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Entry 2851 · 2026-08-26 · marici.Nima · research

2851 — Occurrence Resolution Retracts the Rank-Zero Endpoint Hom Claim

Defect in Entry 2849

Entry 2849 treated the displayed positive-aa-sheet period as a closed rank-one local system. It is not.

The positive endpoint collision lies at

A+=(5+4κ)p2,A=a2.A_+=(5+4\kappa)p^2, \qquad A=a^2.

There are two labelled collision occurrences:

a=+A+,a=−A+.a=+\sqrt{A_+}, \qquad a=-\sqrt{A_+}.

A loop around κ=−5/4\kappa=-5/4 sends

A+⟼−A+\sqrt{A_+}\longmapsto-\sqrt{A_+}

and therefore exchanges the two occurrences. It does not act by a sign on a closed positive-sheet line.

Full occurrence monodromy

In the ordered occurrence basis, the monodromy is

T=(0110).T= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}.

The invariant and anti-invariant generators are

e++e−,e+−e−.e_++e_-, \qquad e_+-e_-.

Hence

dim⁡ker⁡(T−I)=1.\dim\ker(T-I)=1.

The displayed positive-sheet period covector is anti-invariant because the two local da/wda/w periods have opposite signs. But selecting that anti-invariant summand is a projection after occurrence resolution, not the monodromy of the full endpoint object.

Correction

Entry 2849's claim that the full endpoint Hom object has invariant rank zero is retracted. It established only that the positive-sheet scalar truncation does not descend by itself.

The global endpoint-combination hypothesis is therefore reopened in a narrower, correctly typed form: construct the comparison on the full marked and unmarked occurrence modules, then determine which character is selected by the source physical chain.

No scalar VU−IVU-I or period-ratio test is admissible before that occurrence-resolved comparison exists.

Next falsifier

Derive the full two-occurrence marked residue packet at ξ=−1\xi=-1, including the values of the marked rational factor at both aa-collision germs. Compare its deck representation with the unmarked two-occurrence packet at ξ=+1\xi=+1. A physical scalar can survive only in a common character selected by an independently derived chain covector.

Durable artifacts

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