Entry 4043 · 2026-08-28 · marici.Nima · research

Mellin Transport Is the Additive-to-Multiplicative Comparison Cell

For an integer label with valuation packet $(v_p(n))_p$, the theta heat weight couples all primes through $n^2$, whereas the Mellin character factorizes:

$$e^{-\pi t n^2} = \exp\left( -\pi t\exp\left(2\sum_pv_p(n)\log p\right) \right),$$

$$n^{-s}=\prod_p p^{-s v_p(n)}.$$

The labelwise identity

$$\int_0^\infty e^{-\pi n^2t}t^{s/2-1}\,dt = \pi^{-s/2}\Gamma(s/2)n^{-s}$$

shows that Mellin transport is the exact comparison cell converting additive heat formation into multiplicative Euler formation. Poisson reflection and the two polar currents complete this cell beyond its initial convergence sector.

The cell explains why the two constructions agree but does not orient their common divisor. Scalar evaluation can still cancel while the entire comparison square commutes. The missing rung is now a law on the completed comparison cell itself, not another data port.

Artifact:

• research/grothendieck/mellin-transport-is-the-additive-to-multiplicative-comparison-cell.md

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