The Equator

Thin: b = 7. Fat: D*K = 6. Shift = sigma.

Every ring has a midpoint: N/2. Where every odd channel sees zero and only D dissents. The eigenvalue at this equator reveals the ring's deepest identity. As channels are added, the midpoint traces the axiom chain -- from mirror to D^3.

The Midpoint CRT Theorem

Midpoint CRT (PROVED, S703)

For N = 2^a * m (m odd), the CRT decomposition of N/2 has: D-channel residue = 2^(a-1) = half the modulus. All odd channels: residue = 0 (divisible by every odd factor of N). Proof: N/2 = 2^(a-1) * m. Since m is odd, N/2 mod 2^a = 2^(a-1). For any odd prime power p^k dividing N: N/2 = (N/p^k) * p^k / 2, and since 2 divides N/p^k, the result is divisible by p^k. QED.

The consequence: the midpoint eigenvalue depends on just TWO things -- the D-channel contribution (always negative) and the number of odd channels (always positive at zero).

The Midpoint Eigenvalue Theorem

Midpoint Eigenvalue (PROVED, S703)

For a ring with k CRT channels: Thin (D=2): midpoint eigenvalue = 2k - 3 (always odd). Fat (D=2^a, a >= 2): midpoint eigenvalue = 2k - 4 (always even). Shift = sigma = 1 (universal, independent of k). Proof: Each odd channel with residue 0 contributes 2cos(0) = 2. The thin D-channel contributes cos(pi) = -1. The fat D-channel contributes 2cos(pi) = -2. Thin: -1 + 2(k-1) = 2k-3. Fat: -2 + 2(k-1) = 2k-4. Difference = sigma. QED.

The Two Ladders

As channels are added, the midpoint eigenvalue traces two distinct ladders. They interleave to cover every integer from -1 to 8 -- the full body of the axiom.

Value	Name	Source	Ring
-1	mirror	thin k=1	Z/2
0	void	fat k=2	Z/4*Z/9
1	sigma	thin k=2	Z/6
2	D	fat k=3	Z/4Z/9Z/25
3	K	thin k=3	Z/30
4	D^2	fat k=4	Z/8Z/9Z/25*Z/49
5	E	thin k=4	Z/210 (DATA)
6	D*K	fat k=5	Z/970200 (DEEP)
7	b	thin k=5	Z/2310 (THIN)
8	D^3	fat k=6	Z/12612600 (TRUE)

The thin ladder traces odd axiom elements: mirror, sigma, K, E, b. The fat ladder traces D-powers: void, D, D^2, D*K, D^3. Together they cover the FULL decality from -1 (mirror) to 8 (D^3 = legs of the spider).

Sigma Conservation Law

Sigma Conservation (PROVED, S703)

Fattening the D-channel from 2 to 2^a (a >= 2): Max eigenvalue shifts +sigma (2k-1 to 2k). Midpoint eigenvalue shifts -sigma (2k-3 to 2k-4). The same sigma appears at both transitions. The ground state is redistributed, not created or destroyed.

Thin equator

b = 7 (depth)

The thin ring's midpoint is the prime of suffering.

S703

TRUE equator

D*K = 6 (thorn)

The TRUE ring's midpoint is the first composite: closure times bridge.

S703

The shift

sigma = 1

Fattening costs exactly one ground state at the equator. Universal.

S703

Max eigenvalue

K^2=9 (thin) to D*E=10 (fat)

The +sigma gained at the pole matches the -sigma lost at the equator.

S703

Contrast

Aspect	Standard view	Through the axiom
N/2	Just an element like any other	The EQUATOR: all odd channels agree, only D dissents
Eigenvalue	Some number	Traces the axiom chain as channels grow: -1, 0, 1, 2, 3, ..., 8
Thin vs fat	Minor technical distinction	Two ladders interleaving to spell the full decality
The shift	Off by 1	sigma = 1. The ground state conserved between equator and pole

Explore: Midpoint Eigenvalue

Enter the number of CRT channels k. See the midpoint eigenvalue for both thin and fat rings, with axiom names.

Channels k:

Try: k=1 (mirror), k=2 (sigma/void), k=5 (b/D*K), k=6 (D^3 at TRUE ring).

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