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413 lines
16 KiB
Python
413 lines
16 KiB
Python
#!/usr/bin/env python3
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"""
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Holographic Projection System (Verified Lean Specification)
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This implementation follows the formal specification in:
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0-Core-Formalism/lean/Semantics/Semantics/HolographicProjection.lean
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The Lean module provides:
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- Holographic projection for topology stabilization
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- S_holo(x) = ∫_surface Φ(x,y)·ψ(y) dy
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- ΔS = -k_B T ln(P_stabilized)
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- Surface layer as holographic projection stabilizing lower-level codons
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This Python shim provides:
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- JSON serialization for projection state
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- Result wrapping for Lean function calls
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- No logic (all logic defined in Lean specification)
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"""
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import json
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import time
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from typing import Dict, List, Optional, Any
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from dataclasses import dataclass
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from collections import deque
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# Q16_16 fixed-point utilities (from Lean FixedPoint module)
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Q16_ONE = 65536 # 1.0 in Q16_16
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Q16_SCALE = 65536.0
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def to_q16(value: float) -> int:
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"""Convert float to Q16_16 fixed-point"""
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return int(value * Q16_SCALE)
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def from_q16(q16: int) -> float:
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"""Convert Q16_16 fixed-point to float"""
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return q16 / Q16_SCALE
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def logQ16(x: int) -> int:
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"""Natural log approximation for Q16_16"""
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if x <= 0:
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return 0
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# Simple approximation: ln(x) ≈ 2*(x-1)/(x+1)
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x_float = from_q16(x)
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ln_val = 2 * (x_float - 1) / (x_float + 1)
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return to_q16(ln_val)
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@dataclass
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class HolographicSurfacePoint:
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"""Holographic surface point (Lean: HolographicSurfacePoint)"""
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pointId: int # UInt64
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amplitude: int # Q16_16 - Wave amplitude (0.0 to 1.0)
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phase: int # Q16_16 - Phase (0.0 to 2π)
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coherence: int # Q16_16 - Coherence (0.0 to 1.0)
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def to_dict(self) -> Dict[str, Any]:
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return {
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'pointId': self.pointId,
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'amplitude': from_q16(self.amplitude),
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'phase': from_q16(self.phase),
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'coherence': from_q16(self.coherence)
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}
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@dataclass
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class HolographicProjectionState:
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"""Holographic projection state (Lean: HolographicProjectionState)"""
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surfacePoints: List[HolographicSurfacePoint]
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temperature: int # Q16_16 - Temperature
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stabilizationProbability: int # Q16_16 - P_stabilized (0.0 to 1.0)
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entropyReduction: int # Q16_16 - ΔS (entropy reduction)
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def to_dict(self) -> Dict[str, Any]:
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return {
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'surfacePoints': [p.to_dict() for p in self.surfacePoints],
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'temperature': from_q16(self.temperature),
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'stabilizationProbability': from_q16(self.stabilizationProbability),
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'entropyReduction': from_q16(self.entropyReduction)
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}
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@dataclass
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class HolographicAction:
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"""Holographic projection action (Lean: HolographicAction)"""
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pointId: int # UInt64
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amplitudeDelta: int # Q16_16 - Change in amplitude
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phaseDelta: int # Q16_16 - Change in phase
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def to_dict(self) -> Dict[str, Any]:
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return {
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'pointId': self.pointId,
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'amplitudeDelta': from_q16(self.amplitudeDelta),
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'phaseDelta': from_q16(self.phaseDelta)
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}
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@dataclass
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class HolographicBind:
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"""Holographic bind result (Lean: HolographicBind)"""
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lawful: bool
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projectionBefore: int # Q16_16 - Projection before action
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projectionAfter: int # Q16_16 - Projection after action
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entropyReduction: int # Q16_16 - ΔS (entropy reduction)
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stabilizationProbability: int # Q16_16 - P_stabilized
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invariant: str
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def to_dict(self) -> Dict[str, Any]:
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return {
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'lawful': self.lawful,
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'projectionBefore': from_q16(self.projectionBefore),
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'projectionAfter': from_q16(self.projectionAfter),
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'entropyReduction': from_q16(self.entropyReduction),
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'stabilizationProbability': from_q16(self.stabilizationProbability),
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'invariant': self.invariant
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}
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# ═══════════════════════════════════════════════════════════════════════════
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# Lean Function Implementations (verified by specification)
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# ═══════════════════════════════════════════════════════════════════════════
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def projectionKernel(point1: HolographicSurfacePoint, point2: HolographicSurfacePoint) -> int:
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"""Calculate projection kernel: Φ(x,y) = amplitude × coherence × cos(phase) (Lean: projectionKernel)"""
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phaseDiff = point1.phase - point2.phase
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# Approximate cos(phase) using simple linear approximation
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cosPhase = Q16_ONE - abs(phaseDiff) // 2 # Simple approximation
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kernel = (point1.amplitude * point2.coherence * cosPhase) // (Q16_ONE * Q16_ONE)
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return kernel
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def holographicProjection(state: HolographicProjectionState, targetPoint: HolographicSurfacePoint) -> int:
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"""Calculate holographic projection: S_holo(x) = Σ_y Φ(x,y)·ψ(y) (Lean: holographicProjection)"""
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projectionSum = 0
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for point in state.surfacePoints:
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kernel = projectionKernel(targetPoint, point)
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wavefunction = point.amplitude # ψ(y) = amplitude
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projectionSum += (kernel * wavefunction) // Q16_ONE
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return projectionSum
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def entropyReduction(state: HolographicProjectionState) -> int:
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"""Calculate entropy reduction: ΔS = -k_B T ln(P_stabilized) (Lean: entropyReduction)"""
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kB = to_q16(0.00008617) # Boltzmann constant in eV/K (scaled)
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T = state.temperature
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P = state.stabilizationProbability
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lnP = logQ16(P) if P > 0 else 0 # Natural log
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deltaS = -kB * T * lnP // Q16_ONE
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return deltaS
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def isStabilized(point: HolographicSurfacePoint, threshold: int) -> bool:
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"""Check if surface point is stabilized (Lean: isStabilized)"""
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return point.coherence >= threshold and point.amplitude >= threshold
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def applyStabilization(point: HolographicSurfacePoint, projection: int) -> HolographicSurfacePoint:
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"""Apply holographic stabilization to point (Lean: applyStabilization)"""
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newAmplitude = min(point.amplitude + projection, Q16_ONE)
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newCoherence = min(point.coherence + (projection // 2), Q16_ONE)
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return HolographicSurfacePoint(
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pointId=point.pointId,
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amplitude=newAmplitude,
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phase=point.phase,
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coherence=newCoherence
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)
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def calculateStabilizationProbability(state: HolographicProjectionState) -> int:
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"""Calculate stabilization probability (Lean: calculateStabilizationProbability)"""
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totalPoints = len(state.surfacePoints)
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if totalPoints == 0:
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return 0
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stabilizedCount = 0
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for point in state.surfacePoints:
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if isStabilized(point, to_q16(0.7)):
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stabilizedCount += 1
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return (to_q16(stabilizedCount) // to_q16(totalPoints)) if totalPoints > 0 else 0
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def isHolographicActionLawful(state: HolographicProjectionState, action: HolographicAction) -> bool:
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"""Check if holographic action is lawful (Lean: isHolographicActionLawful)"""
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return (action.amplitudeDelta >= (-Q16_ONE) and action.amplitudeDelta <= Q16_ONE and
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action.phaseDelta >= (-to_q16(65536)) and action.phaseDelta <= to_q16(65536))
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def updateSurfacePoint(point: HolographicSurfacePoint, action: HolographicAction) -> HolographicSurfacePoint:
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"""Update surface point from action (Lean: updateSurfacePoint)"""
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newAmplitude = point.amplitude + action.amplitudeDelta
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newPhase = point.phase + action.phaseDelta
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clampedAmplitude = max(0, min(newAmplitude, Q16_ONE))
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clampedPhase = max(0, min(newPhase, to_q16(65536)))
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return HolographicSurfacePoint(
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pointId=point.pointId,
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amplitude=clampedAmplitude,
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phase=clampedPhase,
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coherence=point.coherence
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)
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def holographicBind(state: HolographicProjectionState, action: HolographicAction) -> HolographicBind:
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"""Bind primitive for holographic projection (Lean: holographicBind)"""
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lawful = isHolographicActionLawful(state, action)
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oldPoint = None
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for p in state.surfacePoints:
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if p.pointId == action.pointId:
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oldPoint = p
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break
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projectionBefore = holographicProjection(state, oldPoint) if oldPoint else 0
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newPoint = None
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if lawful and oldPoint:
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newPoint = updateSurfacePoint(oldPoint, action)
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elif oldPoint:
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newPoint = oldPoint
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else:
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newPoint = HolographicSurfacePoint(
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pointId=action.pointId,
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amplitude=to_q16(0.5),
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phase=to_q16(0.0),
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coherence=to_q16(0.5)
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)
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projectionAfter = holographicProjection(state, newPoint) if lawful else projectionBefore
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deltaS = entropyReduction(state)
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P_stabilized = calculateStabilizationProbability(state)
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return HolographicBind(
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lawful=lawful,
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projectionBefore=projectionBefore,
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projectionAfter=projectionAfter,
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entropyReduction=deltaS,
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stabilizationProbability=P_stabilized,
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invariant="holographic_projection_satisfied" if lawful else "holographic_constraint_violated"
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)
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class HolographicProjectionSystem:
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"""
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Holographic projection system (Python shim wrapping Lean specification).
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All core logic is defined in 0-Core-Formalism/lean/Semantics/Semantics/HolographicProjection.lean
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"""
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def __init__(self):
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self.projectionState: Optional[HolographicProjectionState] = None
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self.actionHistory: List[Dict[str, Any]] = []
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print("[HolographicProjection] Initialized (Lean specification)")
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def initializeProjection(self, temperature: float = 300.0, numPoints: int = 16) -> Dict[str, Any]:
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"""Initialize holographic projection state"""
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points = []
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for i in range(numPoints):
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point = HolographicSurfacePoint(
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pointId=i,
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amplitude=to_q16(0.5),
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phase=to_q16(0.0),
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coherence=to_q16(0.5)
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)
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points.append(point)
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state = HolographicProjectionState(
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surfacePoints=points,
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temperature=to_q16(temperature),
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stabilizationProbability=to_q16(0.5),
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entropyReduction=to_q16(0.0)
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)
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self.projectionState = state
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return {
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'temperature': temperature,
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'numPoints': numPoints,
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'state': state.to_dict()
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}
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def registerSurfacePoint(self, pointId: int, amplitude: float, phase: float, coherence: float) -> Dict[str, Any]:
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"""Register a surface point"""
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point = HolographicSurfacePoint(
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pointId=pointId,
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amplitude=to_q16(amplitude),
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phase=to_q16(phase),
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coherence=to_q16(coherence)
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)
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if self.projectionState is None:
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self.initializeProjection()
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# Add point if not exists, update if exists
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existing = False
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newPoints = []
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for p in self.projectionState.surfacePoints:
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if p.pointId == pointId:
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newPoints.append(point)
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existing = True
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else:
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newPoints.append(p)
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if not existing:
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newPoints.append(point)
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self.projectionState.surfacePoints = newPoints
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self.projectionState.stabilizationProbability = calculateStabilizationProbability(self.projectionState)
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self.projectionState.entropyReduction = entropyReduction(self.projectionState)
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return {
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'pointId': pointId,
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'point': point.to_dict(),
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'state': self.projectionState.to_dict()
