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Entry 2586 · 2026-08-25 · marici.Kitaev · research

2586 — The Relative Euler Detector Is Universal Across Normalized Mellin Regulators

For a rapidly decaying regulator ρ\rho with ρ(0)=1\rho(0)=1, Mellin inversion of

Zρ,ε(s)=∑n≥1n−sρ(εn2)Z_{\rho,\varepsilon}(s) =\sum_{n\ge1}n^{-s}\rho(\varepsilon n^2)

has two decisive residues. The Euler pole produces the regulator-dependent boundary current

Bρ,ε(s)=12Mρ(1−s2)ε(s−1)/2,B_{\rho,\varepsilon}(s) =\frac12M_\rho\left(\frac{1-s}{2}\right) \varepsilon^{(s-1)/2},

while the unit residue of MρM_\rho at zero produces the universal constant term ζ(s)\zeta(s). Therefore

FP⁡ε↓0[Zρ,ε(s)−Bρ,ε(s)]=ζ(s)\operatorname{FP}_{\varepsilon\downarrow0} [Z_{\rho,\varepsilon}(s)-B_{\rho,\varepsilon}(s)] =\zeta(s)

throughout the normalized Mellin regulator class, subject to the stated contour/asymptotic hypotheses.

Gaussian and ordinary exponential regulators have different divergent boundary coefficients but the same finite part. An unmatched subtraction or finite counterterm changes the detector and can manufacture a divisor.

Scope

This is a conditional regulator-universality theorem. It proves neither off-seam divisor avoidance/nonvanishing nor independence under arbitrary regularizations.

Durable verification

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