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357 lines
13 KiB
Text
357 lines
13 KiB
Text
/-
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MetaManifoldLanguageMerging.lean — Language Manifold Merging with Geometric Structures
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Extends the InformationManifold taxonomy with:
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- Meta-manifold construction from language manifolds
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- 5D torus topology for routing
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- Menger sponge fractal addressing
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- Gabriel's horn for pathological manifold analysis
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- Mass Number gates for admissibility checking
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Core equations:
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- Language manifold: ℳ_L ⊂ ℝ^d
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- Meta-manifold: ℳ_meta = ⋃_{L∈ℒ} ℳ_L
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- Fold dynamics: ∂_t ℳ = -∇E_fold(ℳ)
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- Mass Number gate: MassLe(m, τ) := A ≤ τ · (R + ε)
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Ref: 17_Meta_Manifold_Language_Merging.md
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-/
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import Semantics.Core.InformationManifold
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import Semantics.Core.MassNumber
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import Semantics.FixedPoint
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namespace Semantics.MetaManifoldLanguageMerging
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open Semantics.Q16_16
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open Semantics.Core.MassNumber
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open Semantics.Core.InformationManifold
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/- ============================================================================
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§0 Language Manifold Structure
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============================================================================ -/
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/-- A language manifold embedded in high-dimensional semantic space. -/
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structure LanguageManifold where
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languageCode : String -- ISO 639 code (e.g., "en", "de", "ja")
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dimensionality : Nat -- Intrinsic dimensionality d_L
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vocabularySize : Nat -- |V_L| number of words
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metric : Matrix (Fin dimensionality) (Fin dimensionality) ℝ -- g_{ij}
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torsion : (Fin dimensionality) → (Fin dimensionality) → (Fin dimensionality) → ℝ -- T^k_{ij}
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anisotropy : Matrix (Fin dimensionality) (Fin dimensionality) ℝ -- M^{ij}
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deriving Repr, Inhabited
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/-- A word as a point on the language manifold. -/
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structure WordPoint where
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word : String
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embedding : ℝ -- Simplified: single coordinate (in practice: ℝ^d)
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manifold : LanguageManifold
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deriving Repr, Inhabited
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/-- The vocabulary of a language as a set of word points. -/
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structure Vocabulary where
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language : LanguageManifold
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words : List WordPoint
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deriving Repr, Inhabited
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/- ============================================================================
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§1 Meta-Manifold Construction
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============================================================================ -/
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/-- The meta-manifold as the union of all language manifolds. -/
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structure MetaManifold where
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languages : List LanguageManifold
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dimensionality : Nat -- d_meta = max d_L + Δd
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unifiedMetric : Matrix (Fin dimensionality) (Fin dimensionality) ℝ
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anchorPoints : List WordPoint -- NSM primes as anchors
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deriving Repr, Inhabited
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/-- Embedding function from language manifold to meta-manifold. -/
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structure Embedding where
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source : LanguageManifold
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target : MetaManifold
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map : WordPoint → WordPoint -- ψ_L: ℳ_L → ℳ_meta
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anchorPreserved : Bool -- ψ_L(φ_L(p)) = φ_meta(p) for all NSM primes p
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localIsometry : Bool -- Preserves local distances
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deriving Repr, Inhabited
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/- ============================================================================
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§2 5D Torus Topology Integration
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============================================================================ -/
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/-- 5D torus topology for parallel processing and routing. -/
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structure FiveDTorus where
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dimensionSizes : List Nat -- [k_0, k_1, k_2, k_3, k_4]
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deriving Repr, Inhabited
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/-- Torus node coordinates. -/
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structure TorusNode where
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coordinates : List Nat -- [i_0, i_1, i_2, i_3, i_4]
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torus : FiveDTorus
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deriving Repr, Inhabited
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/-- Torus distance: d_torus = Σ min(|x_i - y_i|, k_i - |x_i - y_i|). -/
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def torusDistance (n1 n2 : TorusNode) : Nat :=
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let coords1 := n1.coordinates
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let coords2 := n2.coordinates
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let sizes := n1.torus.dimensionSizes
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let rec helper (i : Nat) (acc : Nat) : Nat :=
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if i ≥ 5 then acc
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else
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let diff := Nat.abs (coords1[i]! - coords2[i]!)
