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GOLDBACH MAP PAPER DRAFT

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A Depth-3 Buchstab Minorant Map for Binary Goldbach

Status

Research-map draft.

Not a proof of Goldbach.

This draft records a conditional structural reduction and isolates the remaining analytic wall.

Abstract

We describe a depth-3 Buchstab-minorant route for binary Goldbach. For a large even integer N, set z = N^(1/4) and construct a pointwise prime minorant on the shifted variable m = N-p. The minorant removes visible semiprime obstructions and rebates squarefree triple complements m = r s t through a cheap-pair condition. The resulting positivity problem reduces to a pointwise Type-III shifted-prime estimate on the surface N = p + r s t. We then isolate a sharper analytic wall: phase-sensitive residual shearing for two-prime product clouds modulo medium primes against prime-ratio measures. Existing product-congruence technology appears relevant to first-layer product-cloud flattening, but does not by itself supply the needed transverse non-alignment. The output is an open-problem map, not a proof of Goldbach.

1. Problem

Binary Goldbach asks whether every sufficiently large even integer N has a representation

N = p + q

with p and q prime.

This note studies one possible route: construct a pointwise lower bound for 1_P(N-p) that is strong enough to stay positive after summing over primes p <= N/2.

The point of the note is not to claim that this has been completed. The point is to name the exact analytic theorem that remains.

2. Rough Minorant Setup

Let N be a large even integer and set

z = N^(1/4).

Write P^-(m) for the least prime factor of m. Write 1_P(m) for the prime indicator. Write S(N) for the usual Goldbach singular series; its exact normalization is not important for this map note.

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)).

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

Lemma 2.1 (Pointwise minorant). A direct case check gives

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 elementary combinatorial core.

Sketch of the case check:

The exact triple case is where the Type-III burden enters.

3. Conditional Positivity Target

Proposition 3.1 (Conditional Goldbach implication). The clean conditional statement is:

Suppose there exist beta > 1/2 and delta > 0 such that for every sufficiently large even N, with Y = N^beta,
```text
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 only a conditional implication. It becomes a proof of Goldbach only if the displayed pointwise lower bound is proved uniformly and then combined with the known finite verification range.

4. Decomposition

Expanding the sum gives the schematic form

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

Here:

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 }.

Thus the load-bearing analytic target is a pointwise Type-III lower bound for

N = p + r s t.

In the language of the rest of the bundle, this is the bridge from the combinatorial minorant to the analytic Type-III problem.

5. Open Theorem Target: Residual Product-Cloud Shearing

The Type-III surface can be examined locally modulo medium primes.

Let p ~ P be a medium prime modulus. Let alpha_r be supported on r ~ R, and define the two-prime product coefficient

c_a = sum_{r1 r2 = a} alpha_{r1} alpha_{r2}.

For x in F_p^*, set

C_p(x) = sum_{a == x mod p} c_a,
mu_p = (1/(p-1)) sum_x C_p(x),
E_p(x) = C_p(x) - mu_p.

Let beta_s be supported on s ~ S, and define the prime-ratio measure

H_p(h) = sum_{s2/s1 == h mod p} beta_{s1} conj(beta_{s2}).

Open Theorem 5.1 (Residual shearing target, schematic form). The residual shearing target is a bound of the form

sum_{p ~ P} sum_{h in F_p^*} H_p(h)
  sum_{x in F_p^*} E_p(x) conj(E_p(xh))
  << Error(N; R,S,P),

where Error(N; R,S,P) must be small enough to preserve a positive lower bound in Proposition 3.1 after summing over the relevant dyadic Type-III ranges.

This draft intentionally does not freeze a final numeric exponent for Error(N; R,S,P). That exponent belongs in the formal theorem after the dyadic decomposition and normalization choices are fixed. The open-problem note TYPEIII_RESIDUAL_SHEAR_OPEN_PROBLEM.md records the current target in a reviewer-facing form.

Equivalently, in multiplicative-character language, the model problem has the shape

sum_{p ~ P} sum_{chi != chi0} |A_p(chi)|^4 |B_p(chi)|^2,

after subtracting local terms and removing exact diagonal or deterministic corridor contributions.

This is stronger than plain product-energy control. It asks whether the residual product-cloud error aligns with prime-ratio shears.

The phrase "phase-sensitive" means that one must not simply take absolute values in h and prove a worst-case dangerous-direction estimate. That route asks for an unnecessarily strong deconcentration theorem. The target is cancellation in the weighted average against the actual prime-ratio measure.

6. Import Audit

Existing multiplicative-congruence and product-energy estimates appear relevant to the first-layer problem

r1 r2 == r3 r4 mod p.

The relevant literature neighborhood includes:

These results may plausibly supply product-cloud flattening in the range where R^2 >> P.

They do not by themselves give the final residual shearing estimate, because size control of E_p does not prove phase non-alignment with H_p.

This is the reason the bundle separates:

7. What This Note Does Not Claim

This draft does not claim:

8. Dependency Map

The route has the following dependency structure:

Pointwise minorant lemma
        |
        v
Conditional positivity proposition
        |
        v
Type-III lower bound for N = p + r s t
        |
        v
Product-cloud local model subtraction
        |
        v
Residual shearing estimate against prime ratios

The first two steps are combinatorial.

The later steps are analytic.

The current live wall is the final line.

9. Known Inputs vs Missing Input

| Component | Status | Comment | |---|---|---| | Pointwise minorant Phi_Y <= 1_P | combinatorial | direct case check | | Conditional positivity implication | combinatorial | immediate from the minorant | | Product-cloud flattening | plausibly importable | related to product-congruence estimates | | One-R-match signed kernel | missing | current refined analytic wall | | Residual shearing against prime ratios | missing | broader wall after signed-kernel gate | | Full pointwise Type-III lower bound | missing | depends on signed-kernel and residual-shearing range bookkeeping | | Goldbach conclusion | not obtained | follows only after the missing pointwise theorem |

10. Expert Review Question

The useful expert question is:

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

The desired answer is not encouragement. It is a mathematical fork:

11. Current Conclusion

The combinatorial Buchstab side is clean enough to serve as a map.

The unresolved analytic bridge is now sharpened to the signed-kernel gate inside the one-R-match corridor, before the broader residual-shearing estimate can be attacked honestly.

Until that bridge is proved or imported, the Goldbach project remains a conditional structural reduction and open-problem map.