dyadicPolygonGenus125Component
plain-language theorem explainer
Numeric certificate for the large genus-125 surface in the Phase-36 dyadic polygon gluing: V=8740, E=17976, F=8988, χ=-248, with full cyclic vertex-link audit. Cosmology and foam-interface work cites it as the dominant handle component of the R20 sponge. The body is a pure structure literal; no proof obligations.
Claim. The Phase-36 dyadic polygon-gluing surface component with cell counts $V=8740$, $E=17976$, $F=8988$, Euler characteristic $\chi=-248$, and vertex-link audit $8740$ links all cyclic, realizing a connected surface of genus $125$ (via $\chi=2-2g$).
background
This module builds the algebraic bridge from cubical positive-excursion sets to desingularized regular-neighborhood boundaries. After Phase 25 found nonmanifold edges on the raw cubical boundary, Phase 26 switched to the regular-neighborhood readout. The bridge asserts that if a compact 3D cubical region has Betti triple $(b_0,b_1,b_2)$, then the desingularized boundary has $b_0+b_2$ components, Euler characteristic $2(b_0-b_1+b_2)$, and total genus exactly $b_1$.
Phase 36 supplies a finite polygon-gluing witness: binary edge pairings, cyclic quotient-vertex links, and corrected component Euler data. A PolygonGluingComponent records quotient vertices, split edges, faces, Euler count, and the local vertex-link audit (number of links and how many are single cycles). Phase 37 reduces such a gluing to the Phase-35 component-assembly genus theorem; this is still a combinatorial certificate, not a homeomorphism to the geometric regular-neighborhood boundary.
proof idea
Definition only: a structure literal packing the six integer fields of a polygon-glued component. No tactics, no lemmas. The Euler field is the recorded $\chi=V-E+F=-248$; the link fields assert every quotient vertex has a single cyclic link, matching the Phase-36 cyclic-link hypothesis used downstream.
why it matters
This is the large handle piece of the Phase-36 dyadic sponge. Downstream, dyadicSpongeR20PolygonComponents assembles it with 48 small spheres, 3 medium spheres, and one large sphere as the full polygon-gluing component list. dyadicOrientedGenus125Component wraps the same counts with a full face-orientation assignment and zero contradictions (Phase 38 orientability gate).
In the module arc, Phase 37 says: if a finite polygon gluing has binary edge gluing, cyclic vertex links, and the corrected Euler data, it inherits the Phase-35 theorem that total genus equals $b_1$. This component supplies the genus-125 term in that sum. It does not close the still-open embedded digital-cubical collapse or the regular-neighborhood homeomorphism theorem; it is the numeric witness those later geometric steps must match.
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