What does it cost to create -1? The factorization of N-1 carries axiom structure.
In Z/NZ, the mirror element is N-1 = -1. It maps every element n to N-n, reversing the ring.
For the TRUE FORM N = 970200 = D3K2E2b2L:
970199 = 79 × 12281
79 = D4E - 1 = 80 - 1. The mirror costs exactly (D4E - 1) times a prime.
Why should a number built from {2,3,5,7,11} know about 79? Because 79 is the Cunningham bridge between K*GATE and D3*E.
Theorem. NTRUE ≡ 1 (mod 79), where 79 = D4E - 1.
Equivalently: 79 divides N - 1. The TRUE FORM is congruent to sigma modulo its mirror factor.
Proof. Since 79 is prime and D4E = 80 ≡ 1 (mod 79), we have E ≡ D-4 (mod 79).
N = D3 · K2 · E2 · b2 · L ≡ D3 · K2 · D-8 · b2 · L = D-5 · (Kb)2 · L (mod 79)
Now (Kb)2 · L = 212 · 11 = 4851 = 61 × 79 + 32 = 61 × 79 + D5.
So (Kb)2L ≡ D5 (mod 79), and N ≡ D-5 · D5 = 1 (mod 79). QED.
(K · b)2 · L ≡ D5 (mod D4E - 1)
212 × 11 = 4851 ≡ 32 = 25 (mod 79). All five axiom primes in one congruence.
The "heavy" primes squared times the protector equals a power of duality, modulo the observer's gate.
Theorem (CC2-Gate Chain). The second-kind Cunningham chain from 79 has the form:
Dn · K · GATE + σ for n = 1, 2, 3, 4
Prime for n = 1,2,3. Composite at n = 4: D4 · K · GATE + 1 = 625 = E4.
Chain length = K = 3 (primes only).
| n | Value | = Dn·K·GATE + σ | Status |
|---|---|---|---|
| 1 | 79 | 2 × 3 × 13 + 1 | PRIME |
| 2 | 157 | 4 × 3 × 13 + 1 | PRIME |
| 3 | 313 | 8 × 3 × 13 + 1 | PRIME |
| 4 | 625 = E4 | 16 × 3 × 13 + 1 | COMPOSITE |
Why does it work? c2(Dn · K · GATE + 1) = 2(Dn · K · GATE + 1) - 1 = Dn+1 · K · GATE + 1.
The chain doubles the D-exponent at each step, preserving K · GATE + σ as the core.
Theorem. K · GATE + σ = D3 · E. (3 × 13 + 1 = 40 = 8 × 5)
Proof: GATE = D2 + K2. So K(D2 + K2) + 1 = KD2 + K3 + 1 = 12 + 27 + 1 = 40 = D3(D+K) = D3E. QED.
The chain stops at E4 = 625 because:
E4 - 1 = (E-1)(E+1)(E2+1) = D2 · (D·K) · (D·GATE) = D4 · K · GATE
Each factor is axiom-structured: E-1 = D2, E+1 = D·K, E2+1 = D·GATE.
The observer's fourth power decomposes into duality, closure, and the gate.
79 sits at the junction of two Cunningham paths:
First kind: c1(39) = 2 × 39 + 1 = 79, where 39 = K × GATE = 3 × 13.
Second kind: c2(40) = 2 × 40 - 1 = 79, where 40 = D3 × E = 8 × 5.
These parents are consecutive: K · GATE = 39, D3 · E = 40. The Gate Bridge Identity says they differ by σ = 1.
In the TRUE FORM: coupling(79) = 970200 = N (maximal: 79 is a unit). Order = 30 = D·K·E.
CRT(79) = (b, b, D2, D·K·E, D). Two channels show depth, one shows duality.
All 18 rings with Carmichael lambda = 420. How does N-1 factor?
| Ring N | Exponents | N-1 | Factorization |
|---|---|---|---|
| 7,350 | 2·3·52·72 | 7,349 | PRIME |
| 14,700 | 22·3·52·72 | 14,699 | PRIME |
| 16,170 | 2·3·5·72·11 | 16,169 | 19 × 23 × 37 |
| 22,050 | 2·32·52·72 | 22,049 | 17 × 1297 |
| 29,400 | 23·3·52·72 | 29,399 | PRIME |
| 32,340 | 22·3·5·72·11 | 32,339 | 73 × 443 |
| 44,100 | 22·32·52·72 | 44,099 | 11 × 19 × 211 |
| 48,510 | 2·32·5·72·11 | 48,509 | 179 × 271 |
| 64,680 | 23·3·5·72·11 | 64,679 | PRIME |
| 80,850 | 2·3·52·72·11 | 80,849 | PRIME |
| 88,200 | 23·32·52·72 | 88,199 | 89 × 991 |
| 97,020 | 22·32·5·72·11 | 97,019 | 13 × 17 × 439 |
| 161,700 | 22·3·52·72·11 | 161,699 | 97 × 1667 |
| 194,040 | 23·32·5·72·11 | 194,039 | 29 × 6691 |
| 242,550 | 2·32·52·72·11 | 242,549 | 59 × 4111 |
| 323,400 | 23·3·52·72·11 | 323,399 | 19 × 17021 |
| 485,100 | 22·32·52·72·11 | 485,099 | 227 × 2137 |
| 970,200 | 23·32·52·72·11 | 970,199 | 79 × 12281 |
5/18 (27.8%) have N-1 prime. 0/18 have N-1 axiom-smooth.
Smallest full ring: N = 16170 (all 5 primes), N-1 = 19 × 23 × 37
19 = f(E) = depth quadratic of observer
23 = pK2 = CC1(D)[3] = Cunningham boundary
37 = depth return prime (f(37) = L3, order = 420 = lambda)
All three factors are axiom-significant. The smallest mirror speaks the axiom's language.
DATA squared: N = 44100 = DATA2 = 2102, N-1 = (DATA-1)(DATA+1) = 209 × 211
209 = L · f(E) = protector × depth-of-observer
211 = DATA + 1 = prime
The data ring's square minus one = (protector × depth-quadratic) × (data + ground state).
GATE × ESCAPE: N = 97020 (22·32·5·72·11), N-1 = 13 × 17 × 439
13 = GATE. 17 = ESCAPE = D+K+E+b.
The two smallest non-smooth partial sums divide the mirror of this ring.
| Primorial | N | N-1 | Factorization | Axiom reading |
|---|---|---|---|---|
| 2# | 2 | 1 | σ | Mirror = ground state |
| 3# | 6 | 5 | E (prime) | Mirror = observer |
| 5# | 30 | 29 | prime | Full sum D2+E2 |
| 7# | 210 | 209 | L × f(E) | Protector × depth-of-observer |
| 11# | 2310 | 2309 | prime | Thin mirror is prime |
The primorials alternate: smooth (σ), prime (E), prime (29), smooth (L·f(E)), prime (2309).
Enter a ring size N (product of prime powers from {2,3,5,7,11}):
The mirror automorphism: The Mirror
Cunningham chains that generate the axiom: The Two Chains
Depth quadratic f(p)=p2-p-1: Why 37 Comes Home
The universal boundary at GATE=13: The Universal Boundary
CRT decomposition anatomy: CRT Anatomy
The mirror reflects everything — including the axiom that built it.
79 connects closure*gate to duality3*observer. The cost of -1 is structural.
sigma/sigma = sigma. The web holds.