H_LocalGlobalUnification
This definition packages the hypothesis that vanishing local J-cost variation on every simplex of a ledger forces the recognition potential to its stationary unit value at each simplex. Researchers formalizing coordinate-free unification of local and global costs in simplicial 3-complexes would cite it when stating the empirical bridge between manifold-level minimization and pointwise stationarity. The definition is a direct one-line implication between the local variation predicate and the J-stationary predicate.
claimLet $L$ be a simplicial ledger (a collection of 3-simplices forming a manifold covering) and let $S$ be a simplicial sheaf assigning a potential function to each simplex of $L$. Then $H_{LocalGlobalUnification}(L,S)$ is the proposition that if the local variation of the potential vanishes at every simplex, then the potential equals 1 at every simplex.
background
The module SimplicialLedger formalizes the ledger as a simplicial 3-complex rather than a coordinate-fixed lattice, supplying a coordinate-free sheaf representation that unifies local and global J-cost variations. SimplicialLedger is the structure whose simplices field is a nonempty set of Simplex3 objects together with a manifold-covering property. SimplicialSheaf is the structure that equips each simplex with a recognition potential map from Simplex3 to real numbers together with a consistency placeholder. J_stationary(ψ) is the predicate ψ = 1. local_variation(s,ψ) is the finite-difference approximation to the derivative of J-cost at ψ. Upstream cost definitions appear in MultiplicativeRecognizerL4.cost (derived cost of a comparator on positive ratios) and ObserverForcing.cost (J-cost of a recognition event).
proof idea
The declaration is a definition that directly encodes the implication (∀ s, local_variation s (S.potential s) = 0) → (∀ s, J_stationary (S.potential s)). It is a one-line wrapper packaging the empirical hypothesis as a Prop; no tactics or reductions are applied beyond the arrow constructor.
why it matters in Recognition Science
The definition supplies the hypothesis h that is applied verbatim in the downstream theorem local_global_unification, which asserts that global J-cost stationarity holds if and only if every local J-cost is stationary within its simplicial section. It therefore occupies the local-to-global unification step inside the simplicial ledger layer of the Recognition Science foundation. The surrounding module doc-comment notes that the construction is intended to unify local and global J-cost variations on a manifold covering, touching the empirical hypothesis status flagged in the declaration itself.
scope and limits
- Does not prove that global stationarity forces local stationarity; it only defines the implication as a Prop.
- Does not supply the manifold-covering axioms or simplicial-complex structure required by the ledger.
- Does not address boundary consistency of the sheaf potential across adjacent simplices.
- Does not compute or bound the J-cost values; it only states the stationarity predicates.
Lean usage
theorem local_global_unification (L : SimplicialLedger) (S : SimplicialSheaf L) [Fintype L.simplices] (h : H_LocalGlobalUnification L S) (h_global : ∀ s : L.simplices, local_variation s (S.potential s) = 0) : ∀ s : L.simplices, J_stationary (S.potential s) := h h_globalformal statement (Lean)
68def H_LocalGlobalUnification (L : SimplicialLedger) (S : SimplicialSheaf L) [Fintype L.simplices] : Prop :=
proof body
Definition body.
69 (∀ s : L.simplices, local_variation s (S.potential s) = 0) →
70 ∀ s : L.simplices, J_stationary (S.potential s)
71
72/-- **THEOREM: Local-Global Unification**
73 The global J-cost is stationary if and only if every local J-cost is stationary
74 within its simplicial section. -/