local_section_eq_global
plain-language theorem explainer
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.
Claim. Let $M$ 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
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.
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