dyadicOrientedGenus125Component
plain-language theorem explainer
Concrete Phase-38 certificate packaging the large dyadic genus-125 polygon component with a complete face-orientation assignment and zero sign contradictions. Cosmology auditors cite it when checking that the desingularized foam interface carries an oriented closed surface of genus 125. The definition is a structure literal reusing the Phase-36 cell counts and recording 8988 faces assigned with no orientation failures.
Claim. The oriented dyadic genus-$125$ component is the Phase-36 polygon-gluing component with $8740$ vertices, $17976$ edges, $8988$ faces, Euler characteristic $-248$, and fully cyclic vertex links, equipped with a face-orientation assignment of size $8988$ and zero orientation contradictions.
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 boundary of a regular neighborhood of ${q>0}$. Algebraically one expects boundary components $b_0+b_2$, boundary Euler $2(b_0-b_1+b_2)$, and total desingularized genus exactly $b_1$.
Phase 36 supplies finite polygon-gluing witnesses (binary edge pairing, cyclic quotient-vertex links, corrected Euler data). The large dyadic piece is the genus-125 component: $V=8740$, $E=17976$, $F=8988$, $\chi=-248$, with every vertex link a single cycle.
Phase 38 adds orientability. An oriented polygon-gluing component pairs that cell audit with a face-sign solve: how many faces received an orientation, and how many sign contradictions remain. Success means every face is assigned and the contradiction count is zero.
proof idea
Definitional structure literal of type oriented polygon-gluing component. The polygon field is the existing Phase-36 dyadic genus-125 component; facesAssigned is set to $8988$ (matching the face count); orientationContradictions is set to $0$. No tactics or lemmas are invoked.
why it matters
Phase 39's orientability gate: once every polygon component has a full face-orientation assignment with zero contradictions, the oriented certificate inherits the Phase-37 polygon-gluing genus theorem. This value is the large handle in that gate for the dyadic sponge.
Downstream, the combinatorial-surface theorem unfolds the closed-orientable-surface predicate on this component and discharges it by native decision, establishing a concrete closed orientable surface of genus 125. The Phase-38 oriented component list for the $R=20$ dyadic sponge places this entry first, then stacks the small, medium, and large sphere components.
The module status remains partial through Phase 44 and conditional at Phase 47: arithmetic and numeric certificates are proved, but the embedded digital-cubical collapse and the homeomorphism from corrected cellulations to true regular-neighborhood boundary components stay open. This definition records the orientability witness without closing that geometric gap.
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