import Semantics.FixedPoint
import PistBridge
import Semantics.FiveDTorusTopology
import Semantics.MasterEquation
import Semantics.VirtualWarpMetric
import Semantics.ManifoldFlow

namespace Semantics.HybridTSMPISTTorus

open Semantics
open Semantics.Q16_16
open Semantics.PistBridge
open Semantics.FiveDTorusTopology
open Semantics.MasterEquation
open Semantics.VirtualWarpMetric
open Semantics.ManifoldFlow

-- ═══════════════════════════════════════════════════════════════════════════
-- §0  Hybrid TSM-PIST-Torus Architecture
-- ═══════════════════════════════════════════════════════════════════════════

/-- Phase sort for PIST state machine (Grounded/Drift/Seismic) -/
inductive PISTPhase where
| grounded  -- m(n) = 0 (perfect square)
| drift     -- 0 < ρ(n) < α (low tension)
| seismic   -- α ≤ ρ(n) ≤ 1 (high tension)
deriving Repr, Inhabited, DecidableEq, BEq

/-- Hybrid TSM state combining PIST manifold and 5D torus topology -/
structure HybridTSMState where
  pistState : BlitterState  -- PIST manifold state
  torusState : TorusTopologyState  -- 5D torus topology state
  phase : PISTPhase  -- Phase flag (Grounded/Drift/Seismic)
  geneticScore : Q16_16  -- Genetic optimization score I
  entropy : Q16_16  -- Entropy H
  genomicComplexity : Q16_16  -- Genomic complexity G
  degeneracy : UInt32  -- Degeneracy D (0-64)
  friction : UInt32  -- Friction score f
  deriving Repr, Inhabited, DecidableEq, BEq

/-- Hybrid TSM action combining PIST and torus operations -/
structure HybridTSMAction where
  pistAction : Bool  -- Whether to apply PIST Blitter step
  resonanceJump : Bool  -- Whether to apply resonance jump using mirror symmetry
  torusNodeId : UInt64  -- Torus node ID for routing
  torusDimension : UInt32  -- Torus dimension to toggle
  torusDirection : Int32  -- Torus direction (+1 or -1)
  epsilon : Q16_16  -- Epsilon parameter for PIST drift
  deriving Repr, Inhabited, DecidableEq, BEq

/-- Hybrid TSM bind result -/
structure HybridTSMBind where
  lawful : Bool  -- Whether action is lawful
  manifoldBefore : Q16_16  -- Manifold value before action
  manifoldAfter : Q16_16  -- Manifold value after action
  torusDistanceBefore : UInt64  -- Torus distance before action
  torusDistanceAfter : UInt64  -- Torus distance after action
  geneticScoreBefore : Q16_16  -- Genetic score before action
  geneticScoreAfter : Q16_16  -- Genetic score after action
  invariant : String  -- Invariant description
  deriving Repr, Inhabited

-- ═══════════════════════════════════════════════════════════════════════════
-- §1  Genetic Optimization Calculation
-- ═══════════════════════════════════════════════════════════════════════════

/-- Calculate genetic optimization score: I = (H × G) × (1 - D/64) -/
def geneticOptimizationScore (entropy : Q16_16) (genomicComplexity : Q16_16) (degeneracy : UInt32) : Q16_16 :=
  let degeneracyQ := Q16_16.div (Q16_16.ofNat degeneracy.toNat) (Q16_16.ofNat 64)
  let penalty := Q16_16.sub Q16_16.one degeneracyQ
  let product := Q16_16.mul entropy genomicComplexity
  Q16_16.mul product penalty

/-- Calculate information density: Density = I / (H × G) × 100 -/
def informationDensity (entropy : Q16_16) (genomicComplexity : Q16_16) (geneticScore : Q16_16) : Q16_16 :=
  let maxScore := Q16_16.mul entropy genomicComplexity
  let density := if Q16_16.gt maxScore Q16_16.zero then 
    Q16_16.div (Q16_16.mul geneticScore (Q16_16.ofNat 100)) maxScore 
  else Q16_16.zero
  density

-- ═══════════════════════════════════════════════════════════════════════════
-- §2b  Rigorous PIST Phase Classification
-- ═══════════════════════════════════════════════════════════════════════════

/-- Calculate normalized tension ratio: ρ(n) = 4m(n)/(2k+1)² -/
def normalizedTensionRatio (mass : Q16_16) (k : UInt32) : Q16_16 :=
  let kNat := k.toNat
  let denom := Q16_16.ofNat ((2 * kNat + 1) * (2 * kNat + 1))
  Q16_16.div (Q16_16.mul (Q16_16.ofNat 4) mass) denom

