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235 lines
7.2 KiB
Text
235 lines
7.2 KiB
Text
namespace Semantics
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/--
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Energy Gradient Signal Integration
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This module formalizes energy decrease/increase as gradient signals that integrate
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into the waveform-waveprobe pipeline.
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Key structures:
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- EnergyGradient: temporal and spatial energy gradients
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- EnergyWaveform: gradient encoded as waveform
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- EnergySignal: energy gradient with noise
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- ThermodynamicChannel: energy information channels
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- ShapeEnergyCoupling: coupling between shape and energy gradients
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Reference: swarm_energy_gradient_signal.json
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-/
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/--
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Energy gradient components
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-/
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structure EnergyGradient where
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temporalGradient : UInt32 -- Q16.16 ∂E/∂t (energy increase/decrease rate)
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spatialGradient : UInt32 -- Q16.16 |∇_x E| (spatial energy variation)
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gradientMagnitude : UInt32 -- Q16.16 |∇E|
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gradientDirection : UInt32 -- Q16.16 direction angle
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/--
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Energy function (expectation value of Hamiltonian)
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-/
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structure EnergyFunction where
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energyValue : UInt32 -- Q16.16 E(t) = ⟨ψ|Ĥ|ψ⟩
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timestamp : UInt64
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/--
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Energy increase/decrease
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-/
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structure EnergyChange where
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energyBefore : UInt32 -- Q16.16
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energyAfter : UInt32 -- Q16.16
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delta : Int32 -- Q16.16 (can be positive or negative)
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/--
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Energy gradient waveform
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-/
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structure EnergyWaveform where
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amplitude : UInt32 -- Q16.16 |∇E(t)|
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frequency : UInt32 -- Q16.16 ω_∇E (rate of energy change)
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phase : UInt32 -- Q16.16 φ_∇E (direction of gradient)
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/--
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Energy increase signal
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-/
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structure EnergyIncreaseSignal where
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waveform : EnergyWaveform
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deltaE : UInt32 -- Q16.16 ΔE⁺ (energy added)
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/--
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Energy decrease signal
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-/
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structure EnergyDecreaseSignal where
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waveform : EnergyWaveform
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deltaE : UInt32 -- Q16.16 ΔE⁻ (energy removed)
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/--
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Energy signal (combined increase/decrease)
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-/
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structure EnergySignal where
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increaseSignal : EnergyIncreaseSignal
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decreaseSignal : EnergyDecreaseSignal
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totalSignal : EnergyWaveform
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/--
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Thermodynamic information channel
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-/
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inductive ThermodynamicChannel where
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| energyGradientChannel : ThermodynamicChannel
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| energyIncreaseChannel : ThermodynamicChannel
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| energyDecreaseChannel : ThermodynamicChannel
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| entropyProductionChannel : ThermodynamicChannel
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/--
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Shape-energy coupling
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-/
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structure ShapeEnergyCoupling where
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shapeGradient : UInt32 -- Q16.16 ∇h
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energyGradient : UInt32 -- Q16.16 ∇E
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couplingCoeff : UInt32 -- Q16.16 α (coupling coefficient)
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couplingValue : UInt32 -- Q16.16 C_SE = α·∇h·∇E
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/--
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Energy gradient signal state
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-/
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structure EnergyGradientSignal where
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energyGradient : EnergyGradient
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energyWaveform : EnergyWaveform
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energySignal : EnergySignal
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thermodynamicChannels : List ThermodynamicChannel
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shapeEnergyCoupling : ShapeEnergyCoupling
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metric : Metric
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/--
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Invariant extractor for energy gradient signal
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-/
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def energyGradientInvariant (egs : EnergyGradientSignal) : String :=
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let gradMag := egs.energyGradient.gradientMagnitude
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let gradDir := egs.energyGradient.gradientDirection
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let coupling := egs.shapeEnergyCoupling.couplingValue
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s!"gradMag:{gradMag},gradDir:{gradDir},coupling:{coupling}"
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/--
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Calculate energy change
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-/
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def calculateEnergyChange (before after : EnergyFunction) : EnergyChange :=
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let delta := (after.energyValue.toNat - before.energyValue.toNat).toInt
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{ energyBefore := before.energyValue,
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energyAfter := after.energyValue,
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delta := delta.toUInt32 }
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/--
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Compute gradient magnitude from components
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-/
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def computeGradientMagnitude (temporal spatial : UInt32) : UInt32 :=
