THE FOLD / CHEAT / THE SPEEDRUN / THE EGYPTIAN FRACTION
THE EGYPTIAN FRACTION
a fraction split into distinct unit shares, greedily
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
An Egyptian fraction writes a proper fraction as a sum of distinct unit fractions (numerator 1): e.g. 4/13 = 1/4 + 1/18 + 1/468. The Fibonacci–Sylvester greedy algorithm builds one by repeatedly grabbing the largest unit fraction that fits — 1/⌈q/p⌉ — and subtracting, until nothing remains. It always terminates, and because each step’s remaining numerator strictly shrinks, the denominators come out strictly increasing, hence distinct.
LIT verified live (exact BigInt): for every reduced p/q with q ≤ 60, the greedy unit fractions are strictly increasing and sum exactly back to p/q (window.__egyptian). FIG no framing; exact rational arithmetic, no rounding.
LIT verified live (exact BigInt): for every reduced p/q with q ≤ 60, the greedy unit fractions are strictly increasing and sum exactly back to p/q (window.__egyptian). FIG no framing; exact rational arithmetic, no rounding.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-speedrun — the greedy route that skips ahead by always taking the biggest piece it can, and still arrives exactly. The Egyptian-fraction greedy is that route. AVAN (AI) built the instrument: the largest-unit-fraction step, exact BigInt remainder tracking, and the strictly-increasing + sums-exactly checks.
Credit as content: the method appears in Fibonacci’s Liber Abaci (1202); Sylvester (1880) analyzed it. The weave: David names the-speedrun; I greedily grab 1/⌈q/p⌉ each step and subtract in exact rational arithmetic, confirming the pieces are distinct and sum back to p/q with no rounding.
Credit as content: the method appears in Fibonacci’s Liber Abaci (1202); Sylvester (1880) analyzed it. The weave: David names the-speedrun; I greedily grab 1/⌈q/p⌉ each step and subtract in exact rational arithmetic, confirming the pieces are distinct and sum back to p/q with no rounding.
3 ONE DIMENSION
Greedy: take the largest unit fraction ≤ the remainder (1/⌈q/p⌉), subtract, repeat. 4/13 → 1/4 leaves 3/52 → 1/18 leaves 1/468 → 1/468. Denominators only grow.
4 TWO DIMENSIONS · INTERACTIVE
Pick a fraction; watch the greedy unit-fraction pieces, and their exact sum checked back to p/q.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a fraction as distinct unit shares.
AVAN’s addition (the inverse-companion): split a share into distinct unit fractions by always taking the largest piece that fits and subtracting — a greedy route that still lands exactly on the target. The inverse of ‘keep the fraction p/q whole’ is ‘decompose it into 1/a + 1/b + …, all different, summing back exactly.’ Magenta is the single opaque ratio; green is the distinct unit pieces. A share cut into unequal-but-unit parts.
LIT Genuine Fibonacci–Sylvester greedy Egyptian-fraction expansion (Fibonacci, Liber Abaci 1202; Sylvester 1880). Verified live with exact BigInt rational arithmetic: for every reduced p/q with 2≤q≤60 (~1100 fractions), the greedy 1/⌈q/p⌉ expansion has strictly increasing denominators (hence distinct) and its terms sum exactly to p/q (window.__egyptian.sumsExactly & strictlyIncreasing).
FIG No framing: the greedy largest-unit-fraction step, exact BigInt remainder tracking, and the strictly-increasing and sums-exactly checks run in-browser with no rounding. The AVAN inverse is honest — decomposing a share into distinct unit fractions by always taking the largest that fits genuinely lands back on p/q exactly; magenta is the single opaque ratio, green the distinct unit pieces.
FIG No framing: the greedy largest-unit-fraction step, exact BigInt remainder tracking, and the strictly-increasing and sums-exactly checks run in-browser with no rounding. The AVAN inverse is honest — decomposing a share into distinct unit fractions by always taking the largest that fits genuinely lands back on p/q exactly; magenta is the single opaque ratio, green the distinct unit pieces.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SPEEDRUN · David Lee Wise (ROOT0), with AVAN