# A Depth-3 Buchstab Minorant Reduction for Binary Goldbach

## Honest status

This is a **map-paper note**, not a proof of Goldbach.

The claim is structural:

> Goldbach for all sufficiently large even `N` follows from positivity of a specific `N^(1/4)`-rough pointwise prime minorant.

Within this depth-3 Buchstab framework, the only unresolved analytic input is a **pointwise Type-III shifted-prime estimate** on the surface

`N = p + r s t`.

That is the named wall.

## Setup

Let `N` be a large even integer and set

`z = N^(1/4)`.

For `m in [N/2, N]`, define

`v(m) = #{ r : r | m, r prime, z <= r <= sqrt(m) }`.

Let `c_Y(m) = 1` if

`m = r s t`

with distinct primes

`r < s < t`, `r, s, t >= z`,

and

`r s <= Y`.

Otherwise set `c_Y(m) = 0`.

Define

`Phi_Y(m) = 1_{P^-(m) >= z} ( 1 - v(m) + 2 c_Y(m) )`.

## Pointwise minorant

Because `P^-(m) >= N^(1/4)` and `m < N`, the integer `m` has at most three prime factors.

A direct case check gives the pointwise bound

`Phi_Y(m) <= 1_P(m)`.

Therefore, if

`sum_{p <= N/2, p prime} Phi_Y(N-p) > 0`,

then `N` has a Goldbach representation.

This is the central reduction.

## Conditional Goldbach theorem

The clean conditional theorem is:

> **Theorem.**
> Suppose there exist `beta > 1/2` and `delta > 0` such that for every sufficiently large even `N`, with `Y = N^beta`,
> 
> `sum_{p <= N/2, p prime} Phi_Y(N-p) >= delta * S(N) * N / log^2 N`.
> 
> Then binary Goldbach holds for all sufficiently large even `N`.

This is a conditional implication only. It is not a proof of Goldbach unless the displayed pointwise lower bound is proved uniformly in the stated range and then combined with the known finite verification range.

The implication follows immediately from the pointwise minorant.

## Decomposition

Expanding the sum gives

`A_z(N) - V_z(N) + 2 E_{z,Y}(N) > 0`,

where:

- `A_z(N)` is the `N^(1/4)`-rough shifted-prime mass
- `V_z(N)` is the visible single-factor burden
- `E_{z,Y}(N)` is the cheap-pair rebate on the triple surface

The hard term is

`E_{z,Y}(N) = #{ N = p + r s t : p, r, s, t prime, r < s < t, r,s,t >= N^(1/4), r s <= Y }`.

So the analytic bridge is exactly:

> **pointwise Type-III lower bound for `N = p + r s t`.**

This is the correct load-bearing theorem target for the depth-3 minorant route.

## What this framework does not do

This note should be read together with the existing no-go notes:

- second moments do not eliminate isolated exact zeros
- many-moduli averaging stops at sparsity, not all-even positivity
- Chen-type prime-plus-semiprime near misses do not rigidly upgrade to prime-plus-prime

So the honest statement remains:

`local admissibility + almost-all bridge != all-even Goldbach`.

The remaining load-bearing theorem is not another local filter. It is a genuine pointwise positivity statement for the filtered binary prime count.

## Sharpness claim inside this framework

The right sharpness statement is not universal over all possible Goldbach proofs.

It is local to this framework:

> Within `N^(1/4)`-rough depth-3 Buchstab minorants, any pointwise prime minorant that neutralizes semiprimes must confront squarefree triple complements `N-p = r s t`.
> Neutralizing those triples requires either a cheap-pair rebate or an equivalent Type-III input.

So the Type-III surface is not an artifact of this particular phrasing.
It is structurally forced inside the depth-3 Buchstab minorant route.

## Practical consequence

The combinatorial side is now mapped cleanly enough to separate two tasks:

1. the **map paper**: record the reduction honestly
2. the **analytic expedition**: prove the Type-III shifted-prime estimate

That is a real result shape.

Not “Goldbach solved.”
But:

> We found the structural reduction.
> The remaining load-bearing theorem is a pointwise Type-III shifted-prime estimate.