# Question 76: The Finite Expansion Includes Every k from Zero through n

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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.

P250-0076 · q-20261014-advance-20261109-r05-binomial-theorem-0026



## Original question and givens

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).





## Definition and scope

n is a nonnegative integer and x,y commute. For n=0 use the polynomial empty-product identity.


- [mko-binomial-theorem](https://ocme.evemisslab.com/en/mko/mko-binomial-theorem/)





## Step-by-step derivation

S001 · impliesBefore

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.

Step identifiers and dependenciesRule: finite_product · Uses: G001





S002 · impliesBefore

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.

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





S003 · impliesBefore

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.

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





S004 · impliesBefore

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.

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





S005 · impliesBefore

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.

Step identifiers and dependenciesRule: collect_all_terms · Uses: G001, S004









## Reverse checks


- 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.





## Result

(0, [[0, 1, 0, 0]], constant_one_boundary)

The full expansion consists of all displayed monomials, in the same order as the answer rows.





## This question review and source binding

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 3ab6f1fad6c6b2731837e8c6140d5184db5ed2bb90ec471cbad328ce1196a4d5
- candidate record SHA-256: c4bae52f73f176801dad06167ec0086634c8fc560736f1d4773ccbb50e82e07c
- semantic review receipt SHA-256: 4727d1bd957a9b7f6cadb7972ddb5ab80a31f5259dc5394c1dff2b71944ed36b
- question-first witness SHA-256: 51e18649133238ac992229f6e16542a5f3af53ca73b2262ab0524508ea937dee
- Lean: NOT_RUN · human review: NOT_PERFORMED

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