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let wrapped := sizes[i]! - diff
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let minDist := if diff < wrapped then diff else wrapped
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helper (i + 1) (acc + minDist)
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helper 0 0
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/-- Torus diameter: D_torus = Σ ⌊k_i/2⌋. -/
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def torusDiameter (torus : FiveDTorus) : Nat :=
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let rec helper (sizes : List Nat) (acc : Nat) : Nat :=
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match sizes with
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| [] => acc
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| k :: ks => helper ks (acc + (k / 2))
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helper torus.dimensionSizes 0
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/-- Bisection bandwidth: B = (k_0 · k_1 · k_2 · k_3 · k_4) / 2. -/
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def bisectionBandwidth (torus : FiveDTorus) : Nat :=
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let rec product (sizes : List Nat) : Nat :=
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match sizes with
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| [] => 1
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| k :: ks => k * product ks
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(product torus.dimensionSizes) / 2
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/- ============================================================================
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§3 Menger Sponge Fractal Addressing
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============================================================================ -/
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/-- Menger sponge lattice coordinates. -/
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structure MengerCoord where
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x : Nat
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y : Nat
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z : Nat
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deriving Repr, Inhabited
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/-- Menger sponge lattice state. -/
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structure MengerLattice where
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size : Nat -- N
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hausdorffDim : Q16_16 -- d_H ≈ 2.7268
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occupancyDensity : Q16_16 -- ρ_occ
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deriving Repr, Inhabited
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/-- Menger hash: menger_hash(x,y,z) = x ⊕ (y << 1) ⊕ (z << 2). -/
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def mengerHash (coord : MengerCoord) : Nat :=
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let x := coord.x
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let y := coord.y <<< 1
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let z := coord.z <<< 2
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Nat.xor x (Nat.xor y z)
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/-- Fractal offset: (x + y + z) · d_H / 65536. -/
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def fractalOffset (coord : MengerCoord) (hausdorffDim : Q16_16) : Nat :=
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let sum := coord.x + coord.y + coord.z
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let dim := hausdorffDim.val.toUInt32
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(sum * dim.toNat) / 65536
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/-- Menger address: menger_hash ⊕ fractal_offset. -/
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def mengerAddress (coord : MengerCoord) (hausdorffDim : Q16_16) : Nat :=
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Nat.xor (mengerHash coord) (fractalOffset coord hausdorffDim)
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/-- Fractal occupancy: |P_occ| = ρ_occ · N^{d_H}. -/
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def fractalOccupancy (lattice : MengerLattice) : Nat :=
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let sizeQ := ⟨lattice.size⟩
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let nPowDh := Q16_16.pow sizeQ lattice.hausdorffDim
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let occupancy := lattice.occupancyDensity * nPowDh / Q16_ONE
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occupancy.val.toUInt32.toNat
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/-- State space reduction: R = N^{d_H} / N^3 = N^{d_H - 3}. -/
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def reductionRatio (lattice : MengerLattice) : Q16_16 :=
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let sizeQ := ⟨lattice.size⟩
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let sizeCubed := sizeQ * sizeQ * sizeQ / Q16_ONE
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let sizePowDh := Q16_16.pow sizeQ lattice.hausdorffDim
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sizePowDh / sizeCubed
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/- ============================================================================
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§4 Gabriel's Horn Integration
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============================================================================ -/
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/-- Gabriel's horn parameters. -/
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structure GabrielsHorn where
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xMin : Q16_16 -- Start of horn (typically 1)
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xMax : Q16_16 -- End of horn (truncated, e.g., 1000)
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deriving Repr, Inhabited
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/-- Horn radius at position x: r(x) = 1/x. -/
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def hornRadius (horn : GabrielsHorn) (x : Q16_16) : Q16_16 :=
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Q16_ONE / x
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/-- Horn volume (truncated): V = π ∫_{x_min}^{x_max} (1/x)^2 dx = π(1/x_min - 1/x_max). -/
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def hornVolume (horn : GabrielsHorn) : Q16_16 :=
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let xMinInv := Q16_ONE / horn.xMin
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let xMaxInv := Q16_ONE / horn.xMax
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let pi := ⟨205887⟩ -- π in Q16_16 ≈ 3.14159
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pi * (xMinInv - xMaxInv)
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/-- Horn surface area (truncated approximation). -/
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def hornSurfaceArea (horn : GabrielsHorn) : Q16_16 :=
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-- A = 2π ∫ (1/x) √(1 + 1/x^4) dx
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-- Approximated as 2π · ln(x_max/x_min) for large x
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let ratio := horn.xMax / horn.xMin
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let logRatio := Q16_16.log ratio -- Natural log approximation
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let twoPi := ⟨411774⟩ -- 2π in Q16_16
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twoPi * logRatio
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/- ============================================================================
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§5 Geometric Structure Folding
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============================================================================ -/
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/-- Fold view: which geometric structure is currently active. -/
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inductive FoldView
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| torus
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| menger
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| horn
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deriving BEq, DecidableEq, Inhabited
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/-- Fold energy: E_fold = α E_torus + β E_menger + γ E_horn. -/
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structure FoldEnergy where
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torusEnergy : Q16_16
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mengerEnergy : Q16_16
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hornEnergy : Q16_16
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alpha : Q16_16 -- Weight for torus
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beta : Q16_16 -- Weight for menger
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gamma : Q16_16 -- Weight for horn
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deriving Repr, Inhabited
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/-- Total fold energy. -/
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def totalFoldEnergy (energy : FoldEnergy) : Q16_16 :=
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energy.alpha * energy.torusEnergy +
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energy.beta * energy.mengerEnergy +
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energy.gamma * energy.hornEnergy
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/-- Fold transition: check if transition from view1 to view2 is admissible. -/
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def foldTransitionAdmissible (energy : FoldEnergy) (view1 view2 : FoldView) (threshold : Q16_16) : Bool :=
