Is there no exceptional scale for dyadic Chowla sums?

Status: open · laid 2026-09-14 · type: lead

Statement (as written in the chat, not paraphrased).

> There is no scale Y for which (1/Y)Σ_{Y<n≤2Y} λ(n)λ(n+h) ≫ 1 while the same quantity is o(1) on > average over scales within log-distance (log Y)^{1/2} of Y.

Structurally parallel to "no exceptional character." Data at 10⁷: the dyadic values all sit at 1/√Y, block variance 1.00, no scale stands out.

Why a probe and not a case. The discriminating test is a proof, not a computation. The numerics have already said everything they can — they cannot distinguish "no exceptional scale" from "no exceptional scale below 10⁷". Filed as a lead so the statement is on record with the nature of its test named.

Cross-refs. Companion to P-009 (the two-bets-one-statement probe from the same chat). Source: notes/DIGEST-twin-prime-chat.md §1.

Cross-reference, 2026-09-27 (N-006, N-028, P-026). The expander route to logarithmic Chowla — Helfgott–Radziwiłł's divisibility graph (arXiv:2103.06853; n ~ n+p for p | n, p in a window), on which the log-averaged two-point sum is an edge-average of a mean-zero ±1 labelling and expansion forces it small — is logarithmically averaged by construction: the 1/n weight is the Haar measure of the dilation flow (N-006), and dilation is the map that puts the sum on the graph at all. So expansion "at every N" is expansion of the graph on [1, N], never a statement at one scale Y. This probe asks for a Cesàro statement at a scale. The expander mechanism cannot decide it, and it names what would: whatever removes the logarithmic weight for the two-point correlation — the open "Cesàro from logarithmic" upgrade. P-008 is that problem in the twin-prime chat's clothes; its "discriminating test is a proof" now reads "and the proof is not the expansion theorem".