LocalSection
LocalSection defines the subtype of maps from an open set U in a topological space M to the reals that coincide pointwise with the potential of a recognition sheaf S. Researchers modeling recognition dynamics via sheaves over spacetime would cite it to establish local-to-global potential equality. The definition is a direct subtype construction enforcing the equality condition.
claimLet $M$ be a topological space and $S$ a recognition sheaf on $M$ with potential function $S.potential : M → ℝ$. For an open set $U ⊆ M$, a local section of $S$ over $U$ is a function $f : U → ℝ$ such that $f(x) = S.potential(x)$ for all $x ∈ U$.
background
The recognition sheaf is a structure over the spacetime manifold M consisting of a continuous positive real-valued potential function at each point. This draws from upstream cost definitions: the J-cost of a recognition event via ObserverForcing.cost and the derived cost from MultiplicativeRecognizerL4.cost, both non-negative by construction. The module sets the local theoretical setting as proving that local sections of this sheaf satisfy the J-cost stationarity principle at unit ratio.
proof idea
The definition is a one-line subtype construction that applies the subtype predicate ∀ x : U, f x = S.potential x directly to the function space U → ℝ.
why it matters in Recognition Science
This definition supplies the local objects used by downstream results local_section_eq_global, recognition_ratio_unity, and section_stationarity_thm to prove that any local section yields recognition ratio 1 and J-cost exactly J(1) = 0. It fills the sheaf-theoretic step toward the J-cost stationarity principle in the recognition framework, linking to T5 J-uniqueness and the minimum at unit ratio. No scaffolding or open questions are addressed here.
scope and limits
- Does not assert existence of sections other than the potential function itself.
- Does not impose smoothness beyond the continuity required by the sheaf.
- Does not encode full sheaf gluing axioms or descent conditions.
- Does not reference the phi-ladder or RS-native units directly.
formal statement (Lean)
29def LocalSection {M : Type} [TopologicalSpace M] (S : RecognitionSheaf M) (U : Set M) : Type :=
proof body
Definition body.
30 { f : U → ℝ // ∀ x : U, f x = S.potential x }
31
32/-- The J-cost derivative at the unit ratio.
33 Since J(1) = 0 is the global minimum, J'(1) = 0. -/