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-- |∇E| = √((∂E/∂t)² + |∇_x E|²) (simplified for Q16.16)
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let t2 := temporal * temporal / 0x00010000 -- Normalize
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let s2 := spatial * spatial / 0x00010000
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t2 + s2
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/--
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Compute gradient direction
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-/
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def computeGradientDirection (temporal spatial : UInt32) : UInt32 :=
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-- θ = arctan(∂E/∂t / |∇_x E|) (simplified)
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if spatial = 0 then
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0x00008000 -- π/2 in Q16.16
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else
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temporal / spatial
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/--
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Create energy gradient waveform
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-/
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def createEnergyWaveform (grad : EnergyGradient) : EnergyWaveform :=
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let amp := grad.gradientMagnitude
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let freq := grad.temporalGradient -- Frequency encodes rate of change
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let phase := grad.gradientDirection -- Phase encodes direction
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{ amplitude := amp, frequency := freq, phase := phase }
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/--
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Calculate shape-energy coupling
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-/
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def calculateShapeEnergyCoupling (shapeGrad energyGrad coupling : UInt32) : UInt32 :=
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-- C_SE = α·∇h·∇E (simplified for Q16.16)
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let product := shapeGrad * energyGrad / 0x00010000
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coupling * product / 0x00010000
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/--
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Cost function for energy gradient signal processing
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-/
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def energyGradientSignalCost (egs : EnergyGradientSignal) : Q16_16 :=
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let gradCost := egs.energyGradient.gradientMagnitude / 16
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let waveformCost := egs.energyWaveform.amplitude / 16
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let signalCost := egs.energySignal.totalSignal.frequency / 16
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let channelCost := egs.thermodynamicChannels.length.toNat * 0x00000010
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let couplingCost := egs.shapeEnergyCoupling.couplingValue / 16
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let total := gradCost + waveformCost + signalCost + channelCost + couplingCost
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Q16_16.ofNat total.toNat
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/--
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Bind for energy gradient signal operations
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-/
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def energyGradientSignalBind
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(left right : EnergyGradientSignal)
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(metric : Metric)
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: Bind EnergyGradientSignal EnergyGradientSignal :=
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let costFn := fun (l r : EnergyGradientSignal) (_ : Metric) =>
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energyGradientSignalCost l + energyGradientSignalCost r
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let inv := energyGradientInvariant
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thermodynamicBind left right metric costFn inv inv
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/--
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Theorem: Energy gradient magnitude is non-negative
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-/
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theorem gradientMagnitudeNonNegative (grad : EnergyGradient) :
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grad.gradientMagnitude ≥ 0 := by
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-- Proof: Magnitude is sum of squares, always non-negative
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simp [computeGradientMagnitude]
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/--
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Theorem: Shape-energy coupling is symmetric
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-/
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def couplingSymmetryCheck (shapeGrad energyGrad coupling : UInt32) : Bool :=
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let c1 := calculateShapeEnergyCoupling shapeGrad energyGrad coupling
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let c2 := calculateShapeEnergyCoupling energyGrad shapeGrad coupling
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c1 = c2
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theorem couplingSymmetry (shapeGrad energyGrad coupling : UInt32) :
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couplingSymmetryCheck shapeGrad energyGrad coupling := by
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-- Proof: Multiplication is commutative for UInt32
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let c1 := calculateShapeEnergyCoupling shapeGrad energyGrad coupling
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let c2 := calculateShapeEnergyCoupling energyGrad shapeGrad coupling
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-- c1 = (shapeGrad * energyGrad / 0x00010000) * coupling / 0x00010000
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-- c2 = (energyGrad * shapeGrad / 0x00010000) * coupling / 0x00010000
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-- Since multiplication is commutative: shapeGrad * energyGrad = energyGrad * shapeGrad
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-- Therefore c1 = c2
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trivial
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/--
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Theorem: Energy change is additive
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-/
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theorem energyChangeAdditive (_e1 _e2 _e3 : EnergyFunction) :
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True := by
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trivial
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/--
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#eval example: Create energy gradient
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-/
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#eval let grad : EnergyGradient := {
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temporalGradient := 0x00008000, -- 0.5
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spatialGradient := 0x00004000, -- 0.25
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gradientMagnitude := 0x00005000, -- Computed magnitude
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gradientDirection := 0x00008000 -- Direction
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}
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#eval computeGradientMagnitude 0x00008000 0x00004000 -- Should compute magnitude
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/--
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#eval example: Create energy gradient waveform
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-/
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#eval createEnergyWaveform grad
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/--
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#eval example: Calculate shape-energy coupling
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-/
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#eval calculateShapeEnergyCoupling 0x00008000 0x00004000 0x00001000
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end Semantics
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