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let energy1 := match view1 with
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| FoldView.torus => energy.torusEnergy
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| FoldView.menger => energy.mengerEnergy
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| FoldView.horn => energy.hornEnergy
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let energy2 := match view2 with
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| FoldView.torus => energy.torusEnergy
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| FoldView.menger => energy.mengerEnergy
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| FoldView.horn => energy.hornEnergy
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let energyGain := energy1 - energy2
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let residual := energy2
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let m := mkMassNumber energyGain residual "FOLD" "transition" "FOLD" threshold
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MassLeDefault m
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/- ============================================================================
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§6 Mass Number Gates for Manifold Merging
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============================================================================ -/
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/-- Manifold merging Mass Number. -/
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def manifoldMergeMassNumber (compressionGain : Q16_16) (semanticLoss : Q16_16) (threshold : Q16_16) : MassNumber :=
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mkMassNumber compressionGain semanticLoss "MANIFOLD" "semantic_loss" "MANIFOLD" threshold
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/-- Check if manifold merge is admissible. -/
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def manifoldMergeAdmissible (compressionGain : Q16_16) (semanticLoss : Q16_16) (threshold : Q16_16) : Bool :=
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let m := manifoldMergeMassNumber compressionGain semanticLoss threshold
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MassLeDefault m
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/-- Compression gate using Hutter Prize principles. -/
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def hutterCompressionGateManifold (entropyGain : Q16_16) (reconRisk : Q16_16) (acceptableRatio : Q16_16) : Bool :=
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let m := mkMassNumber entropyGain reconRisk "HUTTER" "entropy" "HUTTER" acceptableRatio
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MassLeDefault m
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/- ============================================================================
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§7 Unified Compression Equation
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============================================================================ -/
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/-- Unified compression: C_unified = α·C_torus + β·C_menger + γ·C_horn. -/
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structure UnifiedCompression where
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torusCompression : Q16_16
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mengerCompression : Q16_16
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hornCompression : Q16_16
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alpha : Q16_16
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beta : Q16_16
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gamma : Q16_16
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deriving Repr, Inhabited
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/-- Total unified compression. -/
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def totalUnifiedCompression (comp : UnifiedCompression) : Q16_16 :=
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comp.alpha * comp.torusCompression +
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comp.beta * comp.mengerCompression +
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comp.gamma * comp.hornCompression
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/- ============================================================================
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§8 Surface Translation (from MassNumberSurfaceTranslation.md)
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============================================================================ -/
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/-- Surface fields for manifold merging. -/
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structure SurfaceFields where
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height : Q16_16 -- Threshold pressure
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ridge : Q16_16 -- Compression ratio where merging becomes forced
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holes : List String -- Forbidden configurations
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seams : List String -- Representation-change boundaries
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flowLines : List String -- Admissible merge routes
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scarField : Q16_16 -- Underverse residue
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compressionGradient : Q16_16
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deriving Repr, Inhabited
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/-- Mass surface packet. -/
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structure MassSurfacePacket where
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surfaceId : String
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sourceMassNumberId : String
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coordinateSystem : String
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fields : SurfaceFields
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invariantContours : List String
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thresholdRidges : List Q16_16
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obstructionHoles : List String
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representationSeams : List String
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proofFlowLines : List String
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validationStatus : String
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deriving Repr, Inhabited
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/- ============================================================================
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§9 #eval Examples
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============================================================================ -/
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#let torus := { dimensionSizes := [16, 8, 8, 8, 4] }
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#let node1 := { coordinates := [0, 0, 0, 0, 0], torus := torus }
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#let node2 := { coordinates := [8, 4, 4, 4, 2], torus := torus }
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#eval torusDistance node1 node2
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#eval torusDiameter torus
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#eval bisectionBandwidth torus
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#let mengerCoord := { x := 10, y := 20, z := 30 }
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#let hausdorffDim := ⟨17910⟩ -- 2.7268 in Q16_16
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#eval mengerHash mengerCoord
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#eval fractalOffset mengerCoord hausdorffDim
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#eval mengerAddress mengerCoord hausdorffDim
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#let mengerLattice := { size := 64, hausdorffDim := hausdorffDim, occupancyDensity := to_q16 0.5 }
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#eval fractalOccupancy mengerLattice
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#eval reductionRatio mengerLattice
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#let horn := { xMin := to_q16 1.0, xMax := to_q16 1000.0 }
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#eval hornRadius horn (to_q16 10.0)
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#eval hornVolume horn
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#eval hornSurfaceArea horn
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#let foldEnergy := {
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torusEnergy := to_q16 0.5,
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mengerEnergy := to_q16 0.161,
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hornEnergy := to_q16 0.072,
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alpha := to_q16 0.4,
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beta := to_q16 0.35,
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gamma := to_q16 0.25
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}
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#eval totalFoldEnergy foldEnergy
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#eval foldTransitionAdmissible foldEnergy FoldView.torus FoldView.menger (to_q16 0.3)
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#eval manifoldMergeAdmissible (to_q16 0.97) (to_q16 0.03) (to_q16 5.0)
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#eval hutterCompressionGateManifold (to_q16 0.868) (to_q16 0.132) (to_q16 6.6)
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end Semantics.MetaManifoldLanguageMerging
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