Goldbach Through the Ring

Every even number is a sum of two primes — but CAN they share a class?

The Same-Class Goldbach Theorem

Classic Goldbach: every even n > 2 equals p + q for primes p, q. The axiom asks a sharper question: can p and q be in the same signature class (same residue mod 210)?

Answer: only if n has at most one null coordinate (divisible by at most one of {2,3,5,7}).

Try It

Even number:

D=2

K=3

E=5

b=7

Enter an even number

Why Does It Work?

If n is divisible by two odd primes k and e (both in {3,5,7}):

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), so p = 0 (mod k). But p is prime, so p = k.

Similarly p = e. But p can't equal both k and e. Contradiction!

With only 1 null, enough room remains for same-class pairs to exist.

The Landscape

Green = same-class possible | Red = impossible | Gold = current selection

Statistics (mod 210)

1-null even residues = phi(210)

2+ null even residues

45.7%

Success rate = 48/105 = SEES/HYDOR

105

Total even residues = HYDOR

Class Map

The 48 unit classes mod 210 (coprime to 210). Primes cluster here. Same-class sums draw from these:

Contrast Panel

Aspect	Classical Goldbach	Same-Class Goldbach
Statement	Every even n>2 = p+q	n = p+q with p,q same class (mod 210)
Condition	None (always works)	n has at most 1 null coordinate
Why it works	Heuristic (unproved)	CRT forces: 2+ nulls make same-class impossible
Success rate	100% (conjectured)	48/105 = phi(210)/HYDOR = 45.7%
Deeper meaning	Primes are dense enough	Ring structure controls which sums are ALLOWED
Source	Goldbach 1742, unproved	CRT channel analysis, verified to 50000