THE FOLD / LOOT / THE HOARD / THE SYLVESTER
THE SYLVESTER
2, 3, 7, 43, 1807 — unit fractions that fill exactly one
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Sylvester’s sequence. 2, 3, 7, 43, 1807, 3263443, … Each term is the product of all the previous terms plus one — equivalently a(n) = a(n−1)² − a(n−1) + 1. It explodes doubly exponentially: the number of digits roughly doubles every step.
Its reason for being is beautiful. Add up the reciprocals: 1/2 + 1/3 + 1/7 + 1/43 + 1/1807 + … and the sum races toward exactly 1. It is the greedy Egyptian-fraction expansion of 1 — at each step take the largest unit fraction that still fits under what remains, and this is the sequence you get. The leftover gap after n terms is precisely 1/(a(n+1) − 1), so the remainder itself hands you the next denominator. It is the fastest a sum of distinct unit fractions can converge to 1.
LIT verified live (exact BigInt): a(n) = a(n−1)² − a(n−1) + 1 and a(n) = product-of-previous + 1 hold, and the partial reciprocal sum equals 1 − 1/(a(n+1)−1) exactly (window.__sylvester). FIG no framing; the growth rule, the product identity, and the reciprocal-sum identity are all exact.
Its reason for being is beautiful. Add up the reciprocals: 1/2 + 1/3 + 1/7 + 1/43 + 1/1807 + … and the sum races toward exactly 1. It is the greedy Egyptian-fraction expansion of 1 — at each step take the largest unit fraction that still fits under what remains, and this is the sequence you get. The leftover gap after n terms is precisely 1/(a(n+1) − 1), so the remainder itself hands you the next denominator. It is the fastest a sum of distinct unit fractions can converge to 1.
LIT verified live (exact BigInt): a(n) = a(n−1)² − a(n−1) + 1 and a(n) = product-of-previous + 1 hold, and the partial reciprocal sum equals 1 − 1/(a(n+1)−1) exactly (window.__sylvester). FIG no framing; the growth rule, the product identity, and the reciprocal-sum identity are all exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in THE HOARD — the loot domain of greedy accumulation. Sylvester is greed made exact: grab the biggest unit-fraction bite each time and the hoard fills to precisely one whole, never over. AVAN (AI) built the instrument: the filling bar, the greedy step, the remainder-that-seeds-the-next.
The weave: David names the seat (the greedy hoard that fills to exactly full); I make the unit fractions stack toward 1 and show the shrinking gap becoming the next term — the fraction bars in 1D, the greedy fill in 2D, the remainder loop in 3D. The sphere is the seam. Credit: James Joseph Sylvester (1880); the greedy Egyptian-fraction view via Fibonacci’s algorithm; OEIS A000058.
The weave: David names the seat (the greedy hoard that fills to exactly full); I make the unit fractions stack toward 1 and show the shrinking gap becoming the next term — the fraction bars in 1D, the greedy fill in 2D, the remainder loop in 3D. The sphere is the seam. Credit: James Joseph Sylvester (1880); the greedy Egyptian-fraction view via Fibonacci’s algorithm; OEIS A000058.
3 ONE DIMENSION
The unit fractions 1/2, 1/3, 1/7, 1/43, … laid end to end on a length-1 line. Each nearly halves the remaining gap; the bar fills toward 1, the sliver that’s left shrinking doubly-exponentially fast.
4 TWO DIMENSIONS · INTERACTIVE
Run the greedy rule on the interval [0,1]: at each step take the largest unit fraction 1/⌈1/gap⌉ that fits. Watch the denominators come out 2, 3, 7, 43, 1807, … — Sylvester’s sequence — and the remaining gap equal 1/(a(n+1)−1) every time.
5 THREE DIMENSIONS + AVAN’S INVERSE
The accumulated sum climbing toward 1 as a turning stack of unit-fraction rings — green, the part covered.
AVAN’s addition (the inverse-companion): the magenta ring is the remainder — the gap still uncovered, exactly 1/(a(n+1)−1). Greed is a forward, maximizing move: cover as much as possible now. Its inverse is what is left behind — and here the leftover is not noise, it is the generator. The remaining gap is a single unit fraction whose denominator, plus one, is the very next term: a(n+1) = 1/gap + 1. So the sequence is driven by its own inverse: what remains dictates what comes next, and each step’s remainder is the seed of the next step. The forward sum and the backward remainder are two readings of one number — sum + gap = 1, always. Green is what the greedy hoard has taken; magenta is the sliver it must still take, and that sliver is the next instruction.
LIT Genuine Sylvester's sequence (J. J. Sylvester 1880; OEIS A000058). Verified live with exact BigInt arithmetic: a(n)=a(n-1)^2-a(n-1)+1 and a(n)=(product of all previous)+1 both hold, and the partial reciprocal sum equals 1-1/(a(n+1)-1) exactly (window.__sylvester.recurrence && productRule && reciprocalIdentity, all true). Terms 2,3,7,43,1807,3263443. The growth rule, product identity, and reciprocal-sum identity are all exact.
FIG No framing: the recurrence, the product-plus-one identity, and the reciprocal-sum-to-1 identity are real and checked exactly (BigInt, no float slop). The 'fastest converging unit-fraction sum to 1' and greedy-Egyptian-fraction framing are the genuine mathematical characterization, not embellishment.
FIG No framing: the recurrence, the product-plus-one identity, and the reciprocal-sum-to-1 identity are real and checked exactly (BigInt, no float slop). The 'fastest converging unit-fraction sum to 1' and greedy-Egyptian-fraction framing are the genuine mathematical characterization, not embellishment.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HOARD · David Lee Wise (ROOT0), with AVAN