A Moving-Cut Correction to the Huang–Li Conditional Goldbach Argument: Consequences for Diagonal Möbius-Twisted Elliott–Halberstam Hypotheses

Abstract

This note identifies and repairs a moving-cut omission in the large-divisor rearrangement used in Huang and Li's conditional argument for the binary Goldbach conjecture. Under the substitution k=(N−n)/d, the condition d>α becomes k<(N−n)/α; after reversing the order of summation, this condition is equivalent to the moving endpoint n<N−αk. The printed rearrangement replaces this moving endpoint by n<N, which is not an exact identity. An explicit finite example shows that the omitted region contributes nonzero terms. We prove that the original Huang–Li conclusion can nevertheless be recovered under their original Möbius-twisted Elliott–Halberstam hypothesis, because that hypothesis is uniform in the truncation parameter y. We introduce a moving-diagonal Möbius-logarithmic Elliott–Halberstam condition adapted to the corrected rearrangement, prove that Huang–Li's original hypothesis implies it, and reconstruct the large-divisor asymptotic while retaining the moving endpoint throughout. We also discuss the consequence of this correction for endpoint-diagonal formulations of Möbius-twisted Elliott–Halberstam hypotheses. In particular, the proof architecture based only on the endpoint y=N requires an additional moving-tail estimate once the exact cutoff is restored. The note does not claim that the Huang–Li theorem is false; rather, it provides a constructive repair under the original assumptions and isolates the precise distributional input required by the corrected argument.