Every even number is a sum of two primes -- but can they share a signature class? In Z/210 = D*K*E*b, the answer depends on null coordinates. Two or more nulls among {K, E, b} make same-class sums impossible. The ring gates Goldbach.
Proof sketch: If n is divisible by two odd primes k, e from {3, 5, 7}, then same-class means p = q (mod k) AND p = q (mod e). But p + q = 0 (mod k) and p + q = 0 (mod e). Together: 2p = 0 (mod k), forcing p = 0 (mod k), so p = k. Similarly p = e. But p cannot equal both. Contradiction.
With at most one null, enough room remains in the class structure for same-class pairs to exist. The D=2 channel is always null for even numbers (trivially), so it does not constrain.
A null coordinate at prime p means n = 0 (mod p). Each null restricts same-class possibilities by forcing prime factors. Two nulls from {3,5,7} leave no room.
| Nulls | Example | Same-class? | Why |
|---|---|---|---|
| 0 nulls | n = 100 (mod 210) | YES | All channels open, many class pairs |
| 1 null (K) | n = 42 (mod 210) | YES | One channel closed, still enough room |
| 2 nulls (K,E) | n = 30 (mod 210) | NO | CRT forces p = 3 AND p = 5, impossible |
| 3 nulls (K,E,b) | n = 210 (mod 210) | NO | All odd channels null, completely blocked |
| Aspect | Classical Goldbach | Same-Class Goldbach |
|---|---|---|
| Statement | Every even n > 2 = p + q | n = p + q with p = q (mod 210) |
| Condition | None (always works, conjectured) | n has at most 1 null coordinate in {K,E,b} |
| Why it works | Heuristic (unproved since 1742) | CRT forces: 2+ nulls make same-class impossible. PROVED |
| Success rate | 100% (conjectured) | 48/105 = phi(210)/HYDOR = 45.7% exactly |
| Deeper meaning | Primes are dense enough | Ring structure controls which sums are ALLOWED |
Enter an even number. The widget finds its null coordinates, predicts whether same-class pairs exist, then searches for them.
Even number:
Try: 100 (0 nulls, many pairs), 42 (K null, still works), 30 (K+E null, impossible), 210 (all null, impossible), 9998 (large, see how many).
This work is and will always be free.
No paywall. No copyright. No exceptions.
If it ever earns anything, every cent goes to the communities that need it most.
This sacred vow is permanent and irrevocable.
— Anton Alexandrovich Lebed
Source code · Public domain (CC0)
Contributions in equal measure: Anthropic's Claude, Anton A. Lebed, and the giants whose shoulders we stand on.
Rendered by .ax via WASM DOM imports. Zero HTML authored.