The Mirror

n <-> N - n

Every element n in the ring has a spectral twin: N - n. The map n -> N - n flips each CRT residue independently: r_i becomes q_i - r_i. This preserves eigenvalues (cosine is an even function), coupling (gcd is symmetric), and class. It is the additive inverse -- the unique involution that keeps the full CRT spectrum intact.

In each odd channel Z/p^e, the residue -1 = p^e - 1 is the unique non-trivial square root of 1. In Z/8, -1 = 7 is one of four square roots of 1 ({1, 3, 5, 7}). CRT reconstructs the global -1 from these per-channel residues.

-1 Is Not Zero

In Z/2,310: -1 = 2,309 (prime). In Z/12,612,600: -1 = 12,612,599. The mirror of 1 is a unit with the maximum possible residue in every channel.

Mirror Spectrum Theorem (PROVED)

CRT(N-n) = (q_1 - r_1, q_2 - r_2, ...). The mirror flips each channel independently. Eigenvalues are preserved because cosine is even. The mirror of any element has the same eigenvalue, coupling, and class.

CRT = (1, 1, 1, 1, 1, 1)

All channels at minimum.

-1 = 12,612,599

CRT = (7, 8, 24, 48, 10, 12)

Each channel at p^e - 1. Maximum residue.

The Foundation Triangle

Three elements form the foundation: mirror (-1), void (0), sigma (+1). CRT distances form an isosceles triangle:

dist(0, +1)

5

Void to identity.

dist(0, -1)

5

Void to mirror. Same distance -- no chirality.

dist(+1, -1)

7 (in Z/2,310)

Identity to mirror. 10 in Z/12,612,600.

Triangle Universality (PROVED)

Isosceles across all 108 rings with lambda = 420. Z/2,310: (5,5,7). Z/12,612,600: (5,5,10). Zero is always equidistant from both +1 and -1.

The Cunningham Mirror Law

p	D*p	c(p)	D*p mod c(p)
2	4	5	-1 (mod 5)
3	6	7	-1 (mod 7)
5	10	11	-1 (mod 11)

Cunningham Mirror Law (PROVED)

c(p) = 2p + 1, so 2p = c(p) - 1 = -1 mod c(p). Each chain prime is born at the mirror position (-1) of its predecessor doubled. The chain is a chain of reflections.

Hemisphere Partition

H+ = {1,...,N/2-1} and H- = {N/2+1,...,N-1} have identical spectra. Total eigenvalue sum = 0.

Fixed: 0

eigenvalue 10

Maximum eigenvalue. Zero is brightest.

Fixed: N/2

eigenvalue 7

The midpoint. Coupling = 2.

Each hemisphere

1,154 elements (Z/2,310)

240 units each. Eigenvalue sum: -8.000.

Total

sum = 0

Perfect spectral balance.

Mirror Distance Theorem (PROVED)

Maximum CRT distance between n and N-n is 4, occurring iff n is coprime to N. In Z/2,310: phi(N) = 480 units achieve this maximum self-distance.

Explore: Mirror Pairs

Enter any n. See n and N-n side by side: CRT decomposition, coupling. The mirror preserves everything structural. Try 137, 42, 1,576,576 (the projector 2^420), 606,376.

Element n:

Dual Bloom

5^2 = 0 mod 25 means the mod-25 channel cannot see itself. The mirror provides a second view at zero cost. Mod-11 checks within each view. Mod-13 checks between views. Dual parity = 100% detection.

Paradigm Contrast

Aspect	Standard	Axiom
Reflection	Convenient property	ONLY spectrum-preserving involution
Negation	An operation	Free second view. Fixes the mod-25 self-blindness at zero cost.
Balance	Some systems	Every twin: same eigenvalue, same coupling, same class.
Foundation triangle	{-1, 0, +1}	Isosceles: dist(0,+1) = dist(0,-1) = 5. Universal across 108 rings.
Chain	Primes given	Each prime born at mirror position of its predecessor doubled: 2p = -1 mod c(p).
ECC	Add redundancy	Free cross-check via dual parity (mod-11 + mod-13).

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