dyadicSpongeR20_surfaceTypeGenusTotal
plain-language theorem explainer
The ordered inventory of standard surface types for the R=20 dyadic sponge has total genus 125: one genus-125 surface and fifty-two spheres. Cosmology auditors of the desingularized foam interface cite this as the Phase-42 numeric certificate that the polygon-component genus budget matches the regular-neighborhood readout. The proof is a one-shot native decision on a finite integer sum.
Claim. The sum of genera over the standard-surface type list for the dyadic sponge at radius $R=20$ equals $125$. Explicitly, if that list is one closed surface of genus $125$ followed by fifty-two spheres, then $\sum_i g(S_i)=125$.
background
In this cosmology module the raw cubical boundary of the positive excursion set is replaced by the boundary of a regular neighborhood, which removes nonmanifold edges. For a compact 3D cubical region with Betti triple $(b_0,b_1,b_2)$, that desingularized boundary has $b_0+b_2$ components, Euler characteristic $2(b_0-b_1+b_2)$, and total genus exactly $b_1$.
Standard surface types package each boundary component as a closed oriented surface of fixed genus (sphere genus $0$, or a single high-genus model). The total-genus functional sums the genus field over a finite list of such types. The dyadic sponge inventory at $R=20$ is defined as one genus-$125$ surface concatenated with fifty-two standard spheres, so the arithmetic target is forced once the list is fixed.
proof idea
The surface-type list is a concrete closed term: one standard genus-$125$ surface followed by List.replicate 52 spheres. Total genus is the sum of the mapped genus integers over that list. Each sphere contributes $0$ and the single high-genus entry contributes $125$, so the sum equals $125$ by finite arithmetic. The proof is a one-line native_decide that evaluates the integer equality in the kernel; no lemmas beyond the definitions of the list and the sum are required.
why it matters
Phase 42 of the cosmogenesis pipeline needs a certificate that the ordered polygon Euler signature and the aggregate surface inventory match the regular-neighborhood boundary. This lemma pins the genus total of the dyadic sponge components at $125$, aligning the combinatorial surface budget with the module's genus bridge (total desingularized boundary genus equals $b_1$).
No Lean used_by edges are wired yet; the certificate is aimed at the foam-interface desingularization scripts and at later phase wrappers that compare polygon-gluing and orientability witnesses against the same numeric target. The geometric homeomorphism of corrected cellulations to the regular-neighborhood boundary remains open: this file supplies arithmetic inventory, not realization.
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