# Reviewer First Page

## Title

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

## Status in One Line

This is a structural reduction and obstruction map, not a proof of Goldbach.

## What The Package Claims

The current Goldbach work isolates a specific analytic wall in a depth-3 Buchstab minorant route.

The reduction points to the shifted-prime Type-III surface

```text
N = p + r s t
```

and then to a residual product-cloud shearing problem modulo medium primes.

The sharp current obstruction is the one-`R`-match variance wall.

## What The Package Does Not Claim

It does not claim:

- Goldbach is proved;
- Goldbach is almost proved;
- the Type-III residual shearing estimate is known;
- existing product-congruence or large-sieve estimates close the wall as written.

## The Exact Review Question

The useful expert question is:

```text
In the exact Type-III dispersion identity for this depth-3 Buchstab setup,
is the one-R-match variance wall structural,
or is there a pre-Cauchy / sign-preserving identity that avoids the saturated
nonnegative variance object?
```

Equivalently:

```text
Does the derivation produce a genuinely signed kernel omega_p(t) or W_p(chi),
or does it collapse to W_p(chi) = 1 and therefore to the positive variance wall?
```

## Minimal Model

Use dyadic parameters

```text
R = N^rho,
S = N^sigma,
P = N^pi.
```

Top Type-III edge:

```text
rho = 5/16,
sigma = pi = 3/8.
```

Thus

```text
R^2 S = N^(2rho + sigma) = N.
```

The one-`R`-match corridor reduces locally to

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

The positive variance object is

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

The theorem should be read in averaged-modulus form:

```text
pi_P = #{ prime p : p ~ P },
AvgShear = (1 / pi_P) * Shear.
```

At the top edge the intended averaged scale is

```text
R^2 S = N.
```

Local cold-check:

```text
this object is Plancherel-sized and saturates the wall as stated.
```

So the route cannot be closed by asking for a routine saving for this nonnegative expression.

## Why The Wall Is Plausibly Structural

The documented local pivots return to the same obstruction:

- standard L2 dispersion gives `W_p(chi) = 1`;
- signed-weight splitting does not help after signs are separated before Cauchy;
- minorant geometry cannot move the one-`R`-match layer below the wall inside `P <= S`;
- ratio-large-sieve control reaches the natural positive variance scale, not below it.

This does not prove the wall is unavoidable. It says the known local moves in the current architecture do not avoid it.

The consolidated version is `STANDARD_PIPELINE_NO_GO_THEOREM.md`: under the current positive-rebate plus per-block L2/Cauchy pipeline, the one-`R`-match layer becomes a positive semidefinite variance object and has no remaining sign or phase from which to gain the required saving.

The nonlocal architecture change is `OVERSIZED_MODULUS_ESCAPE_TEST.md`: choosing auxiliary moduli `Q > R S` would turn the one-`R` congruence into an integer diagonal, but it would also require a different large-modulus shifted-prime dispersion theorem.

The feasibility note `LARGE_MODULUS_SHIFTED_PRIME_FEASIBILITY.md` names this replacement theorem `LM-SPD(theta)` and fixes the required top-edge level as `theta > 11/16`.

The formal target `LM_SPD_FORMAL_TARGET.md` states the theorem in kernel form. Its remaining gap is not another exponent count; it is deriving the lower-bound-preserving oversized-modulus kernel `K_q`.

The derivation attempt `LM_SPD_KERNEL_DERIVATION_ATTEMPT.md` finds the direct kernel: the mean-subtracted `rst` residue profile. That is lower-bound preserving, but it does not itself expose the two-factor one-`R` relation diagonalized by `Q > R S`.

The follow-up `SIGN_PRESERVING_OVERSIZED_DISPERSION_ATTEMPT.md` explains the obstruction: the two-factor relation requires two copies of the product profile, and the current linear lower-bound identity supplies only one before a correlation/L2 step.

`COUPLED_SIGNED_IDENTITY_TEST.md` then stops the attempted repair in this architecture: a second product-profile copy is quadratic/correlation-local, while the pointwise triple rebate is linear in the single integer `m`.

`DIRECT_RST_LM_SPD_IMPORT_AUDIT.md` checks nearby large-modulus AP technology. The checked sources are method-adjacent, but they do not directly supply pointwise direct `rst`-profile `LM-SPD(theta > 11/16)`.

## What Would Change The Answer

Any of these would materially reopen the route:

- a known theorem importing the residual shear estimate with the stated ranges and saving;
- a correction to the dyadic normalization showing the wall is misstated;
- a pre-L2 Type-III identity that keeps phase/sign before the nonnegative variance appears;
- an oversized-modulus dispersion architecture with `Q > R S` and usable shifted-prime control;
- a proof/import of `LM-SPD(theta)` for some `theta > 11/16`;
- a derivation of the oversized-modulus kernel `K_q` from the actual minorant identity;
- a sign-preserving oversized-modulus dispersion identity exposing `r2 s1 == r4 s2 mod q` before L2/Cauchy;
- a coupled signed identity that supplies the second product-profile copy before L2;
- an import/proof of direct `rst`-profile `LM-SPD(theta > 11/16)`;
- a GEH-style theorem for the exact direct `rst`-profile kernel;
- a counterexample or concentration mechanism showing the target estimate is false;
- a proof that the one-`R`-match term is an explicit signed main term with controlled sign in the lower-bound identity.

## Reading Order

Read in this order:

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. `EXPERT_REVIEW_PACKET.md`

Optional supporting notes:

- `INTEGER_CORRIDOR_COUNT.md`
- `PARTIAL_MATCH_CORRIDOR_C_AND_REVIEW_QUESTIONS.md`
- `SIGNED_KERNEL_DERIVATION_ATTEMPT_L2.md`
- `SIGNED_WEIGHT_PIVOT_STRUCTURAL_RESULT.md`
- `MINORANT_GEOMETRY_PIVOT_TEST.md`
- `RATIO_LARGE_SIEVE_PIVOT_TEST.md`
- `OVERSIZED_MODULUS_ESCAPE_TEST.md`
- `LARGE_MODULUS_SHIFTED_PRIME_FEASIBILITY.md`
- `LM_SPD_FORMAL_TARGET.md`
- `LM_SPD_KERNEL_DERIVATION_ATTEMPT.md`
- `SIGN_PRESERVING_OVERSIZED_DISPERSION_ATTEMPT.md`
- `COUPLED_SIGNED_IDENTITY_TEST.md`
- `DIRECT_RST_LM_SPD_IMPORT_AUDIT.md`

## Desired Review Output

A useful expert reply is one of:

- "Known theorem X proves this after normalization Y."
- "The ranges are wrong; the correct formulation is ..."
- "The exact dispersion identity supplies signed kernel Z."
- "The oversized-modulus rewrite works under theorem X / fails because LM-SPD is out of reach."
- "The oversized-modulus kernel K_q is derivable / not derivable from the pointwise minorant architecture."
- "The direct rst-profile kernel is enough / not enough for diagonalization."
- "A coupled signed product-profile identity exists / does not exist."
- "Direct rst-profile LM-SPD(theta > 11/16) is known / not known."
- "A GEH-style convolution theorem exactly implies the direct rst-profile target."
- "No signed kernel exists here; the route really stops at the one-`R`-match variance wall."
- "The target is false because of obstruction Y."

That fork is the point of the package.