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R0030-15988 · q-20261014-advance-20261031-r04-finite-quotient-ring-0088
Use the parent ring Z_16 with residues 0 through 15 and the proper two-sided ideal I of multiples of d=4, 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=4, b_prime=5 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_16 with residues 0 through 15 and the proper two-sided ideal I of multiples of d=4, 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=4, b_prime=5 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?).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.
Before
quotient ring parent modulus divided by ideal generator := div(16, 4)
After
quotient ring parent modulus divided by ideal generator = 4
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.
Rule: exact_rational_arithmetic · Uses: G001
Before
same declared parent Z_16;generator d=4;n/d=4;ideal multiples=[0, 4, 8, 12]
After
I=kernel of residue map Z_16 to Z_4;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.
Rule: quotient_ring_actual_parent_proper_two_sided_ideal · Uses: G001, S001
Before
complete parent residues=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15];same ideal I=[0, 4, 8, 12]
After
all additive cosets in representative order0..d-1=[[0, 4, 8, 12], [1, 5, 9, 13], [2, 6, 10, 14], [3, 7, 11, 15]];quotient size=4;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.
Rule: quotient_ring_full_cosets_not_representative_carrier · Uses: G001, S002
Before
representative slot0 input class exact quotient fraction := div(0, 4)
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.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
representative slot0 input class:integer value=0;positive modulus=4;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.
Rule: modular_Euclidean_floor_not_rational_division_result · Uses: G001, S004
Before
representative slot0 input class quotient multiple := mul(0, 4)
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.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
representative slot1 input class exact quotient fraction := div(1, 4)
After
representative slot1 input class exact quotient fraction = 1/4
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
representative slot1 input class:integer value=1;positive modulus=4;exact ratio=1/4
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.
Rule: modular_Euclidean_floor_not_rational_division_result · Uses: G001, S008
Before
representative slot1 input class quotient multiple := mul(0, 4)
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.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
representative slot2 input class exact quotient fraction := div(4, 4)
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.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
representative slot2 input class:integer value=4;positive modulus=4;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.
Rule: modular_Euclidean_floor_not_rational_division_result · Uses: G001, S012
Before
representative slot2 input class quotient multiple := mul(1, 4)
After
representative slot2 input class quotient multiple = 4
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
representative slot2 input class canonical nonnegative residue := sub(4, 4)
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.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
representative slot3 input class exact quotient fraction := div(5, 4)
After
representative slot3 input class exact quotient fraction = 5/4
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
representative slot3 input class:integer value=5;positive modulus=4;exact ratio=5/4
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.
Rule: modular_Euclidean_floor_not_rational_division_result · Uses: G001, S016
Before
representative slot3 input class quotient multiple := mul(1, 4)
After
representative slot3 input class quotient multiple = 4
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
representative slot3 input class canonical nonnegative residue := sub(5, 4)
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.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
first representative change := sub(0, 4)
After
first representative change = -4
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
second representative change := sub(1, 5)
After
second representative change = -4
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
first representative change divided by d := div(-4, 4)
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.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
second representative change divided by d := div(-4, 4)
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.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
original representative choices=[0, 1, 4, 5];class labels in slot order=[0, 1, 0, 1];changes a-a_prime=-4;b-b_prime=-4
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.
Rule: quotient_well_defined_input_cosets_are_premise · Uses: G001, S023
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
representative choice0 addition parent mod n exact quotient fraction := div(1, 16)
After
representative choice0 addition parent mod n exact quotient fraction = 1/16
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
representative choice0 addition parent mod n:integer value=1;positive modulus=16;exact ratio=1/16
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.
Rule: modular_Euclidean_floor_not_rational_division_result · Uses: G001, S026
Before
representative choice0 addition parent mod n quotient multiple := mul(0, 16)
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.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
representative choice0 addition quotient class mod d exact quotient fraction := div(1, 4)
After
representative choice0 addition quotient class mod d exact quotient fraction = 1/4
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
representative choice0 addition quotient class mod d:integer value=1;positive modulus=4;exact ratio=1/4
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.
Rule: modular_Euclidean_floor_not_rational_division_result · Uses: G001, S030
Before
representative choice0 addition quotient class mod d quotient multiple := mul(0, 4)
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.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S032
Before
representative choice0 addition:original parent representatives=[0, 1];integer result=1;parent modulus=16
After
parent result=1;quotient class label=1;class modulus=4
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.
Rule: quotient_operation_parent_result_separate_from_class · Uses: G001, S033
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
representative choice0 multiplication parent mod n exact quotient fraction := div(0, 16)
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.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
representative choice0 multiplication parent mod n:integer value=0;positive modulus=16;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.
Rule: modular_Euclidean_floor_not_rational_division_result · Uses: G001, S036
Before
representative choice0 multiplication parent mod n quotient multiple := mul(0, 16)
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.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S038
Before
representative choice0 multiplication quotient class mod d exact quotient fraction := div(0, 4)
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.
Rule: exact_rational_arithmetic · Uses: G001, S039
Before
representative choice0 multiplication quotient class mod d:integer value=0;positive modulus=4;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.
Rule: modular_Euclidean_floor_not_rational_division_result · Uses: G001, S040
Before
representative choice0 multiplication quotient class mod d quotient multiple := mul(0, 4)
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.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S042
Before
representative choice0 multiplication:original parent representatives=[0, 1];integer result=0;parent modulus=16
After
parent result=0;quotient class label=0;class modulus=4
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.
