THE FOLD / BOSS / THE GATEKEEPER / THE PROTH
THE PROTH
one witness decides a Proth prime
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Proth’s theorem gives a fast, exact primality test for Proth numbers — those of the form N = k·2n + 1 with k odd and k < 2n. It says: N is prime if and only if there exists an integer a with a(N−1)/2 ≡ −1 (mod N). Such an a is a witness to primality; for a prime Proth number, half of all bases work, so a small random search finds one fast. It powers the search for many of the largest known primes.
LIT verified live: over thousands of Proth numbers, “a witness exists” matches primality (by trial division) exactly, and known Proth primes are witnessed (window.__proth). FIG no framing; exact modular arithmetic vs a trial-division oracle.
LIT verified live: over thousands of Proth numbers, “a witness exists” matches primality (by trial division) exactly, and known Proth primes are witnessed (window.__proth). FIG no framing; exact modular arithmetic vs a trial-division oracle.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-gatekeeper — the guard that admits a Proth number as prime only if a single witness raises −1; one modular power decides it. AVAN (AI) built the instrument: the Proth-number generator, the a(N−1)/2 ≡ −1 witness search, and the match against a trial-division oracle.
Credit as content: François Proth (1878). The weave: David names the-gatekeeper; I search for a base a whose modular power lands on −1, and confirm that a witness exists exactly when the Proth number is prime — an iff test, checked against factoring.
Credit as content: François Proth (1878). The weave: David names the-gatekeeper; I search for a base a whose modular power lands on −1, and confirm that a witness exists exactly when the Proth number is prime — an iff test, checked against factoring.
3 ONE DIMENSION
N = k·2n+1, k odd, k < 2n. Prime ⇔ some a has a(N−1)/2 ≡ −1 (mod N). 3, 5, 13, 17, 41, 97, 113, 193, 241 … are Proth primes.
4 TWO DIMENSIONS · INTERACTIVE
A Proth number, the search for a witness a, and its verdict checked against trial division.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: primality by a single witness.
AVAN’s addition (the inverse-companion): certify a Proth number prime by finding one witness, not by factoring — a base a whose power raises −1 proves primality outright (and half of all bases work for a prime). The inverse of ‘trial-divide N up to √N’ is ‘search a few bases for a(N−1)/2 ≡ −1 — one witness certifies it.’ Magenta is the factor hunt; green is the witness found. Primality proven by one base.
LIT Genuine Proth's theorem (François Proth 1878). Verified live: over ~2600 Proth numbers N=k·2ⁿ+1 (k odd, k<2ⁿ) below a few million, the existence of a base a with a^((N−1)/2) ≡ −1 (mod N) matches N being prime (by trial division) exactly (window.__proth.witnessMatchesPrimality), and the known Proth primes 3,5,13,17,41,97,113,193,241 are all witnessed (knownWitnessed).
FIG No framing: the Proth-number generator, the a^((N−1)/2)≡−1 witness search, and the match against a trial-division oracle run in-browser with exact modular arithmetic and agree. The AVAN inverse is honest — certifying a Proth number prime by exhibiting one witness base (whose power raises −1) proves primality outright, replacing trial division to √N; magenta is that factor hunt, green the witness found. Primality proven by one base.
FIG No framing: the Proth-number generator, the a^((N−1)/2)≡−1 witness search, and the match against a trial-division oracle run in-browser with exact modular arithmetic and agree. The AVAN inverse is honest — certifying a Proth number prime by exhibiting one witness base (whose power raises −1) proves primality outright, replacing trial division to √N; magenta is that factor hunt, green the witness found. Primality proven by one base.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GATEKEEPER · David Lee Wise (ROOT0), with AVAN