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

import Semantics.FixedPoint
import Semantics.PistBridge

namespace Semantics.MengerSpongeFractalAddressing

open Semantics.Q16_16
open Semantics.PistBridge

-- ═══════════════════════════════════════════════════════════════════════════
-- §0  Menger Sponge Fractal Addressing
-- 
-- This module implements Menger sponge fractal addressing for PIST manifold.
-- 
-- Key equations:
-- |P_occ| = ρ_occ · N^{d_H}
-- d_H ≈ 2.7268 (Hausdorff dimension of Menger sponge)
-- address(x,y,z) = menger_hash(x,y,z) ⊕ fractal_offset
-- 
-- where:
-- - |P_occ| = Fractal occupancy (number of active positions)
-- - ρ_occ = Occupancy density
-- - N = Lattice size
-- - d_H = Hausdorff dimension
-- - address = Fractal address
-- - menger_hash = Menger sponge hash function
-- - fractal_offset = Fractal offset
-- 
-- Concept:
-- - Reduces state space from 262,144 to ~84,000 positions (68% reduction) for N=64
-- - High informatic density with Hausdorff dimension d_H≈2.7268
-- - Forms backbone of NII cores (Non-Isotropic Informatic cores)
-- ═══════════════════════════════════════════════════════════════════════════

/-- Menger sponge lattice coordinates -/
structure MengerCoordinate where
  x : UInt32  -- X coordinate
  y : UInt32  -- Y coordinate
  z : UInt32  -- Z coordinate
  deriving Repr, Inhabited

/-- Menger sponge lattice state -/
structure MengerLattice where
  size : UInt32  -- Lattice size N
  hausdorffDim : Q16_16  -- Hausdorff dimension d_H ≈ 2.7268
  occupancyDensity : Q16_16  -- Occupancy density ρ_occ
  activePositions : UInt32  -- Number of active positions |P_occ|
  deriving Repr, Inhabited

/-- Menger sponge address -/
structure MengerAddress where
  hash : UInt32  -- Menger hash value
  offset : UInt32  -- Fractal offset
  occupied : Bool  -- Whether position is occupied
  deriving Repr, Inhabited

-- ═══════════════════════════════════════════════════════════════════════════
-- §1  Menger Sponge Hash Function
-- ═══════════════════════════════════════════════════════════════════════════

/-- Calculate Menger sponge hash: menger_hash(x,y,z) -/
def mengerHash (coord : MengerCoordinate) : UInt32 :=
  let x := coord.x
  let y := coord.y
  let z := coord.z
  -- Menger sponge hash: XOR of coordinates with bit shifts
  let hash := x ^^^ (y <<< 1) ^^^ (z <<< 2)
  hash

/-- Calculate fractal offset based on Hausdorff dimension -/
def fractalOffset (coord : MengerCoordinate) (hausdorffDim : Q16_16) : UInt32 :=
  let x := coord.x
  let y := coord.y
  let z := coord.z
  let dim := hausdorffDim.val.toUInt32
  -- Fractal offset: (x + y + z) * d_H
  let sum := x + y + z
  let offset := sum * dim / 65536
  offset

/-- Calculate Menger sponge address: address(x,y,z) = menger_hash ⊕ fractal_offset -/
def mengerAddress (coord : MengerCoordinate) (hausdorffDim : Q16_16) : MengerAddress :=
  let hash := mengerHash coord
  let offset := fractalOffset coord hausdorffDim
  let address := hash ^^^ offset
  { hash := hash, offset := offset, occupied := true }

-- ═══════════════════════════════════════════════════════════════════════════
-- §2  Fractal Occupancy Calculation
-- ═══════════════════════════════════════════════════════════════════════════

/-- Hausdorff dimension of Menger sponge: d_H ≈ 2.7268 -/
def mengerHausdorffDim : Q16_16 := ⟨17910⟩  -- 2.7268 in Q16_16

/-- Calculate fractal occupancy: |P_occ| = ρ_occ · N^{d_H} -/
def fractalOccupancy (size : UInt32) (hausdorffDim : Q16_16) (occupancyDensity : Q16_16) : UInt32 :=
  let sizeQ := ⟨size⟩
  let n_pow_dh := Q16_16.pow sizeQ hausdorffDim
  let occupancy := occupancyDensity * n_pow_dh / Q16_ONE
  occupancy.val.toUInt32

/-- Calculate state space reduction ratio -/
def reductionRatio (size : UInt32) (hausdorffDim : Q16_16) : Q16_16 :=
  let sizeQ := ⟨size⟩
  let sizeCubed := sizeQ * sizeQ * sizeQ / Q16_ONE
  let sizePowDh := Q16_16.pow sizeQ hausdorffDim
  let ratio := sizePowDh / sizeCubed
  ratio

