Open Problem: Residual Product-Cloud Shearing in the Goldbach Type-III Range
Short statement
Prove phase-sensitive cancellation between the residual residue distribution of two-prime products and prime-ratio shears over medium prime moduli.
This is the current analytic wall in the depth-3 Buchstab minorant reduction for binary Goldbach.
Ranges
Let N be large and let
N^(1/4) <= R <= N^(5/16),
N^(5/16) <= S <= N^(3/8),
S^(1/2+epsilon) <= P <= S.
Let p ~ P run over primes.
Let alpha_r be prime-supported or sieve-supported on r ~ R, and let beta_s be prime-supported or sieve-supported on s ~ S.
Notation conventions
All sums are dyadic unless explicitly stated otherwise.
Weights may be smooth, prime-supported, or sieve-supported, provided their divisor and size bounds match the Type-III application.
The exact normalization of Fourier transforms over F_p^* is not load-bearing in this note. Any missing factor of p-1 should be absorbed into the final stated theorem when the estimate is formalized.
The ratio convention h = s2 / s1 mod p may be inverted depending on where the conjugate is placed. The open problem is invariant under replacing h by h^{-1}.
Product cloud
Define the two-prime product coefficient
c_a = sum_{r1 r2 = a} alpha_{r1} alpha_{r2}.
For each p, define
C_p(x) = sum_{a == x mod p} c_a
on F_p^*, after removing forced-zero cases.
Let
mu_p = (1/(p-1)) sum_{x in F_p^*} C_p(x),
and
E_p(x) = C_p(x) - mu_p.
Prime-ratio measure
Define
H_p(h) = sum_{s2/s1 == h mod p} beta_{s1} conj(beta_{s2}),
for h in F_p^*.
Equivalently, if
B_p(chi) = sum_{s ~ S} beta_s chi(s),
then H_p is the multiplicative autocorrelation of the S-side prime sequence.
Desired estimate
Prove 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),
with enough logarithmic saving for the depth-3 Buchstab minorant application.
In character form, this is
sum_{p ~ P} sum_{chi != chi0} |Ehat_p(chi)|^2 |B_p(chi)|^2 << Error(N; R,S,P).
In the pure product model, Ehat_p(chi) is the nonprincipal part of
A_p(chi)^2,
where
A_p(chi) = sum_{r ~ R} alpha_r chi(r).
Thus the problem is a phase-sensitive mixed energy estimate for
|A_p(chi)|^4 |B_p(chi)|^2,
but after subtracting the product-cloud mean/local terms.
Equivalent congruence form
The same problem is the local-model-subtracted six-variable congruence
r1 r2 s1 == r3 r4 s2 mod p.
After averaging over p, this can be written as divisor switching:
r1 r2 s1 - r3 r4 s2 = p k.
The desired theorem is not a count of this equation alone; it is a cancellation estimate after subtracting:
- exact integer diagonals;
- forced-zero residue cases;
- additive/multiplicative model corrections;
- smooth product-cloud main mass;
- any explicitly classified deterministic corridor families.
What known product-energy estimates seem to cover
Existing multiplicative-congruence technology appears relevant to the first-layer estimate
r1 r2 == r3 r4 mod p.
This likely gives product-cloud flattening in ranges where R^2 >> P.
Relevant literature neighborhood:
- Ayyad--Cochrane--Zheng-type estimates for
x1 x2 == x3 x4 mod p; - 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 control product-energy size. The missing issue is transverse alignment with prime ratios.
What is not enough
The following route is not acceptable:
- take absolute values in
h; - bound dangerous autocorrelation level sets;
- require a weighted eighth-moment asymptotic for
A_p(chi).
That path has already been rejected because it asks for a stronger pointwise deconcentration theorem than the phase-sensitive problem should require.
The open problem is specifically to keep the phase-sensitive average against H_p(h).
Why it matters for Goldbach
The depth-3 Buchstab minorant gives a pointwise route toward binary Goldbach by reducing positivity to a Type-III shifted-prime estimate on
N = p + r s t.
The unresolved analytic bridge is exactly the medium-prime product-residue energy above.
If this residual shear estimate is proved with the required uniformity, it would close the sharpest currently identified obstruction in this route.
Without it, the Goldbach work remains a conditional structural reduction and open-problem map, not a proof.
Honest status
This is a real open theorem target.
It may already be accessible to experts in bilinear forms, large sieve variants, and multiplicative congruences over finite fields, but it has not been derived in the present notes.
The next mathematically honest step is expert review or a focused proof attempt on this exact statement.