dyadicPolygonSmallSphereComponent
plain-language theorem explainer
Records the cell counts of one six-face cube-sphere surface component in the Phase-36 dyadic polygon gluing: 8 vertices, 12 edges, 6 faces, Euler characteristic 2, with a full eight-cycle vertex-link audit. Cosmologists citing the dyadic sponge R20 assembly or the Phase-38 oriented certificate use this constant. It is a pure structure literal, not a proved identity.
Claim. The dyadic Phase-36 small-sphere polygon component is the surface cellulation with $V=8$, $E=12$, $F=6$, Euler characteristic $\chi=2$, eight vertex links, and eight vertex-link cycles (the combinatorial type of a cube boundary).
background
This module builds the algebraic bridge from a compact cubical 3-region with Betti triple $(b_0,b_1,b_2)$ to the genus of its regular-neighborhood boundary. After raw cubical boundaries showed nonmanifold edges, the readout switched to the desingularized regular-neighborhood surface; the target identity is that total boundary genus equals $b_1$.
Phase 36 supplies finite polygon-gluing witnesses: each component is a PolygonGluingComponent recording quotient vertices, split edges, faces, Euler count, and a local vertex-link audit (link count versus cyclic-link count). Binary edge gluing plus cyclic quotient-vertex links let the component reduce to the Phase-35 assembly arithmetic.
The present value is the cube-sphere atom in that catalogue: six faces, twelve edges, eight vertices, $\chi=2$, with every vertex link a single cycle. It is the small spherical building block later replicated in the dyadic sponge list.
proof idea
No proof. The declaration is a structure literal that fills the six fields of a polygon-gluing component with the integer tuple $(8,12,6,2)$ and the link audit $(8,8)$. Euler is recorded as $2$, matching $V-E+F=8-12+6$.
why it matters
Phase 37 wraps polygon-gluing components into the Phase-35 genus assembly: if binary edge gluing, cyclic vertex links, and corrected Euler data hold, total genus is forced to $b_1$. This six-face sphere is one of the atomic certificates in that witness list.
Downstream, the dyadic sponge R20 component list replicates it forty-eight times alongside one genus-125 piece, three medium spheres, and one large sphere. The Phase-38 oriented wrapper then attaches a full six-face orientation assignment with zero contradictions, so the oriented certificate inherits the Phase-37 genus theorem.
Within the broader RS cosmology track this is pure combinatorial scaffolding for the foam-interface desingularization scripts. It does not close the still-open geometric realization or homeomorphism to the true regular-neighborhood boundary; it only feeds the finite numeric certificate chain (Phases 36–39).
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