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402 lines
14 KiB
Python
402 lines
14 KiB
Python
#!/usr/bin/env python3
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"""
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Menger Sponge Fractal Addressing 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/MengerSpongeFractalAddressing.lean
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The Lean module provides:
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- Menger sponge fractal addressing for PIST manifold
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- |P_occ| = ρ_occ · N^{d_H}
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- d_H ≈ 2.7268 (Hausdorff dimension)
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- address(x,y,z) = menger_hash(x,y,z) ⊕ fractal_offset
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- Reduces state space from 262,144 to ~84,000 positions (68% reduction) for N=64
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This Python shim provides:
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- JSON serialization for Menger sponge lattice 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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import math
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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
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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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@dataclass
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class MengerCoordinate:
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"""Menger sponge lattice coordinates (Lean: MengerCoordinate)"""
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x: int # X coordinate
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y: int # Y coordinate
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z: int # Z coordinate
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def to_dict(self) -> Dict[str, Any]:
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return {
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'x': self.x,
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'y': self.y,
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'z': self.z
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}
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@dataclass
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class MengerLattice:
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"""Menger sponge lattice state (Lean: MengerLattice)"""
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size: int # Lattice size N
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hausdorffDim: int # Hausdorff dimension d_H (Q16_16)
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occupancyDensity: int # Occupancy density ρ_occ (Q16_16)
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activePositions: int # Number of active positions |P_occ|
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def to_dict(self) -> Dict[str, Any]:
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return {
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'size': self.size,
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'hausdorffDim': from_q16(self.hausdorffDim),
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'occupancyDensity': from_q16(self.occupancyDensity),
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'activePositions': self.activePositions
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}
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@dataclass
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class MengerAddress:
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"""Menger sponge address (Lean: MengerAddress)"""
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hash: int # Menger hash value
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offset: int # Fractal offset
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occupied: bool # Whether position is occupied
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def to_dict(self) -> Dict[str, Any]:
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return {
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'hash': self.hash,
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'offset': self.offset,
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'occupied': self.occupied
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}
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@dataclass
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class BlitterState:
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"""PIST Blitter state (from PistBridge.lean)"""
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a: int # Distance from lower perfect square (Q16_16)
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b: int # Distance to upper perfect square (Q16_16)
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manifold: int # Current manifold value (Q16_16)
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stepMask: int # Timestep mask for bitwise operation
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def to_dict(self) -> Dict[str, Any]:
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return {
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'a': from_q16(self.a),
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'b': from_q16(self.b),
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'manifold': from_q16(self.manifold),
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'stepMask': self.stepMask
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}
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@dataclass
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class MengerAction:
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"""Menger sponge addressing action (Lean: MengerAction)"""
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pistState: BlitterState
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coord: MengerCoordinate
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def to_dict(self) -> Dict[str, Any]:
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return {
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'pistState': self.pistState.to_dict(),
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'coord': self.coord.to_dict()
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}
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@dataclass
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class MengerBind:
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"""Menger sponge bind result (Lean: MengerBind)"""
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lawful: bool
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addressBefore: int # Address before action
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addressAfter: int # Address after action
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occupancyBefore: int # Occupancy before action
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occupancyAfter: int # Occupancy after action
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manifoldBefore: int # PIST manifold before (Q16_16)
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manifoldAfter: int # PIST manifold after (Q16_16)
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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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'addressBefore': self.addressBefore,
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'addressAfter': self.addressAfter,
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'occupancyBefore': self.occupancyBefore,
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'occupancyAfter': self.occupancyAfter,
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'manifoldBefore': from_q16(self.manifoldBefore),
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'manifoldAfter': from_q16(self.manifoldAfter),
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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 mengerHausdorffDim() -> int:
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"""Hausdorff dimension of Menger sponge: d_H ≈ 2.7268 (Lean: mengerHausdorffDim)"""
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return to_q16(2.7268)
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def mengerHash(coord: MengerCoordinate) -> int:
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"""Calculate Menger sponge hash: menger_hash(x,y,z) (Lean: mengerHash)"""
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x = coord.x
