PolygonGluingComponent
plain-language theorem explainer
Records one polygon-glued surface component from the Phase-36 combinatorial witness: quotient vertices, split edges, faces, Euler characteristic, and a vertex-link audit (link count vs cyclic links). Cosmology certificates cite it as the cell-count carrier for horizon and dyadic sponge gluings. It is a plain data structure with no proof obligations.
Claim. A polygon-gluing surface component is a 6-tuple $(V,E,F,\chi,L,C)$ of integers (with $L,C\in\mathbb{N}$) recording quotient vertex count $V$, split-edge count $E$, face count $F$, Euler characteristic $\chi$, number of vertex links $L$, and number of cyclic vertex links $C$.
background
This module builds the algebraic bridge from a compact 3D cubical region's Betti triple $(b_0,b_1,b_2)$ to the genus of the boundary of a regular neighborhood of the positive excursion set. After raw cubical boundaries showed nonmanifold edges, the canonical readout became that desingularized boundary: component count $b_0+b_2$, Euler $2(b_0-b_1+b_2)$, and total genus $b_1$.
Phase 36 supplies a finite polygon-gluing witness (binary edge pairing, cyclic quotient-vertex links, corrected component Euler data). Phase 37 wraps that witness so it reduces to the Phase-35 component-assembly genus theorem. The structure here is the per-component cell-count record used by those certificates: vertices after quotient, edges after splitting, faces, the recorded Euler number, and a local audit that every vertex link is a single cycle.
The geometric homeomorphism of these cellulations to true regular-neighborhood boundary components remains open; only the arithmetic bridge and numeric certificates are proved in-file.
proof idea
No proof: this is a structure declaration (data carrier). Fields are bare integer counts plus a vertex-link audit pair. Downstream definitions inhabit it by supplying concrete literals (e.g. genus-125 and sphere components). Any theorem that a component's cell counts match its recorded Euler characteristic is a separate lemma on top of this type.
why it matters
Phase 37's polygon-gluing genus wrapper needs a uniform type for each glued surface piece. This structure is that type. Concrete inhabitants feed the horizon annulus handle list (torus plus sphere) and the dyadic sponge $R_{20}$ list (one genus-125 component, 48 small cube-spheres, three medium spheres, one large sphere).
Those lists are the combinatorial input to the algebraic reduction: binary edge gluing, cyclic vertex links, and corrected Euler data imply total genus $b_1$, matching the regular-neighborhood bridge. The declaration does not close the embedded digital-cubical collapse or the homeomorphism to the geometric regular neighborhood; it only standardizes the Phase-36 witness payload so later phases (orientability gate, numeric certificates) can cite one shape of component.
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