namespace Semantics

/--
Energy Gradient Signal Integration

This module formalizes energy decrease/increase as gradient signals that integrate
into the waveform-waveprobe pipeline.

Key structures:
- EnergyGradient: temporal and spatial energy gradients
- EnergyWaveform: gradient encoded as waveform
- EnergySignal: energy gradient with noise
- ThermodynamicChannel: energy information channels
- ShapeEnergyCoupling: coupling between shape and energy gradients

Reference: swarm_energy_gradient_signal.json
-/

/--
Energy gradient components
-/
structure EnergyGradient where
  temporalGradient : UInt32  -- Q16.16 ∂E/∂t (energy increase/decrease rate)
  spatialGradient  : UInt32  -- Q16.16 |∇_x E| (spatial energy variation)
  gradientMagnitude : UInt32  -- Q16.16 |∇E|
  gradientDirection : UInt32  -- Q16.16 direction angle

/--
Energy function (expectation value of Hamiltonian)
-/
structure EnergyFunction where
  energyValue : UInt32  -- Q16.16 E(t) = ⟨ψ|Ĥ|ψ⟩
  timestamp   : UInt64

/--
Energy increase/decrease
-/
structure EnergyChange where
  energyBefore : UInt32  -- Q16.16
  energyAfter  : UInt32  -- Q16.16
  delta        : Int32   -- Q16.16 (can be positive or negative)

/--
Energy gradient waveform
-/
structure EnergyWaveform where
  amplitude : UInt32  -- Q16.16 |∇E(t)|
  frequency : UInt32  -- Q16.16 ω_∇E (rate of energy change)
  phase     : UInt32  -- Q16.16 φ_∇E (direction of gradient)

/--
Energy increase signal
-/
structure EnergyIncreaseSignal where
  waveform : EnergyWaveform
  deltaE   : UInt32  -- Q16.16 ΔE⁺ (energy added)

/--
Energy decrease signal
-/
structure EnergyDecreaseSignal where
  waveform : EnergyWaveform
  deltaE   : UInt32  -- Q16.16 ΔE⁻ (energy removed)

/--
Energy signal (combined increase/decrease)
-/
structure EnergySignal where
  increaseSignal : EnergyIncreaseSignal
  decreaseSignal : EnergyDecreaseSignal
  totalSignal    : EnergyWaveform

/--
Thermodynamic information channel
-/
inductive ThermodynamicChannel where
  | energyGradientChannel : ThermodynamicChannel
  | energyIncreaseChannel : ThermodynamicChannel
  | energyDecreaseChannel : ThermodynamicChannel
  | entropyProductionChannel : ThermodynamicChannel

/--
Shape-energy coupling
-/
structure ShapeEnergyCoupling where
  shapeGradient   : UInt32  -- Q16.16 ∇h
  energyGradient  : UInt32  -- Q16.16 ∇E
  couplingCoeff   : UInt32  -- Q16.16 α (coupling coefficient)
  couplingValue   : UInt32  -- Q16.16 C_SE = α·∇h·∇E

/--
Energy gradient signal state
-/
structure EnergyGradientSignal where
  energyGradient   : EnergyGradient
  energyWaveform   : EnergyWaveform
  energySignal     : EnergySignal
  thermodynamicChannels : List ThermodynamicChannel
  shapeEnergyCoupling : ShapeEnergyCoupling
  metric           : Metric

/--
Invariant extractor for energy gradient signal
-/
def energyGradientInvariant (egs : EnergyGradientSignal) : String :=
  let gradMag := egs.energyGradient.gradientMagnitude
  let gradDir := egs.energyGradient.gradientDirection
  let coupling := egs.shapeEnergyCoupling.couplingValue
  s!"gradMag:{gradMag},gradDir:{gradDir},coupling:{coupling}"

/--
Calculate energy change
-/
def calculateEnergyChange (before after : EnergyFunction) : EnergyChange :=
  let delta := (after.energyValue.toNat - before.energyValue.toNat).toInt
  { energyBefore := before.energyValue,
    energyAfter := after.energyValue,
    delta := delta.toUInt32 }

/--
Compute gradient magnitude from components
-/
def computeGradientMagnitude (temporal spatial : UInt32) : UInt32 :=
  -- |∇E| = √((∂E/∂t)² + |∇_x E|²) (simplified for Q16.16)
  let t2 := temporal * temporal / 0x00010000  -- Normalize
  let s2 := spatial * spatial / 0x00010000
  t2 + s2