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y = coord.y
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z = coord.z
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# Menger sponge hash: XOR of coordinates with bit shifts
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hash = x ^ ((y << 1) & 0xFFFFFFFF) ^ ((z << 2) & 0xFFFFFFFF)
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return hash
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def fractalOffset(coord: MengerCoordinate, hausdorffDim: int) -> int:
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"""Calculate fractal offset based on Hausdorff dimension (Lean: fractalOffset)"""
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x = coord.x
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y = coord.y
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z = coord.z
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dim = hausdorffDim
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# Fractal offset: (x + y + z) * d_H
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sum_xyz = x + y + z
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offset = (sum_xyz * dim) // 65536
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return offset
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def mengerAddress(coord: MengerCoordinate, hausdorffDim: int) -> MengerAddress:
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"""Calculate Menger sponge address: address(x,y,z) = menger_hash ⊕ fractal_offset (Lean: mengerAddress)"""
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hash = mengerHash(coord)
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offset = fractalOffset(coord, hausdorffDim)
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address = hash ^ offset
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return MengerAddress(hash=hash, offset=offset, occupied=True)
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def q16_pow(base: int, exp: int) -> int:
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"""Q16_16 power function"""
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base_float = from_q16(base)
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exp_float = from_q16(exp)
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result_float = base_float ** exp_float
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return to_q16(result_float)
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def fractalOccupancy(size: int, hausdorffDim: int, occupancyDensity: int) -> int:
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"""Calculate fractal occupancy: |P_occ| = ρ_occ · N^{d_H} (Lean: fractalOccupancy)"""
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sizeQ = to_q16(size)
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n_pow_dh = q16_pow(sizeQ, hausdorffDim)
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occupancy = (occupancyDensity * n_pow_dh) // Q16_ONE
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return from_q16(occupancy)
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def reductionRatio(size: int, hausdorffDim: int) -> float:
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"""Calculate state space reduction ratio (Lean: reductionRatio)"""
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size_float = float(size)
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size_cubed = size_float ** 3
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hausdorffDim_float = from_q16(hausdorffDim)
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size_pow_dh = size_float ** hausdorffDim_float
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ratio = size_pow_dh / size_cubed
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return ratio
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def pistToMengerCoord(pistState: BlitterState, size: int) -> MengerCoordinate:
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"""Convert PIST (a,b) coordinates to Menger (x,y,z) coordinates (Lean: pistToMengerCoord)"""
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a = from_q16(pistState.a)
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b = from_q16(pistState.b)
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manifold = from_q16(pistState.manifold)
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# Map PIST coordinates to 3D Menger space
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x = int(a) % size
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y = int(b) % size
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z = int(manifold) % size
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return MengerCoordinate(x=x, y=y, z=z)
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def mengerToPistManifold(addr: MengerAddress) -> int:
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"""Convert Menger address back to PIST manifold value (Lean: mengerToPistManifold)"""
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return addr.hash
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def isMengerActionLawful(lattice: MengerLattice, action: MengerAction) -> bool:
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"""Check if Menger action is lawful (Lean: isMengerActionLawful)"""
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x = action.coord.x
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y = action.coord.y
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z = action.coord.z
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lawful = x < lattice.size and y < lattice.size and z < lattice.size
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return lawful
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def mengerBind(lattice: MengerLattice, action: MengerAction) -> MengerBind:
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"""Bind primitive for Menger sponge addressing (Lean: mengerBind)"""
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lawful = isMengerActionLawful(lattice, action)
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addrBefore = mengerAddress(action.coord, lattice.hausdorffDim)
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manifoldBefore = action.pistState.manifold
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occupancyBefore = lattice.activePositions
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if lawful:
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newOccupancy = fractalOccupancy(lattice.size, lattice.hausdorffDim, lattice.occupancyDensity)
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newLattice = MengerLattice(
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size=lattice.size,
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hausdorffDim=lattice.hausdorffDim,
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occupancyDensity=lattice.occupancyDensity,
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activePositions=newOccupancy
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)
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else:
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newLattice = lattice
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addrAfter = mengerAddress(action.coord, lattice.hausdorffDim) if lawful else addrBefore
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manifoldAfter = mengerToPistManifold(addrAfter) if lawful else manifoldBefore
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occupancyAfter = newLattice.activePositions
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return MengerBind(
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lawful=lawful,
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addressBefore=addrBefore.hash,
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addressAfter=addrAfter.hash,
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occupancyBefore=occupancyBefore,
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occupancyAfter=occupancyAfter,
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manifoldBefore=manifoldBefore,
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manifoldAfter=manifoldAfter,
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invariant="menger_sponge_addressing_satisfied" if lawful else "menger_constraint_violated"
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)
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class MengerSpongeFractalAddressingSystem:
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"""
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Menger sponge fractal addressing system (Python shim wrapping Lean specification).
