local_global_unification
plain-language theorem explainer
The theorem shows that if global J-cost variation vanishes on every simplex of a simplicial ledger equipped with a sheaf, then the recognition potential equals one at each simplex. Researchers modeling Recognition Science via simplicial complexes cite it to justify reducing global stationarity checks to local per-simplex conditions. The proof is a direct one-line application of the defining hypothesis that encodes the global-to-local implication.
Claim. Let $L$ be a collection of 3-simplices forming a manifold covering and let $S$ assign a real-valued potential to each simplex. If the local J-cost variation vanishes at every simplex, then the potential equals 1 at every simplex.
background
The module develops a coordinate-free representation of the ledger as a simplicial 3-complex. A SimplicialLedger consists of a nonempty set of 3-simplices together with a manifold-covering property. A SimplicialSheaf on such a ledger assigns a recognition potential to each simplex. J-stationary holds precisely when this potential equals 1. The local setting is the unification of local and global J-cost variations on the simplicial structure.
proof idea
The proof is a one-line wrapper that applies the hypothesis H_LocalGlobalUnification directly to the supplied assumption that local variations vanish on all simplices.
why it matters
The declaration records the local-global unification step inside the simplicial ledger, matching the statement in the theorem comment. It supports the sheaf-theoretic unification of J-cost stationarity required for the Recognition Science ledger at D=3. No downstream theorems are recorded yet; the result remains available for later use in recognition-loop constructions.
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