finite_resolution_pos

formal source

 140  exact Finset.card_pos.mpr ((Set.Finite.toFinset_nonempty hfin).mpr hne)
 141
 142/-- Finite resolution neighborhoods have positive event count -/
 143theorem finite_resolution_pos (c : C) (h : HasFiniteLocalResolution L r c) :
 144    ∃ U ∈ L.N c, ∃ hfin : (r.R '' U).Finite, 0 < eventCountFinite (r.R '' U) hfin := by
 145  obtain ⟨U, hU, hfin⟩ := h
 146  use U, hU, hfin
 147  apply eventCountFinite_pos
 148  exact ⟨r.R c, c, L.mem_of_mem_N c U hU, rfl⟩
 149
 150/-! ## Physical Interpretation -/
 151
 152/-- **Physical Interpretation**: In Recognition Science, finite resolution
 153    corresponds to the 8-tick atomicity. The number of distinguishable
 154    states in any local region is bounded by the number of ledger updates
 155    that can occur in that region.
 156
 157    This is not an approximation or limitation of measurement—it is a
 158    fundamental feature of reality as described by recognition geometry. -/
 159theorem physical_interpretation_finite_resolution
 160    (h : IsLocallyDiscrete L r) :
 161    ∀ c, ∃ U ∈ L.N c, (r.R '' U).Finite := by
 162  intro c
 163  exact h c
 164
 165/-! ## Module Status -/
 166
 167def finite_resolution_status : String :=
 168  "✓ HasFiniteLocalResolution defined (RG4)\n" ++
 169  "✓ HasFiniteResolution: global finite resolution\n" ++
 170  "✓ finite_resolution_event_in_finite\n" ++
 171  "✓ finite_resolution_mono: inheritance by smaller neighborhoods\n" ++
 172  "✓ no_injection_on_infinite_finite: key non-injection theorem\n" ++
 173  "✓ finite_resolution_not_injective: corollary\n" ++