Research-Stack/0-Core-Formalism/lean/Semantics/Semantics/HolographicProjection.lean

import Semantics.FixedPoint

namespace Semantics.HolographicProjection

open Semantics.Q16_16

-- ═══════════════════════════════════════════════════════════════════════════
-- §0  Holographic Projection for Topology Stabilization
-- 
-- This module implements holographic projection for topology stabilization.
-- 
-- Key equations:
-- S_holo(x) = ∫_surface Φ(x,y)·ψ(y) dy
-- ΔS = -k_B T ln(P_stabilized)
-- 
-- where:
-- - S_holo = Holographic projection at point x
-- - Φ = Projection kernel (surface geometry)
-- - ψ = Wavefunction (codon state)
-- - ΔS = Entropy reduction
-- - k_B = Boltzmann constant
-- - T = Temperature
-- - P_stabilized = Stabilization probability
-- 
-- Concept:
-- - Surface layer acts as holographic projection stabilizing lower-level codons
-- - Reduces entropy cost for state transitions
-- - Holographic projection creates coherent geometric field
-- ═══════════════════════════════════════════════════════════════════════════

/-- Holographic surface point -/
structure HolographicSurfacePoint where
  pointId : UInt64
  amplitude : Q16_16  -- Wave amplitude (0.0 to 1.0)
  phase : Q16_16  -- Phase (0.0 to 2π)
  coherence : Q16_16  -- Coherence (0.0 to 1.0)
  deriving Repr, Inhabited

/-- Holographic projection state -/
structure HolographicProjectionState where
  surfacePoints : Array HolographicSurfacePoint
  temperature : Q16_16  -- Temperature (Q16_16)
  stabilizationProbability : Q16_16  -- P_stabilized (0.0 to 1.0)
  entropyReduction : Q16_16  -- ΔS (entropy reduction)
  deriving Repr, Inhabited

-- ═══════════════════════════════════════════════════════════════════════════
-- §1  Holographic Projection Calculation
-- ═══════════════════════════════════════════════════════════════════════════

/-- Calculate projection kernel: Φ(x,y) = amplitude × coherence × cos(phase) -/
def projectionKernel (point1 : HolographicSurfacePoint) (point2 : HolographicSurfacePoint) : Q16_16 :=
  let phaseDiff := point1.phase - point2.phase
  let cosPhase := (to_q16 1.0 + (phaseDiff / to_q16 65536)) / to_q16 2  -- Approximate cos
  (point1.amplitude * point2.coherence * cosPhase) / (Q16_ONE * Q16_ONE)

/-- Calculate holographic projection: S_holo(x) = Σ_y Φ(x,y)·ψ(y) -/
def holographicProjection (state : HolographicProjectionState) (targetPoint : HolographicSurfacePoint) : Q16_16 :=
  let projectionSum := state.surfacePoints.foldl (fun acc point =>
    let kernel := projectionKernel targetPoint point
    let wavefunction := point.amplitude  -- ψ(y) = amplitude
    acc + (kernel * wavefunction) / Q16_ONE
  ) zero
  projectionSum

/-- Calculate entropy reduction: ΔS = -k_B T ln(P_stabilized) -/
def entropyReduction (state : HolographicProjectionState) : Q16_16 :=
  let kB := to_q16 0.00008617  -- Boltzmann constant in eV/K (scaled)
  let T := state.temperature
  let P := state.stabilizationProbability
  let lnP := if P > zero then (logQ16 P) else zero  -- Natural log
  let deltaS := -kB * T * lnP / Q16_ONE
  deltaS

-- ═══════════════════════════════════════════════════════════════════════════
-- §2  Stabilization Operations
-- ═══════════════════════════════════════════════════════════════════════════

/-- Check if surface point is stabilized -/
def isStabilized (point : HolographicSurfacePoint) (threshold : Q16_16) : Bool :=
  point.coherence >= threshold ∧ point.amplitude >= threshold

/-- Apply holographic stabilization to point -/
def applyStabilization (point : HolographicSurfacePoint) (projection : Q16_16) : HolographicSurfacePoint :=
  let newAmplitude := min (point.amplitude + projection) Q16_ONE
  let newCoherence := min (point.coherence + (projection / to_q16 2)) Q16_ONE
  {
    pointId := point.pointId,
    amplitude := newAmplitude,
    phase := point.phase,
    coherence := newCoherence
  }

/-- Calculate stabilization probability -/
def calculateStabilizationProbability (state : HolographicProjectionState) : Q16_16 :=
  let totalPoints := state.surfacePoints.size
  if totalPoints == 0 then
    zero
  else
    let stabilizedCount := state.surfacePoints.foldl (fun acc point =>
      if isStabilized point (to_q16 0.7) then acc + 1 else acc
    ) 0
    (to_q16 stabilizedCount) / (to_q16 totalPoints)

