H_LocalGlobalUnification
plain-language theorem explainer
H_LocalGlobalUnification defines the empirical hypothesis that vanishing local J-cost variation on every simplex of a simplicial ledger forces every potential to the stationary value 1. Researchers formalizing coordinate-free recognition costs on 3-complexes cite it when connecting local derivatives to global stationarity. The declaration is introduced directly as the implication between the two predicates.
Claim. For a simplicial ledger $L$ and sheaf $S$ over $L$, the hypothesis asserts: if the local variation of the potential vanishes on every simplex, then the potential equals 1 on every simplex.
background
The module develops a simplicial 3-complex representation of the ledger that supplies a coordinate-free sheaf for unifying local and global J-cost variations. SimplicialLedger collects 3-simplices into a manifold covering. SimplicialSheaf assigns a recognition potential to each simplex in the ledger. J_stationary(psi) is the predicate psi = 1. local_variation approximates the derivative of Jcost with respect to the potential at each simplex.
proof idea
The declaration is a direct definition of the implication from zero local variation on all simplices to J_stationary potentials on all simplices.
why it matters
The hypothesis is invoked by the theorem local_global_unification, which states that global J-cost stationarity holds if and only if every local J-cost is stationary within its simplicial section. It encodes the empirical claim that global minimization on the simplicial manifold forces local potentials to unit value, addressing the local-global unification step in the simplicial ledger formulation.
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