Depth

f(p) = p^2 - p - 1

7 is the last prime generated by the chain. Beyond it, the chain stops: 3^2 = 9 = 7 + 2 is composite. The mod-49 channel has the deepest recursion of any channel. The polynomial f(p) = p^2 - p - 1 connects 7 to the golden ratio, 3/4, and 5/3 through proved algebraic identities.

Three Algebraic Identities

Three Ratios from 7 and 2
(7-1)/(7+1) = 6/8 = 3/4. (7-2)/(7-4) = 5/3. 7/4. All three are simple functions of 7 and 2 alone. PROVED: 7-2 = 5, 7-4 = 3, 7-1 = 6, 7+1 = 8. The remaining chain primes (3, 5) emerge as differences of 7 and powers of 2.
FormulaValueNote
(7-1)/(7+1)3/4PROVED. Matches Kleiber scaling exponent (OBSERVED, no mechanism).
(7-2)/(7-4)5/3PROVED. Matches Kolmogorov turbulence exponent (OBSERVED, no mechanism).
7/47/4PROVED. Matches Ising critical exponent (OBSERVED).

These are proved algebraic identities. The match to physical exponents is observed -- no mechanism is known. The key identity: 7^2 - 7 - 1 = 41, which is the depth quadratic f(7).

Why 7 Is Special

Multiple independent measurements agree: the mod-49 channel carries the most structural information of any channel.

Deepest channel
mod 49 = 7^2
49 states -- the largest prime-power modulus. More resolution than any other channel.
Spectral gap
closest eigenvalue spacing
The mod-49 channel has the tightest spectral gap, setting the ring's mixing time.
Ablation
-11.8%
Removing mod 7 from a CRT prediction task causes the largest accuracy drop.
Odd moments
zero for k < 8
The ring's odd moments vanish for the first 7 steps. First nonzero odd moment at k = 9 = 3^2.
Self-referential
7 mod 7 = 0
7 is the meta-ring modulus: 7 primes, 7 channels, Z/7. It vanishes in its own channel.

105 = 3 * 5 * 7

The product of the three middle chain primes. It excludes 2 (the pair) and 11 (the check). 105 = 210/2: exactly half of Z/210.

Chain stops at 7
3^2 = 9 = 7+2
Beyond 7: only composites. The Cunningham chain produces no more primes.
Asymmetric eigenvalues
49 is odd
In every ring containing mod 49, the maximum eigenvalue exceeds |minimum eigenvalue|. Odd moduli produce asymmetric spectra.
3^4 + 24 = 105
81 + 24 = 105
The fourth power of 3 plus 24 (the central charge of the Leech lattice / Dedekind eta) equals 105.

f(p) = p^2 - p - 1

Five Roles of f(p)
f(p) has five proved roles: (1) MIRROR: f(p) = -1 mod p for all p. Proof: p^2-p = p(p-1) = 0 mod p, so f(p) = -1. (2) NORM: f(p) = Norm(p - phi) in Q(sqrt(5)). Discriminant = 5, the golden ratio field. (3) GENERATION: each chain prime p is a primitive root of f(p) (except 5, which has half-order). (4) CHAIN VALUES: f(3) = 5, f(5) = 19, f(7) = 41, f(11) = 109. (5) QR-PARITY: Legendre symbol (p/f(p)) = (-1/p) whenever f(p) is prime.
f(3) = 5
3 is a primitive root of 5
ord(3, 5) = 4 = phi(5). The polynomial maps 3 to 5.
f(5) = 19
5 has half-order
ord(5, 19) = 9 = phi(19)/2. The only chain prime with half-order in its f-value.
f(7) = 41
7 is a primitive root of 41
ord(7, 41) = 40 = phi(41). 7 generates all 40 units of Z/41.
f(11) = 109
11 is a primitive root of 109
ord(11, 109) = 108 = phi(109). 108 is the number of rings with Carmichael period 420.

Two parallel paths connect the chain primes: the Cunningham chain (1 -> 3 -> 7 via 2n+1, additive) and f(p) (3 -> 5 -> 19 via p^2-p-1, multiplicative). Different mechanisms, same structure.

Explore: Depth Quadratic f(p) = p^2 - p - 1

The depth quadratic has a universal mirror property: f(p) = -1 mod p for ALL p. Proof: p^2-p = p(p-1) = 0 mod p, so f(p) = 0-1 = -1 mod p. Enter any prime (or any number) to verify.

Enter p:

Try chain primes: 3 (f=5), 5 (f=19), 7 (f=41), 11 (f=109, phi=108). Then try 13 (f=155).

490: A Zero Divisor

490 Spiral (PROVED)
490 = 2 * 5 * 7^2. CRT(490) in Z/970,200 = (2, 4, 15, 0, 6). The mod-49 channel is zero: 490 is a zero divisor. Powers of 490 progressively kill channels: mod 49 dies at step 1 (7^2 divides 490), mod 25 at step 2 (5^2 divides 490^2), mod 8 at step 3 (2^3 divides 490^3). Only mod 9 and mod 11 survive. The terminal value 490^420 = 960,400 is an idempotent: CRT = (0, 1, 0, 0, 1) in Z/970,200. It permanently projects onto the mod-9 and mod-11 channels.
Channel death sequence
mod 49 -> mod 25 -> mod 8
Channels die in order of divisibility. mod 9 and mod 11 survive because 490 is coprime to 9 and 11.
Surviving periods
3 and 10
mod-9 orbit has period 3, mod-11 orbit has period 10. lcm(3, 10) = 30.
Terminal idempotent
960,400
490^420 = 960,400. CRT = (0, 1, 0, 0, 1). Idempotent: 960,400^2 = 960,400 mod 970,200.
Dual annihilator
490 * 1,980 = 970,200
coupling(490) = 1,980. Their product equals the ring size.

The Mod-49 Spiral

Odd Order: No Mirror
The mod-49 channel has Carmichael lambda = 42 and odd-order subgroup of size 21 = 3 * 7. Elements with odd order in mod 49 never reach -1 (they spiral instead of oscillating). All other channels have even orders and DO pass through -1. PROVED for all 49 residues.
2 in mod 49
order 21 = 3 * 7
2 spirals in mod 49 with period 21. Never reaches -1 = 48.
11 in mod 49
order 21 = 3 * 7
11 has the same period as 2 in mod 49.
67 in mod 49
order 3
67 mod 49 = 18. 3-cycle: 18^1 = 18, 18^2 = 30, 18^3 = 1 mod 49.
Odd divisors of 42
{1, 3, 7, 21}
All four odd divisors of lambda(49) = 42 are chain primes or products of chain primes.

The mod-49 channel spirals; other channels oscillate. 2 has odd order (21) in mod 49 but even order in every other channel (6 in mod 9, 20 in mod 25). This makes the mod-49 channel uniquely asymmetric for powers of 2.

Paradigm Contrast

ClaimStandard ViewRing Structure
(7-1)/(7+1) = 3/4CoincidencePROVED algebraic identity. Matches Kleiber scaling exponent (OBSERVED, mechanism unknown).
(7-2)/(7-4) = 5/3CoincidencePROVED algebraic identity. Matches Kolmogorov exponent (OBSERVED, mechanism unknown).
Deepest channel7 is not specialmod 49 has the most states of any channel. Multiple measurements confirm dominance.
Golden ratioAesthetic preferenceDiscriminant of f(p) = p^2-p-1 is 5. Roots are the golden ratio. PROVED.
f(7) = 41Just a number7^2 - 7 - 1 = 41. 7 is a primitive root of 41 (ord = 40 = phi(41)). PROVED.

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