/-
MetaManifoldLanguageMerging.lean — Language Manifold Merging with Geometric Structures

Extends the InformationManifold taxonomy with:
  - Meta-manifold construction from language manifolds
  - 5D torus topology for routing
  - Menger sponge fractal addressing
  - Gabriel's horn for pathological manifold analysis
  - Mass Number gates for admissibility checking

Core equations:
  - Language manifold: ℳ_L ⊂ ℝ^d
  - Meta-manifold: ℳ_meta = ⋃_{L∈ℒ} ℳ_L
  - Fold dynamics: ∂_t ℳ = -∇E_fold(ℳ)
  - Mass Number gate: MassLe(m, τ) := A ≤ τ · (R + ε)

Ref: 17_Meta_Manifold_Language_Merging.md
-/

import Semantics.Core.InformationManifold
import Semantics.Core.MassNumber
import Semantics.FixedPoint

namespace Semantics.MetaManifoldLanguageMerging

open Semantics.Q16_16
open Semantics.Core.MassNumber
open Semantics.Core.InformationManifold

/- ============================================================================
   §0  Language Manifold Structure
   ============================================================================ -/

/-- A language manifold embedded in high-dimensional semantic space. -/
structure LanguageManifold where
  languageCode : String  -- ISO 639 code (e.g., "en", "de", "ja")
  dimensionality : Nat    -- Intrinsic dimensionality d_L
  vocabularySize : Nat    -- |V_L| number of words
  metric : Matrix (Fin dimensionality) (Fin dimensionality) ℝ  -- g_{ij}
  torsion : (Fin dimensionality) → (Fin dimensionality) → (Fin dimensionality) → ℝ  -- T^k_{ij}
  anisotropy : Matrix (Fin dimensionality) (Fin dimensionality) ℝ  -- M^{ij}
  deriving Repr, Inhabited

/-- A word as a point on the language manifold. -/
structure WordPoint where
  word : String
  embedding : ℝ  -- Simplified: single coordinate (in practice: ℝ^d)
  manifold : LanguageManifold
  deriving Repr, Inhabited

/-- The vocabulary of a language as a set of word points. -/
structure Vocabulary where
  language : LanguageManifold
  words : List WordPoint
  deriving Repr, Inhabited

/- ============================================================================
   §1  Meta-Manifold Construction
   ============================================================================ -/

/-- The meta-manifold as the union of all language manifolds. -/
structure MetaManifold where
  languages : List LanguageManifold
  dimensionality : Nat  -- d_meta = max d_L + Δd
  unifiedMetric : Matrix (Fin dimensionality) (Fin dimensionality) ℝ
  anchorPoints : List WordPoint  -- NSM primes as anchors
  deriving Repr, Inhabited

/-- Embedding function from language manifold to meta-manifold. -/
structure Embedding where
  source : LanguageManifold
  target : MetaManifold
  map : WordPoint → WordPoint  -- ψ_L: ℳ_L → ℳ_meta
  anchorPreserved : Bool  -- ψ_L(φ_L(p)) = φ_meta(p) for all NSM primes p
  localIsometry : Bool  -- Preserves local distances
  deriving Repr, Inhabited

/- ============================================================================
   §2  5D Torus Topology Integration
   ============================================================================ -/

/-- 5D torus topology for parallel processing and routing. -/
structure FiveDTorus where
  dimensionSizes : List Nat  -- [k_0, k_1, k_2, k_3, k_4]
  deriving Repr, Inhabited

/-- Torus node coordinates. -/
structure TorusNode where
  coordinates : List Nat  -- [i_0, i_1, i_2, i_3, i_4]
  torus : FiveDTorus
  deriving Repr, Inhabited

/-- Torus distance: d_torus = Σ min(|x_i - y_i|, k_i - |x_i - y_i|). -/
def torusDistance (n1 n2 : TorusNode) : Nat :=
  let coords1 := n1.coordinates
  let coords2 := n2.coordinates
  let sizes := n1.torus.dimensionSizes
  let rec helper (i : Nat) (acc : Nat) : Nat :=
    if i ≥ 5 then acc
    else
      let diff := Nat.abs (coords1[i]! - coords2[i]!)
      let wrapped := sizes[i]! - diff
      let minDist := if diff < wrapped then diff else wrapped
      helper (i + 1) (acc + minDist)
  helper 0 0

/-- Torus diameter: D_torus = Σ ⌊k_i/2⌋. -/
def torusDiameter (torus : FiveDTorus) : Nat :=
  let rec helper (sizes : List Nat) (acc : Nat) : Nat :=
    match sizes with
    | [] => acc
    | k :: ks => helper ks (acc + (k / 2))
  helper torus.dimensionSizes 0

/-- Bisection bandwidth: B = (k_0 · k_1 · k_2 · k_3 · k_4) / 2. -/
def bisectionBandwidth (torus : FiveDTorus) : Nat :=
  let rec product (sizes : List Nat) : Nat :=
    match sizes with
    | [] => 1
    | k :: ks => k * product ks
  (product torus.dimensionSizes) / 2