Rule: quotient_operation_parent_result_separate_from_class · Uses: G001, S043
Before
representative choice1 addition integer addition := sum(4, 5)
After
representative choice1 addition integer addition = 9
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.
Rule: exact_rational_arithmetic · Uses: G001, S044
Before
representative choice1 addition parent mod n exact quotient fraction := div(9, 16)
After
representative choice1 addition parent mod n exact quotient fraction = 9/16
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.
Rule: exact_rational_arithmetic · Uses: G001, S045
Before
representative choice1 addition parent mod n:integer value=9;positive modulus=16;exact ratio=9/16
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.
Rule: modular_Euclidean_floor_not_rational_division_result · Uses: G001, S046
Before
representative choice1 addition parent mod n quotient multiple := mul(0, 16)
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.
Rule: exact_rational_arithmetic · Uses: G001, S047
Before
representative choice1 addition parent mod n canonical nonnegative residue := sub(9, 0)
After
representative choice1 addition parent mod n canonical nonnegative residue = 9
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.
Rule: exact_rational_arithmetic · Uses: G001, S048
Before
representative choice1 addition quotient class mod d exact quotient fraction := div(9, 4)
After
representative choice1 addition quotient class mod d exact quotient fraction = 9/4
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.
Rule: exact_rational_arithmetic · Uses: G001, S049
Before
representative choice1 addition quotient class mod d:integer value=9;positive modulus=4;exact ratio=9/4
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.
Rule: modular_Euclidean_floor_not_rational_division_result · Uses: G001, S050
Before
representative choice1 addition quotient class mod d quotient multiple := mul(2, 4)
After
representative choice1 addition quotient class mod d quotient multiple = 8
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.
Rule: exact_rational_arithmetic · Uses: G001, S051
Before
representative choice1 addition quotient class mod d canonical nonnegative residue := sub(9, 8)
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.
Rule: exact_rational_arithmetic · Uses: G001, S052
Before
representative choice1 addition:original parent representatives=[4, 5];integer result=9;parent modulus=16
After
parent result=9;quotient class label=1;class modulus=4
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.
Rule: quotient_operation_parent_result_separate_from_class · Uses: G001, S053
Before
representative choice1 multiplication integer multiplication := mul(4, 5)
After
representative choice1 multiplication integer multiplication = 20
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.
Rule: exact_rational_arithmetic · Uses: G001, S054
Before
representative choice1 multiplication parent mod n exact quotient fraction := div(20, 16)
After
representative choice1 multiplication parent mod n exact quotient fraction = 5/4
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.
Rule: exact_rational_arithmetic · Uses: G001, S055
Before
representative choice1 multiplication parent mod n:integer value=20;positive modulus=16;exact ratio=5/4
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.
Rule: modular_Euclidean_floor_not_rational_division_result · Uses: G001, S056
Before
representative choice1 multiplication parent mod n quotient multiple := mul(1, 16)
After
representative choice1 multiplication parent mod n quotient multiple = 16
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.
Rule: exact_rational_arithmetic · Uses: G001, S057
Before
representative choice1 multiplication parent mod n canonical nonnegative residue := sub(20, 16)
After
representative choice1 multiplication parent mod n canonical nonnegative residue = 4
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.
Rule: exact_rational_arithmetic · Uses: G001, S058
Before
representative choice1 multiplication quotient class mod d exact quotient fraction := div(4, 4)
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.
Rule: exact_rational_arithmetic · Uses: G001, S059
Before
representative choice1 multiplication quotient class mod d:integer value=4;positive modulus=4;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.
Rule: modular_Euclidean_floor_not_rational_division_result · Uses: G001, S060
Before
representative choice1 multiplication quotient class mod d quotient multiple := mul(1, 4)
After
representative choice1 multiplication quotient class mod d quotient multiple = 4
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,signs,denominators and the operation's carrier.
Rule: exact_rational_arithmetic · Uses: G001, S061
Before
representative choice1 multiplication quotient class mod d canonical nonnegative residue := sub(4, 4)
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.
Rule: exact_rational_arithmetic · Uses: G001, S062
Before
representative choice1 multiplication:original parent representatives=[4, 5];integer result=20;parent modulus=16
After
parent result=4;quotient class label=0;class modulus=4
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.
Rule: quotient_operation_parent_result_separate_from_class · Uses: G001, S063
Before
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.
Rule: quotient_ring_universal_representative_independence_not_two_samples · Uses: G001, S064
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.([0, 1, 4, 5], [1, 1], [0, 0], Yes, Yes, Yes)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: c4a19d22fcd3933de30db7674910c32b202740cda873e3d3262704209fb2639acandidate record SHA-256: 9ecca87d5914f62173d3ec1c47a71ede110ce7b06cd7a7c63ebc1b0039b21ff0semantic review receipt SHA-256: 73bc08066ffb1a81eacb2d4e288d03f522aeefd3949fa0d97da41b484ed737bcquestion-first witness SHA-256: ac120452a208b257b8270f38351a09dcd832cdbc8d9db6b69933f83e43cc9336Lean: NOT_RUN · human review: NOT_PERFORMED