# Expert Review Packet

## Title

`A Depth-3 Buchstab Minorant Map for Binary Goldbach`

## Status

This is a research map and open-problem packet.

It is not a proof of Goldbach.

## One-Paragraph Summary

The note isolates a depth-3 Buchstab minorant route for binary Goldbach. The combinatorial reduction gives a pointwise prime minorant whose positivity would imply a Goldbach representation. After the semiprime burden is neutralized, the remaining hard surface is `N = p + r s t`. The broad analytic wall is a phase-sensitive Type-III residual shearing estimate. The current sharper gate is the one-`R`-match layer: after subtracting integer and ratio-equidistribution corridors, the naive nonprincipal character variance saturates the wall. The documented local pivots all return to this same obstruction, so the package is currently an obstruction map, not an active near-proof.

## What Needs Review

The specific question is:

> Is the one-`R`-match variance wall a real structural obstruction in this depth-3 Buchstab architecture, or is there a pre-Cauchy/sign-preserving Type-III identity that avoids the saturated nonnegative variance object?

## Minimal Technical Target

At the top Type-III edge

```text
R = N^(5/16),
S = P = N^(3/8),
```

the live one-`R`-match corridor reduces to a ratio relation

```text
r4/r2 == s1/s2 mod p.
```

The nonnegative variance target

```text
(1/(p-1)) sum_{chi != chi0} |A_p(chi)|^2 |B_p(chi)|^2
```

saturates the wall. The useful review target is therefore the stop decision and the failed local pivots, not a request to import a routine estimate.

Normalization note:

The intended theorem is an averaged-modulus estimate. If

```text
pi_P = #{ prime p : p ~ P },
```

then the target object is

```text
AvgShear = (1 / pi_P) * Shear.
```

At the top edge, the intended averaged scale is `R^2 S = N`. The equivalent unaveraged aggregate statement carries a factor `pi_P`.

## What Is Already Separated

- Full integer corridors have the expected wall-scale main term and must be treated as structured, not random.
- The one-`R`-match layer is live in the current model and dominates Family A in the raw taxonomy.
- The direct bilinear ratio variance bound is saturated as stated.
- The standard L2 dispersion expansion gives no useful signed kernel.
- A signed-weight split does not help unless the signed pieces remain coupled before L2/Cauchy.
- The minorant-geometry pivot fails inside `P <= S`.
- The ratio-large-sieve pivot would need to beat the natural positive variance scale.
- `STANDARD_PIPELINE_NO_GO_THEOREM.md` consolidates these checks into a conditional no-go result for the current positive-rebate plus per-block L2 pipeline.
- `OVERSIZED_MODULUS_ESCAPE_TEST.md` identifies a real architecture change: moduli `Q > R S` would diagonalize the one-`R` congruence, but would replace the current wall by a large-modulus shifted-prime dispersion theorem.
- `LARGE_MODULUS_SHIFTED_PRIME_FEASIBILITY.md` names that replacement theorem `LM-SPD(theta)` and fixes the required top-edge threshold as `theta > 11/16`.
- `LM_SPD_FORMAL_TARGET.md` states the replacement theorem in kernel form and records the remaining gap: derive the lower-bound-preserving oversized-modulus kernel `K_q`.
- `LM_SPD_KERNEL_DERIVATION_ATTEMPT.md` derives the direct kernel as the mean-subtracted `rst` residue profile, but shows that this direct kernel does not automatically expose the two-factor one-`R` relation diagonalized by `Q > R S`.
- `SIGN_PRESERVING_OVERSIZED_DISPERSION_ATTEMPT.md` shows why: the two-factor relation requires two product-profile copies, which the current linear lower-bound identity does not contain before a correlation/L2 step.
- `COUPLED_SIGNED_IDENTITY_TEST.md` stops the attempted fix inside the current pointwise minorant: a second product-profile copy is quadratic/correlation-local, while the triple rebate is linear and pointwise-local.
- `DIRECT_RST_LM_SPD_IMPORT_AUDIT.md` checks nearby large-modulus AP technology and finds no direct import for pointwise direct `rst`-profile `LM-SPD(theta > 11/16)`.
- The current stop state is the one-`R`-match variance wall.

## What Not To Review As A Claim

Do not read this packet as claiming:

- Goldbach is proved.
- Goldbach is almost proved.
- The residual shearing estimate is already proved.
- Existing product-congruence estimates automatically close the Type-III wall.

## Reading Order

0. `REVIEWER_FIRST_PAGE.md`
1. `TYPEIII_RESIDUAL_SHEAR_OPEN_PROBLEM.md`
2. `FORMAL_DYADIC_MODEL.md`
3. `EXPONENT_BUDGET.md`
4. `GOLDBACH_NORMALIZATION_AUDIT.md`
5. `GOLDBACH_CORRIDOR_MODEL_AUDIT.md`
6. `ONE_R_MATCH_LOCAL_MODEL.md`
7. `STANDARD_PIPELINE_NO_GO_THEOREM.md`
8. `OVERSIZED_MODULUS_ESCAPE_TEST.md`
9. `LARGE_MODULUS_SHIFTED_PRIME_FEASIBILITY.md`
10. `LM_SPD_FORMAL_TARGET.md`
11. `LM_SPD_KERNEL_DERIVATION_ATTEMPT.md`
12. `SIGN_PRESERVING_OVERSIZED_DISPERSION_ATTEMPT.md`
13. `COUPLED_SIGNED_IDENTITY_TEST.md`
14. `DIRECT_RST_LM_SPD_IMPORT_AUDIT.md`
15. `POST_SATURATION_BRANCH_DECISION.md`
16. `PUBLIC_MAP_STOP_DECISION.md`
17. `DEPTH3_BUCHSTAB_MINORANT_REDUCTION.md`

## Desired Expert Answer

The ideal review is one of:

- "This estimate follows from theorem X after these normalizations."
- "This estimate is false because of obstruction Y."
- "The exact dispersion identity supplies signed kernel Z."
- "The oversized-modulus route reduces to known theorem X / fails because LM-SPD is too strong."
- "The oversized-modulus kernel K_q can/cannot be derived from the minorant without losing the lower-bound identity."
- "The direct rst-profile kernel is enough / is not enough; a sign-preserving two-factor dispersion identity is needed."
- "A coupled signed identity supplies the second product-profile copy before L2 / no such identity exists."
- "Direct rst-profile LM-SPD(theta > 11/16) is importable / out of reach."
- "A GEH-style theorem for the exact rst-profile kernel implies the target / does not."
- "No signed kernel exists in this formulation; the route stops at the one-`R`-match variance wall."
- "The ranges are misstated; the right formulation is ..."

That is the useful mathematical fork.