Goldbach Ring

A binary Goldbach ring is the quotient ring obtained from the polynomial ring over a field by imposing the quadratic relations whenever (Nelson 2026). The variables have bidegrees , recording the sum and number of primes represented by a monomial. These relations need not permit cancellation, so the ring is not automatically the toric quotient obtained by identifying every pair of monomials of the same bidegree.

Nelson (2026) proved that every map from the odd primes to any Abelian group that preserves equal two-prime sums satisfies

Equivalently, the universal Abelian relation group is . Including the prime 2 adds an independent generator, and gives , rather than the same affine conclusion on all primes.

The ring also has quantitative graded Betti numbers. Write for the number of generators in homological position , additive weight , and polynomial degree in its minimal bigraded free resolution. Each fixed bidegree stabilizes along finite-support truncations, making these numbers well-defined for the infinite ring. For fixed and admissible weights , Nelson (2026) obtained

where counts ordered -tuples of odd primes summing to , with repetitions allowed. The leading asymptotic holds over every field, without asserting that individual Betti numbers are independent of the characteristic. For the finite quotient on odd primes at most , with all binary relations retained and variables, the total linear-strand numbers satisfy for fixed . Keeping only relations whose common sum is at most instead gives the leading coefficient .

These results do not prove the binary Goldbach conjecture, connectedness of every binary-relation fiber, or the integral-domain property of the full ring. Nelson (2026) credits GPT-6 Astra with proposals and initial proofs of the principal additive and asymptotic results, subsequently checked and revised by the author.