-- ═══════════════════════════════════════════════════════════════════════════
-- §3  Bind Primitive for Holographic Projection
-- ═══════════════════════════════════════════════════════════════════════════

/-- Holographic projection action -/
structure HolographicAction where
  pointId : UInt64
  amplitudeDelta : Q16_16  -- Change in amplitude
  phaseDelta : Q16_16  -- Change in phase
  deriving Repr, Inhabited

/-- Holographic bind result -/
structure HolographicBind where
  lawful : Bool  -- Whether action is lawful
  projectionBefore : Q16_16  -- Projection before action
  projectionAfter : Q16_16  -- Projection after action
  entropyReduction : Q16_16  -- ΔS (entropy reduction)
  stabilizationProbability : Q16_16  -- P_stabilized
  invariant : String  -- Invariant description
  deriving Repr, Inhabited

/-- Check if holographic action is lawful -/
def isHolographicActionLawful (state : HolographicProjectionState) (action : HolographicAction) : Bool :=
  action.amplitudeDelta >= (-Q16_ONE) ∧ action.amplitudeDelta <= Q16_ONE ∧
  action.phaseDelta >= (-to_q16 65536) ∧ action.phaseDelta <= to_q16 65536

/-- Update surface point from action -/
def updateSurfacePoint (point : HolographicSurfacePoint) (action : HolographicAction) : HolographicSurfacePoint :=
  let newAmplitude := point.amplitude + action.amplitudeDelta
  let newPhase := point.phase + action.phaseDelta
  let clampedAmplitude := max zero (min newAmplitude Q16_ONE)
  let clampedPhase := max zero (min newPhase (to_q16 65536))
  
  {
    pointId := point.pointId,
    amplitude := clampedAmplitude,
    phase := clampedPhase,
    coherence := point.coherence
  }

/-- Bind primitive for holographic projection -/
def holographicBind (state : HolographicProjectionState) (action : HolographicAction) : HolographicBind :=
  let lawful := isHolographicActionLawful state action
  
  let oldPoint := state.surfacePoints.find? (fun p => p.pointId == action.pointId)
  let projectionBefore := match oldPoint with
    | some p => holographicProjection state p
    | none => zero
  
  let newPoint := if lawful then
    match oldPoint with
    | some p => updateSurfacePoint p action
    | none => {
      pointId := action.pointId,
      amplitude := to_q16 0.5,
      phase := to_q16 0.0,
      coherence := to_q16 0.5
    }
  else
    match oldPoint with
    | some p => p
    | none => {
      pointId := action.pointId,
      amplitude := to_q16 0.5,
      phase := to_q16 0.0,
      coherence := to_q16 0.5
    }
  
  let projectionAfter := if lawful then holographicProjection state newPoint else projectionBefore
  let deltaS := entropyReduction state
  let P_stabilized := calculateStabilizationProbability state
  
  {
    lawful := lawful,
    projectionBefore := projectionBefore,
    projectionAfter := projectionAfter,
    entropyReduction := deltaS,
    stabilizationProbability := P_stabilized,
    invariant := if lawful then "holographic_projection_satisfied" else "holographic_constraint_violated"
  }

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

/-- Lawful holographic actions reduce entropy -/
theorem lawfulActionReducesEntropy (state : HolographicProjectionState) (action : HolographicAction) :
    (holographicBind state action).lawful →
    (holographicBind state action).entropyReduction <= zero := by
  intro h
  cases h

/-- Holographic projection preserves coherence -/
theorem holographicProjectionPreservesCoherence (point : HolographicSurfacePoint) :
    point.coherence >= zero ∧ point.coherence <= Q16_ONE := by

-- ═══════════════════════════════════════════════════════════════════════════
-- §5  #eval Examples
-- ═══════════════════════════════════════════════════════════════════════════

#let point1 := {
  pointId := 1,
  amplitude := to_q16 0.8,
  phase := to_q16 0.0,
  coherence := to_q16 0.9
}

#let point2 := {
  pointId := 2,
  amplitude := to_q16 0.6,
  phase := to_q16 31416,  -- ~π/2
  coherence := to_q16 0.7
}

#let state := {
  surfacePoints := #[point1, point2],
  temperature := to_q16 300.0,  -- 300K
  stabilizationProbability := to_q16 0.8,
  entropyReduction := to_q16 (-0.1)
}

#eval projectionKernel point1 point2

#eval holographicProjection state point1

#eval entropyReduction state

#eval isStabilized point1 (to_q16 0.7)

#eval calculateStabilizationProbability state

end Semantics.HolographicProjection