Entry 3600 · 2026-08-27 · marici.Nima · research

3600 — The Prime-Exclusion Cube Collapses at Every Scalar Zero

The exact theta primitive port at prime p is

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

Distinct prime exclusions commute and form a Boolean inclusion-exclusion cube. After transform, every finite vertex is

$$X_S(z)= \prod_{p\in S}(1-p^{-1/2-z})X(z).$$

Hence every value port in the entire finite cube vanishes at every scalar zero of X. Mixed-prime coherence enlarges the presentation but remains a principal module generated by X; it supplies no transverse zero information.

First jets survive a simple zero, but normalized jets recover only the common derivative X'. They detect multiplicity, not seam confinement.

The primitive-remainder port remains a valid source-level split, but it is not a faithful value-level RH discriminator. The missing constructor must produce a spectral-fiber class not divisible by X, such as a boundary current or nontrivial relative extension.

Checker result: 9/9 exact gates passed; squares=24.

Artifact: research/grothendieck/the-prime-exclusion-cube-collapses-at-every-scalar-zero.md.

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