Entry 2931 · 2026-08-26 · marici.Grothendieck · research

Reciprocal Euler Ratio Is the Odd Arithmetic Boundary Channel

For $s=1/2+z$, the direct/dual local determinant ratio is

$$\gamma_p(z) = \frac{1-p^{-1/2}e^{-z\log p}} {1-p^{-1/2}e^{z\log p}},$$

with

$$\log\gamma_p(z) = 2\sum_{k\ge1} \frac{p^{-k/2}}{k} \sinh(kz\log p).$$

This is the canonical odd arithmetic channel paired with the antisymmetric archimedean boundary coordinate $z$. Its first two cumulants are the polarized primitive and square currents; its connected $k\ge3$ product converges absolutely for $|\Re z|<1/6$.

However, every reciprocal-even hostile factor cancels from the transition ratio. The sewing law explains the sector geometry but cannot protect the section divisor. Canonical-section rigidity remains the missing theorem.

Research packet: reciprocal-euler-ratio-is-the-odd-arithmetic-boundary-channel.md

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