dyadicSpongeR20SingularComponents
plain-language theorem explainer
Records the Phase-30 singular-edge inventory for the dyadic sponge R20: twenty-four V2-E1 components and twenty-one V4-E3 components. Cosmology certificates cite it as the fixed singular-graph input to the half-vertex quotient. The body is a pure list concatenation of the two component templates.
Claim. The Phase-30 singular-edge graph of the dyadic sponge is the finite list consisting of $24$ copies of the two-vertex one-edge component together with $21$ copies of the four-vertex three-edge component.
background
This module builds the algebraic bridge from a compact cubical region's Betti triple $(b_0,b_1,b_2)$ to the genus of its regular-neighborhood boundary. After Phase 29 pairs four-face edges, the remaining singular edges form a graph whose connected pieces are recorded only by cell counts, not by an embedded geometry.
A singular graph component is a pair of natural numbers (vertices, edges). Two templates appear here: the two-vertex one-edge piece and the four-vertex three-edge piece. Each component contributes a half-vertex quotient equal to half its vertex count (one lift per singular-vertex pair), which Phase 30 uses to close the missing vertex budget on the desingularized boundary.
The local setting is the dyadic-sponge numeric certificate path: edge-only Euler data from Phase 29 is corrected by this singular inventory so that the half-vertex-adjusted Euler matches the canonical CW Euler of the regular-neighborhood boundary.
proof idea
Definitional construction, not a proof. The value is the concatenation of List.replicate 24 of the two-vertex one-edge template with List.replicate 21 of the four-vertex three-edge template. Downstream theorems discharge equalities about this list by native_decide.
why it matters
Feeds the three Phase-30/31 dyadic-sponge certificates: the half-vertex delta equals $24\cdot 1 + 21\cdot 2 = 66$; correcting the Phase-29 edge-only Euler $-210$ by that delta recovers the canonical CW Euler $-144$; and the half-vertex quotient closes under the recorded Betti data. Those results are the algebraic half of the vertex-link half-quotient bridge described in the module status for Phase 31.
They do not discharge the still-open embedded digital-cubical collapse or the geometric realization of the regular-neighborhood boundary. Within Recognition cosmology this is a numeric witness on the foam-interface desingularization path, not a forcing-chain (T0–T8) step.
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