-- MetaSum.lean (QuantumAP)
-- Formal definition of MetaSum and Sovereign Shift for the Quantum AP Orchestrator
-- Source: Ahmad Ali Parr 2026-08-16
--
-- θ = 89/2462: the closed loop parameter
-- MetaSum = phase-weighted direct sum on NC Torus T²_θ
-- Dream Cycle = self-healing phase crystallization

import Mathlib.Analysis.SpecialFunctions.Log.Basic

open Real

-- Sovereign Shift
def sovereignShiftNum : ℕ := 89
def sovereignShiftDen : ℕ := 2462

theorem sovereign_shift_coprime : Nat.Coprime sovereignShiftNum sovereignShiftDen := by
  native_decide

theorem sovereignShiftNum_prime : Nat.Prime sovereignShiftNum := by
  native_decide

-- Period: V^2462 commutes with U
theorem lateral_period : sovereignShiftDen = 2462 := rfl

-- Winding: 89 full phase rotations per cycle
theorem phase_winding : sovereignShiftNum = 89 := rfl

-- The Weyl relation structure
structure NCTorus where
  q : ℕ  -- dimension (= 2462)
  p : ℕ  -- winding (= 89)
  coprime : Nat.Coprime p q

-- MetaSum validity condition: entropy ≤ 0.20
structure MetaSumValid where
  magnitude : ℝ
  entropy : ℝ
  mag_positive : magnitude > 0
  entropy_bounded : entropy ≤ 0.20

-- Dream Cycle: triggered when magnitude < threshold
structure DreamCycleResult where
  recovered_magnitude : ℝ
  recovery_achieved : recovered_magnitude > 512  -- N/2