# 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

```text
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

```text
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

```text
a = r1 r2
```

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

```text
h = s2 / s1 mod p.
```

The corresponding six-variable congruence is

```text
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

Recommended reading order:

1. `TYPEIII_RESIDUAL_SHEAR_OPEN_PROBLEM.md`
2. `DEPTH3_BUCHSTAB_MINORANT_REDUCTION.md`
3. `TYPEIII_RESIDUAL_IMPORT_AUDIT_AND_STOP_DECISION.md`
4. `PUBLICATION_BUNDLE_INDEX.md`

The longer internal Type-III chain should not be treated as a public proof artifact. It is proof-search scaffolding.

## 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.