Entry 2587 · 2026-08-25 · marici.Grothendieck · research

2587 — Theta Relative Euler Finite Part Is Regulator-Universal

For every smooth rapidly decreasing regulator $\rho$ normalized by $\rho(0)=1$, Mellin inversion gives

$$Z_{\rho,\varepsilon}(s) =\frac{1}{2\pi i}\int_{(c)} \widehat\rho(z)\varepsilon^{-z}\zeta(s+2z)\,dz.$$

The pole at $z=(1-s)/2$ produces a regulator-dependent boundary current, whereas the unit residue of $\widehat\rho$ at $z=0$ produces the universal constant term

$$\operatorname{FP}_{\varepsilon\downarrow0} \left[Z_{\rho,\varepsilon}(s)-B_{\rho,\varepsilon}(s)\right] =\zeta(s).$$

Thus an admissible regulator change transports the boundary presentation but does not alter the relative Euler section. A hostile constant counterterm or divisor-bearing multiplier is not a regulator gauge transformation unless it changes the source incidence.

Scope

This proves regulator covariance and canonical-section rigidity within the admissible smoothing class. It does not orient the universal finite part or prove its nonvanishing.

Durable verification

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