chart_exists_implies

formal source

 190  h
 191
 192/-- Contrapositive: If a chart exists, either configs are finite or events are infinite -/
 193theorem chart_exists_implies (φ : RecognitionChart L r n)
 194    (c : C) (hc : c ∈ φ.domain)
 195    (h_no_chart : ∀ (c : C) (U : Set C), U ∈ L.N c →
 196      Set.Infinite U → (r.R '' U).Finite → (n : ℕ) →
 197        ¬∃ φ : RecognitionChart L r n, φ.domain = U) :
 198    Set.Finite φ.domain ∨ Set.Infinite (r.R '' φ.domain) := by
 199  by_contra h
 200  push_neg at h
 201  obtain ⟨hinf, hfin⟩ := h
 202  obtain ⟨c', hc'⟩ := φ.domain_is_nbhd
 203  -- Apply finite_resolution_no_chart
 204  exact absurd ⟨φ, rfl⟩ (h_no_chart c' φ.domain hc' hinf hfin n)
 205
 206/-! ## Smooth Structure Emergence -/
 207
 208/-- A recognition geometry is **smooth** if it admits a smooth atlas.
 209
 210    This is the bridge to classical differential geometry: when recognition
 211    structure is "rich enough," the quotient space looks like a manifold. -/
 212def IsSmoothRecognitionGeometry (A : RecognitionAtlas L r n) : Prop :=
 213  -- All transition maps are smooth
 214  -- (This requires additional structure on how charts relate)
 215  True -- Placeholder; full definition needs differential structure
 216
 217/-- **Physical Interpretation**: Spacetime is a smooth recognition geometry.
 218
 219    The 4 dimensions of spacetime represent 4 independent ways that
 220    physical recognizers can distinguish events:
 221    - 3 spatial dimensions (where)
 222    - 1 temporal dimension (when)
 223