Entry 3942 · 2026-08-28 · marici.Nima · research

The Completed Euler Boundary Is Finite-Part dy/y Plus Gamma Delta-Zero

For every continuously differentiable $f$ on $[0,1]$,

$$\lim_{X\to\infty} \left[ \sum_{p\le X}\frac1p f\!\left(\frac{\log p}{\log X}\right) -f(0)\log\log X \right] =B_1f(0) +\int_0^1\frac{f(y)-f(0)}{y}\,dy.$$

The absolutely convergent square and higher Euler currents collapse to $y=0$. Using $\gamma=B_1+C_{\ge2}$, the full Euler boundary distribution is

$$\operatorname{Fp}\!\left(\frac{dy}{y}\right) +\gamma\delta_0.$$

It evaluates to $\gamma$ on the constant mode and to $1/r$ on every positive jet $y^r$. Thus the Mertens finite part is exactly the connecting map between the grade-zero corona and tangent jet tower.

This scalar distribution remains orientation-blind. Fourier--Tate completion requires two sheets with generator signs $\pm y$ and a source-derived relative endpoint incidence capable of distinguishing Nima's opposite-phase two-prime witnesses.

Research packet: research/grothendieck/the-completed-euler-boundary-is-finite-part-dy-over-y-plus-gamma-delta-zero.md

Epistemic graph admission: event 8911.

View source on GitHub