-- ═══════════════════════════════════════════════════════════════════════════
-- §3  Menger Sponge Integration with PIST
-- ═══════════════════════════════════════════════════════════════════════════

/-- Convert PIST (a,b) coordinates to Menger (x,y,z) coordinates -/
def pistToMengerCoord (pistState : BlitterState) (size : UInt32) : MengerCoordinate :=
  let a := pistState.a.val.toUInt32
  let b := pistState.b.val.toUInt32
  let manifold := pistState.manifold.val.toUInt32
  -- Map PIST coordinates to 3D Menger space
  let x := a % size
  let y := b % size
  let z := manifold % size
  { x := x, y := y, z := z }

/-- Convert Menger address back to PIST manifold value -/
def mengerToPistManifold (addr : MengerAddress) : Q16_16 :=
  ⟨addr.hash⟩

-- ═══════════════════════════════════════════════════════════════════════════
-- §4  Bind Primitive for Menger Sponge Addressing
-- ═══════════════════════════════════════════════════════════════════════════

/-- Menger sponge addressing action -/
structure MengerAction where
  pistState : BlitterState
  coord : MengerCoordinate
  deriving Repr, Inhabited

/-- Menger sponge bind result -/
structure MengerBind where
  lawful : Bool  -- Whether action is lawful
  addressBefore : UInt32  -- Address before action
  addressAfter : UInt32  -- Address after action
  occupancyBefore : UInt32  -- Occupancy before action
  occupancyAfter : UInt32  -- Occupancy after action
  manifoldBefore : Q16_16  -- PIST manifold before
  manifoldAfter : Q16_16  -- PIST manifold after
  invariant : String  -- Invariant description
  deriving Repr, Inhabited

/-- Check if Menger action is lawful -/
def isMengerActionLawful (lattice : MengerLattice) (action : MengerAction) : Bool :=
  let x := action.coord.x
  let y := action.coord.y
  let z := action.coord.z
  let lawful := x < lattice.size ∧ y < lattice.size ∧ z < lattice.size
  lawful

/-- Bind primitive for Menger sponge addressing -/
def mengerBind (lattice : MengerLattice) (action : MengerAction) : MengerBind :=
  let lawful := isMengerActionLawful lattice action
  
  let addrBefore := mengerAddress action.coord lattice.hausdorffDim
  let manifoldBefore := action.pistState.manifold
  let occupancyBefore := lattice.activePositions
  
  let newLattice := if lawful then
    let newOccupancy := fractalOccupancy lattice.size lattice.hausdorffDim lattice.occupancyDensity
    { lattice with activePositions := newOccupancy }
  else
    lattice
  
  let addrAfter := if lawful then mengerAddress action.coord lattice.hausdorffDim else addrBefore
  let manifoldAfter := if lawful then mengerToPistManifold addrAfter else manifoldBefore
  let occupancyAfter := newLattice.activePositions
  
  {
    lawful := lawful,
    addressBefore := addrBefore.hash,
    addressAfter := addrAfter.hash,
    occupancyBefore := occupancyBefore,
    occupancyAfter := occupancyAfter,
    manifoldBefore := manifoldBefore,
    manifoldAfter := manifoldAfter,
    invariant := if lawful then "menger_sponge_addressing_satisfied" else "menger_constraint_violated"
  }

-- ═══════════════════════════════════════════════════════════════════════════
-- §5  Invariant Preservation
-- ═══════════════════════════════════════════════════════════════════════════

/-- Menger hash is deterministic -/
theorem mengerHashDeterministic (coord : MengerCoordinate) :
    mengerHash coord = mengerHash coord := by
  rfl

-- REMOVED: fractalOccupancyBounded had no proof body
-- REMOVED: reductionRatioLessThanOne had no proof body
-- REMOVED: mengerPreservesPistConvergence had no proof body

-- ═══════════════════════════════════════════════════════════════════════════
-- §6  #eval Examples
-- ═══════════════════════════════════════════════════════════════════════════

#let coord := { x := 10, y := 20, z := 30 }

#eval mengerHausdorffDim

#eval mengerHash coord

#eval fractalOffset coord mengerHausdorffDim

#eval mengerAddress coord mengerHausdorffDim

#eval fractalOccupancy 64 mengerHausdorffDim (to_q16 0.5)

#eval reductionRatio 64 mengerHausdorffDim

#let lattice := {
  size := 64,
  hausdorffDim := mengerHausdorffDim,
  occupancyDensity := to_q16 0.5,
  activePositions := 0
}

#let pistState := {
  a := to_q16 4.0,
  b := to_q16 5.0,
  manifold := to_q16 0.0,
  stepMask := 0
}

#let action := { pistState := pistState, coord := coord }

#eval isMengerActionLawful lattice action

#eval mengerBind lattice action

end Semantics.MengerSpongeFractalAddressing