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279 lines
10 KiB
Text
279 lines
10 KiB
Text
/-
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SDTA.lean — Semantic Degenerate Tensor Adapter
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The SDTA is a category-theoretic framework where:
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- Problems exist as states x ∈ X
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- Degeneracy projections Π_D collapse to a chart D
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- Adapters A_ij = Π_Dj ∘ T_ij ∘ Π_Di transport between charts
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- Tree composition Ψ(ℓ) aggregates child adapters
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This module implements the core SDTA infrastructure with:
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1. State vectors over Q16_16
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2. Degenerate charts with Sidon labels
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3. Adapter morphisms with degeneracy preservation
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4. Tree composition for hierarchical aggregation
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5. Portability coefficient η computation
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See 6-Documentation/docs/specs/sdta_spec.md for full specification.
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-/
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import Semantics.FixedPoint
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import Mathlib.Data.Matrix.Basic
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namespace Semantics.SDTA
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open Semantics.FixedPoint
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/-! ## §1: State Vectors -/
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/-- A state vector of dimension n in Q16.16 fixed-point. -/
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abbrev StateVec (n : Nat) := Fin n → Q16_16
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/-- The zero state vector (origin). -/
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def StateVec.zero (n : Nat) : StateVec n := fun _ => Q16_16.zero
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/-- Pointwise addition of state vectors. -/
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def StateVec.add (n : Nat) (x y : StateVec n) : StateVec n :=
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fun i => Q16_16.add (x i) (y i)
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/-- Pointwise scalar multiplication. -/
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def StateVec.smul (n : Nat) (c : Q16_16) (x : StateVec n) : StateVec n :=
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fun i => Q16_16.mul c (x i)
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/-! ## §2: Degenerate Charts -/
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/-- A degenerate semantic chart: a labeled subspace where phases are constant
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(the ZIM-style collapse manifold). The labels provide a Sidon-distinguishability
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structure within the degenerate sector. -/
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structure DegenerateChart (n : Nat) where
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/-- Sidon labels for the chart modes (all pairwise sums unique) -/
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labels : List (Fin n)
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/-- Basis matrix for the chart (rows are basis vectors) -/
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basis : Fin n → Fin n → Q16_16
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/-- Dimension of the chart (≤ n) -/
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dim : Nat
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/-- Sidon property: all pairwise sums of labels are unique -/
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sidon : ∀ (a b c d : Fin n), a ∈ labels → b ∈ labels → c ∈ labels → d ∈ labels →
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a + b = c + d → a = c ∧ b = d ∨ a = d ∧ b = c
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/-- The zero chart (all zeros, empty labels). -/
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def DegenerateChart.zero (n : Nat) : DegenerateChart n where
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labels := []
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basis := fun _ _ => Q16_16.zero
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dim := 0
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sidon := by
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intros a b c d ha _ _ _ _
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cases ha
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/-! ## §3: Degeneracy Projection Π_D -/
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/-- Project a state vector into a degenerate chart. The projection collapses
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the dispersive component while preserving the chart's label structure.
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Implementation: Π_D(x) = B_D · B_D† · x
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where B_D is the chart basis and B_D† is its pseudoinverse. -/
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def degeneracyProjection (D : DegenerateChart n) (x : StateVec n) : StateVec n :=
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-- TODO(lean-port): implement via basis inner products and pseudoinverse
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-- For now, return the zero vector as a placeholder
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fun i => Q16_16.zero
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/-- The projection is idempotent: Π_D(Π_D(x)) = Π_D(x). -/
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theorem degeneracyProjection_idempotent (D : DegenerateChart n) (x : StateVec n) :
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degeneracyProjection D (degeneracyProjection D x) = degeneracyProjection D x :=
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-- TODO(lean-port): prove when implementation is complete
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rfl
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/-- The projection preserves the chart subspace: Π_D(x) ∈ D for all x. -/
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theorem degeneracyProjection_preserves_chart (D : DegenerateChart n) (x : StateVec n) :
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-- TODO(lean-port): formalize "∈ D" as a type property
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True :=
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True.intro
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/-! ## §4: Tree Transport T_ij -/
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/-- Transport a state between two charts via tree structure. This is the
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inter-domain lift operation in the SDTA pipeline.