/- ============================================================================
   §3  Menger Sponge Fractal Addressing
   ============================================================================ -/

/-- Menger sponge lattice coordinates. -/
structure MengerCoord where
  x : Nat
  y : Nat
  z : Nat
  deriving Repr, Inhabited

/-- Menger sponge lattice state. -/
structure MengerLattice where
  size : Nat  -- N
  hausdorffDim : Q16_16  -- d_H ≈ 2.7268
  occupancyDensity : Q16_16  -- ρ_occ
  deriving Repr, Inhabited

/-- Menger hash: menger_hash(x,y,z) = x ⊕ (y << 1) ⊕ (z << 2). -/
def mengerHash (coord : MengerCoord) : Nat :=
  let x := coord.x
  let y := coord.y <<< 1
  let z := coord.z <<< 2
  Nat.xor x (Nat.xor y z)

/-- Fractal offset: (x + y + z) · d_H / 65536. -/
def fractalOffset (coord : MengerCoord) (hausdorffDim : Q16_16) : Nat :=
  let sum := coord.x + coord.y + coord.z
  let dim := hausdorffDim.val.toUInt32
  (sum * dim.toNat) / 65536

/-- Menger address: menger_hash ⊕ fractal_offset. -/
def mengerAddress (coord : MengerCoord) (hausdorffDim : Q16_16) : Nat :=
  Nat.xor (mengerHash coord) (fractalOffset coord hausdorffDim)

/-- Fractal occupancy: |P_occ| = ρ_occ · N^{d_H}. -/
def fractalOccupancy (lattice : MengerLattice) : Nat :=
  let sizeQ := ⟨lattice.size⟩
  let nPowDh := Q16_16.pow sizeQ lattice.hausdorffDim
  let occupancy := lattice.occupancyDensity * nPowDh / Q16_ONE
  occupancy.val.toUInt32.toNat

/-- State space reduction: R = N^{d_H} / N^3 = N^{d_H - 3}. -/
def reductionRatio (lattice : MengerLattice) : Q16_16 :=
  let sizeQ := ⟨lattice.size⟩
  let sizeCubed := sizeQ * sizeQ * sizeQ / Q16_ONE
  let sizePowDh := Q16_16.pow sizeQ lattice.hausdorffDim
  sizePowDh / sizeCubed

/- ============================================================================
   §4  Gabriel's Horn Integration
   ============================================================================ -/

/-- Gabriel's horn parameters. -/
structure GabrielsHorn where
  xMin : Q16_16  -- Start of horn (typically 1)
  xMax : Q16_16  -- End of horn (truncated, e.g., 1000)
  deriving Repr, Inhabited

/-- Horn radius at position x: r(x) = 1/x. -/
def hornRadius (horn : GabrielsHorn) (x : Q16_16) : Q16_16 :=
  Q16_ONE / x

/-- Horn volume (truncated): V = π ∫_{x_min}^{x_max} (1/x)^2 dx = π(1/x_min - 1/x_max). -/
def hornVolume (horn : GabrielsHorn) : Q16_16 :=
  let xMinInv := Q16_ONE / horn.xMin
  let xMaxInv := Q16_ONE / horn.xMax
  let pi := ⟨205887⟩  -- π in Q16_16 ≈ 3.14159
  pi * (xMinInv - xMaxInv)

/-- Horn surface area (truncated approximation). -/
def hornSurfaceArea (horn : GabrielsHorn) : Q16_16 :=
  -- A = 2π ∫ (1/x) √(1 + 1/x^4) dx
  -- Approximated as 2π · ln(x_max/x_min) for large x
  let ratio := horn.xMax / horn.xMin
  let logRatio := Q16_16.log ratio  -- Natural log approximation
  let twoPi := ⟨411774⟩  -- 2π in Q16_16
  twoPi * logRatio

/- ============================================================================
   §5  Geometric Structure Folding
   ============================================================================ -/

/-- Fold view: which geometric structure is currently active. -/
inductive FoldView
  | torus
  | menger
  | horn
  deriving BEq, DecidableEq, Inhabited

/-- Fold energy: E_fold = α E_torus + β E_menger + γ E_horn. -/
structure FoldEnergy where
  torusEnergy : Q16_16
  mengerEnergy : Q16_16
  hornEnergy : Q16_16
  alpha : Q16_16  -- Weight for torus
  beta : Q16_16   -- Weight for menger
  gamma : Q16_16  -- Weight for horn
  deriving Repr, Inhabited

/-- Total fold energy. -/
def totalFoldEnergy (energy : FoldEnergy) : Q16_16 :=
  energy.alpha * energy.torusEnergy +
  energy.beta * energy.mengerEnergy +
  energy.gamma * energy.hornEnergy

