The fifth prime doesn't compute. It checks. Every element in the ring carries its own error-detecting code -- for free. No extra bits. No overhead. 11 = 1 + 2 + 3 + 5. Add mod-13 for dual parity: 100% detection and correction.
Every number in Z/214,414,200 decomposes into 7 channels via CRT: (mod 8, mod 9, mod 25, mod 49, mod 11, mod 13, mod 17). The first four channels carry data. The last three (mod 11, mod 13, mod 17) provide triple-parity ECC -- rate 4/7, 100% correction. Error correction is structural, automatic, free.
Encode: take any element n. CRT gives 7 residues. The first four (mod 8, 9, 25, 49) are data -- they determine n within Z/88,200. The last three (mod 11, 13, 17) are parity. Start with the simplest check: n mod 11.
Detect: reconstruct n from the first 4 channels. Compute n mod 11. If it matches channel 5: intact. If not: corruption detected. 100% for prime channels (mod < 11). Prime-power channels (mod 25, 49): ~92% -- errors at multiples of 11 are invisible to mod-11 alone. Dual-check with mod-13 closes the gap.
Correct: if you know which channel was hit, try all values in that channel (at most 49 trials for mod-49). For prime channels: exactly one value passes the mod-11 check. For prime-power channels: multiple candidates may pass. Dual-check with mod-11 + mod-13 narrows to exactly one (lcm(11,13) = 143 > 49).
Proof sketch: CRT lifting idempotent ei satisfies ei mod p != 0 (from gcd condition). Corruption delta in channel i shifts the mod-p residue by delta * (ei mod p). Since ei mod p != 0, the shift vanishes iff delta = 0 mod p. With p = 11: prime channels have max delta < 11, so no blind spot. Prime-power channels (mod 25, 49) have some deltas divisible by 11. Dual-check with mod 13 eliminates these: lcm(11,13) = 143 > 49.
Three rings, three ECC stories. Raising prime exponents increases data capacity but changes detection rates:
| Ring | Channels | Detection | Notes |
|---|---|---|---|
| Z/210 | 2, 3, 5, 7 (no 11) | 0% | Unprotected. Fast but fragile. |
| Z/2,310 | 2, 3, 5, 7, 11 | 100% + 100% corr | All channels < 11. Perfect ECC. |
| Z/970,200 | 8, 9, 25, 49, 11 | 100% prime, ~92% pp | Prime-power channels leak 1/12. |
Z/2,310 has perfect protection: every data channel (mod 2, 3, 5, 7) is smaller than 11, so the mod-11 check catches everything. Raising exponents to (mod 8, 9, 25, 49) means mod 25 and mod 49 are larger than 11. Some corruptions in those channels produce the same mod-11 residue.
The solution to prime-power channel leakage: add 13 as a second check channel. A corruption is invisible only if delta mod lcm(11, 13) = delta mod 143 = 0. The maximum delta in any channel is 48 (from mod 49). Since 48 < 143: no corruption can be invisible to both checks.
The 490 split goes further: {2, 5, 7} = data (8 * 25 * 49 = 9800 values), {3, 11, 13} = parity (9 * 11 * 13 = 1287 values). 9800 * 1287 = 12,612,600. Three parity channels for three data channels: a (6,3,3) code that corrects any single-channel error. See the live demo on the ECC Live page.
When the ring raises exponents from Z/2,310 to Z/12,612,600, every data prime's exponent increases: 2 -> 2^3, 3 -> 3^2, 5 -> 5^2, 7 -> 7^2. But 11 stays at exponent 1. It is the only channel unchanged by raising exponents -- the fixed reference against which data channels grow.
Enter any element n (0 to 970,199). See its CRT decomposition and mod-11 syndrome. The mod-11 channel acts as a free checksum -- if it mismatches after corruption, the error is detected.
Enter n:
Try: 137, 42, 210. Corrupt one channel mentally and see if mod-11 catches it.
| Aspect | Standard (INT8) | Ring (CRT + mod 11) |
|---|---|---|
| Error detection | 0% -- bit flips are silent | 100% with dual parity mod 11 + mod 13 (proved) |
| Error correction | 0% -- no mechanism exists | 100% known-location with dual parity |
| Levels | 256 (8 bits, arbitrary) | 210 or 970200 (structured) |
| Cost | 0 overhead, 0 protection | 0.3 bits SQNR -- free protection |
| On corruption | Wrong value, no warning | Detected -> located -> corrected |
| Paradigm | Pray nothing goes wrong | Ring self-heals |
Reed-Solomon, LDPC, Turbo codes all ADD redundancy to protect data. The CRT ring has redundancy built in. 11 is not bolted on -- it is the fifth prime in the number system. Protection is a structural property of the ring's CRT decomposition.
Three ESP32 boards running .ax WASM firmware on the TRANS ring Z/214,414,200. Seven channels. Triple-parity ECC (rate 4/7). Packets broadcast via ESP-NOW radio every 3 seconds. Error injection via serial command corrupts one CRT channel per packet. The receiver corrects it.
Source code · Public domain (CC0)
.ax source compiled to WASM via self-hosting compiler. Zero HTML authored.