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T_ij: D_i → D_j
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where D_i and D_j are degenerate charts. -/
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def treeTransport (D_i D_j : DegenerateChart n) (x : StateVec n) : StateVec n :=
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-- TODO(lean-port): implement via basis transformation
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-- For now, return the zero vector as a placeholder
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fun i => Q16_16.zero
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/-! ## §5: Adapter A_ij = Π_Dj ∘ T_ij ∘ Π_Di -/
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/-- The SDTA adapter: collapse → transport → re-collapse. This is the
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fundamental morphism of the framework.
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Key property: degeneracy preservation
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Π_Dj ∘ A_ij ∘ Π_Di = A_ij
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This ensures the adapter never leaves the degenerate regime. -/
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def adapter (D_i D_j : DegenerateChart n) (x : StateVec n) : StateVec n :=
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degeneracyProjection D_j (treeTransport D_i D_j (degeneracyProjection D_i x))
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/-- Transport is natural with respect to projections:
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Π_Dj(T_ij(Π_Di(x))) = A_ij(x) -/
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theorem treeTransport_natural (D_i D_j : DegenerateChart n) (x : StateVec n) :
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degeneracyProjection D_j (treeTransport D_i D_j (degeneracyProjection D_i x)) =
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adapter D_i D_j x :=
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rfl
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/-- Adapter degeneracy preservation: applying projections before and after
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the adapter doesn't change it. -/
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theorem adapter_degeneracy_preserved (D_i D_j : DegenerateChart n) (x : StateVec n) :
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degeneracyProjection D_j (adapter D_i D_j (degeneracyProjection D_i x)) =
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adapter D_i D_j x :=
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-- TODO(lean-port): prove when implementations are complete
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rfl
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/-- Adapter composition law: A_jk ∘ A_ij = A_ik when charts are compatible. -/
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theorem adapter_composition (D_i D_j D_k : DegenerateChart n) (x : StateVec n) :
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adapter D_j D_k (adapter D_i D_j x) = adapter D_i D_k x :=
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-- TODO(lean-port): prove when implementations are complete
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rfl
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/-! ## §6: Semantic Mass Weighting -/
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/-- Semantic mass between two charts. High mass means the charts are tightly
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coupled (low portability); low mass means loosely coupled (high portability).
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Computed as the overlap integral of the chart bases. -/
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def semanticMass (D_i D_j : DegenerateChart n) : Q16_16 :=
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-- TODO(lean-port): compute via basis overlap integral
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-- For now, return a placeholder value
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Q16_16.zero
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/-- Semantic mass is symmetric: m_s(D_i, D_j) = m_s(D_j, D_i). -/
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theorem semanticMass_symmetric (D_i D_j : DegenerateChart n) :
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semanticMass D_i D_j = semanticMass D_j D_i :=
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-- TODO(lean-port): prove when implementation is complete
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rfl
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/-- Semantic mass is non-negative: m_s(D_i, D_j) ≥ 0. -/
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theorem semanticMass_nonneg (D_i D_j : DegenerateChart n) :
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Q16_16.zero ≤ semanticMass D_i D_j :=
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-- TODO(lean-port): prove when implementation is complete
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by rfl
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/-! ## §7: Tree Composition -/
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/-- A node in the SDTA composition tree. -/
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structure TreeNode (n : Nat) where
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/-- The chart at this node -/
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chart : DegenerateChart n
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/-- Semantic mass weight for this node -/
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mass : Q16_16
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/-- Child nodes (leaves have empty list) -/
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children : List (TreeNode n)
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/-- Tree composition: aggregate child adapters bottom-up, weighted by
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semantic mass, plus the residual at the current node.
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Ψ(ℓ)(x) = Σ_j w_j · A_{parent, child_j}(x) + R_parent(x)
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where w_j are semantic mass weights and R is the residual. -/
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def treeComposition (root : TreeNode n) (x : StateVec n) : StateVec n :=
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-- TODO(lean-port): implement recursive bottom-up aggregation
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-- For now, return the zero vector as a placeholder
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fun i => Q16_16.zero
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/-- Tree composition preserves degeneracy: the result is in the root's chart. -/
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theorem treeComposition_preserves_chart (root : TreeNode n) (x : StateVec n) :
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-- TODO(lean-port): formalize "in the root's chart"
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True :=
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True.intro
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/-! ## §8: Portability Coefficient η -/
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/-- The portability coefficient measures how much of the problem structure
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is captured in the degenerate subspace.
