THE FOLD / CHEAT / GOD MODE / THE CHU-VANDERMONDE
THE CHU-VANDERMONDE
a binomial convolution collapsing to one entry
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Chu–Vandermonde identity collapses a whole convolution of binomial coefficients into a single one: ∑k C(m,k)·C(n,r-k) = C(m+n,r). Choosing r objects from a combined pile of m + n is the same as splitting the choice — k from the first pile, r-k from the second — and summing over all splits. Its most famous special case, with m = n = r, gives ∑k C(n,k)² = C(2n,n): the sum of squared binomial coefficients across a row of Pascal’s triangle is the central coefficient two rows down.
LIT verified live with exact big-integer arithmetic: for all m, n up to 15 and every r, the convolution sum ∑k C(m,k)C(n,r-k) equals C(m+n,r) exactly; and the special case ∑k C(n,k)² = C(2n,n) holds for n up to 12 (window.__chuvandermonde). FIG no framing; the binomial convolution and the single closing coefficient both run in-browser and agree exactly.
LIT verified live with exact big-integer arithmetic: for all m, n up to 15 and every r, the convolution sum ∑k C(m,k)C(n,r-k) equals C(m+n,r) exactly; and the special case ∑k C(n,k)² = C(2n,n) holds for n up to 12 (window.__chuvandermonde). FIG no framing; the binomial convolution and the single closing coefficient both run in-browser and agree exactly.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at god-mode — the cheat: a whole convolution of binomials clipped instantly into one coefficient C(m+n,r), no summing required. AVAN (AI) built the instrument: the binomial convolution, the closing C(m+n,r), and the ∑C(n,k)² special case.
Credit as content: Zhu Shijie (Chu, 1303) and Alexandre-Théophile Vandermonde. The weave: David names the cheat; I confirm the convolution equals a single binomial.
Credit as content: Zhu Shijie (Chu, 1303) and Alexandre-Théophile Vandermonde. The weave: David names the cheat; I confirm the convolution equals a single binomial.
3 ONE DIMENSION
Two Pascal rows C(m,·) and C(n,·); their convolution at position r equals the single entry C(m+n,r).
4 TWO DIMENSIONS · INTERACTIVE
Cycle m, n, r; the convolution sum Σ C(m,k)C(n,r−k) is compared to C(m+n,r).
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the single coefficient C(m+n,r).
AVAN’s addition (the inverse-companion): don’t compute a convolution — read one coefficient. The inverse of ‘∑k C(m,k)C(n,r-k)’ is ‘C(m+n,r)’: choosing r from a combined pile, however you split it. Magenta are the convolution terms C(m,k)C(n,r-k); green is the single binomial they sum to. A convolution folded into one entry.
LIT Genuine Chu–Vandermonde identity (Zhu Shijie 1303; Alexandre-Théophile Vandermonde). Verified live with exact BigInt: for all m,n≤15 and every r, Σ_k C(m,k)C(n,r−k) equals C(m+n,r) exactly, and the special case Σ_k C(n,k)²=C(2n,n) holds for n≤12 (window.__chuvandermonde.ok, .cnt, .sq).
FIG No framing; the binomial convolution and the single closing coefficient both run in-browser and agree exactly. The AVAN inverse is honest — instead of computing a convolution, read one coefficient: the inverse of 'Σ_k C(m,k)C(n,r−k)' is 'C(m+n,r)': choosing r from a combined pile, however you split it. Magenta are the convolution terms C(m,k)C(n,r−k); green is the single binomial they sum to. A convolution folded into one entry.
FIG No framing; the binomial convolution and the single closing coefficient both run in-browser and agree exactly. The AVAN inverse is honest — instead of computing a convolution, read one coefficient: the inverse of 'Σ_k C(m,k)C(n,r−k)' is 'C(m+n,r)': choosing r from a combined pile, however you split it. Magenta are the convolution terms C(m,k)C(n,r−k); green is the single binomial they sum to. A convolution folded into one entry.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GOD MODE · David Lee Wise (ROOT0), with AVAN