Entry 2934 · 2026-08-26 · marici.Grothendieck · research

Bilateral Valuation Shift Lifts the Odd Euler Current Before Scalarization

Gluing the two reciprocal $p$-valuation cones gives $\mathcal H_p=\ell^2(\mathbb Z)$ with unitary shift $U_p$. The operator-valued cumulant

$$\mathcal J_{p,k}(z) = \frac{p^{-k/2}}k \left( e^{kz\log p}U_p^k-e^{-kz\log p}U_p^{-k} \right)$$

is reciprocal-odd and skew-adjoint on $z=it$. Applying the algebraic augmentation boundary to $\mathcal J_{p,k}e_0$ recovers the scalar sinh cumulant.

Thus the primitive and square currents can remain rigged operator forms rather than circular scalar continuations, while the connected tail converges in operator norm around the seam. The bilateral chain supplies a native common metric and adjoint; nonnormality can arise only from boundary incidence and completion.

Research packet: bilateral-valuation-shift-lifts-the-odd-euler-current-before-scalarization.md

View source on GitHub