# 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 ```text 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 ```text 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 ```text v(m) = #{ r : r | m, r prime, z <= r <= sqrt(m) }. ``` Let `c_Y(m) = 1` if ```text m = r s t ``` with distinct primes ```text r < s < t, r,s,t >= z, ``` and ```text r s <= Y. ``` Otherwise set `c_Y(m) = 0`. Define ```text 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 ```text Phi_Y(m) <= 1_P(m). ``` Therefore, if ```text 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: - If `m` is prime and `m >= z`, then `v(m) = 0` and `c_Y(m) = 0`, so `Phi_Y(m) = 1`. - If `m` is a product of two primes both at least `z`, then `v(m) = 1` and `c_Y(m) = 0`, so `Phi_Y(m) = 0`. - If `m` is a product of three primes all at least `z`, then `v(m)` detects the visible small or middle factor burden. The rebate term `2c_Y(m)` is designed to restore only the controlled cheap-pair triples. - If `m` has any prime factor below `z`, then the roughness cutoff makes `Phi_Y(m) = 0`. 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 ```text A_z(N) - V_z(N) + 2 E_{z,Y}(N) > 0. ``` Here: - `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 ```text 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 ```text 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 ```text c_a = sum_{r1 r2 = a} alpha_{r1} alpha_{r2}. ``` For `x in F_p^*`, set ```text 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 ```text 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 ```text 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 ```text 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 ```text r1 r2 == r3 r4 mod p. ``` The relevant literature neighborhood includes: - Ayyad--Cochrane--Zheng type product-congruence estimates. - Bryce Kerr, *On the congruence x1x2 == x3x4 mod q*, Journal of Number Theory 180 (2017), 154--168. - Bourgain--Garaev--Konyagin--Shparlinski, *Multiplicative Congruences with Variables from Short Intervals*. - Cilleruelo--Garaev, *Congruences involving product of intervals and sets with small multiplicative doubling modulo a prime and applications*. 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: - first-layer product-cloud flattening; and - residual shearing against prime ratios. ## 7. What This Note Does Not Claim This draft does not claim: - Goldbach is proved. - Goldbach is almost proved. - Existing product-congruence estimates close the Type-III wall. - The residual shearing estimate is already known. - The finite verification range is relevant unless the asymptotic pointwise theorem is first proved. ## 8. Dependency Map The route has the following dependency structure: ```text 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: - signed kernel `X`; - no signed kernel, giving obstruction `Y`; - normalization correction `Z`; - or confirmation that this is a genuine Type-III dispersion gate. ## 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.