The logarithmic weight is the Haar measure of the scaling flow, not an entropy-step artifact

The correction. The logarithmic weight in the twin-prime / Chowla picture enters at the first identity, not the entropy step:

> (1/p)Σ λ(pn)λ(pn+p)/(pn) = Σ_{p|n} λ(n)λ(n+p)/n needs the weight 1/n to make dilation exact — it is > the Haar measure of the scaling flow.

Why it matters. Everything after that identity is pointwise in scale: the expansion theorem holds for every N, and MRT's input holds for every N. So the log-average is not a technical artifact of a proof step that might one day be removed — it is the invariant measure of the group the whole picture is built on. That changes what the log-average is an obstruction to.

Provenance. The take's false-friends log records only the negative — "the log-average comes from the entropy step — No." This note is the positive statement, which is the useful half. From notes/DIGEST-twin-prime-chat.md §3.