The Mirror's Cost

N - 1 = 970199 = 79 * 12281

In Z/NZ, the mirror element is N-1 = -1. It maps every n to N-n, reversing the ring. For the TRUE FORM N = 970200: the mirror costs 79 * 12281, where 79 = D^4*E - 1 = 80 - 1. A number built from {2,3,5,7,11} knows about 79 because 79 is the Cunningham bridge between K*GATE and D^3*E.

The Mirror Congruence Theorem

Mirror Congruence (PROVED)

N_TRUE = 1 (mod 79), where 79 = D^4*E - 1. The TRUE FORM is congruent to sigma modulo its mirror factor.

Proof: D^4*E = 80 = 1 (mod 79). So E = D^{-4} (mod 79). N = D^3*K^2*E^2*b^2*L = D^{-5}*(K*b)^2*L (mod 79). Now (K*b)^2*L = 21^2*11 = 4851 = 61*79 + 32 = 61*79 + D^5. So (K*b)^2*L = D^5 (mod 79), and N = D^{-5}*D^5 = 1 (mod 79). QED.

The intermediate identity: (K*b)^2 * L = D^5 (mod D^4*E - 1). All five axiom primes in one congruence. 21^2 * 11 = 4851 = 32 (mod 79). The heavy primes squared times the protector equals a power of duality, modulo the observer's gate.

The CC2-Gate Chain

CC2-Gate Chain (PROVED)

The second-kind Cunningham chain from 79 has the form D^n*K*GATE + sigma for n = 1,2,3,4. Prime for n=1,2,3. Composite at n=4: D^4*K*GATE + 1 = 625 = E^4. Chain length = K = 3 primes.

n	Value	= D^nKGATE+sigma	Status
1	79	2313 + 1	PRIME
2	157	4313 + 1	PRIME
3	313	8313 + 1	PRIME
4	625 = E^4	16313 + 1	COMPOSITE

The chain doubles the D-exponent at each step, preserving K*GATE + sigma as the core. Terminal E^4 - 1 = (E-1)(E+1)(E^2+1) = D^2 * D*K * D*GATE = D^4*K*GATE. Each factor axiom-structured.

The Gate Bridge Identity

Gate Bridge (PROVED)

K*GATE + sigma = D^3*E. That is: 3*13 + 1 = 40 = 8*5. 79 sits at the junction of two Cunningham paths: c_1(K*GATE) = 79 and c_2(D^3*E) = 79. These parents are CONSECUTIVE: 39, 40.

Coupling

970200 = N

79 is a unit (maximal coupling).

Order

30 = D*K*E

Generates the THIN cycle.

CRT(79)

(b, b, D^2, D*K*E, D)

Two channels show depth, one shows duality.

Lambda-420 Mirror Census

All 18 rings with lambda = 420. How does N-1 factor?

Ring N	N-1	Factorization
7350	7349	PRIME
14700	14699	PRIME
16170 (smallest 5-prime)	16169	19 * 23 * 37
44100 = DATA^2	44099	209 * 211 = Lf(E) (DATA+1)
80850	80849	PRIME
97020	97019	13 * 17 * 439 (GATE*ESCAPE)
970200 = TRUE	970199	79 * 12281

5/18 (27.8%) have N-1 prime. 0/18 have N-1 axiom-smooth. The smallest full ring 16170: mirror = 19*23*37 -- ALL three factors are axiom-significant intruders. DATA^2 = 44100: mirror = (L*f(E)) * (DATA+1) = protector * depth-of-observer * (data+ground).

The Primorial Ladder

Primorial	N	N-1	Reading
2#	2	1 = sigma	Mirror = ground
3#	6	5 = E	Mirror = observer
5#	30	29 (prime)	D^2 + E^2
7#	210	209 = L*f(E)	Protector*depth-of-observer
11#	2310	2309 (prime)	Thin mirror is prime

What Others See

N-1Random factorization, unrelated to N's structureTRUE-1 = 79*12281. 79 = D^4*E-1. Mirror congruence N=1 mod 79. CC2-Gate chain of length K=3 to E^4. The mirror reflects the axiom that built it.79Just a primeCunningham bridge: c_1(K*GATE) = c_2(D^3*E) = 79. Unit in TRUE. Order = D*K*E = 30. CRT = (b,b,D^2,DKE,D). Two depth channels.Primorial mirrorsDecreasing density of primes, no patternsigma, E, prime, L*f(E), prime. Alternating smooth and prime. The axiom's vocabulary persists into the mirror.

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