dyadicPolygonMediumSphereComponent
plain-language theorem explainer
Fixed combinatorial certificate for one medium sphere in the dyadic Phase-36 polygon gluing: 24 vertices, 44 edges, 22 faces, Euler characteristic 2, with a full 24-cycle vertex-link audit. Cosmology scripts and the Phase-37/38 genus wrappers cite it as a numeric witness. The body is a pure structure literal; no proof obligations.
Claim. The medium sphere component of the dyadic Phase-36 polygon gluing is the surface cellulation with $V=24$, $E=44$, $F=22$, Euler characteristic $\chi=2$, and vertex-link audit $(24,24)$ (every quotient vertex has a single cyclic link).
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 Phase 25 found nonmanifold edges on the raw positive excursion set, Phase 26 switched to the desingularized readout; later phases supply finite combinatorial certificates that the corrected boundary genus equals $b_1$.
Phase 36 packages each corrected surface piece as a PolygonGluingComponent: quotient vertex count, split-edge count, face count, recorded Euler number, and a local vertex-link audit (number of links versus number of cyclic links). Binary edge gluing plus cyclic links lets Phase 37 reduce the gluing to the Phase-35 component-assembly theorem.
The present value is one numeric witness in that family: a 22-face sphere ($\chi=2$) appearing three times in the dyadic sponge $R_{20}$ component list.
proof idea
Definition only: a structure literal assigning the six integer fields of a polygon-gluing component. No tactics, no lemmas, no arithmetic obligations at this site. Downstream lemmas that consume the list check Euler consistency and the vertex-link audit against these constants.
why it matters
Fills the medium-sphere slot in the Phase-36 dyadic polygon-gluing witness. Downstream, dyadicSpongeR20PolygonComponents assembles one genus-125 piece, 48 small spheres, three copies of this medium sphere, and one large sphere; that list is the finite certificate Phase 37 wraps into the component-assembly genus theorem. The oriented Phase-38 twin records a full face-orientation assignment with zero contradictions on the same 22 faces, so the orientability gate inherits the same cell counts.
Within the module arc this is arithmetic scaffolding for the desingularized foam interface used by the cosmogenesis scripts, not the missing geometric realization or homeomorphism to the regular-neighborhood boundary. It does not touch the T0–T8 forcing chain or the mass ladder; it sits entirely in the cosmology boundary-genus bridge (Phases 36–39).
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.