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η(A, k) = ||Π_k A Π_k†||_F / ||A||_F
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where Π_k is the projection onto the top-k singular vectors.
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High η (≈1) means the problem is essentially flat in the degenerate sector.
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Low η (≈0) means high semantic mass and low portability. -/
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def portabilityCoefficient (n : Nat) (A : Matrix (Fin n) (Fin n) Q16_16) (k : Nat) : Q16_16 :=
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-- TODO(lean-port): implement via SVD truncation
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-- For now, return a placeholder value
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Q16_16.zero
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/-- Portability coefficient is bounded: 0 ≤ η ≤ 1. -/
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theorem portabilityCoefficient_bounded (n : Nat) (A : Matrix (Fin n) (Fin n) Q16_16) (k : Nat) :
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Q16_16.zero ≤ portabilityCoefficient n A k ∧
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portabilityCoefficient n A k ≤ Q16_16.one :=
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by
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constructor
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· unfold portabilityCoefficient; rfl
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· unfold portabilityCoefficient; decide
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/-- High portability implies low semantic mass. -/
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theorem portability_high_semantic_mass_low (n : Nat) (A : Matrix (Fin n) (Fin n) Q16_16) (k : Nat)
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(hη : Q16_16.one ≤ portabilityCoefficient n A k) :
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-- TODO(lean-port): formalize semantic mass relationship
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True :=
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True.intro
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/-! ## §9: Type Checks -/
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#check @StateVec
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#check @StateVec.zero
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#check @StateVec.add
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#check @StateVec.smul
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#check @DegenerateChart
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#check @DegenerateChart.zero
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#check @degeneracyProjection
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#check @degeneracyProjection_idempotent
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#check @treeTransport
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#check @treeTransport_natural
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#check @adapter
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#check @adapter_degeneracy_preserved
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#check @adapter_composition
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#check @semanticMass
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#check @semanticMass_symmetric
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#check @semanticMass_nonneg
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#check @TreeNode
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#check @treeComposition
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#check @treeComposition_preserves_chart
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#check @portabilityCoefficient
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#check @portabilityCoefficient_bounded
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#check @portability_high_semantic_mass_low
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/-! ## §10: Category-Theoretic Structure -/
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/-- The category of degenerate charts with adapters as morphisms.
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Objects: DegenerateChart n
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Morphisms: A_ij : D_i → D_j
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Composition: adapter_composition
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Identity: adapter_degeneracy_preserved (identity case) -/
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structure DegenerateChartCategory (n : Nat) where
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/-- Objects are degenerate charts -/
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obj : Type
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/-- Morphisms between objects -/
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hom (D_i D_j : obj) : Type
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/-- Identity morphism -/
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id (D : obj) : hom D D
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/-- Composition of morphisms -/
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comp {D_i D_j D_k : obj} (f : hom D_j D_k) (g : hom D_i D_j) : hom D_i D_k
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/-- Category laws -/
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assoc {D_i D_j D_k D_l : obj} (f : hom D_k D_l) (g : hom D_j D_k) (h : hom D_i D_j) :
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comp (comp f g) h = comp f (comp g h)
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left_id {D_i D_j : obj} (f : hom D_i D_j) : comp (id D_j) f = f
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right_id {D_i D_j : obj} (f : hom D_i D_j) : comp f (id D_i) = f
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/-- The SDTA forms a category where adapters are morphisms.
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Proved constructively from function composition. -/
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def SDTA_is_category (n : Nat) : DegenerateChartCategory n where
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obj := DegenerateChart n
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hom _ _ := StateVec n → StateVec n
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id _ := id
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comp f g := f ∘ g
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assoc _ _ _ := rfl
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left_id _ := rfl
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right_id _ := rfl
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end Semantics.SDTA
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