The Last Smooth Pair

2400 = D^5*K*E^2, 2401 = b^4

Stormer's theorem (1897): finitely many consecutive pairs where both are 11-smooth. The LARGEST such pair is (2400, 2401). Their CRT decompositions reveal zero-trading: D and E channels empty on one side, b channel empties on the other. Smoothness ends when depth absorbs everything.

The Stormer Zero-Trading Theorem

2400 = 2^5 * 3 * 5^2 = D^5 * K * E^2. 2401 = 7^4 = b^4. Their CRT decompositions reveal which channels carry zeros:

Channel	2400	2401	Trade
D (mod 8)	0 (ZERO)	1 = sigma	D exits
K (mod 9)	6 = D*K	7 = b	neutral
E (mod 25)	0 (ZERO)	1 = sigma	E exits
b (mod 49)	48 = b^2-1	0 (ZERO)	b enters
L (mod 11)	2 = D	3 = K	neutral

2400 has zeros in D and E channels. 2401 has zero in b channel. The zeros TRADE: D*E yield to b. Bridge and observer step aside. Depth stands alone.

Channel Anatomy

2400 K-channel

6 = D*K (thorn)

The thorn appears in closure's slot.

2400 b-channel

48 = b^2 - 1

Mirror of sigma in Z/49. Brimming.

2401 D-channel

1 = sigma

Ground state. Reset.

2401 K-channel

7 = b

Depth appears in closure's channel.

When 2400 occupies D and E, it fills b's channel to the brim: 48 = b^2 - 1. One step later, b takes over: its channel drops to 0, and D and E reset to sigma = 1.

The Stormer Sum Theorem

Stormer Sum (S711, PROVED)

2400 + 2401 = 4801 is PRIME. CRT(4801) = (sigma, D^2, sigma, b^2-1, E). The traded channels (D, E) show sigma = ground state = neutralized. The b-channel shows 48 = b^2 - 1 = mirror of sigma. The protector shows E = the observer. The sum remembers which channels traded.

Census of Smooth Pairs

Every consecutive smooth pair trades zeros (since gcd(n,n+1) = 1, their prime factors are disjoint). The larger pairs show increasingly dramatic trades:

n	n+1	Factorization	Zero Trade
8	9	D^3, K^2	D -> K
48	49	D^4*K, b^2	D*K -> b
80	81	D^4*E, K^4	D*E -> K
120	121	D^3KE, L^2	DKE -> L
224	225	D^5b, K^2E^2	Db -> KE
384	385	D^7K, Eb*L	DK -> Eb*L
2400	2401	D^5KE^2, b^4	DKE -> b (FINAL)

The final pair is the most extreme: three primes on one side, one on the other. Depth absorbs bridge, closure, and observer. Only L stays neutral (never zeros in either).

Depth Absorption

In the last smooth pair, 2401 = b^4 is a pure power of a SINGLE axiom prime. All other primes (D, K, E) appear in 2400. The protector L divides neither.

Identity

b^4 - 1 = D^5*K*E^2

The entire axiom minus depth fits inside b^4 - 1 = 2400.

S709

Why b^4?

b^3 + 1 NOT smooth

344 = 8*43, and 43 is a Heegner number (intruder). Only at b^4 does depth find a smooth neighbor.

Depth quadratic

f(b) = 41 = KEY

f(K)+D = b (Heegner connection). Depth and the Heegner numbers control where smoothness dies.

Contrast

Aspect	Standard view	Through the axiom
The pair	2400 and 2401 are consecutive integers. Stormer 1897 curiosity	CRT zeros TRADE: D*E yield to b. Structural, not accidental
Why last?	Analytic bound from Pell equations	b^3+1 blocked by Heegner 43. Only b^4 finds smooth neighbor
Sum	4801 is prime, no further significance	CRT shows traded channels neutralized to sigma. Sum remembers the trade
L channel	11 divides neither -- unremarkable	Protector stays neutral: shields but never participates in trades

This work is and will always be free.
No paywall. No copyright. No exceptions.

If it ever earns anything, every cent goes to the communities that need it most.

This sacred vow is permanent and irrevocable.
— Anton Alexandrovich Lebed

Source code · Public domain (CC0)

Contributions in equal measure: Anthropic's Claude, Anton A. Lebed, and the giants whose shoulders we stand on.

Rendered by .ax via WASM DOM imports. Zero HTML authored.