EdgeCovered
plain-language theorem explainer
EdgeCovered specifies that a vertex set S covers an edge e precisely when at least one endpoint belongs to S. Researchers formalizing the vertex cover decision problem in the Recognition Science complexity layer cite this predicate when constructing covering relations over graph instances. The definition is a direct disjunction on the two endpoint membership checks.
Claim. For a vertex set $S$ and edge $e=(u,v)$, the edge is covered if $u$ belongs to $S$ or $v$ belongs to $S$.
background
This definition sits in the Complexity.VertexCover module, whose local setting treats complexity pairs as functions of input size. It rests on the InCover predicate, which holds exactly when a vertex number appears in the list S. The module imports upstream enumerations of narrative plot families, kinship systems, nuclear density tiers, and ledger factorizations, placing the covering condition inside the unified Recognition Science framework.
proof idea
The definition is a one-line wrapper that applies the logical disjunction to the two InCover conditions on the first and second components of the edge pair. No lemmas are invoked.
why it matters
EdgeCovered supplies the atomic condition for the Covers predicate, which requires every edge of an instance to be covered. It occupies a foundational slot in the complexity domain and inherits structure from the forcing chain via dependencies on phi-forcing and spectral emergence. The predicate supports formalization of covering problems that align with the discrete phi-ladder structures elsewhere in the framework.
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