rs_pp_condition

formal source

 164/-- The RS framework recovers the classical Hodge (p,p) type condition:
 165    z_charge ≥ 0 is the RS version of the (p,p) Hodge condition.
 166    Classical (p,p) classes have equal holomorphic and antiholomorphic degree. -/
 167theorem rs_pp_condition (L : DefectBoundedSubLedger)
 168    (cls : CoarseGrainingStableClass L) :
 169    0 ≤ cls.z_charge := cls.symmetric
 170
 171/-- A non-trivial DefectBoundedSubLedger exists (so the conjecture is non-vacuous). -/
 172theorem sub_ledger_exists : ∃ L : DefectBoundedSubLedger,
 173    L.events.length > 0 := by
 174  use {
 175    events := [⟨0, 1, 1, by norm_num⟩]
 176    defect := 0
 177    defect_bounded := by
 178      have : (0 : ℝ) < phi ^ (44 : ℕ) := pow_pos phi_pos 44
 179      linarith
 180    defect_nonneg := le_refl _
 181  }
 182  simp
 183
 184/-! ## Part 5: Coarse-Graining Flow (Defect Budget) -/
 185
 186/-- A coarse-graining flow on a sub-ledger: zooming out decreases the apparent defect.
 187    This is the RS analog of the heat flow in classical Hodge theory. -/
 188structure CoarseGrainingFlow (L : DefectBoundedSubLedger) where
 189  /-- Defect at scale 2^n (coarsened n times) -/
 190  coarsened_defect : ℕ → ℝ
 191  /-- At scale 0, we see the full defect -/
 192  at_zero : coarsened_defect 0 = L.defect
 193  /-- Coarse-graining is monotone: zooming out can only decrease apparent defect -/
 194  monotone_decrease : ∀ n, coarsened_defect (n + 1) ≤ coarsened_defect n
 195  /-- Defect is always nonneg -/
 196  nonneg : ∀ n, 0 ≤ coarsened_defect n
 197