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P250-0076 · q-20261014-advance-20261109-r05-binomial-theorem-0026
Use natural exponent n=0, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).
G001 — Use natural exponent n=0, including0, in a commutative coefficient setting. Give (n, coefficient_exponent_rows, boundary).n is a nonnegative integer and x,y commute. For n=0 use the polynomial empty-product identity.
Before
n=0≥0; x,y commute
After
(x+y)^0 is a finite product of 0 factors
Conditions: none
Choose x or y from each factor; commutativity permits grouping products by the number of y choices.
Rule: finite_product · Uses: G001
Before
a product with zero factors in the polynomial ring
After
empty product = multiplicative identity = 1
Conditions: none
A finite product of zero factors is defined as the multiplicative identity, so the entire polynomial is the constant 1.
Rule: empty_product_convention · Uses: G001, S001
Before
choose exactly 0 of 0 factors to contribute y
After
C(0,0)=1/(1·1)=1
Conditions: none
Count orderings of 0 positions and divide by reorderings within the 0 y positions and 0 x positions, giving coefficient 1.
Rule: subset_count · Uses: G001, S002
Before
C(0,0)=1; choose y 0 times
After
1·x^0·y^0
Conditions: none
The x exponent is 0-0=0; the two exponents still sum to 0.
Rule: group_like_terms · Uses: G001, S003
Before
all k=0 through 0 have been accounted for
After
(x+y)^0=1·x^0·y^0
Conditions: none
For n=0 the empty product is 1. The table's x^0 y^0 denotes the constant monomial, not an arithmetic evaluation of 0^0 at x=y=0.
Rule: collect_all_terms · Uses: G001, S004
S005 · premise_and_result_audit — There are n+1 coefficient positions, with the complete range of k retained.S004 · premise_and_result_audit — The answer row is [k,coefficient,x exponent,y exponent]; do not swap the exponent columns.S003 · premise_and_result_audit — Check the coefficient independently with the integer Pascal recurrence, not approximate factorials.S002 · premise_and_result_audit — This polynomial empty-product convention does not assign a separate arithmetic value to the number 0^0.S001 · premise_and_result_audit — This is not an infinite expansion for a noninteger power or a claim for noncommuting multiplication.(0, [[0, 1, 0, 0]], constant_one_boundary)
The full expansion consists of all displayed monomials, in the same order as the answer rows.
Review date: 2026-10-24
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
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