R0028-14754 · q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0054
Original question and givens
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0], [0, 2, 1, 3], [0, 1, 2, 3], [0, 3, 3, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[0, 3, 0, 0], right=[2, 0, 0, 0]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
G001 — Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0], [0, 2, 1, 3], [0, 1, 2, 3], [0, 3, 3, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[0, 3, 0, 0], right=[2, 0, 0, 0]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).
Step-by-step derivation
S001 · implies
Before
same carrier Fin(4);addition=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]];multiplication=[[0, 0, 0, 0], [0, 2, 1, 3], [0, 1, 2, 3], [0, 3, 3, 0]]
After
actual additive zero=0;actual multiplicative one=2;additive inverses=[0, 2, 1, 3]
Conditions: none
Bind coefficients and domain reasoning to the same complete parent tables.Identities are actual elements,not presumed labels0 or1.
Step identifiers and dependencies
Rule: commutative_parent_actual_tables_and_identities · Uses: G001
S002 · implies
Before
all additive/multiplicative pairs and triples checked;first failures=[null, null, null, null]
After
additive commute=Yes;multiplicative commute=Yes;zero differs from one=Yes
Conditions: none
Qualify the whole additive abelian group,multiplicative associativity/commutativity,both distributive laws and distinct identities.Domain and polynomial obligations follow this premise,not sampled successes.
Step identifiers and dependencies
Rule: complete_commutative_unital_parent_qualification · Uses: G001, S001
S003 · implies
Before
left polynomial:raw ascending coefficient labels=[0, 3, 0, 0];actual ring zero=0
After
normalized finite support=[0, 3];last-index degree field=1
Conditions: none
Remove only trailing actual-zero coefficients,not low-degree or internal zeros.Empty/all-zero inputs become[zero].Here degree means last index and returns0 for the zero polynomial,not the usual mathematical zero-polynomial degree convention.
Step identifiers and dependencies
Rule: polynomial_trailing_actual_zero_protocol · Uses: G001, S002
S004 · implies
Before
right polynomial:raw ascending coefficient labels=[2, 0, 0, 0];actual ring zero=0
After
normalized finite support=[2];last-index degree field=0
Conditions: none
Remove only trailing actual-zero coefficients,not low-degree or internal zeros.Empty/all-zero inputs become[zero].Here degree means last index and returns0 for the zero polynomial,not the usual mathematical zero-polynomial degree convention.
Step identifiers and dependencies
Rule: polynomial_trailing_actual_zero_protocol · Uses: G001, S003
S005 · implies
Before
convolution coefficient0 times coefficient0:operation=multiplication;left label=0;right label=2;original row=[0, 0, 0, 0]
After
result label=0
Conditions: none
Use the original operation/order lookup.Labels are not integer sums/products,and modular guesses do not define the parent ring.
Step identifiers and dependencies
Rule: same_parent_operation_not_integer_labels · Uses: G001, S004
S006 · implies
Before
accumulate product at degree0:operation=addition;left label=0;right label=0;original row=[0, 1, 2, 3]
After
result label=0
Conditions: none
Use the original operation/order lookup.Labels are not integer sums/products,and modular guesses do not define the parent ring.
Step identifiers and dependencies
Rule: same_parent_operation_not_integer_labels · Uses: G001, S005
S007 · implies
Before
convolution coefficient1 times coefficient0:operation=multiplication;left label=3;right label=2;original row=[0, 3, 3, 0]
After
result label=3
Conditions: none
Use the original operation/order lookup.Labels are not integer sums/products,and modular guesses do not define the parent ring.
Step identifiers and dependencies
Rule: same_parent_operation_not_integer_labels · Uses: G001, S006
S008 · implies
Before
accumulate product at degree1:operation=addition;left label=0;right label=3;original row=[0, 1, 2, 3]
After
result label=3
Conditions: none
Use the original operation/order lookup.Labels are not integer sums/products,and modular guesses do not define the parent ring.
Step identifiers and dependencies
Rule: same_parent_operation_not_integer_labels · Uses: G001, S007
S009 · implies
Before
product result:raw ascending coefficient labels=[0, 3];actual ring zero=0
After
normalized finite support=[0, 3];last-index degree field=1
Conditions: none
Remove only trailing actual-zero coefficients,not low-degree or internal zeros.Empty/all-zero inputs become[zero].Here degree means last index and returns0 for the zero polynomial,not the usual mathematical zero-polynomial degree convention.
Step identifiers and dependencies
Rule: polynomial_trailing_actual_zero_protocol · Uses: G001, S008
S010 · implies
Before
normalized product=[0, 3];last-index degree field=1
After
degree_additivity_not_claimed_with_zero_divisors
Conditions: none
Convolve finite supports using parent-ring operations,then trim trailing zeros.Leading coefficients may annihilate through zero divisors,and two nonzero polynomials may multiply to zero.No unconditional degree-additivity or field property is claimed.
Step identifiers and dependencies
Rule: polynomial_convolution_zero_divisor_boundary · Uses: G001, S009