Two classical sequences independently produce the ring's primes. Cyclotomic polynomials evaluated at 2 yield {1, 3, 5, 7, 11, 13} -- all six chain primes. The Fibonacci sequence has a fixed point at F(5) = 5, and its periods modulo chain primes have lcm = 240 = |roots(E8)|. Two independent paths, same destination.
Cyclotomic Polynomials at x = 2
Cyclotomic Generation (PROVED)
Phi_n(2) is 11-smooth for exactly n in {1, 2, 3, 4, 6, 10} -- six values. The distinct outputs are {1, 3, 5, 7, 11} (3 appears twice, from n=2 and n=6). The next index n = 12 gives Phi_12(2) = 13, breaking smoothness.
n
Phi_n(x)
Phi_n(2)
11-smooth?
1
x - 1
1
Yes
2
x + 1
3
Yes
3
x^2 + x + 1
7
Yes
4
x^2 + 1
5
Yes
5
x^4 + ... + 1
31
No (31 > 11)
6
x^2 - x + 1
3
Yes
10
x^4 - x^3 + ... + 1
11
Yes
12
x^4 - x^2 + 1
13
Wall: first value > 11
The input 2 itself is not an output -- it is the generator. All other chain primes emerge from evaluating cyclotomic polynomials at 2. The first Mersenne number outside the chain: Phi_5(2) = 31 = 2^5 - 1.
Zsygmondy Characterization
Zsygmondy Smooth Zone (PROVED)
Phi_n(2) is 11-smooth if and only if every Zsygmondy primitive prime divisor of 2^n - 1 is at most 11. The primitive primes at indices n = 2, 3, 4, 10 are exactly 3, 7, 5, 11 (the chain primes). Index n = 6 has no primitive prime (the unique Zsygmondy exception at base 2). Index n = 12 introduces 13, which breaks smoothness.
Chain prime
Zsygmondy index n
Index
3
n = 2
2
5
n = 4
4
7
n = 3
3
11
n = 10
10
13 (wall)
n = 12
12 = lambda(Z/210)
The Twin Gate Theorem
f(x) = Phi_6(x) - 2 (PROVED)
The polynomial f(x) = x^2 - x - 1 (golden ratio zeros, discriminant 5) equals Phi_6(x) - 2. Also: Phi_3(2) = 7 and Phi_3(3) = 13. The same cyclotomic polynomial maps 2 to 7 and 3 to 13.
Twin Gate (PROVED)
Two classical sequences hit the same wall at 13 from opposite sides. Divisor sums: sigma(n) is 11-smooth for n = 1..8, breaks at n = 9 where sigma(9) = 1+3+9 = 13. Fibonacci: F(n) is 11-smooth for n = 0..6, breaks at n = 7 where F(7) = 13. Note: 9 - 7 = 2 and 9 + 7 = 16 = 2^4.
Fibonacci at Chain Primes
n
F(n)
Prime?
0
0
-
3
2
Yes
5
5 (fixed point!)
Yes
7
13
Yes
11
89
Yes
13
233
Yes
Fibonacci Fixed Point (PROVED)
F(5) = 5. The prime 5 is a fixed point of the Fibonacci sequence. F(7) = 13: the prime 7 maps to 13. F(p) is prime for all chain primes p in {3,5,7,11,13}. First composite: F(19) = 37 * 113, where 19 = f(5) = 5^2-5-1.
Pisano Periods and 240
Pisano-E8 (PROVED)
The Pisano period pi(p) (period of Fibonacci mod p) at chain primes: pi(2)=3, pi(3)=8, pi(5)=20, pi(7)=16, pi(11)=10. Their lcm: lcm(3, 8, 20, 16, 10) = 240 = |roots(E8)|. Also: pi(210) = 240.
p
pi(p)
Factorization
11-smooth?
2
3
3
Yes
3
8
2^3
Yes
5
20
4*5
Yes
7
16
2^4
Yes
11
10
2*5
Yes
13
28
4*7
Yes
37
76
4*19
No (first break)
All Pisano periods at primes up to 31 are 11-smooth. The first non-smooth period occurs at p = 37, where pi(37) = 76 = 4*19. The Legendre symbol (5|p) determines the divisibility pattern: pi(p) divides 2(p+1) when (5|p) = -1, and pi(p) divides (p-1) when (5|p) = 1.
Cyclotomic Order
Each non-D axiom prime p satisfies p = Phi_{ord(2,p)}(2), where ord(2,p) is the multiplicative order of 2 mod p. The orders are all axiom-native: {2, 4, 3, 10, 12, 8} for {3, 5, 7, 11, 13, 17}. The identity 2^12 - 1 = 4095 = 9*5*7*13 follows from the cyclotomic factorization at lambda(DATA) = 12.
Cyclotomic Order (PROVED)
Every non-D axiom prime p = Phi_{ord(2,p)}(2). Orders: ord(2,3)=2, ord(2,5)=4, ord(2,7)=3, ord(2,11)=10, ord(2,13)=12, ord(2,17)=8. Order sum = 39 = 3*13. The cyclotomic factorization 2^12 - 1 = 4095 = 9*5*7*13 uses all DATA primes. Extension primes {11, 17} have orders {10, 8} not dividing 12: this is WHY they are extensions. Intruder contamination: Phi_18(2) = 57 = 3*19 (composite). The first 7 primes form the maximal cyclotomic-prime initial segment at x = 2.