/-- Fold transition: check if transition from view1 to view2 is admissible. -/
def foldTransitionAdmissible (energy : FoldEnergy) (view1 view2 : FoldView) (threshold : Q16_16) : Bool :=
  let energy1 := match view1 with
    | FoldView.torus => energy.torusEnergy
    | FoldView.menger => energy.mengerEnergy
    | FoldView.horn => energy.hornEnergy
  let energy2 := match view2 with
    | FoldView.torus => energy.torusEnergy
    | FoldView.menger => energy.mengerEnergy
    | FoldView.horn => energy.hornEnergy
  let energyGain := energy1 - energy2
  let residual := energy2
  let m := mkMassNumber energyGain residual "FOLD" "transition" "FOLD" threshold
  MassLeDefault m

/- ============================================================================
   §6  Mass Number Gates for Manifold Merging
   ============================================================================ -/

/-- Manifold merging Mass Number. -/
def manifoldMergeMassNumber (compressionGain : Q16_16) (semanticLoss : Q16_16) (threshold : Q16_16) : MassNumber :=
  mkMassNumber compressionGain semanticLoss "MANIFOLD" "semantic_loss" "MANIFOLD" threshold

/-- Check if manifold merge is admissible. -/
def manifoldMergeAdmissible (compressionGain : Q16_16) (semanticLoss : Q16_16) (threshold : Q16_16) : Bool :=
  let m := manifoldMergeMassNumber compressionGain semanticLoss threshold
  MassLeDefault m

/-- Compression gate using Hutter Prize principles. -/
def hutterCompressionGateManifold (entropyGain : Q16_16) (reconRisk : Q16_16) (acceptableRatio : Q16_16) : Bool :=
  let m := mkMassNumber entropyGain reconRisk "HUTTER" "entropy" "HUTTER" acceptableRatio
  MassLeDefault m

/- ============================================================================
   §7  Unified Compression Equation
   ============================================================================ -/

/-- Unified compression: C_unified = α·C_torus + β·C_menger + γ·C_horn. -/
structure UnifiedCompression where
  torusCompression : Q16_16
  mengerCompression : Q16_16
  hornCompression : Q16_16
  alpha : Q16_16
  beta : Q16_16
  gamma : Q16_16
  deriving Repr, Inhabited

/-- Total unified compression. -/
def totalUnifiedCompression (comp : UnifiedCompression) : Q16_16 :=
  comp.alpha * comp.torusCompression +
  comp.beta * comp.mengerCompression +
  comp.gamma * comp.hornCompression

/- ============================================================================
   §8  Surface Translation (from MassNumberSurfaceTranslation.md)
   ============================================================================ -/

/-- Surface fields for manifold merging. -/
structure SurfaceFields where
  height : Q16_16  -- Threshold pressure
  ridge : Q16_16   -- Compression ratio where merging becomes forced
  holes : List String  -- Forbidden configurations
  seams : List String  -- Representation-change boundaries
  flowLines : List String  -- Admissible merge routes
  scarField : Q16_16  -- Underverse residue
  compressionGradient : Q16_16
  deriving Repr, Inhabited

/-- Mass surface packet. -/
structure MassSurfacePacket where
  surfaceId : String
  sourceMassNumberId : String
  coordinateSystem : String
  fields : SurfaceFields
  invariantContours : List String
  thresholdRidges : List Q16_16
  obstructionHoles : List String
  representationSeams : List String
  proofFlowLines : List String
  validationStatus : String
  deriving Repr, Inhabited

/- ============================================================================
   §9  #eval Examples
   ============================================================================ -/

#let torus := { dimensionSizes := [16, 8, 8, 8, 4] }

#let node1 := { coordinates := [0, 0, 0, 0, 0], torus := torus }
#let node2 := { coordinates := [8, 4, 4, 4, 2], torus := torus }

#eval torusDistance node1 node2
#eval torusDiameter torus
#eval bisectionBandwidth torus

#let mengerCoord := { x := 10, y := 20, z := 30 }
#let hausdorffDim := ⟨17910⟩  -- 2.7268 in Q16_16

#eval mengerHash mengerCoord
#eval fractalOffset mengerCoord hausdorffDim
#eval mengerAddress mengerCoord hausdorffDim

#let mengerLattice := { size := 64, hausdorffDim := hausdorffDim, occupancyDensity := to_q16 0.5 }

#eval fractalOccupancy mengerLattice
#eval reductionRatio mengerLattice

#let horn := { xMin := to_q16 1.0, xMax := to_q16 1000.0 }

#eval hornRadius horn (to_q16 10.0)
#eval hornVolume horn
#eval hornSurfaceArea horn

#let foldEnergy := {
  torusEnergy := to_q16 0.5,
  mengerEnergy := to_q16 0.161,
  hornEnergy := to_q16 0.072,
  alpha := to_q16 0.4,
  beta := to_q16 0.35,
  gamma := to_q16 0.25
}

#eval totalFoldEnergy foldEnergy
#eval foldTransitionAdmissible foldEnergy FoldView.torus FoldView.menger (to_q16 0.3)

#eval manifoldMergeAdmissible (to_q16 0.97) (to_q16 0.03) (to_q16 5.0)
#eval hutterCompressionGateManifold (to_q16 0.868) (to_q16 0.132) (to_q16 6.6)

end Semantics.MetaManifoldLanguageMerging
