=42;parent modulus=36

After

parent result=6;quotient class label=0;class modulus=6

Conditions: none

Operate in the specified parent modulo n,then take the quotient class modulo d.These fields can differ;do not conflate their residues or confuse a class label with its full coset.

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





S065 · impliesBefore

ANY changes a-a_prime,b-b_prime in same two-sided ideal I;actual addition classes=[1, 1];actual multiplication classes=[0, 0]

After

(a+b)-(a_prime+b_prime)=(a-a_prime)+(b-b_prime) in I;ab-a_prime*b_prime=a*(b-b_prime)+(a-a_prime)*b_prime in I;both universal operation obligations=Yes

Conditions: none

Closure of I handles the additive difference;left/right absorption and addition handle the multiplicative difference.Thus any legal representative changes preserve both cosets.The two recomputations instantiate the result;universal well-definedness comes from this ideal argument,not merely two matching samples.

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









## Reverse checks


- 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S063 · premise_and_result_audit — Recompute representative choice1 multiplication quotient class mod d canonical nonnegative residue with exact rational arithmetic, not rounded decimals.
- S062 · premise_and_result_audit — Recompute representative choice1 multiplication quotient class mod d quotient multiple with exact rational arithmetic, not rounded decimals.
- S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S060 · premise_and_result_audit — Recompute representative choice1 multiplication quotient class mod d exact quotient fraction with exact rational arithmetic, not rounded decimals.
- S059 · premise_and_result_audit — Recompute representative choice1 multiplication parent mod n canonical nonnegative residue with exact rational arithmetic, not rounded decimals.
- S058 · premise_and_result_audit — Recompute representative choice1 multiplication parent mod n quotient multiple with exact rational arithmetic, not rounded decimals.
- S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S056 · premise_and_result_audit — Recompute representative choice1 multiplication parent mod n exact quotient fraction with exact rational arithmetic, not rounded decimals.
- S055 · premise_and_result_audit — Recompute representative choice1 multiplication integer multiplication 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 representative choice1 addition quotient class mod d canonical nonnegative residue with exact rational arithmetic, not rounded decimals.
- S052 · premise_and_result_audit — Recompute representative choice1 addition quotient class mod d quotient multiple with exact rational arithmetic, not rounded decimals.
- S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S050 · premise_and_result_audit — Recompute representative choice1 addition quotient class mod d exact quotient fraction with exact rational arithmetic, not rounded decimals.
- S049 · premise_and_result_audit — Recompute representative choice1 addition parent mod n canonical nonnegative residue with exact rational arithmetic, not rounded decimals.
- S048 · premise_and_result_audit — Recompute representative choice1 addition parent mod n quotient multiple with exact rational arithmetic, not rounded decimals.
- S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S046 · premise_and_result_audit — Recompute representative choice1 addition parent mod n exact quotient fraction with exact rational arithmetic, not rounded decimals.
- S045 · premise_and_result_audit — Recompute representative choice1 addition integer addition with exact rational arithmetic, not rounded decimals.
- S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S043 · premise_and_result_audit — Recompute representative choice0 multiplication quotient class mod d canonical nonnegative residue with exact rational arithmetic, not rounded decimals.
- S042 · premise_and_result_audit — Recompute representative choice0 multiplication quotient class mod d quotient multiple with exact rational arithmetic, not rounded decimals.
- S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S040 · premise_and_result_audit — Recompute representative choice0 multiplication quotient class mod d exact quotient fraction with exact rational arithmetic, not rounded decimals.
- S039 · premise_and_result_audit — Recompute representative choice0 multiplication parent mod n canonical nonnegative residue with exact rational arithmetic, not rounded decimals.
- S038 · premise_and_result_audit — Recompute representative choice0 multiplication parent mod n quotient multiple with exact rational arithmetic, not rounded decimals.
- 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 representative choice0 multiplication parent mod n exact quotient fraction with exact rational arithmetic, not rounded decimals.
- S035 · premise_and_result_audit — Recompute representative choice0 multiplication integer multiplication 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 — Recompute representative choice0 addition quotient class mod d canonical nonnegative residue with exact rational arithmetic, not rounded decimals.
- S032 · premise_and_result_audit — Recompute representative choice0 addition quotient class mod d quotient multiple with exact rational arithmetic, not rounded decimals.
- S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S030 · premise_and_result_audit — Recompute representative choice0 addition quotient class mod d exact quotient fraction with exact rational arithmetic, not rounded decimals.
- S029 · premise_and_result_audit — Recompute representative choice0 addition parent mod n canonical nonnegative residue with exact rational arithmetic, not rounded decimals.
- S028 · premise_and_result_audit — Recompute representative choice0 addition parent mod n quotient multiple with exact rational arithmetic, not rounded decimals.
- S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S026 · premise_and_result_audit — Recompute representative choice0 addition parent mod n exact quotient fraction with exact rational arithmetic, not rounded decimals.
- S025 · premise_and_result_audit — Recompute representative choice0 addition integer addition with exact rational arithmetic, not rounded decimals.
- S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S023 · premise_and_result_audit — Recompute second representative change divided by d with exact rational arithmetic, not rounded decimals.
- S022 · premise_and_result_audit — Recompute first representative change divided by d with exact rational arithmetic, not rounded decimals.
- S021 · premise_and_result_audit — Recompute second representative change with exact rational arithmetic, not rounded decimals.
- S020 · premise_and_result_audit — Recompute first representative change with exact rational arithmetic, not rounded decimals.
- S019 · premise_and_result_audit — Recompute representative slot3 input class canonical nonnegative residue with exact rational arithmetic, not rounded decimals.
- S018 · premise_and_result_audit — Recompute representative slot3 input class quotient multiple with exact rational arithmetic, not rounded decimals.
- S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S016 · premise_and_result_audit — Recompute representative slot3 input class exact quotient fraction with exact rational arithmetic, not rounded decimals.
- S015 · premise_and_result_audit — Recompute representative slot2 input class canonical nonnegative residue with exact rational arithmetic, not rounded decimals.
- S014 · premise_and_result_audit — Recompute representative slot2 input class quotient multiple 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 representative slot2 input class exact quotient fraction with exact rational arithmetic, not rounded decimals.
- S011 · premise_and_result_audit — Recompute representative slot1 input class canonical nonnegative residue with exact rational arithmetic, not rounded decimals.
- S010 · premise_and_result_audit — Recompute representative slot1 input class quotient multiple 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 representative slot1 input class exact quotient fraction with exact rational arithmetic, not rounded decimals.
- S007 · premise_and_result_audit — Recompute representative slot0 input class canonical nonnegative residue with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Recompute representative slot0 input class quotient multiple 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 representative slot0 input class exact quotient fraction with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Recompute quotient ring parent modulus divided by ideal generator with exact rational arithmetic, not rounded decimals.





## Result

([0, 1, 6, 7], [1, 1], [0, 0], 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: e5b6ef90652d18ee25768a6ecb3503ecf65b32c83440b41465ca88027d826703
- candidate record SHA-256: 6660de519942e47e3a51373d42dbcfe68e2732fb8a2d4484d03a0cc5f01494aa
- semantic review receipt SHA-256: 73bc08066ffb1a81eacb2d4e288d03f522aeefd3949fa0d97da41b484ed737bc
- question-first witness SHA-256: ac120452a208b257b8270f38351a09dcd832cdbc8d9db6b69933f83e43cc9336
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261031-r04-finite-quotient-ring-0100.json)