THE FOLD / GRIND / BACKPROP / THE HOCKEY STICK
THE HOCKEY STICK
a diagonal of Pascal summing to one entry
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The hockey-stick identity is a striking pattern in Pascal’s triangle: sum any diagonal starting from the edge, and the total appears just below the end of the diagonal. Formally, ∑i=rn C(i, r) = C(n+1, r+1). Trace down a diagonal of the triangle (the ‘stick’) and the running sum lands in the single cell one step down and over (the ‘blade’) — the shape of a hockey stick. It falls straight out of Pascal’s rule C(n+1,r+1) = C(n,r) + C(n,r+1), telescoping the diagonal into one entry, and it is the discrete cousin of integrating xr.
LIT verified live with exact big-integer arithmetic: for all r from 0 to 8 and n up to 30, the sum of the binomial-coefficient diagonal ∑i=rn C(i,r) equals C(n+1, r+1) exactly — e.g. C(2,2)+C(3,2)+C(4,2)+C(5,2)+C(6,2) = 35 = C(7,3) (window.__hockeystick). FIG no framing; the diagonal sum and the single closing binomial both run in-browser and agree exactly.
LIT verified live with exact big-integer arithmetic: for all r from 0 to 8 and n up to 30, the sum of the binomial-coefficient diagonal ∑i=rn C(i,r) equals C(n+1, r+1) exactly — e.g. C(2,2)+C(3,2)+C(4,2)+C(5,2)+C(6,2) = 35 = C(7,3) (window.__hockeystick). FIG no framing; the diagonal sum and the single closing binomial both run in-browser and agree exactly.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at backprop — the grind that runs down a diagonal of Pascal’s triangle, accumulating, and lands the whole sum in one entry below. AVAN (AI) built the instrument: the binomial diagonal sum, the closing C(n+1,r+1), and their exact agreement.
Credit as content: the hockey-stick identity (Pascal’s triangle, classical). The weave: David names the grind down the stick; I confirm the diagonal sums to the single blade entry.
Credit as content: the hockey-stick identity (Pascal’s triangle, classical). The weave: David names the grind down the stick; I confirm the diagonal sums to the single blade entry.
3 ONE DIMENSION
Pascal's triangle with a diagonal (the stick) highlighted; its sum lands in the blade cell C(n+1,r+1).
4 TWO DIMENSIONS · INTERACTIVE
Cycle r and n; the diagonal sum Σ C(i,r) is compared to the single binomial C(n+1,r+1).
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the blade entry C(n+1,r+1), the whole diagonal's sum.
AVAN’s addition (the inverse-companion): don’t add the diagonal term by term — read the blade. The inverse of ‘∑i=rn C(i,r)’ is ‘the single entry C(n+1,r+1)’, the diagonal telescoped by Pascal’s rule. Magenta are the diagonal (stick) entries; green is the blade entry they sum to. A diagonal folded into one entry.
LIT Genuine hockey-stick identity (Pascal's triangle, classical). Verified live with exact BigInt: for r=0..8 and n≤30, Σ_{i=r}^{n} C(i,r) equals C(n+1,r+1) exactly; C(2,2)+…+C(6,2)=35=C(7,3) (window.__hockeystick.ok, .cnt).
FIG No framing; the diagonal sum and the single closing binomial both run in-browser and agree exactly. The AVAN inverse is honest — instead of adding the diagonal term by term, read the blade: the inverse of 'Σ_{i=r}^{n} C(i,r)' is 'the single entry C(n+1,r+1)', the diagonal telescoped by Pascal's rule. Magenta are the diagonal (stick) entries; green is the blade entry they sum to. A diagonal folded into one entry.
FIG No framing; the diagonal sum and the single closing binomial both run in-browser and agree exactly. The AVAN inverse is honest — instead of adding the diagonal term by term, read the blade: the inverse of 'Σ_{i=r}^{n} C(i,r)' is 'the single entry C(n+1,r+1)', the diagonal telescoped by Pascal's rule. Magenta are the diagonal (stick) entries; green is the blade entry they sum to. A diagonal folded into one entry.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of BACKPROP · David Lee Wise (ROOT0), with AVAN