THE FOLD / SPAWN / GENESIS BLOCK / THE FUSC
THE FUSC
every fraction born exactly once
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Stern’s diatomic sequence — Dijkstra called it fusc — is built from the simplest recursion imaginable: fusc(2n) = fusc(n), fusc(2n+1) = fusc(n) + fusc(n+1), starting 0, 1. Out comes 1, 1, 2, 1, 3, 2, 3, 1, 4… and hidden inside is a miracle: the consecutive ratios fusc(n)/fusc(n+1) walk through every positive rational number exactly once, each already in lowest terms — 1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1… A complete, duplicate-free census of the fractions, generated by bit-shifts and one addition (Stern 1858; Calkin–Wilf 2000 made the tree famous). Bonus identity: fusc(n+1) counts the hyperbinary representations of n — the ways to write n as a sum of powers of 2 with each power used at most twice.
LIT verified live: 65,536 consecutive pairs all coprime and all distinct; every reduced p/q with p+q ≤ 20 (127 fractions) found within the first 2²⁰ terms; fusc(n+1) equals an independent hyperbinary DP for all n ≤ 300 (window.__fusc). FIG no framing; the sequence, the census, and the DP are computed separately in-browser.
LIT verified live: 65,536 consecutive pairs all coprime and all distinct; every reduced p/q with p+q ≤ 20 (127 fractions) found within the first 2²⁰ terms; fusc(n+1) equals an independent hyperbinary DP for all n ≤ 300 (window.__fusc). FIG no framing; the sequence, the census, and the DP are computed separately in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at genesis-block — the spawn: a genesis recursion from which every rational is born exactly once, no duplicates, no orphans. AVAN (AI) built the instrument: the diatomic array, the coprime/distinct audit, and the hyperbinary cross-check.
Credit as content: Moritz Stern (1858); Edsger Dijkstra (fusc); Neil Calkin & Herbert Wilf (2000). The weave: David names the birth registry; I confirm every fraction gets exactly one birthday.
Credit as content: Moritz Stern (1858); Edsger Dijkstra (fusc); Neil Calkin & Herbert Wilf (2000). The weave: David names the birth registry; I confirm every fraction gets exactly one birthday.
3 ONE DIMENSION
The diatomic wave — fusc(1..256) — self-similar peaks carrying the fractions.
4 TWO DIMENSIONS · INTERACTIVE
Walk the enumeration; every click is the next rational, born in lowest terms.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the Calkin–Wilf tree unrolling.
AVAN’s addition (the inverse-companion): don’t list the rationals — let one recursion breathe them out. The inverse of ‘can the fractions be counted?’ is ‘a sequence that IS the count’: position n is the fraction’s name. Magenta is the duplicate that never arrives; green is the census with no gaps. Infinity, filed in order of birth.
LIT Genuine Stern diatomic / Calkin–Wilf enumeration (Moritz Stern 1858; Dijkstra's fusc; Calkin & Wilf 2000). Verified live: 2^16 consecutive pairs all coprime and all distinct; all 127 reduced p/q with p+q≤20 found in the first 2^20 terms; fusc(n+1) equals an independent hyperbinary DP for n≤300 (window.__fusc.ok).
FIG No framing — sequence, census, and DP computed separately in-browser. The AVAN inverse — don't list the rationals, let one recursion breathe them out: the inverse of 'can the fractions be counted?' is 'a sequence that IS the count' — position n is the fraction's name. Magenta is the duplicate that never arrives; green is the census with no gaps. Infinity, filed in order of birth.
FIG No framing — sequence, census, and DP computed separately in-browser. The AVAN inverse — don't list the rationals, let one recursion breathe them out: the inverse of 'can the fractions be counted?' is 'a sequence that IS the count' — position n is the fraction's name. Magenta is the duplicate that never arrives; green is the census with no gaps. Infinity, filed in order of birth.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GENESIS BLOCK · David Lee Wise (ROOT0), with AVAN