Alternative GPY weights: exact perturbation geometry and a radial escape

Definitions and mathematical statement

For $\mathcal H=\{h_1,\ldots,h_k\}$, $R>1$, and $\log^+x=\max(0,\log x)$, the source GPY weight is \[ \Lambda_R(n;\mathcal H)=\sum_{d\mid\prod_i(n+h_i)}\lambda_d, \qquad\lambda_d=\mu(d)\frac{(\log^+(R/d))^{k+\ell}}{(k+\ell)!}. \] J. B. Conrey asks whether changing the divisor coefficients improves the method quantitatively, including smooth choices \[ \lambda_d=\mu(d)\prod_{p\mid d}f\left(\frac{\log p}{\log R}\right) Q(\log^+(R/d)). \] The source does not specify an optimization objective, distribution assumptions, admissibility of $\mathcal H$, or endpoint/support conditions on $f,Q$. In particular $Q(0)\ne0$ would not automatically impose a cutoff $d\le R$. Vaughan suggests other traditional sieve weights; another suggestion combines upper and lower sieves in Chen's style, with Soundararajan warning about negative contributions.

The supplied finite-level reduction says that adjoining a candidate coefficient vector to a baseline improves a Rayleigh quotient exactly when the orthogonalized candidate either couples to the baseline or has larger quotient. In an orthonormal two-plane the matrix is $\begin{pmatrix}\rho&b\\\bar b&\sigma\end{pmatrix}$, with optimum \[ \frac{\rho+\sigma+\sqrt{(\rho-\sigma)^2+4|b|^2}}2. \] This exceeds the baseline $\rho$ iff $b\ne0$ or $\sigma>\rho$. A zero new direction adds nothing; degeneracy of a finite Gram form must be removed before this orthogonalization. The report does not provide the complete arithmetic Gram forms or a prime-gap improvement.

Its separate candidate theorem uses the standard Maynard simplex $\Delta_k=\{t_i\ge0:\sum_i t_i\le1\}$. For real $F\in L^2(\Delta_k)$, let \[ I(F)=\int_{\Delta_k}F^2,\qquad J_i(F)=\int\left(\int_0^{1-\sum_{j\ne i}t_j}F(t)\,dt_i\right)^2dt_{-i}, \quad\mathcal R(F)=\frac{\sum_iJ_i(F)}{I(F)}, \] where the outer integral is over $t_{-i}\ge0$, $\sum_{j\ne i}t_j\le1$. Let $A$ be the self-adjoint operator satisfying $\langle AF,F\rangle=\sum_iJ_i(F)$. For every $k\ge2$ and nonzero continuous radial profile $F(t)=\phi(\sum_i t_i)$, the report asserts that the residual $r=AF-\mathcal R(F)F$ is nonzero. Since $r\perp F$, the two-plane formula with $\rho=\mathcal R(F)$, $b=\|r\|/\|F\|$, $\sigma=\langle Ar,r\rangle/\|r\|^2$ gives a strict quantified improvement. At a profile stationary under all radial variations, the residual is genuinely nonradial. This is a variational statement in the later simplex framework, not a direct completed optimization of every proposed workshop weight.

The report also claims a signed-mixture lemma identifying a correction needed for upper-minus-lower sieve positivity. Its exact correction formula is absent, so a signed expression is not silently treated as nonnegative and no Chen-style positivity conclusion is inferred.

Short English statement

Can alternative GPY divisor weights give quantitative gains? The supplied report gives exact two-direction optimization and a strict escape from continuous radial profiles in the Maynard functional, while leaving arithmetic implementation and signed-sieve corrections separate.

Sources

• Local UnsolvedMath registry, permanent ID $20000750$, source code catalogue entry; AIM analytic number theory corpus, aim-analytic-number-theory-notes.json, notes[113]. Attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0.
• AIM Gaps between primes, Problem 15, "Alternative weights in the GPY method": https://aimath.org/WWN/primegaps/WorkshopProblems.pdf. The text names J. B. Conrey, R. C. Vaughan, Chen and K. Soundararajan. The supplied verification identifies the PDF as authority for fraction/line-break repair; no new inspection is claimed. Original remarks and literature fields are empty.
• The report invokes the standard Maynard simplex and finite Gram/Rayleigh methods without full primary citations, arithmetic forms or signed-mixture formula.
• Registry dated August 14, 2026: statement exact, status partially_solved, classification PARTIAL-PROGRESS; attempt 1 labelled partial result, rigor proved, novelty confidence low. The perturbation and radial-residual results are machine-generated research aids, not peer reviewed, requiring independent verification before citation.