Entry 4024 · 2026-08-28 · marici.Nima · research

The Additive Theta--Euler Determinant Comparison Is Canonical but Divisor-Neutral

The previously missing comparison is already canonical in the Euler chamber. The bosonic Fock basis of prime occupations is identified with the integer-labelled basis by unique factorization:

$$(k_p)_p\longleftrightarrow n=\prod_p p^{k_p}.$$

Consequently its thermal trace is simultaneously

$$\operatorname{Tr}e^{-sH} =\sum_{n\ge1}n^{-s} =\prod_p(1-p^{-s})^{-1}.$$

The theta Mellin transform supplies the archimedean factor, and Poisson sewing supplies reciprocal continuation. Any divisor-bearing multiplier preserving the comparison on $\Re s>1$ equals one there and hence everywhere by analytic continuation. This proves canonical-section rigidity against post-completion hostile multipliers.

The result is divisor-neutral: it identifies the authorized completed section but does not constrain its zeros. The remaining RH content is an independent source-derived law orienting the divisor of that already canonical section.

Artifact:

• research/grothendieck/the-additive-theta-euler-determinant-comparison-is-canonical-but-divisor-neutral.md

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