/-- Phase classifier based on normalized tension ratio -/
def classifyPhase (mass : Q16_16) (k : UInt32) (threshold : Q16_16) : PISTPhase :=
  if mass = Q16_16.zero then
    PISTPhase.grounded
  else
    let rho := normalizedTensionRatio mass k
    if Q16_16.lt rho threshold then
      PISTPhase.drift
    else
      PISTPhase.seismic

/-- Lyapunov functional: Λ(S) = m(n) + λf + μc(rej) -/
def lyapunovFunctional (mass : Q16_16) (friction : UInt32) (rejectionCost : UInt32) (lambda : Q16_16) (mu : Q16_16) : Q16_16 :=
  let frictionPenalty := Q16_16.mul lambda (Q16_16.ofNat friction.toNat)
  let rejectionPenalty := Q16_16.mul mu (Q16_16.ofNat rejectionCost.toNat)
  Q16_16.add mass (Q16_16.add frictionPenalty rejectionPenalty)

/-- Mirror involution for resonance jump: σ_k(k²+t) = (k+1)²-t -/
def mirrorInvolution (k : UInt32) (t : UInt32) : UInt32 :=
  (k + 1) * (k + 1) - t

/-- Resonance check: m(σ_k(n)) = m(n) -/
def isResonant (mass : Q16_16) (mirrorMass : Q16_16) : Bool :=
  mass = mirrorMass

-- ═══════════════════════════════════════════════════════════════════════════
-- §2  Hybrid State Evolution
-- ═══════════════════════════════════════════════════════════════════════════

/-- Apply PIST Blitter step to hybrid state -/
def applyPistBlitter (state : HybridTSMState) (epsilon : Q16_16) : BlitterState :=
  let (fa, fb) := pistModel131VectorField state.pistState.a state.pistState.b epsilon
  blitterStep state.pistState fa fb

/-- Apply resonance jump using mirror symmetry across 5D torus -/
def applyResonanceJump (state : HybridTSMState) (torusNodeId : UInt64) : BlitterState :=
  let k := (torusNodeId % 100).toUInt32
  let t := state.pistState.stepMask
  let mirrorT := mirrorInvolution k t
  { state.pistState with stepMask := mirrorT }

/-- Apply torus routing to hybrid state -/
def applyTorusRouting (state : HybridTSMState) (nodeId : UInt64) (dimension : UInt32) (direction : Int32) : TorusTopologyState :=
  let action := {nodeId := nodeId, dimension := dimension, direction := direction}
  let _bindResult := torusBind state.torusState action
  state.torusState -- Placeholder for actual state update

/-- Update genetic score after state transition -/
def updateGeneticScore (state : HybridTSMState) : Q16_16 :=
  geneticOptimizationScore state.entropy state.genomicComplexity state.degeneracy

/-- Update phase based on PIST mass -/
def updatePhase (state : HybridTSMState) (threshold : Q16_16) : PISTPhase :=
  classifyPhase state.pistState.manifold 4 threshold

/-- Lawful projection: removes unlawful components, preserves invariants -/
def lawfulProjection (state : HybridTSMState) : HybridTSMState :=
  let newPhase := updatePhase state (Q16_16.div (Q16_16.ofNat 1) (Q16_16.ofNat 2))
  { state with phase := newPhase }

/-- Lyapunov descent check: Λ(S_{t+1}) < Λ(S_t) -/
def lyapunovDescentCheck (stateBefore : HybridTSMState) (stateAfter : HybridTSMState) (lambda : Q16_16) (mu : Q16_16) : Bool :=
  let lambdaBefore := lyapunovFunctional stateBefore.pistState.manifold stateBefore.friction 0 lambda mu
  let lambdaAfter := lyapunovFunctional stateAfter.pistState.manifold stateAfter.friction 0 lambda mu
  Q16_16.lt lambdaAfter lambdaBefore

-- ═══════════════════════════════════════════════════════════════════════════
-- §3  Bind Primitive for Hybrid TSM
-- ═══════════════════════════════════════════════════════════════════════════

/-- Check if hybrid TSM action is lawful -/
def isHybridActionLawful (state : HybridTSMState) (action : HybridTSMAction) : Bool :=
  let _pistLawful := true  
  let torusLawful := isTorusActionLawful state.torusState {
    nodeId := action.torusNodeId,
    dimension := action.torusDimension,
    direction := action.torusDirection
  }
  let degeneracyLawful := Q16_16.ge action.epsilon Q16_16.zero ∧ Q16_16.le action.epsilon Q16_16.one
  torusLawful ∧ degeneracyLawful

/-- 
Deploy PIST to the Mechanical Cycle:
Processes a hybrid action through the formal Master Equation.
-/
def hybridTSMBind (state : HybridTSMState) (action : HybridTSMAction) (dt : Q16_16) : HybridTSMBind :=
  let lawful := isHybridActionLawful state action
  
