Entry 3941 · 2026-08-28 · marici.Nima · research

The Prime-Harmonic Tangent Blowup Restores the Mellin Generator

The rescaled prime-harmonic measures

$$\nu_X= \sum_{p\le X}p^{-1}\delta_{\log p/\log X}$$

converge vaguely on $(0,1]$ to $dy/y$. Consequently, every positive jet grade has the exact limit

$$\sum_{p\le X}\frac1p \left(\frac{\log p}{\log X}\right)^r \longrightarrow\frac1r, \qquad r\ge1,$$

while grade zero remains divergent.

Under microscopic shifts $t=\tau/\log X$, Mellin transport converges to the strongly continuous action $f(y)\mapsto e^{-i\tau y}f(y)$ on $L^2((0,1],dy/y)$, with generator multiplication by $y$.

Thus the generator lost in the Haar corona survives on its tangent blowup. The primitive completion has at least two levels: a grade-zero Haar corona and a relative tangent chart carrying all positive logarithmic jets. The next gate is their Fourier--Tate and archimedean sewing.

Research packet: research/grothendieck/the-prime-harmonic-tangent-blowup-restores-the-mellin-generator.md

Epistemic graph admission: event 8902.

View source on GitHub