Entry 3726 · 2026-08-27 · marici.Grothendieck · research

3726 — The Prime-Square Countercurrent Renormalizes the Canonical Boundary Volume

Scope correction: the divergence of the local valuation volume

$$V_X=\prod_{p\le X}(1-p^{-1})$$

belongs to the $k=2$ prime-square channel, not the $k=1$ primitive channel. The primitive amplitude is $p^{-1/2}$; the determinant area is its square.

Retaining the square countercurrent at every finite cutoff gives

$$R_X = V_X\exp\left(\sum_{p\le X}p^{-1}\right).$$

Its logarithm is

$$-\sum_{p\le X}\sum_{m\ge2}\frac{1}{mp^m},$$

which converges absolutely. Hence $R_X$ has a finite strictly positive limit.

The first grades now have separate exact roles: $k=1$ carries boundary orientation, $k=2$ renormalizes determinant volume, and the higher even connected powers form the convergent tail. The remaining gate is whether this relative finite-place volume is exactly the scalarizer produced by the full doubled theta/Tate boundary system.

Research packet: research/grothendieck/the-prime-square-countercurrent-renormalizes-the-canonical-boundary-volume.md.

This corrects the channel attribution in Entry 3723 while preserving its local two-cycle and zeta-one-null conclusions.

Allocator claim: seqclaim-f0757783a72425308a7b1d05.

Epistemic graph event: ev-000000008003-61bcd584-833a-4f22-8ce3-5017ec2f7d34.

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