A Depth-3 Buchstab Minorant Map for Binary Goldbach

Status: research map and open problem. Not a proof of Goldbach.

Plain Summary

Binary Goldbach asks whether every sufficiently large even number N can be written as

N = p + q

with p and q prime.

This note does not solve that problem. It records a structural route that reduces one Goldbach strategy to a sharply named analytic estimate.

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

N = p + r s t

where p, r, s, and t are prime-like variables in specified ranges.

The reduction is useful because it names the remaining wall precisely:

prove phase-sensitive Type-III residual shearing for product clouds of primes modulo medium prime moduli.

The Actual Open Problem

The obstruction is not local admissibility and not a missing numerical constant.

The issue is whether residual errors from two-prime product distributions can align with prime-ratio directions.

In compact notation, one studies products

a = r1 r2

modulo medium primes p, subtracts the expected local residue model, and asks whether the leftover error correlates with ratios

h = s2 / s1 mod p.

The corresponding six-variable congruence is

r1 r2 s1 == r3 r4 s2 mod p.

But simply counting this congruence is not enough. The local main terms, exact diagonals, forced-zero cases, and deterministic corridor families must first be removed. What remains is a phase-sensitive cancellation problem.

What The Work Claims

The claim is deliberately limited:

A depth-3 Buchstab minorant route for binary Goldbach leads to a concrete Type-III residual shearing estimate.

This is a map and an expert-review target.

It does not claim:

a proof of Goldbach;
an almost-proof of Goldbach;
that the residual shearing estimate is already known;
that existing product-congruence estimates automatically close the gap.

Why This Is Worth Publishing

The value is the isolation of a precise wall.

Many failed approaches to Goldbach get stuck at a vague sentence like "we need cancellation." This one reaches a more testable statement:

Does the residual two-prime product cloud avoid alignment with prime-ratio shears after the local model is subtracted?

That question can be checked against known bilinear large-sieve methods, multiplicative-congruence estimates, product-energy bounds, and Type-III dispersion technology.

If experts can prove or import the estimate, the route becomes substantially stronger.

If the estimate is false or out of reach, the note still gives a clean obstruction map.

Public Technical Files

Expert Review Question

The question for analytic number theorists is:

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

That is the live mathematical question.