Research map / open problem

A Depth-3 Buchstab Minorant Map for Binary Goldbach

We do not prove Goldbach. We isolate a concrete analytic wall: a phase-sensitive Type-III residual shearing estimate for product clouds of primes modulo medium prime moduli. The current refined gate is the one-factor-match variance wall: in the documented architecture, standard L2, signed-weight, minorant-geometry, and ratio-large-sieve pivots all return to the same obstruction.

Status: public-review candidate for the coming-week homepage pass; not a proof. This page replaces an older Alpha proof-claim presentation. The current work is deliberately framed as a structural reduction and expert-review target.

The Map

Binary Goldbach asks whether every sufficiently large even number N can be written as N = p + q with both terms prime.

The current route uses a depth-3 Buchstab minorant. After decomposing the non-prime side into controlled rough pieces, the hard shifted-prime surface becomes:

N = p + r s t

The value of the reduction is that the remaining obstruction is no longer vague. It is a specific Type-III residual shearing estimate, currently sharpened to a signed-kernel question inside the one-R-match corridor.

The Remaining Analytic Wall

One studies two-prime products a = r1 r2 modulo medium primes p, subtracts the expected local residue model, and asks whether the leftover error can align with prime-ratio directions.

h = s2 / s1 mod p

Equivalently, after local terms, exact diagonals, forced-zero cases, and deterministic corridor families are removed, the live congruence has the shape:

r1 r2 s1 == r3 r4 s2 mod p

Counting this congruence is not enough. The open target is phase-sensitive cancellation after subtracting the local model. The latest internal audit shows that the direct nonprincipal variance target is saturated. The tested local pivots do not remove that wall, so the honest public object is now an obstruction map unless a new pre-L2 mechanism is supplied.

Expert Review Question

Does the exact Type-III dispersion identity produce a nontrivial signed one-R-match kernel omega_p(t) or W_p(chi), or does the route collapse to the saturated nonnegative variance wall?

The current local answer is negative for the standard L2 path, signed-weight split, minorant-geometry adjustment, and ratio-large-sieve import. A new answer would need to change the architecture before the positive variance object appears.

Public Technical Notes

The longer internal Type-III chain is proof-search scaffolding and is intentionally not presented as a public proof artifact.