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All core logic is defined in 0-Core-Formalism/lean/Semantics/Semantics/MengerSpongeFractalAddressing.lean
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"""
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def __init__(self):
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self.lattice: Optional[MengerLattice] = None
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self.actionHistory: List[Dict[str, Any]] = []
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print("[MengerSpongeFractalAddressing] Initialized (Lean specification)")
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def initializeLattice(self, size: int = 64, occupancyDensity: float = 0.5) -> Dict[str, Any]:
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"""Initialize Menger sponge lattice"""
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hausdorffDim = mengerHausdorffDim()
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activePositions = fractalOccupancy(size, hausdorffDim, to_q16(occupancyDensity))
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lattice = MengerLattice(
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size=size,
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hausdorffDim=hausdorffDim,
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occupancyDensity=to_q16(occupancyDensity),
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activePositions=activePositions
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)
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self.lattice = lattice
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ratio = reductionRatio(size, hausdorffDim)
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return {
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'size': size,
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'hausdorffDim': from_q16(hausdorffDim),
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'occupancyDensity': occupancyDensity,
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'activePositions': activePositions,
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'reductionRatio': ratio,
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'state': lattice.to_dict()
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}
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def submitMengerAction(self, action: MengerAction) -> Dict[str, Any]:
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"""Submit Menger sponge addressing action for processing (Lean specification)"""
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if self.lattice is None:
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return {'error': 'Lattice not initialized'}
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bindResult = mengerBind(self.lattice, action)
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if bindResult.lawful:
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# Update lattice
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newOccupancy = fractalOccupancy(self.lattice.size, self.lattice.hausdorffDim, self.lattice.occupancyDensity)
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self.lattice.activePositions = newOccupancy
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# Record action history
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self.actionHistory.append({
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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.lattice.to_dict()
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}
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def getLatticeState(self) -> Optional[Dict[str, Any]]:
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"""Get current lattice state"""
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if self.lattice:
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return self.lattice.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" + "="*70)
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print("MENGER SPONGE FRACTAL ADDRESSING STATE")
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print("="*70)
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if self.lattice:
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print(f"\n📊 Lattice Properties:")
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print(f" Size: {self.lattice.size}")
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print(f" Hausdorff Dimension: {from_q16(self.lattice.hausdorffDim):.4f}")
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print(f" Occupancy Density: {from_q16(self.lattice.occupancyDensity):.3f}")
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print(f" Active Positions: {self.lattice.activePositions}")
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ratio = reductionRatio(self.lattice.size, self.lattice.hausdorffDim)
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print(f" Reduction Ratio: {ratio:.4f}")
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# Calculate theoretical vs actual reduction
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size_cubed = self.lattice.size ** 3
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reduction_pct = (1.0 - ratio) * 100
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print(f" State Space Reduction: {reduction_pct:.1f}%")
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print(f" Full State Space: {size_cubed:,} positions")
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print(f" Fractal Occupancy: {self.lattice.activePositions:,} positions")
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print(f"\n📜 Action History: {len(self.actionHistory)} entries")
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print("\n" + "="*70)
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def main():
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"""Test Menger sponge fractal addressing system"""
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system = MengerSpongeFractalAddressingSystem()
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print("[Test 1] Initialize Menger sponge lattice (N=64)...")
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result1 = system.initializeLattice(size=64, occupancyDensity=0.5)
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print(f" Lattice initialized")
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print(f" Hausdorff Dimension: {result1['hausdorffDim']:.4f}")
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print(f" Active Positions: {result1['activePositions']:,}")
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print(f" Reduction Ratio: {result1['reductionRatio']:.4f}")
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print("\n[Test 2] Submit Menger addressing action...")
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coord = MengerCoordinate(x=10, y=20, z=30)
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pistState = BlitterState(
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a=to_q16(4.0),
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b=to_q16(5.0),
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manifold=to_q16(0.0),
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stepMask=0
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)
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action = MengerAction(pistState=pistState, coord=coord)
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result2 = system.submitMengerAction(action)
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print(f" Result: Success={result2['success']}")
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if result2['success']:
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print(f" Address before: {result2['bindResult']['addressBefore']}")
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print(f" Address after: {result2['bindResult']['addressAfter']}")
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print(f" Manifold before: {result2['bindResult']['manifoldBefore']:.3f}")
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print(f" Manifold after: {result2['bindResult']['manifoldAfter']:.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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