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 Z/970,200 (N = 970,200): the mirror costs 79 * 12,281, where 79 = 2^4*5 - 1 = 80 - 1. CRT(-1) = (7,8,24,48,10,12): all channels at maximum. A number built from {2,3,5,7,11} knows about 79 because 79 sits at the junction of 3*13 = 39 and 2^3*5 = 40.

The Mirror Congruence Theorem

Mirror Congruence (PROVED)
12,612,600 = 1 (mod 79), where 79 = 2^4*5 - 1. The ring Z/12,612,600 is congruent to 1 modulo its mirror factor.

Proof: 2^4*5 = 80 = 1 (mod 79). So 5 = 2^{-4} (mod 79). N = 2^3*3^2*5^2*7^2*11 = 2^{-5}*(3*7)^2*11 (mod 79). Now (3*7)^2*11 = 21^2*11 = 4851 = 61*79 + 32 = 61*79 + 2^5. So (3*7)^2*11 = 2^5 (mod 79), and N = 2^{-5}*2^5 = 1 (mod 79). QED.

The intermediate identity: (3*7)^2 * 11 = 2^5 (mod 2^4*5 - 1). All five primes {2,3,5,7,11} in one congruence. 21^2 * 11 = 4851 = 32 (mod 79), where 32 = 2^5.

The Second-Kind Chain from 79

CC2 Chain from 79 (PROVED)
The second-kind Cunningham chain from 79 has the form 2^n*3*13 + 1 for n = 1,2,3,4. Prime for n=1,2,3. Composite at n=4: 2^4*3*13 + 1 = 625 = 5^4. Chain length = 3 primes.
nValue= 2^n*3*13 + 1Status
1792*3*13 + 1PRIME
21574*3*13 + 1PRIME
33138*3*13 + 1PRIME
4625 = 5^416*3*13 + 1COMPOSITE

The chain doubles the power of 2 at each step, preserving 3*13 + 1 as the core. Terminal: 5^4 - 1 = 624 = (5-1)(5+1)(5^2+1) = 4*6*26 = 2^4*3*13. Each factor decomposes into the ring's primes.

The 39-40 Junction

39-40 Junction (PROVED)
3*13 + 1 = 40 = 2^3*5. The numbers 39 and 40 are consecutive, and 79 sits at their junction: c_1(39) = 79 and c_2(40) = 79, linking two Cunningham paths.
Coupling
970,200
79 is a unit (maximal coupling).
Order
30 = 2*3*5
Generates the Z/2,310 cycle.
CRT(79)
(7, 7, 4, 30, 2)
Two channels show 7, one shows 2.

Lambda-420 Mirror Census

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

Ring NN-1Factorization
73507349PRIME
1470014699PRIME
16170 (smallest 5-prime)1616919 * 23 * 37
44100 = 210^244099209 * 211 = (11*19) * 211
8085080849PRIME
970209701913 * 17 * 439
970200 (Z/970,200)97019979 * 12281

5/18 (27.8%) have N-1 prime. 0/18 have N-1 smooth over {2,3,5,7,11}. The smallest full ring 16170: mirror = 19*23*37 -- all three factors are significant intruders. 210^2 = 44100: mirror = (11*19) * 211.

The Primorial Ladder

PrimorialNN-1Reading
2#21Mirror = 1
3#65Mirror = 5
5#3029 (prime)4 + 25 = 29
7#210209 = 11*1911 * 19
11#23102309 (prime)Mirror is prime

Explore: Mirror Cost

Enter any N to factorize N-1 (the mirror element). Check the mirror congruence mod 79.

Enter N:

Try: N=970200 (Z/970,200, 79*12281), N=2310 (Z/2,310, 2309 prime), N=210 (Z/210, 209=11*19), N=6 (mirror=5).

Contrast Table

N-1Random factorization, unrelated to N's structure970,199 = 79*12,281. 79 = 2^4*5-1. Mirror congruence N=1 mod 79. Second-kind Cunningham chain of length 3 to 5^4 = 625. The mirror reflects the structure that built it.79Just a primeCunningham junction: c_1(39) = c_2(40) = 79. Unit in Z/12,612,600. Order = 2*3*5 = 30. CRT = (7,7,4,30,2). Two channels show 7.Primorial mirrorsDecreasing density of primes, no pattern1, 5, prime, 11*19, prime. Alternating smooth and prime. The ring's primes persist into the mirror.

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