horizonPolygonTorusComponent
plain-language theorem explainer
Concrete Phase-36 horizon polygon-gluing certificate for the torus component at radius R=20: V=1632, E=3264, F=1632, χ=0, with a full cyclic vertex-link audit. Cosmology readers cite it as the numeric torus half of the desingularized annulus-handle surface. It is a pure structure literal, not a derived proof.
Claim. The Phase-36 horizon polygon-glued torus component at radius $R=20$ is the surface cellulation with $V=1632$ quotient vertices, $E=3264$ split edges, $F=1632$ faces, Euler characteristic $\chi=V-E+F=0$, and $1632$ cyclic vertex links (one cycle per vertex).
background
This module builds the algebraic bridge for the desingularized boundary of a regular neighborhood of the positive excursion set ${q>0}$ in cubical cosmology. After Phase 25 found nonmanifold edges on the raw cubical boundary, Phase 26 switched to the regular-neighborhood readout; later phases supply combinatorial certificates that the total boundary genus equals the first Betti number $b_1$ of the region.
A PolygonGluingComponent records one Phase-36 polygon-glued surface piece: quotient vertex count, split-edge count, face count, recorded Euler characteristic, and a local vertex-link audit (number of links versus number of cyclic links). Binary edge gluing plus cyclic quotient-vertex links is the finite combinatorial witness that Phase 37 reduces to the Phase-35 component-assembly genus theorem.
The horizon annulus handle at $R=20$ splits into a torus piece ($\chi=0$) and a sphere piece; this definition is the torus half of that pair.
proof idea
Definition by structure literal. The six fields are hard-coded integers from the Phase-36 horizon polygon-gluing run: vertices 1632, edges 3264, faces 1632, euler 0, and matching vertex-link counts 1632/1632. No lemmas are applied; the values are the exported certificate.
why it matters
Supplies the torus entry in the Phase-36 horizon component list consumed by the annulus-handle polygon assembly, and is the underlying polygon data for the Phase-38 oriented torus wrapper (1632 faces assigned, zero orientation contradictions). That oriented certificate then feeds the Phase-39 orientability gate, which inherits the Phase-37 polygon-gluing genus theorem once every component is fully oriented.
In the module arc this is a finite combinatorial witness, not the missing geometric realization: the file proves arithmetic bridges and horizon/dyadic numeric certificates, while the embedded digital-cubical collapse and homeomorphism of corrected cellulations to the true regular-neighborhood boundary remain open. It sits downstream of the genus-equals-$b_1$ program (Phases 27–35) and upstream of the orientability and assembly wrappers (Phases 37–39).
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