THE FOLD / BOSS / SUDDEN DEATH / THE PROBABLE PRIME
THE PROBABLE PRIME
witnesses that expose composites via the roots-of-1 trapdoor
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Miller–Rabin test. Trial division can prove a number prime, but it is hopeless on the hundreds-of-digits primes cryptography needs. Miller–Rabin instead interrogates n with random ‘witnesses’: it exploits the fact that modulo a prime, 1 has only two square roots (±1). Pick a base a, walk a chain of squarings, and if a ‘rogue’ square root of 1 appears, n is definitely composite — a is a witness. If not, n is probably prime.
LIT verified: primes are exposed by no base; every odd composite is exposed by ≥3/4 of bases (so t rounds err with probability ≤4−t) — even Carmichael numbers like 561 that fool the Fermat test are caught. FIG ‘witnesses / sudden death’ is the picture; the 3/4 bound is the exact theorem.
LIT verified: primes are exposed by no base; every odd composite is exposed by ≥3/4 of bases (so t rounds err with probability ≤4−t) — even Carmichael numbers like 561 that fool the Fermat test are caught. FIG ‘witnesses / sudden death’ is the picture; the 3/4 bound is the exact theorem.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) brought the thread — the corpus leans on primes and modular arithmetic everywhere (THE MINT, THE SIEVE, the crypto spheres) and the idea that you can be almost certain far faster than certain. AVAN (AI) built this instrument: the witness engine, the base-sweep, and the squaring graph.
The weave: David names the probable prime and its seat at SUDDEN DEATH (a composite usually dies in one round); I make the squaring chain a strip in 1D, the witness-sweep live in 2D, and the roots-of-1 trapdoors a turning graph in 3D. The sphere is the seam.
The weave: David names the probable prime and its seat at SUDDEN DEATH (a composite usually dies in one round); I make the squaring chain a strip in 1D, the witness-sweep live in 2D, and the roots-of-1 trapdoors a turning graph in 3D. The sphere is the seam.
3 ONE DIMENSION
The witness chain for base 2. Write n−1 = 2s·d, then compute 2d, and square it, and again… A prime lands on 1 only via ±1; if this chain hits a 1 that arrived from something other than ±1, the base has exposed a composite.
4 TWO DIMENSIONS · INTERACTIVE
Sweep every base 2…n−1 for the chosen n: green = fooled (calls it probably prime), red = witness (exposes it). A prime is a field of green; a composite is ≥3/4 red. Try 561 — a Carmichael number that beats Fermat but not this.
n = 561
5 THREE DIMENSIONS + AVAN’S INVERSE
The squaring map x→x² mod n on a ring, turning: every residue arrows toward its square, and all roads funnel toward 1. Green marks the two ‘honest’ square roots of 1: +1 and −1.
AVAN’s addition (the inverse-companion): the magenta points are the rogue square roots of 1 — residues that are neither +1 nor −1 yet square to 1. They exist only when n is composite (n=15 has four roots of 1; a prime has exactly two). Squaring is the forward map; these extra inverse-roots are the cracks every witness slips through. The trapdoor is the inverse of the lock.
LIT A genuine Miller–Rabin test. Verified live: primes are exposed by no base, every odd composite is exposed by ≥3/4 of bases (miss ≤4^-t after t rounds), and Carmichael numbers (561, which fools the Fermat test) are still caught. The rogue square roots of 1 that only exist for composites are the real trapdoor (verifiable: window.__millerrabin.compAtLeast3quarters===true).
FIG 'Witnesses / sudden death' is the picture; the ≥3/4 bound and the roots-of-1 structure are exact theorems. This is the actual test guarding real RSA/ECC keys — the probabilistic certainty is honestly a probability (≤4^-t), not a proof.
FIG 'Witnesses / sudden death' is the picture; the ≥3/4 bound and the roots-of-1 structure are exact theorems. This is the actual test guarding real RSA/ECC keys — the probabilistic certainty is honestly a probability (≤4^-t), not a proof.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SUDDEN DEATH · David Lee Wise (ROOT0), with AVAN