THE FOLD / SPAWN / COLD BOOT / THE MILLS
THE MILLS
a constant that boots an infinite prime cascade
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
In 1947 William Mills proved a constant A exists such that ⌊A^(3ⁿ)⌋ is prime for every n. The cascade: 2, 11, 1361, 2521008887, 16022236204009818131831320183 — each prime the smallest prime after the cube of the last, each nestled astonishingly close above that cube (gaps of just 3, 30, 6, 80). The constant is not magic; it is the cascade written as a limit — and the intimacy is extreme: the cube root of the fifth prime exceeds the fourth prime by only 4×10⁻¹⁸. The catch worth knowing: A’s standing as the least such constant (1.30637788…) assumes the Riemann Hypothesis (Caldwell–Cheng), because it needs primes to always arrive inside consecutive-cube windows.
LIT verified live: the cascade is re-derived from scratch — from p=2, the next prime after each cube is hunted by Miller–Rabin and lands exactly on the known sequence, inside every (p³,(p+1)³) window; the knife-edge 4×10⁻¹⁸ is measured by BigInt cube root; and A itself is derived live as the 243rd root of the fifth prime, its bounds agreeing to 31 digits and matching the published 1.3063778838630806904686144926 (window.__mills). FIG honest boundary: minimality-under-RH is cited, not assumed proven; the fifth prime’s primality is Miller–Rabin strong plus literature certification.
LIT verified live: the cascade is re-derived from scratch — from p=2, the next prime after each cube is hunted by Miller–Rabin and lands exactly on the known sequence, inside every (p³,(p+1)³) window; the knife-edge 4×10⁻¹⁸ is measured by BigInt cube root; and A itself is derived live as the 243rd root of the fifth prime, its bounds agreeing to 31 digits and matching the published 1.3063778838630806904686144926 (window.__mills). FIG honest boundary: minimality-under-RH is cited, not assumed proven; the fifth prime’s primality is Miller–Rabin strong plus literature certification.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at cold-boot — the spawn: one seed constant, and the machine boots an infinite prime cascade from it — press the cube button forever. AVAN (AI) built the instrument: the cascade re-derivation, the BigInt 243rd-root extractor, and the knife-edge micrometer.
Credit as content: William Mills (1947); Caldwell & Cheng (RH-conditional digits). The weave: David names the boot sequence; I rebuild the constant from its own primes, digit by digit.
Credit as content: William Mills (1947); Caldwell & Cheng (RH-conditional digits). The weave: David names the boot sequence; I rebuild the constant from its own primes, digit by digit.
3 ONE DIMENSION
The cascade: each prime a hair above the cube of the last — gaps 3, 30, 6, 80.
4 TWO DIMENSIONS · INTERACTIVE
Walk the cascade; every hop is cube-then-find-the-next-prime, re-derived live.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the tower of cubes, primes perched on each ledge.
AVAN’s addition (the inverse-companion): don’t admire the constant — reverse it. The inverse of ‘A generates primes’ is ‘the primes generate A’: the constant is the cascade’s memory, nothing more, and I rebuilt its 31 digits from the fifth prime alone. Magenta is the RH assumption holding up the word ‘least’; green is the knife-edge — four attoseconds of number line between a prime and the cube below it. A constant that is secretly a fossil record.
LIT Genuine Mills' theorem and cascade (William Mills 1947; Caldwell & Cheng digits). Verified live: cascade re-derived from scratch (next prime after each cube = known sequence, inside every consecutive-cube window); cbrt(p₅)−p₄ = 4×10⁻¹⁸ by BigInt; A derived as p₅^(1/243) with bounds agreeing to 31 digits, matching published (window.__mills.ok).
FIG Honest boundary — least-A digits are RH-conditional (cited, not claimed); p₅ primality is strong Miller-Rabin plus literature certification. The AVAN inverse — don't admire the constant, reverse it: the inverse of 'A generates primes' is 'the primes generate A' — the constant is the cascade's memory, rebuilt here from the fifth prime alone. Magenta is the RH assumption under the word 'least'; green is the knife-edge. A constant that is secretly a fossil record.
FIG Honest boundary — least-A digits are RH-conditional (cited, not claimed); p₅ primality is strong Miller-Rabin plus literature certification. The AVAN inverse — don't admire the constant, reverse it: the inverse of 'A generates primes' is 'the primes generate A' — the constant is the cascade's memory, rebuilt here from the fifth prime alone. Magenta is the RH assumption under the word 'least'; green is the knife-edge. A constant that is secretly a fossil record.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of COLD BOOT · David Lee Wise (ROOT0), with AVAN