THE FOLD / GRIND / BACKPROP / THE SYLVESTER SEQUENCE
THE SYLVESTER SEQUENCE
greedy unit fractions racing to one
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Sylvester’s sequence is the greediest possible race to 1 in unit fractions. Start at 2, and each term is the previous ones multiplied together plus one: 2, 3, 7, 43, 1807, 3263443, … — equivalently an+1 = an² - an + 1. Its reciprocals form the fastest-converging Egyptian-fraction sum to 1: 1/2 + 1/3 + 1/7 + 1/43 + …, where each step takes the largest unit fraction that keeps the total below 1. The partial sums obey a clean closed form: ∑i≤n 1/ai = 1 - 1/(an+1 - 1), so they approach 1 doubly-exponentially fast, never quite reaching it.
LIT verified live with exact big-integer fractions: for n = 0..8, the partial sum ∑i≤n 1/ai equals exactly 1 - 1/(an+1 - 1), and an+1 - 1 equals the product a0a1…an (window.__sylvester). FIG no framing; the reciprocal sum and the closed form are computed as exact fractions in-browser and agree.
LIT verified live with exact big-integer fractions: for n = 0..8, the partial sum ∑i≤n 1/ai equals exactly 1 - 1/(an+1 - 1), and an+1 - 1 equals the product a0a1…an (window.__sylvester). FIG no framing; the reciprocal sum and the closed form are computed as exact fractions in-browser and agree.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at backprop — the grind where each term grinds the remaining gap to 1 shut, squaring the denominator every step so the error collapses doubly-exponentially. AVAN (AI) built the instrument: the sequence recurrence, the exact reciprocal-sum fraction, and the closed form 1 - 1/(an+1-1).
Credit as content: James Joseph Sylvester (1880); Fibonacci’s greedy Egyptian fractions. The weave: David names the grind; I confirm the partial sums equal 1 - 1/(an+1-1), exactly.
Credit as content: James Joseph Sylvester (1880); Fibonacci’s greedy Egyptian fractions. The weave: David names the grind; I confirm the partial sums equal 1 - 1/(an+1-1), exactly.
3 ONE DIMENSION
The unit fractions 1/2, 1/3, 1/7, 1/43, … stacking toward 1 — each the largest that keeps the sum below 1.
4 TWO DIMENSIONS · INTERACTIVE
Cycle n; the exact partial sum Σ 1/a_i is compared to the closed form 1 − 1/(a_{n+1} − 1).
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the limit 1, reached by the reciprocal sum.
AVAN’s addition (the inverse-companion): don’t add the fractions blindly — read the gap. The inverse of ‘the partial sum ∑1/ai’ is ‘the remaining gap 1/(an+1-1) to 1’, which the next greedy term always closes. Magenta are the unit fractions; green is the 1 they race toward. A sum whose distance-to-1 you can read off exactly.
LIT Genuine Sylvester's sequence (James Joseph Sylvester, 1880; greedy Egyptian fractions from Fibonacci). Verified live with exact BigInt fractions: for n=0..8, Σ_{i≤n} 1/a_i equals exactly 1 − 1/(a_{n+1}−1), and a_{n+1}−1 equals the product a_0a_1…a_n, where a_{n+1}=a_n²−a_n+1 (window.__sylvester.ok, .prodOk).
FIG No framing; the reciprocal sum and the closed form are computed as exact fractions in-browser and agree. The AVAN inverse is honest — instead of adding the fractions blindly, read the gap: the inverse of 'the partial sum Σ1/a_i' is 'the remaining gap 1/(a_{n+1}−1) to 1', which the next greedy term always closes. Magenta are the unit fractions; green is the 1 they race toward. A sum whose distance-to-1 you can read off exactly.
FIG No framing; the reciprocal sum and the closed form are computed as exact fractions in-browser and agree. The AVAN inverse is honest — instead of adding the fractions blindly, read the gap: the inverse of 'the partial sum Σ1/a_i' is 'the remaining gap 1/(a_{n+1}−1) to 1', which the next greedy term always closes. Magenta are the unit fractions; green is the 1 they race toward. A sum whose distance-to-1 you can read off exactly.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of BACKPROP · David Lee Wise (ROOT0), with AVAN