derivation-mko-binomial-theorem-01 · ocme-new142-reader-adoption-20261021
Establish coefficient counting and expand n=4 completely.
Nonnegative integer n and commuting x,y; no noninteger exponents.
workG1 — n belongs to N0 and x,y are in a commutative real polynomial ring; the instance is n=4.k — number of y-factorsS — positions choosing ySome rule statements have reader clarifications in this revision: they are paraphrases, not verbatim quotations. Referenced original source bytes and hashes are unchanged.
mko_definition · cited — For n∈N_0, (x+y)^n=sum over k=0..n of C(n,k)*x^(n−k)*y^k. Coefficients are exact, all indices and both endpoint terms are retained, and the exponents must not be interchanged. At n=0 the polynomial is constant 1. A combinatorial witness chooses k y-factors from n factors. Negative/noninteger exponents and noncommuting variables are outside this finite-polynomial slice. · mko-binomial-theoremwork · prerequisite — Finite distributivity, commutative multiplication and the count comb(n,k)=n!/[k!(n-k)!] of size-k subsets; the empty product is 1.Before
(x+y)^n=\prod_{i=1}^{n}(x+y)After
\sum_{S\subseteq\{i\in\mathbb Z:1\le i\le n\}}x^{n-|S|}y^{|S|}Conditions: none
Rule: work · Uses: G1, work
Leaf basis: work
Choose x or y in each factor, indexing the full distributive expansion by positions S choosing y.
Before
|S|=k
After
\#\{S:|S|=k\}=\binom nk,\quad x^{n-|S|}y^{|S|}=x^{n-k}y^kConditions: none
Rule: work · Uses: G1, S01, work
Leaf basis: work
Group by the number k of y-factors; each group has comb(n,k) terms.
Before
k\in\{j\in\mathbb Z:0\le j\le n\}After
(x+y)^n=\sum_{k=0}^{n}\binom nk x^{n-k}y^kConditions: none
Rule: work · Uses: G1, S02, work
Leaf basis: work
Subset sizes partition every choice, including endpoints k=0 and k=n.
Before
n=4,\quad k=0,1,2,3,4
After
\binom40,\binom41,\binom42,\binom43,\binom44=1,4,6,4,1
Conditions: none
Rule: work · Uses: G1, S03, work
Leaf basis: work
Use the factorial combination formula for all five coefficients.
Before
(x+y)^4
After
x^4+4x^3y+6x^2y^2+4xy^3+y^4
Conditions: none
Rule: work · Uses: G1, S04, work
Leaf basis: work
Pair every coefficient with the correct exponents.
Before
x=2,\quad y=-1
After
(2-1)^4=16-32+24-8+1=1
Conditions: none
Rule: work · Uses: G1, S05, work
Leaf basis: work
An exact evaluation at negative y checks signs and endpoint terms.
Before
n=0
After
(x+y)^0=1=\binom00x^0y^0
Conditions: none
Rule: work · Uses: G1, S06, work
Leaf basis: work
The empty product and sole empty subset give constant 1.
Before
xy\ne yx\quad\text{(outside scope)}After
(x+y)^2=x^2+xy+yx+y^2\text{ need not equal }x^2+2xy+y^2Conditions: none
Rule: work · Uses: G1, S07, work
Leaf basis: work
Display the noncommuting boundary, identifying the assumption needed for grouping.
No substeps in this guide.
S08 · premise_audit · G1, S07, work — Do not extend the commutative proof to arbitrary matrices.S07 · premise_audit · G1, S06, work — This is a polynomial identity with the constant-polynomial convention even at x+y=0.S06 · premise_audit · G1, S05, work — Odd y-powers are negative and even powers positive.S05 · premise_audit · G1, S04, work — Each monomial has total degree4.S04 · premise_audit · G1, S03, work — The middle coefficient is 6, not 4.S03 · premise_audit · G1, S02, work — The y-exponent is the selected count k; roles are not exchanged.S02 · premise_audit · G1, S01, work — Commutativity permits choices at different positions to merge into one monomial.S01 · premise_audit · G1, work — The index set contains every integer1≤i≤n; no factor choices are omitted.k y-choices correspond to x^(n-k)y^k.
Keep all five terms including endpoints; the exact polynomial grid checks the n=4 identity.
reject — At x=y=1 the left side is 16 and the right2.reject — The scope requires n in N0; a negative power is not this polynomial.reject — For noncommuting matrices they can differ.Grouping all subsets proves the binomial formula; at n=4 coefficients are 1,4,6,4,1.
2 declared checks for this guide; only the listed expressions are covered.
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