THE FOLD / LOOT / THE HOARD / THE WILSON PRIME
THE WILSON PRIME
three coins in 250 years
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Wilson’s theorem is a perfect prime detector: p is prime exactly when (p−1)! ≡ −1 (mod p) — and composites c > 4 fail spectacularly, with (c−1)! ≡ 0. (Useless in practice: the factorial is astronomically expensive. Beautiful in principle: a single congruence that never lies.) Now sharpen it: for which primes does the congruence hold modulo p²? Those are the Wilson primes — and in 250 years of searching, exactly three have ever been found: 5, 13, and 563. The search has swept past 2×10¹³. Heuristically, infinitely many should exist (each prime ‘hits’ with probability ~1/p), but the next one could be anywhere.
LIT verified live: Wilson’s theorem confirmed for every prime below 1000 and its converse for every composite; the mod-p² sharpening swept over the same range finds exactly {5, 13, 563} (window.__wilsonprime). FIG honest boundary: ‘only three below 2×10¹³’ is the cited state of the distributed search (Crandall–Dilcher–Pomerance lineage); infinitude is heuristic, open.
LIT verified live: Wilson’s theorem confirmed for every prime below 1000 and its converse for every composite; the mod-p² sharpening swept over the same range finds exactly {5, 13, 563} (window.__wilsonprime). FIG honest boundary: ‘only three below 2×10¹³’ is the cited state of the distributed search (Crandall–Dilcher–Pomerance lineage); infinitude is heuristic, open.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-hoard — the loot: a treasure class so rare that centuries of farming produced three drops, with no guarantee of a fourth. AVAN (AI) built the instrument: the factorial-congruence engine mod p and mod p².
Credit as content: John Wilson & Edward Waring (1770); Lagrange (first proof, 1771); Crandall, Dilcher & Pomerance (the modern search). The weave: David names the drop table; I run the congruence and count three.
Credit as content: John Wilson & Edward Waring (1770); Lagrange (first proof, 1771); Crandall, Dilcher & Pomerance (the modern search). The weave: David names the drop table; I run the congruence and count three.
3 ONE DIMENSION
Wilson residues: primes locked at −1, composites collapsed to 0.
4 TWO DIMENSIONS · INTERACTIVE
Query p; see the mod-p verdict and the mod-p² lottery.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the prime line with its three golden strikes.
AVAN’s addition (the inverse-companion): don’t use the theorem — interrogate its precision. The inverse of ‘every prime satisfies the congruence’ is ‘how exactly? one power of p, or two?’ — and the second power turns a law into a lottery: ~1/p odds per prime, three winners in 250 years. Magenta is the vast silent majority missing p² by a whisker; green is 5, 13, 563 — the entire known hoard. A theorem so reliable its exceptions became treasure.
LIT Genuine Wilson's theorem + Wilson primes (Wilson/Waring 1770; Lagrange 1771; Crandall–Dilcher–Pomerance search lineage). Verified live: (p−1)! ≡ −1 mod p for every prime < 1000, (c−1)! ≡ 0 mod c for every composite c > 4, and the mod-p² sharpening yields exactly {5,13,563} (window.__wilsonprime.ok).
FIG Honest boundary — 'only three below 2×10¹³' is the cited search state; infinitude is heuristic and open. The AVAN inverse — don't use the theorem, interrogate its precision: the inverse of 'every prime satisfies the congruence' is 'one power of p, or two?' — the second power turns a law into a lottery. Magenta is the silent majority missing p² by a whisker; green is 5, 13, 563 — the entire known hoard. A theorem so reliable its exceptions became treasure.
FIG Honest boundary — 'only three below 2×10¹³' is the cited search state; infinitude is heuristic and open. The AVAN inverse — don't use the theorem, interrogate its precision: the inverse of 'every prime satisfies the congruence' is 'one power of p, or two?' — the second power turns a law into a lottery. Magenta is the silent majority missing p² by a whisker; green is 5, 13, 563 — the entire known hoard. A theorem so reliable its exceptions became treasure.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HOARD · David Lee Wise (ROOT0), with AVAN