horizonPolygonSphereComponent
plain-language theorem explainer
Numeric certificate for the spherical polygon-glued surface component of the Phase-36 horizon at radius R=20: 2484 vertices, 4964 edges, 2482 faces, Euler characteristic 2, with full cyclic vertex-link audit. Cosmology and discrete-geometry workers cite it when assembling the desingularized horizon boundary. The body is a pure structure literal, not a derived proof.
Claim. The Phase-36 horizon sphere component is the polygon-glued surface with $V=2484$, $E=4964$, $F=2482$, Euler characteristic $\chi=2$, and $2484$ vertex links each forming a single cycle.
background
This module builds the algebraic bridge from cubical positive-excursion sets to desingularized regular-neighborhood boundaries (Phases 27–47). Raw cubical boundaries can carry nonmanifold edges; the canonical readout is the boundary of a regular neighborhood of ${q>0}$. For a compact 3D cubical region with Betti triple $(b_0,b_1,b_2)$, that boundary has $b_0+b_2$ components, Euler characteristic $2(b_0-b_1+b_2)$, and total genus $b_1$.
Phase 36 supplies finite polygon-gluing witnesses. A PolygonGluingComponent records quotient vertex count, split-edge count, face count, Euler number, and a local vertex-link audit (number of links and how many are single cycles). Binary edge gluing plus cyclic quotient-vertex links lets the component reduce to the Phase-35 assembly theorem.
The present value is the spherical piece of the horizon certificate at $R=20$ (Euler characteristic 2). It sits beside a toroidal handle component in the same phase.
proof idea
No proof obligations: the declaration is a structure literal packing the Phase-36 numeric certificate. Fields are set to the measured cell counts and link audit for the sphere component ($V=2484$, $E=4964$, $F=2482$, $\chi=2$, full cyclic links). Downstream lemmas treat these integers as fixed data when checking Euler identities and genus formulas.
why it matters
Feeds the Phase-36 horizon component list (sphere plus torus handle) and the Phase-38 oriented sphere certificate, which records a full face-orientation assignment with zero contradictions. Those wrappers inherit the Phase-37 polygon-gluing genus theorem and the Phase-39 orientability gate.
In the Recognition cosmology stack this is a finite combinatorial witness, not the missing geometric homeomorphism from corrected cellulations to true regular-neighborhood boundary components. The module status remains partial through Phase 44 and conditional at Phase 47: arithmetic bridges and numeric certificates are closed; the embedded digital-cubical collapse and homeomorphism theorem stay open. The sphere piece ($\chi=2$) anchors the genus bookkeeping so the handle's genus can be isolated against the $b_1$ target.
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