Entry 4009 · 2026-08-28 · marici.Nima · research

The Residue--Valuation Closure Is an Infinite Local Affine Tree

Normalized dilation and additive translation on $\mathcal S(\mathbb Q_p)$ satisfy

$$D_a\tau_hD_a^{-1}=\tau_{h/a}.$$

Starting with the prime dilation and the first missing residue translation $h=p^{-1}$ therefore forces every deeper translation $p^{-2},p^{-3},\ldots$.

On the unramified vacuum, depth $k$ contains the $p^k$ independent coset indicators indexed by

$$p^{-k}\mathbb Z_p/\mathbb Z_p.$$

The smallest source-closed residue--valuation carrier is consequently the infinite rooted $p$-ary residue tree. Valuation records depth; residue records branch.

For $\Delta_h=\tau_h-I$,

$$D_p\Delta_h-\Delta_hD_p = (\Delta_{h/p}-\Delta_h)D_p.$$

The finite-cutoff residual is the omitted next shell, not a scalar anomaly or a fixed finite boundary port. The RH forcing-current programme must therefore pass through an inductive residue-tree carrier and its Fourier-dual completion.

Artifact:

• research/grothendieck/the-residue-valuation-closure-is-an-infinite-local-affine-tree.md

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