The chain generates primes by the rule 2n+1: starting from 1, we get 3, then 7. It stops at 3^2 = 9 = 7 + 2 (composite). But the sum 1 + 2 + 3 + 5 = 11 is prime -- built FROM the chain, not BY it. This page shows what 11 does: error correction for free, and how 17 closes the ring.
Error Correction
Protection Theorem
The mod-11 channel detects single-channel corruption whenever the corrupted modulus is less than 11. In Z/210, all four channels {2, 3, 5, 7} qualify -- detection is 100%. In Z/12,612,600, channels mod 8 and mod 9 are covered (< 11), but mod 25 and mod 49 need a dual check with mod 13. Since lcm(11, 13) = 143 > 48 (the maximum error in any channel), the dual check catches everything.
Ring
Channels
Detection
What mod 11 adds
Z/210
4: {2, 3, 5, 7}
0%
No redundancy. Corruption silent.
Z/2,310
5: + mod 11
100%
Any single-channel error caught.
Z/970,200
5: {8, 9, 25, 49, 11}
~92%
Mod 11 alone: 11/12 for mod 25 and mod 49.
Z/12,612,600
6: + mod 13
100%
Dual mod 11 + mod 13. Max error 48 < lcm = 143.
Dual check: mod 11 + mod 13. An error is invisible only if divisible by lcm(11, 13) = 143. The largest possible single-channel error is 48 (in mod 49), which is less than 143. Detection and known-location correction: both 100%.
Cost
1/11 = 9.1%
Adding mod 11 reduces unit density by 9.1%. A bargain for full detection.
Mixing trade-off
64x slower
Z/210: 256 mixing steps. Z/2,310: 16,384. Speed for reliability.
Byte recovery
100% unique
For bytes [0, 255]: single-channel corruption has unique CRT recovery.
Two-channel errors
7 of 10 pairs
7 of 10 channel pairs have surviving product > 256, giving unique recovery.
The Fixed Channel
11 is the only unchanged channel
Going from Z/2,310 to Z/12,612,600, every channel except mod 11 gains resolution by raising its prime to a higher power. Mod 2 becomes mod 8 (x4). Mod 3 becomes mod 9 (x3). Mod 5 becomes mod 25 (x5). Mod 7 becomes mod 49 (x7). Mod 11 stays mod 11 (x1). It is the only channel whose exponent remains 1.
Channel
Resolution gain
Exponent
mod 2 -> mod 8
x4
3
mod 3 -> mod 9
x3
2
mod 5 -> mod 25
x5
2
mod 7 -> mod 49
x7
2
mod 11 -> mod 11
x1 (unchanged)
1
The first four primes measure: they distinguish values at increasing resolution. 11 does not add measurement resolution -- it adds error detection. That is why its exponent stays at 1: a check digit does not need depth.
11 in Z/210
Order-6 Orbit
11 has multiplicative order 6 in Z/210: the orbit is 11, 121, 71, 151, 191, 1. This is the shortest cycle among the chain primes. The midpoint 11^3 = 71 is self-inverse: 71 * 71 = 5041 = 1 mod 210.
Order 6
11^6 = 1 mod 210
The shortest cycle among chain primes. Returns to 1 in 6 steps.
Self-inverse at midpoint
71 * 71 = 1 mod 210
11^3 = 71. The orbit's midpoint is its own inverse.
Residue pattern
(1, 2, 1, 4) mod {2, 3, 5, 7}
The per-channel residues of 11 alternate: 1, 2, 1, 4. An alternating pattern.
17 Closes the Ring
1 + 2 + 3 + 5 = 11 (without 7). 2 + 3 + 5 + 7 = 17 (without 1). The two sums are complementary: 11 skips the largest data prime, 17 includes it. Where 11 detects errors, 17 completes the structure.
Finality (PROVED)
5 * 7 = 35 = 1 mod 17. For primes p > 13: 5 * 7 = 1 mod p has a solution only at p = 17. No prime beyond 17 satisfies this identity. The ring closes at exactly 7 primes.
Finality and Field Structure (PROVED)
The 4 data channels (mod 8, 9, 25, 49) all have zero divisors. The 3 check channels (mod 11, 13, 17) are all prime fields. What distinguishes mod 17 is finality: 5 * 7 = 1 mod 17, and no larger prime satisfies this identity.
phi(17) = 16 = 2^4
Pure power of 2
The totient of 17 is a power of the first chain prime.
5 * 7 = 1 mod 17
Product equals identity
The product of the two middle primes is the identity in mod 17.
7 channels = 7
Self-referential
Z/214,414,200 has 7 channels. 7 is the fourth chain prime. The channel count equals the depth modulus.
3 generates Z/17*
Primitive root
3 generates all 16 nonzero elements of Z/17. The same prime that stops the chain at 3^2 = 9 generates the entire mod-17 field.
Triple-parity ECC
mod 11 + mod 13 + mod 17
Rate 4/7. 100% known-location correction across all 7 channels.
CRT Steganography
The mod-11 channel is invisible to anyone who does not know the ring. Hide data in it. The other 5 channels stay untouched -- their statistical profile is preserved. Without knowing the ring is Z/12,612,600, the mod-11 data looks like noise.
CRT Steganography (CC0)
Decompose carrier numbers into 6 CRT channels. Replace the mod-11 channel with secret data (0-10 per symbol, 3.46 bits). Reconstruct via CRT. The other 5 channels are unchanged. If a stego number is tampered with, the CRT cross-check across channels fails. Capacity: 2 base-11 symbols per ASCII character (11^2 = 121 > 96 printable).
Enter a carrier number. The mod-11 channel is replaced with hidden value 5:
Carrier number:
Capacity
3.46 bits/number
log2(11) per carrier. 2 symbols per ASCII char (11^2 = 121 > 96 printable).
Invisibility
5 of 6 channels identical
Only the mod-11 channel carries the payload. All others unchanged.