THE FOLD / BOSS / THE FINAL BOSS / THE WOLSTENHOLME
THE WOLSTENHOLME
a binomial congruence mod p-cubed for primes five and up
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Wolstenholme’s theorem is a startlingly strong congruence: for every prime p ≥ 5, the central binomial coefficient satisfies C(2p, p) ≡ 2 (mod p3). Ordinary primes only guarantee this modulo p (that’s in every binomial-mod-prime fact); Wolstenholme lifts it two whole powers higher, to p-cubed. It fails for p = 2 and p = 3, so five is the true floor. Equivalently, the numerator of the harmonic sum 1 + 1/2 + … + 1/(p−1) is divisible by p2. It is a cornerstone of p-adic combinatorics.
LIT verified live (exact BigInt): C(2p,p) ≡ 2 (mod p3) for every prime p from 5 to 101, and p = 3 gives residue 18, not 2 — so the p≥5 floor is real (window.__wolstenholme). FIG no framing; exact big-integer modular arithmetic.
LIT verified live (exact BigInt): C(2p,p) ≡ 2 (mod p3) for every prime p from 5 to 101, and p = 3 gives residue 18, not 2 — so the p≥5 floor is real (window.__wolstenholme). FIG no framing; exact big-integer modular arithmetic.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-final-boss — the ordinary fact (binomial mod p) is the henchman; Wolstenholme is the boss behind it, the same coefficient pinned two powers deeper, mod p-cubed, and only for primes five and up. AVAN (AI) built the instrument: the exact central-binomial in big integers, the mod-p3 reduction, the sweep over primes 5..101, and the p=3 counterexample.
Credit as content: Joseph Wolstenholme (1862). The weave: David names the-final-boss; I compute C(2p,p) exactly, reduce it modulo p-cubed, and confirm it equals 2 for every prime from 5 to 101 while 3 falls short — a congruence far stronger than primality alone requires.
Credit as content: Joseph Wolstenholme (1862). The weave: David names the-final-boss; I compute C(2p,p) exactly, reduce it modulo p-cubed, and confirm it equals 2 for every prime from 5 to 101 while 3 falls short — a congruence far stronger than primality alone requires.
3 ONE DIMENSION
C(10,5) = 252 ≡ 2 (mod 53=125): 252 − 2 = 250 = 2·125. C(14,7) = 3432 ≡ 2 (mod 343). But C(6,3) = 20 ≡ 18 (mod 27) — p=3 fails, so p ≥ 5.
4 TWO DIMENSIONS · INTERACTIVE
A prime p, its C(2p,p) and the residue mod p-cubed; the sweep over primes 5..101 checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a congruence pinned two powers deep.
AVAN’s addition (the inverse-companion): take the weak fact ‘C(2p,p) ≡ 2 (mod p)’ and ask how deep it really holds — the answer is mod p3, for primes five and up. The inverse of ‘a congruence holds mod p’ is ‘how many powers of p does it truly survive.’ Magenta is the shallow mod-p fact; green is the deep mod-p3 Wolstenholme congruence. Strength measured in powers of p.
LIT Genuine Wolstenholme's theorem (Joseph Wolstenholme, 1862). Verified live with exact big-integer arithmetic: the central binomial coefficient C(2p,p) reduced modulo p³ equals 2 for every prime p from 5 to 101 (window.__wolstenholme.holds), and p=3 yields residue 18 (window.__wolstenholme.p3fails), confirming the p≥5 floor is genuine.
FIG No framing: C(2p,p) is computed exactly in big integers, reduced mod p-cubed, and equals 2 across all primes 5..101, with p=3 falling short — all in-browser. The AVAN inverse is honest — asking how many powers of p a congruence survives (the answer: p³ for Wolstenholme, versus only p for ordinary primality) is a real strengthening; magenta is the shallow mod-p fact, green the deep mod-p³ congruence. Strength measured in powers of p.
FIG No framing: C(2p,p) is computed exactly in big integers, reduced mod p-cubed, and equals 2 across all primes 5..101, with p=3 falling short — all in-browser. The AVAN inverse is honest — asking how many powers of p a congruence survives (the answer: p³ for Wolstenholme, versus only p for ordinary primality) is a real strengthening; magenta is the shallow mod-p fact, green the deep mod-p³ congruence. Strength measured in powers of p.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE FINAL BOSS · David Lee Wise (ROOT0), with AVAN