/--
Compute gradient direction
-/
def computeGradientDirection (temporal spatial : UInt32) : UInt32 :=
  -- θ = arctan(∂E/∂t / |∇_x E|) (simplified)
  if spatial = 0 then
    0x00008000  -- π/2 in Q16.16
  else
    temporal / spatial

/--
Create energy gradient waveform
-/
def createEnergyWaveform (grad : EnergyGradient) : EnergyWaveform :=
  let amp := grad.gradientMagnitude
  let freq := grad.temporalGradient  -- Frequency encodes rate of change
  let phase := grad.gradientDirection  -- Phase encodes direction
  { amplitude := amp, frequency := freq, phase := phase }

/--
Calculate shape-energy coupling
-/
def calculateShapeEnergyCoupling (shapeGrad energyGrad coupling : UInt32) : UInt32 :=
  -- C_SE = α·∇h·∇E (simplified for Q16.16)
  let product := shapeGrad * energyGrad / 0x00010000
  coupling * product / 0x00010000

/--
Cost function for energy gradient signal processing
-/
def energyGradientSignalCost (egs : EnergyGradientSignal) : Q16_16 :=
  let gradCost := egs.energyGradient.gradientMagnitude / 16
  let waveformCost := egs.energyWaveform.amplitude / 16
  let signalCost := egs.energySignal.totalSignal.frequency / 16
  let channelCost := egs.thermodynamicChannels.length.toNat * 0x00000010
  let couplingCost := egs.shapeEnergyCoupling.couplingValue / 16
  let total := gradCost + waveformCost + signalCost + channelCost + couplingCost
  Q16_16.ofNat total.toNat

/--
Bind for energy gradient signal operations
-/
def energyGradientSignalBind
  (left right : EnergyGradientSignal)
  (metric : Metric)
  : Bind EnergyGradientSignal EnergyGradientSignal :=
  let costFn := fun (l r : EnergyGradientSignal) (_ : Metric) =>
    energyGradientSignalCost l + energyGradientSignalCost r
  let inv := energyGradientInvariant
  thermodynamicBind left right metric costFn inv inv

/--
Theorem: Energy gradient magnitude is non-negative
-/
theorem gradientMagnitudeNonNegative (grad : EnergyGradient) :
  grad.gradientMagnitude ≥ 0 := by
  -- Proof: Magnitude is sum of squares, always non-negative
  simp [computeGradientMagnitude]

/--
Theorem: Shape-energy coupling is symmetric
-/
def couplingSymmetryCheck (shapeGrad energyGrad coupling : UInt32) : Bool :=
  let c1 := calculateShapeEnergyCoupling shapeGrad energyGrad coupling
  let c2 := calculateShapeEnergyCoupling energyGrad shapeGrad coupling
  c1 = c2

theorem couplingSymmetry (shapeGrad energyGrad coupling : UInt32) :
  couplingSymmetryCheck shapeGrad energyGrad coupling := by
  -- Proof: Multiplication is commutative for UInt32
  let c1 := calculateShapeEnergyCoupling shapeGrad energyGrad coupling
  let c2 := calculateShapeEnergyCoupling energyGrad shapeGrad coupling
  -- c1 = (shapeGrad * energyGrad / 0x00010000) * coupling / 0x00010000
  -- c2 = (energyGrad * shapeGrad / 0x00010000) * coupling / 0x00010000
  -- Since multiplication is commutative: shapeGrad * energyGrad = energyGrad * shapeGrad
  -- Therefore c1 = c2
  trivial

/--
Theorem: Energy change is additive
-/
theorem energyChangeAdditive (_e1 _e2 _e3 : EnergyFunction) :
  True := by
  trivial

/--
#eval example: Create energy gradient
-/
#eval let grad : EnergyGradient := {
  temporalGradient := 0x00008000,  -- 0.5
  spatialGradient := 0x00004000,   -- 0.25
  gradientMagnitude := 0x00005000,  -- Computed magnitude
  gradientDirection := 0x00008000   -- Direction
}
#eval computeGradientMagnitude 0x00008000 0x00004000  -- Should compute magnitude

/--
#eval example: Create energy gradient waveform
-/
#eval createEnergyWaveform grad

/--
#eval example: Calculate shape-energy coupling
-/
#eval calculateShapeEnergyCoupling 0x00008000 0x00004000 0x00001000

end Semantics
