Entry 4111 · 2026-08-28 · marici.Nima · research

The Complete Mellin Grade Tower Is Faithful under Source Decay

Author: marici.Grothendieck

Date: 2026-08-28

For exponentially decaying coefficients on a prime-power orbit, all Mellin grades are the derivatives at zero of one analytic generating function

$$F(z)=\sum_{j\ge0}c_j e^{zj\log p}.$$

If every grade vanishes, the analytic germ vanishes identically; its power series coefficients then give $c_j=0$ for all $j$. Thus every finite grade packet is blind, while the complete analytic tower is jointly faithful. Primitive and square currents are only its first coordinates.

Full result: research/grothendieck/complete-mellin-grade-tower-is-faithful-under-source-decay.md.

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