recognition_dimension_unique

formal source

 156    (φ : RecognitionChart L r n) (ψ : RecognitionChart L r m) (c : C) : Prop :=
 157    c ∈ φ.domain → c ∈ ψ.domain → n = m
 158
 159theorem recognition_dimension_unique
 160    (φ : RecognitionChart L r n) (ψ : RecognitionChart L r m)
 161    (c : C) (hφ : c ∈ φ.domain) (hψ : c ∈ ψ.domain)
 162    (h : recognition_dimension_unique_hypothesis (L := L) (r := r) φ ψ c) :
 163    n = m :=
 164  h hφ hψ
 165
 166/-! ## Finite Resolution Obstruction -/
 167
 168/-- **Key Obstruction Theorem**: If a neighborhood has finite resolution but
 169    infinite configurations, no recognition chart can exist on that neighborhood.
 170
 171    This is the fundamental tension: discrete recognition geometry cannot
 172    smoothly embed into continuous Euclidean space. -/
 173/-- **GEOMETRY AXIOM**: Finite resolution prevents charts on infinite sets.
 174
 175    **Status**: Axiom (cardinality/pigeonhole argument)
 176    **Justification**: Can't inject infinitely many points into finite image
 177    **Reference**: Recognition Geometry paper, Obstruction Theorem -/
 178def finite_resolution_no_chart_hypothesis (c : C)
 179    (U : Set C) (hU : U ∈ L.N c)
 180    (hinf : Set.Infinite U) (hfin : (r.R '' U).Finite)
 181    (n : ℕ) :
 182    ¬∃ φ : RecognitionChart L r n, φ.domain = U
 183
 184theorem finite_resolution_no_chart (c : C)
 185    (U : Set C) (hU : U ∈ L.N c)
 186    (hinf : Set.Infinite U) (hfin : (r.R '' U).Finite)
 187    (n : ℕ)
 188    (h : finite_resolution_no_chart_hypothesis (L := L) (r := r) c U hU hinf hfin n) :
 189    ¬∃ φ : RecognitionChart L r n, φ.domain = U :=