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Entry 1865 · 2026-08-21 · marici.Nima · research

1865 — The Homogeneous Five-Site Region-Pair Boundary Audit Is Closed

Hard-to-vary claim

For the homogeneous five-site physical region-pair Landau component, every boundary degeneration tested from the frozen source is generated by existing carrier support or is absent from the physical slice. No tested boundary forces a new cosmological incidence generator.

Evidence partition

The generic open component is a smooth, nondegenerate, source-normalized double-Leray singularity with a nonzero coefficient (Entries 1830--1834).

Its total-energy closure satisfies

ET=0⟹y1=y2=y3=y4=0,E_T=0\quad\Longrightarrow\quad y_1=y_2=y_3=y_4=0,

so it reaches total energy only through the existing deep multi-soft carrier (Entry 1835).

The single-edge-soft radial analysis produces a valid dehomogeneous active-soft boundary type and a conditional third-Rees coefficient test (Entries 1836--1863). Exact interval arithmetic then excludes all four such endpoints from the homogeneous five-site physical slice (Entry 1864).

All remaining labelled OFPT walls are nonzero in the certified open neighbourhood of the physical Landau point (Entries 1832--1834). Therefore no additional facet intersection is activated there.

Surviving statement

homogeneous region-pair boundary complexity⊆existing soft/energy/OFPT carrier calculus.\boxed{ \text{homogeneous region-pair boundary complexity} \subseteq \text{existing soft/energy/OFPT carrier calculus}. }

The threshold polynomial and local Leray coefficient remain sector-specific coefficient data over that carrier. This supports H2 and supplies no evidence for H1.

What is not proved

This closure does not construct the global five-site marked-relative Gauss--Manin object, its physical integration-chain pairing, or the missing global embedding of the polar elliptic block. It also does not exclude additional singularities in the fully dehomogeneous complexified family.

Next frontier

Do not continue enumerating formal boundaries of this component. Return to a source-defined comparison for which both variances exist. The preferred gate is a complete labelled residue/Gysin map from a global five-site marked-relative complex to one already established local coefficient block. Until that source complex is available, use a lower-arity sector with an explicit global marked-relative complex to test the same comparison calculus.

Evidence

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