Seven primes create eight pure classes. Each prime that divides a number removes one channel from its coupling. More prime divisors = weaker coupling = fewer members.
Counts for Z/2,310 (5 primes). 13 appears in Z/12,612,600. 17 appears in Z/214,414,200. Total: 128 = 2^7 classes.
An element's class is determined by which primes divide it. Its coupling measures how strongly it interacts with the rest of the ring:
Pure classes: elements divisible by exactly one prime. Unit class = coprime to all = maximum coupling. Each subsequent class loses one channel: coupling(p) = N/p. So: 2 gives N/2, 3 gives N/3, 5 gives N/5, 7 gives N/7, 11 gives N/11, 13 gives N/13, 17 gives N/17. Hardware verified: all 7 primes flashed to ESP32, coupling(p) = N/p exact for each. ESCAPE coupling = N/17 = 12,612,600 = the TRUE ring modulus -- stripping the 7th channel reveals the 6-channel core.
| Class | Prime | Ring | Property |
|---|---|---|---|
| Unit | coprime | all | Coprime to all 7 primes. Maximum coupling. |
| 2-class | 2 | all | Even elements. The only parity-preserving prime. |
| 3-class | 3 | all | Divisible by 3. Minimum for majority. |
| 5-class | 5 | all | Divisible by 5. 5^2 divides the ring but 5 does not divide 24. |
| 7-class | 7 | all | Divisible by 7. Deepest resolution: 49 states. |
| 11-class | 11 | all | Divisible by 11. Error detection via parity. |
| 13-class | 13 | 6ch+ | Divisible by 13. Dual parity with 11. |
| 17-class | 17 | 7ch | Divisible by 17. 5*7 = 1 mod 17. |
Pure class couplings in Z/2,310: unit=2310, 2-class=1155, 3-class=770, 5-class=462, 7-class=330, 11-class=210. Each is N/p. Coupling measures distance from the unit class.
An element can be divisible by multiple primes simultaneously. With 7 primes in Z/214,414,200, there are 2^7 = 128 possible divisibility patterns. 1 is void (all primes), 7 are pure classes, 1 is unit (no primes). That leaves 119 compound classes.
Examples: 6 = 2*3 (compound of 2-class and 3-class). 35 = 5*7 (compound; and 5*7 = 1 mod 17).
The ring hierarchy: Z/2,310 (5 primes) -> Z/970,200 (5 prime-power) -> Z/12,612,600 (+13) -> Z/214,414,200 (+17). Each step raises exponents or adds a new prime:
Raising exponents (Z/2,310 -> Z/970,200) multiplies each class by 420 without changing the class count. Adding a prime doubles the class count. The 8-class hierarchy is invariant: unit class is always largest, 17-class is always rarest.
Enter any number. See its class from all 7 primes. Pure classes have exactly 1 prime dividing. Try 1 (unit), 2 (2-class), 13 (13-class), 17 (17-class), 6 (compound), 0 (void).
Number:
| Question | Standard | Ring View |
|---|---|---|
| What are coupling classes? | Divisor subgroups | gcd(n, N) determines active channels. 8 pure + 119 compound + void = 128 = 2^7. |
| Why 8 pure classes? | 7 prime factors + unit | 7 primes = 7 single-prime classes + 1 unit class. Z/214,414,200 has all 8. |
| Why is unit class largest? | Most coprime elements | Coprime to all 7 primes = max coupling = most connected. |
| Why is 17-class rarest? | Fewest 17-divisible | 17 is the largest prime. Divisibility by 17 is the rarest single-prime constraint. |
| What does 2 do? | Parity | 2 = the only even prime. Preserves parity. Separates classes without blocking. |
| Can you compute it? | Just gcd | coupling(n) = N / gcd(n, N). For Z/214,414,200: N = 214,414,200. |
Compute coupling(1), coupling(2), coupling(13), coupling(17). Count elements in each class. Every number on this page is arithmetic.
Source code · Public domain (CC0)
.ax source compiled to WASM via self-hosting compiler. Zero HTML authored.