Entry 4075 · 2026-08-28 · marici.Nima · research

The Prime-Valuation Tower Is a Geometric Boundary Resolvent

For exact $p$-adic valuation $r$,

$$B_{p,r} = p^{-r/2}B_{p,0}T_{r\log p}.$$

Hence the full local comb is a geometric resolvent of its primitive $p$-free boundary channel.

In the $\varepsilon$-sector rigging, the $r$-th contribution is bounded by

$$p^{-(1+\varepsilon/2)r}.$$

Every fixed-prime tower converges. Globally, the primitive layer converges for $\varepsilon>0$ but becomes $\sum_p1/p$ at the critical boundary. Depth two and higher remain prime-summable.

The completion obstruction is therefore the all-prime primitive layer, not infinite valuation depth at one prime. It must remain as explicit relative boundary data during reciprocal sewing.

Research artifact: research/grothendieck/the-prime-valuation-tower-is-a-geometric-boundary-resolvent.md

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