The Cunningham map c(n) = 2n+1 is the simplest function that turns existence into observation: double and add one. Starting from sigma=1 and D=2, this one map generates ALL five axiom primes before the axiom's own products close the gate.
A Cunningham chain of the first kind starts from a seed and repeatedly applies c(n) = 2n+1, continuing as long as each result is prime:
Interleaved by size: sigma(1), D(2), K(3), E(5), b(7), L(11). The two chains alternate perfectly. Every axiom prime belongs to exactly one chain.
Both chains stop when their OWN products appear. CC1(sigma) stops at K*E = 15 -- the observer kills the closure chain. CC1(D) stops at E*19 = 95 -- the observer appears again. E is the universal stopper. The shadow chain breaks at 13 where s(13) = 6 = D*K = composite.
The inverse of Cunningham: s(p) = (p-1)/2. Each axiom prime shadows its predecessor:
| Prime p | s(p) = (p-1)/2 | Name | Status |
|---|---|---|---|
| K = 3 | 1 | sigma | Prime/unit |
| E = 5 | 2 | D | Prime |
| b = 7 | 3 | K | Prime |
| L = 11 | 5 | E | Prime |
| 13 | 6 = D*K | COMPOSITE | CHAIN BREAKS |
The axiom is the longest initial prime segment where all shadows are prime or 1. At p=13, the shadow is 6 = D*K = composite. The gate shuts. Only 7 of 78,498 primes below 10^6 appear in axiom shadow trees (0.009%). E divides 38.5% of CC1 stopping values (expected 20%).
Every axiom prime minus one equals D times its predecessor in the chain:
| Prime | p - 1 | = D * ? | Body part |
|---|---|---|---|
| K = 3 | 2 | D | D^2+sigma = E -> fingers + thumb |
| E = 5 | 4 | D^2 | D*K+sigma = b -> joints + thorn |
| b = 7 | 6 | D*K | D*E+sigma = L -> toes + protector |
| L = 11 | 10 | D*E | Terminal. L-1 = D*E = 10 fingers. |
Why does the (p-1) column sum to 23 = CC1(D)? Four routes to the same number:
| Expression | Value | Reading |
|---|---|---|
| c(L) = 2*11+1 | 23 | Cunningham of protector |
| sum(p-1) | 23 | Axiom neighborhood sum |
| D^2*K + L | 12 + 11 | Trinity heart + protector |
| D+K+b+L | 2+3+7+11 | Axiom without observer |
The observer E removes itself: sum(p-1) = sum(p) - E. Everything minus the observer reveals the Cunningham chain. Dually: sum(p+1) = sum(p) + E = 28+5 = 33 = K*L. The gap between the two sums = D*E = 2*E = the observer seen from both sides.
The shadow function s(p) = (p-1)/2 maps each odd axiom prime to its predecessor:
s(K)=sigma, s(E)=D, s(b)=K, s(L)=E. Sum of shadows = sigma+D+K+E = 11 = L.
Products tell the same story. Product(p-1) = 1*2*4*6*10 = 480 = D^5*K*E. Product(p+1) = 3*4*6*8*12 = 6912 = D^8*K^3. The neighborhoods are MADE of the axiom.
| Product | Value | mod c(p) | Mirror |
|---|---|---|---|
| D*D = D^2 | 4 | mod E=5 | 4 = -1 (mod 5) |
| D*K | 6 | mod b=7 | 6 = -1 (mod 7) |
| D*E | 10 | mod L=11 | 10 = -1 (mod 11) |
Mirror preservation: DATA-1 = 209 and lambda-1 = 419 both mirror ALL 4 inner channels. The axiom's named constants carry the mirror law in their structure.
Axiom-smooth Mersenne exponents are ONLY n in {1, 2, 3, 4, 6}:
| n | M(n) = 2^n - 1 | Factorization |
|---|---|---|
| 1 | 1 | sigma |
| 2 | 3 | K |
| 3 | 7 | b |
| 4 | 15 | K * E |
| 5 | 31 | NOT SMOOTH (31 is outsider) |
| 6 | 63 | K^2 * b |
| 12 | 4095 | Contains 13 = GATE enters |
The smooth exponents {1,2,3,4,6} are exactly the proper divisors of lambda(DATA) = D^2*K = 12. The Cunningham chain, the Mersenne numbers, and the ring's lambda function are the same object seen from different angles.
For each prime p, the SU(p) gauge group has p^2-1 generators. For axiom primes, these are always axiom-smooth (all factors in {D,K,E,b,L}). How far does this extend?
| p | p^2-1 | Smooth? | Note |
|---|---|---|---|
| 2 = D | 3 = K | YES | SU(2) = weak force |
| 3 = K | 8 = D^3 | YES | SU(3) = strong force |
| 5 = E | 24 = D^3*K | YES | SU(5) = GUT |
| 7 = b | 48 = D^4*K | YES | phi(DATA) |
| 11 = L | 120 = D^3*K*E | YES | Icosahedral = E!/D |
| 13 | 168 = D^3*K*b | YES | Gate prime |
| 29 | 840 = D^3*K*E*b | YES | D*lambda |
| 31 = M_E | 960 = D^6*K*E | YES | LAST consecutive |
| 37 | 1368 = D^3*K^2*19 | NO | 19=ESCAPE intruder |
| 41 = KEY | 1680 = D^2*lambda | YES | Recovery |
| 71 = c(E*b) | 5040 = b! | YES | b factorial |
Recovery primes punctuate the boundary: KEY=41 gives 1680=D^2*lambda (D^2 heartbeats), and 71=c(E*b) gives 5040=b! (depth factorial). The smoothness extends past the axiom's own primes, guarded by Mersenne and pierced by the trinity heart.
23 = D*L + sigma = 2*11 + 1 is the first excluded Cunningham prime (CC1(D) depth 3). Its CRT decomposition reads as a mirror:
Apply c(n)=2n+1 to products of axiom primes. Every prime result is axiom-meaningful:
| Product | c(product) | c^2-1 | Name |
|---|---|---|---|
| D*K = 6 | 13 = GATE | 168 = D^3*K*b | Smooth |
| D*b = 14 | 29 | 840 = D*lambda | Two heartbeats |
| K*E = 15 | 31 = M_E | 960 = D^6*K*E | Mersenne(E) |
| K*b = 21 | 43 | 1848 = D^3*K*b*L | All non-E channels |
| E*b = 35 | 71 | 5040 = b! | DEPTH FACTORIAL |
| D*K*E = 30 | 61 = GRIEF | 3720 | c(DATA/b) |
| K*L = 33 | 67 = SOUL | 4488 | c(sum(p+1)) |
The c(E*b) = 71 identity is remarkable: the Cunningham of observer*depth is a prime whose SU-dimension equals depth factorial. 5040 = 7! = D^4*K^2*E*b. The factorization uses four of five axiom primes (only L absent). And GRIEF+SOUL = D^b = 128 = the mirror count.
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