RecognitionSheaf
RecognitionSheaf equips a topological manifold M with a continuous positive real-valued potential that serves as the recognition field. Dynamics researchers cite the structure when deriving that local sections achieve J-cost stationarity at unit ratio. The declaration is a direct structure definition that records the potential map together with its continuity and strict positivity axioms.
claimA recognition sheaf on a topological space $M$ is a continuous function $ψ : M → ℝ$ such that $ψ(x) > 0$ for all $x ∈ M$.
background
Recognition Science derives the J-cost functional J(x) = (x + x^{-1})/2 - 1 from the forcing chain T0-T8, with its global minimum J(1) = 0 at the self-similar fixed point. The module defines the recognition sheaf ℛ over spacetime manifold M so that local sections can be shown to satisfy the stationarity principle. Upstream structures supply the RS-native units (c = 1) and the phi-ladder calibration of J from PhiForcingDerived.of.
proof idea
The declaration is a direct structure definition that introduces the three fields for the potential, continuity, and positivity without invoking any lemmas or tactics.
why it matters in Recognition Science
The structure supplies the carrier for the downstream theorems section_stationarity_thm, recognition_ratio_unity, and sheaf_gluing, which prove that every local section evaluates to J(1) = 0. It fills the module objective of establishing J-cost stationarity for sections of the recognition sheaf and connects to the T5 J-uniqueness step in the forcing chain.
scope and limits
- Does not impose differentiability on the potential beyond continuity.
- Does not encode the sheaf gluing axiom on overlaps.
- Does not identify the potential with any concrete physical field such as metric or electromagnetic potential.
formal statement (Lean)
21structure RecognitionSheaf (M : Type) [TopologicalSpace M] where
22 potential : M → ℝ
23 -- Smoothness requirement (scaffold; would use Mathlib's Smooth in full version)
24 is_continuous : Continuous potential
25 potential_pos : ∀ x, 0 < potential x
26
27/-- **DEFINITION: Local Section**
28 A local section of the recognition sheaf on an open set U. -/