Mod 2: eigenvalue 7. Mod 2^a: eigenvalue 6. Shift = 1.
Every ring has a midpoint: N/2. Where every odd channel sees zero and only the mod-2 channel dissents. The eigenvalue at this equator reveals the ring's identity. As channels are added, the midpoint walks through every integer from -1 to 8.
The Midpoint CRT Theorem
Midpoint CRT (PROVED)
For N = 2^a * m (m odd), the CRT decomposition of N/2 has: mod-2^a 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 mod-2 channel contribution (always negative) and the number of odd channels (always positive at zero).
The Midpoint Eigenvalue Theorem
Midpoint Eigenvalue (PROVED)
For a ring with k CRT channels: Mod 2 (squarefree): midpoint eigenvalue = 2k - 3 (always odd). Mod 2^a (a >= 2): midpoint eigenvalue = 2k - 4 (always even). Shift = 1 (universal, independent of k). Proof: Each odd channel with residue 0 contributes 2cos(0) = 2. The mod-2 channel contributes cos(pi) = -1. The mod-2^a channel contributes 2cos(pi) = -2. Mod 2: -1 + 2(k-1) = 2k-3. Mod 2^a: -2 + 2(k-1) = 2k-4. Difference = 1. 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.
Value
Factorization
Source
Ring
-1
(negative)
mod 2, k=1
Z/2
0
0
mod 2^a, k=2
Z/4*Z/9
1
1
mod 2, k=2
Z/6
2
2
mod 2^a, k=3
Z/4*Z/9*Z/25
3
3
mod 2, k=3
Z/30
4
2^2
mod 2^a, k=4
Z/8*Z/9*Z/25*Z/49
5
5
mod 2, k=4
Z/210
6
2*3
mod 2^a, k=5
Z/970,200
7
7
mod 2, k=5
Z/2,310
8
2^3
mod 2^a, k=6
Z/12,612,600
The mod-2 ladder traces odd values: -1, 1, 3, 5, 7. The mod-2^a ladder traces even values: 0, 2, 4, 6, 8. Together they cover every integer from -1 to 8 = 2^3, spanning from the mirror to the fattening ceiling.
Identity Conservation Law
Identity Conservation (PROVED)
Raising the 2-channel from mod 2 to mod 2^a (a >= 2): max eigenvalue shifts +1 (from 2k-1 to 2k). Midpoint eigenvalue shifts -1 (from 2k-3 to 2k-4). The same 1 appears at both transitions. The identity element is redistributed, not created or destroyed.
Z/2,310 equator
7
The squarefree ring's midpoint eigenvalue is 7 (at k=5).
Z/970,200 equator
6 = 2*3
The prime-power ring's midpoint eigenvalue is 6 = 2*3 (at k=5).
The shift
1
Raising the 2-channel costs exactly 1 at the equator. Universal.
Max eigenvalue
9 (mod 2) to 10 (mod 2^a)
The +1 gained at the pole matches the -1 lost at the equator.
Contrast
Aspect
Standard view
Ring structure
N/2
Just an element like any other
The equator: all odd channels agree, only the mod-2 channel dissents
Eigenvalue
Some number
Traces the chain as channels grow: -1, 0, 1, 2, 3, ..., 8
Mod 2 vs mod 2^a
Minor technical distinction
Two ladders interleaving to cover all integers from -1 to 8
The shift
Off by 1
The identity (= 1) is conserved between equator and pole
Explore: Midpoint Eigenvalue
Enter the number of CRT channels k. See the midpoint eigenvalue for both mod-2 and mod-2^a rings.
Channels k:
Try: k=1 (value -1), k=2 (value 1 or 0), k=5 (value 7 or 6), k=6 (value 8 at Z/12,612,600).