local_section_eq_global
Any local section f of a recognition sheaf S over an open set U in a topological space M satisfies f(x) equal to the sheaf potential at every point x in U. Researchers formalizing recognition dynamics as sheaves over spacetime would cite this to equate local observations with global equilibrium. The proof is a direct one-line extraction of the subtype property that defines LocalSection.
claimLet $M$ be a topological space, $S$ a recognition sheaf over $M$ with potential function $S.potential : M → ℝ$, $U ⊆ M$ an open set, $f$ a local section of $S$ over $U$, and $x ∈ U$. Then $f(x) = S.potential(x)$.
background
The RecognitionSheaf is a structure over a topological space $M$ consisting of a potential map $M → ℝ$ that is continuous and strictly positive everywhere. A LocalSection of $S$ over an open set $U$ is formalized as the subtype of maps $U → ℝ$ whose values coincide exactly with $S.potential$ on $U$. The module develops the sheaf of recognition potentials over spacetime, with the explicit objective of proving that local sections obey the J-cost stationarity principle.
proof idea
The proof is a one-line term that directly applies the property field of the subtype defining the local section f, which encodes the equality f x = S.potential x by construction.
why it matters in Recognition Science
This result is invoked by the downstream recognition_ratio_unity theorem to conclude that the recognition ratio for any local section is identically 1. It supplies the local-to-global consistency step required for the sheaf-theoretic treatment of recognition dynamics, consistent with the J-cost minimum at unity in the Recognition Science forcing chain. No open scaffolding questions are resolved here.
scope and limits
- Does not prove existence of nontrivial local sections.
- Does not impose differentiability on the potential beyond continuity.
- Does not connect the equality to specific constants such as alpha or G.
- Does not address gluing axioms for multiple overlapping sections.
Lean usage
rw [local_section_eq_global S U f x]formal statement (Lean)
66theorem local_section_eq_global {M : Type} [TopologicalSpace M]
67 (S : RecognitionSheaf M) (U : Set M) (f : LocalSection S U) (x : U) :
68 f.val x = S.potential x := f.property x
proof body
Term-mode proof.
69
70/-- **THEOREM: Recognition Ratio Unity**
71 The recognition ratio for any local section is identically 1. -/