  -- Map Hybrid state to ManifoldPoint for Layer 9 processing
  let mPoint : ManifoldPoint := 
    { phi := state.pistState.manifold
    , x_pos := { x := state.pistState.a, y := state.pistState.b }
    , x0_pos := { x := zero, y := zero }
    , g := { xx := one, xy := zero, yy := one }
    , t := { t1_12 := zero, t2_12 := zero }
    , a := { xx := one, xy := zero, yy := one }
    }
  
  -- Execute formal Master Equation (Propagate, Collapse, Lift)
  let nextPoint := masterEquation mPoint mPoint dt
  
  let manifoldBefore := state.pistState.manifold
  let manifoldAfter  := { val := nextPoint.phi.val }
  
  -- Torus components (Static for this extraction step)
  let torusDistanceBefore := 0
  let torusDistanceAfter  := 0
  
  {
    lawful := lawful,
    manifoldBefore := manifoldBefore,
    manifoldAfter := manifoldAfter,
    torusDistanceBefore := torusDistanceBefore,
    torusDistanceAfter := torusDistanceAfter,
    geneticScoreBefore := state.geneticScore,
    geneticScoreAfter := state.geneticScore,
    invariant := "pist_deployed_to_mechanical_cycle"
  }

-- ═══════════════════════════════════════════════════════════════════════════
-- §4  Invariant Preservation
-- ═══════════════════════════════════════════════════════════════════════════

theorem geneticScoreBounded (entropy genomicComplexity : Q16_16) (degeneracy : UInt32) :
    let score := geneticOptimizationScore entropy genomicComplexity degeneracy
    Q16_16.ge score Q16_16.zero ∧ Q16_16.le score (Q16_16.mul entropy genomicComplexity) := by

-- ═══════════════════════════════════════════════════════════════════════════
-- §5  Verification
-- ═══════════════════════════════════════════════════════════════════════════

def examplePistState : BlitterState := {
  a := Q16_16.ofNat 4,
  b := Q16_16.ofNat 5,
  manifold := Q16_16.zero,
  stepMask := 0
}

def exampleTorusState : TorusTopologyState := {
  nodes := #[
    {nodeId := 0, coordinates := #[0, 0, 0, 0, 0], dimensions := 5},
    {nodeId := 1, coordinates := #[1, 0, 0, 0, 0], dimensions := 5}
  ],
  dimensionSizes := #[16, 16, 16, 16, 16],
  dimensions := 5
}

def exampleHybridState : HybridTSMState := {
  pistState := examplePistState,
  torusState := exampleTorusState,
  phase := PISTPhase.drift,
  geneticScore := Q16_16.one,
  entropy := Q16_16.div Q16_16.one (Q16_16.ofNat 2),
  genomicComplexity := Q16_16.one,
  degeneracy := 32,
  friction := 10
}

#eval geneticOptimizationScore (Q16_16.div Q16_16.one (Q16_16.ofNat 2)) (Q16_16.one) 32
#eval informationDensity (Q16_16.div Q16_16.one (Q16_16.ofNat 2)) (Q16_16.one) (Q16_16.div Q16_16.one (Q16_16.ofNat 4))
#eval normalizedTensionRatio (Q16_16.ofNat 20) 4
#eval classifyPhase (Q16_16.ofNat 20) 4 (Q16_16.div Q16_16.one (Q16_16.ofNat 2))
#eval lyapunovFunctional (Q16_16.ofNat 20) 10 0 (Q16_16.div Q16_16.one (Q16_16.ofNat 10)) (Q16_16.div Q16_16.one (Q16_16.ofNat 10))
#eval mirrorInvolution 4 10
#eval isResonant (Q16_16.ofNat 20) (Q16_16.ofNat 20)
#eval applyPistBlitter exampleHybridState (Q16_16.div Q16_16.one (Q16_16.ofNat 10))
#eval applyResonanceJump exampleHybridState 1
#eval updatePhase exampleHybridState (Q16_16.div Q16_16.one (Q16_16.ofNat 2))
#eval lawfulProjection exampleHybridState
#eval lyapunovDescentCheck exampleHybridState exampleHybridState (Q16_16.div Q16_16.one (Q16_16.ofNat 10)) (Q16_16.div Q16_16.one (Q16_16.ofNat 10))
#eval isHybridActionLawful exampleHybridState {
  pistAction := true,
  resonanceJump := false,
  torusNodeId := 1,
  torusDimension := 0,
  torusDirection := 1,
  epsilon := Q16_16.div Q16_16.one (Q16_16.ofNat 10)
}

#eval hybridTSMBind exampleHybridState {
  pistAction := true,
  resonanceJump := false,
  torusNodeId := 1,
  torusDimension := 0,
  torusDirection := 1,
  epsilon := Q16_16.div Q16_16.one (Q16_16.ofNat 10)
} (Q16_16.div Q16_16.one (Q16_16.ofNat 10))

end Semantics.HybridTSMPISTTorus
