Question 457: Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem

Original question and givens

Use boundary exponent=-7. Give (exponent, finite_polynomial_claim, boundary).

• G001 — Use boundary exponent=-7. Give (exponent, finite_polynomial_claim, boundary).

Definition and scope

The finite binomial expansion needs a nonnegative integer exponent. Only that finite-polynomial claim is withheld; other series are not disproved.

• mko-binomial-theorem

Step-by-step derivation

exponent=-7

finite polynomial formula requires a nonnegative integer n

exponent=-7

not a permitted finite-product exponent

finite product with n factors

cannot enumerate a negative/noninteger number of factors

this finite theorem is inapplicable

no claim about a different generalized binomial series

Reverse checks

• S004 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S003 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S002 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.
• S001 · premise_and_result_audit — Count coefficients in the integers first; do not invert possibly nonunit factorials in a general coefficient ring.

Result

(-7, No, finite_binomial_polynomial_requires_natural_zero_exponent_and_commutative_multiplication)

The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings.

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.