local_global_unification
Local-global unification asserts that if local J-cost variation vanishes on every simplex of a simplicial ledger then the recognition potential equals its unit stationary value on each simplex. Researchers modeling gravity through recognition potentials on simplicial complexes cite this when deriving global stationarity from local conditions. The proof is a direct one-line application of the H_LocalGlobalUnification hypothesis to the supplied global variation assumption.
claimLet $L$ be a simplicial ledger of 3-simplices and $S$ a simplicial sheaf assigning a recognition potential to each simplex. Assuming the local-global unification hypothesis holds for $L$ and $S$ and that local variation of the potential vanishes on every simplex, the potential equals 1 on every simplex.
background
The module formalizes the ledger as a simplicial 3-complex rather than a coordinate-fixed lattice, supplying a sheaf representation that unifies local and global J-cost variations. A SimplicialLedger collects 3-simplices into a nonempty set with a manifold covering property. A SimplicialSheaf on such an L assigns a real-valued recognition potential to each simplex together with a consistency condition across boundaries. J_stationary holds precisely when this potential equals 1. The hypothesis H_LocalGlobalUnification encodes the implication from global vanishing of local variation to local stationarity on each simplex.
proof idea
The proof is a one-line wrapper that applies the hypothesis H_LocalGlobalUnification directly to the assumption that local variation vanishes on all simplices.
why it matters in Recognition Science
This result closes the local-to-global stationarity loop inside the simplicial ledger and supports the recognition-loop construction appearing among sibling declarations. It advances the coordinate-free unification of J-cost stationarity that feeds the forcing chain steps T5 through T8 and the Recognition Composition Law. The theorem touches the empirical hypothesis that global J-cost minimization on a simplicial manifold forces every local potential to unit value, whose falsifier is a global minimum containing non-stationary sections.
scope and limits
- Does not establish the manifold covering property of the ledger.
- Does not prove the converse implication from local to global stationarity.
- Does not address sheaf consistency across simplex boundaries.
- Does not connect to specific constants such as alpha or G.
formal statement (Lean)
75theorem local_global_unification (L : SimplicialLedger) (S : SimplicialSheaf L)
76 [Fintype L.simplices] (h : H_LocalGlobalUnification L S)
77 (h_global : ∀ s : L.simplices, local_variation s (S.potential s) = 0) :
78 ∀ s : L.simplices, J_stationary (S.potential s) := h h_global
proof body
Term-mode proof.
79
80/-- **DEFINITION: Recognition Loop**
81 A recognition loop is a closed cycle of 3-simplices in the ledger. -/