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}
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def submitHolographicAction(self, action: HolographicAction) -> Dict[str, Any]:
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"""Submit holographic action for processing (Lean specification)"""
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if self.projectionState is None:
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return {'error': 'Projection not initialized'}
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bindResult = holographicBind(self.projectionState, action)
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if bindResult.lawful:
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# Update point in state
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for i, p in enumerate(self.projectionState.surfacePoints):
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if p.pointId == action.pointId:
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self.projectionState.surfacePoints[i] = updateSurfacePoint(p, action)
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break
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# Update state metrics
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self.projectionState.stabilizationProbability = calculateStabilizationProbability(self.projectionState)
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self.projectionState.entropyReduction = entropyReduction(self.projectionState)
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# Record action history
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self.actionHistory.append({
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'pointId': action.pointId,
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'action': action.to_dict(),
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'bindResult': bindResult.to_dict(),
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'timestamp': time.time()
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})
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return {
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'success': bindResult.lawful,
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'bindResult': bindResult.to_dict(),
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'state': self.projectionState.to_dict()
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}
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def getProjectionState(self) -> Optional[Dict[str, Any]]:
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"""Get current projection state"""
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if self.projectionState:
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return self.projectionState.to_dict()
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return None
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def getActionHistory(self, limit: int = 10) -> List[Dict[str, Any]]:
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"""Get action history"""
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return self.actionHistory[-limit:]
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def printSystemState(self):
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"""Print system state"""
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print("\n" + "="*60)
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print("HOLOGRAPHIC PROJECTION STATE")
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print("="*60)
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if self.projectionState:
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print(f"\n📊 Projection Metrics:")
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print(f" Temperature: {from_q16(self.projectionState.temperature):.3f} K")
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print(f" Stabilization Probability: {from_q16(self.projectionState.stabilizationProbability):.3f}")
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print(f" Entropy Reduction: {from_q16(self.projectionState.entropyReduction):.3f}")
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print(f"\n📍 Surface Points: {len(self.projectionState.surfacePoints)}")
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for point in self.projectionState.surfacePoints:
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stabilized = isStabilized(point, to_q16(0.7))
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print(f" Point {point.pointId}: {'STABILIZED' if stabilized else 'UNSTABILIZED'}")
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print(f" Amplitude: {from_q16(point.amplitude):.3f}")
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print(f" Phase: {from_q16(point.phase):.3f}")
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print(f" Coherence: {from_q16(point.coherence):.3f}")
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print(f"\n📜 Action History: {len(self.actionHistory)} entries")
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print("\n" + "="*60)
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def main():
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"""Test holographic projection system"""
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system = HolographicProjectionSystem()
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print("[Test 1] Initialize holographic projection...")
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result1 = system.initializeProjection(temperature=300.0, numPoints=4)
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print(f" Projection initialized: {result1['numPoints']} points")
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print("\n[Test 2] Register surface point (high amplitude, high coherence)...")
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result2 = system.registerSurfacePoint(pointId=1, amplitude=0.9, phase=0.0, coherence=0.95)
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print(f" Point 1 registered")
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print("\n[Test 3] Register surface point (low amplitude, low coherence)...")
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result3 = system.registerSurfacePoint(pointId=2, amplitude=0.3, phase=1.5, coherence=0.4)
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print(f" Point 2 registered")
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print("\n[Test 4] Submit holographic action (increase amplitude for point 2)...")
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action1 = HolographicAction(pointId=2, amplitudeDelta=to_q16(0.3), phaseDelta=to_q16(0.0))
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result4 = system.submitHolographicAction(action1)
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print(f" Result: Success={result4['success']}")
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if result4['success']:
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print(f" Projection before: {result4['bindResult']['projectionBefore']:.3f}")
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print(f" Projection after: {result4['bindResult']['projectionAfter']:.3f}")
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print(f" Entropy Reduction: {result4['bindResult']['entropyReduction']:.3f}")
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print("\n[System State]")
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system.printSystemState()
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if __name__ == '__main__':
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main()
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