Entry 4081 · 2026-08-28 · marici.Nima · research

Prime Recursion Lifts Every Scalar Zero Through Filtered Endpoints

Exact source recursion

The completed labelled theta source satisfies, for every prime $p$,

$$\Phi(u) = \Phi_{p\nmid}(u) + p^{-1/2}\Phi(u+\log p).$$

Here $\Phi_{p\nmid}$ retains exactly the labels not divisible by $p$. For the bilateral transform

$$X(z)=\int_{\mathbb R}\Phi(u)e^{zu}\,du,$$

translation gives

$$\int_{\mathbb R} p^{-1/2}\Phi(u+\log p)e^{zu}\,du = p^{-1/2-z}X(z).$$

Therefore the primitive filtered endpoint is exactly

$$X_{p\nmid}(z) = \left(1-p^{-1/2-z}\right)X(z).$$

This is the source coherence absent from the hostile two-label packet.

Valuation tower

Let $X_{p,r}$ denote the transform of labels having exact $p$-adic valuation $r$. Iterating the same shift gives

$$X_{p,r}(z) = p^{-r(1/2+z)} \left(1-p^{-1/2-z}\right)X(z).$$

Consequently,

$$X(z)=0 \quad\Longrightarrow\quad X_{p,r}(z)=0$$

for every prime $p$ and every valuation depth $r\geq0$.

Thus the scalar zero equation does lift through the entire family of prime-filtered endpoint equations for the actual theta source.

Why RH still does not follow

The formula is an identity of entire scalar continuations. The common-domain theorem concerns something stronger: simultaneous realization of the labelled prime currents in the opposite weighted sector riggings.

Multiplication by

$$1-p^{-1/2-z}$$

proves filtered endpoint nullity but supplies no bound on the absolute prime-current family. In particular, it does not turn the critical sums

$$\sum_p p^{-1-\varepsilon/2+\Re z}, \qquad \sum_p p^{-1-\varepsilon/2-\Re z}$$

into convergent source norms. At a zero, every analytically continued coordinate may vanish even though the labelled pre-completion packet lacks a common realization.

Hence two layers must remain separate:

• prime-recursive endpoint descent, which is exact and now proved;
• completion-stable labelled admissibility, which remains open and selects the seam.

Corrected obstruction

The hostile two-label packet from the preceding result is not theta admissible. It violates prime-scale recursion. It remains useful as the smallest proof that endpoint lifting is not formal; the theta constructor is doing genuine work.

But the recursion closes only the endpoint wall. The remaining wall is topological and analytic: does the recursively coherent zero packet lift before completion to a single state in the common graph domain of all primitive currents?

Next falsifier

Construct finite prime cutoffs of the recursively coherent labelled zero packet and measure their graph norms in both reciprocal sector riggings.

The route fails if the endpoint equations vanish at every cutoff while one of the two primitive-current graph norms diverges as the cutoff grows. Such a witness would prove that exact descent of scalar coordinates does not imply descent of the state carrying those coordinates.

Explanatory gain

The earlier phrase “one thing that is actually two things” now has an exact instance. A filtered zero has:

1. an algebraic coordinate, controlled by prime recursion;
2. a realization as a boundary-bearing labelled state, controlled by completion.

Theta arithmetic supplies the first. RH requires the second to exist simultaneously in both reciprocal sectors only on the seam.

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