# Question 458: Negative or Noninteger Exponents Are Outside This Finite Polynomial Theorem

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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.

P1000-0458 · q-20261014-advance-20261109-r05-binomial-theorem-0083



## Original question and givens

Use boundary exponent=-8. Give (exponent, finite_polynomial_claim, boundary).


- G001 — Use boundary exponent=-8. 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](https://ocme.evemisslab.com/en/mko/mko-binomial-theorem/)





## Step-by-step derivation

S001 · impliesBefore

exponent=-8

After

finite polynomial formula requires a nonnegative integer n

Conditions: none

The finite-product theorem requires a nonnegative integer exponent.

Step identifiers and dependenciesRule: natural_exponent · Uses: G001





S002 · impliesBefore

exponent=-8

After

not a permitted finite-product exponent

Conditions: none

A negative or noninteger exponent is outside this finite expansion's declared domain.

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





S003 · impliesBefore

finite product with n factors

After

cannot enumerate a negative/noninteger number of factors

Conditions: none

Consequently the finite coefficient list k=0 through n cannot be applied directly.

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





S004 · impliesBefore

this finite theorem is inapplicable

After

no claim about a different generalized binomial series

Conditions: none

This does not disprove generalized series under other convergence hypotheses, and it does not drop commutativity.

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









## 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

(-8, 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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 555c6ee6b2c5f774c438d790221953eb789b95d0ecf5fe4ca9aa1d9d96687db7
- candidate record SHA-256: 0ce16394329a8bd5d8e5bad11c523082ccd12816883e230cbc54749e13a7a35c
- semantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772
- question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261109-r05-binomial-theorem-0083.json)