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

/-
SDTA.lean — Semantic Degenerate Tensor Adapter

The SDTA is a category-theoretic framework where:
- Problems exist as states x ∈ X
- Degeneracy projections Π_D collapse to a chart D
- Adapters A_ij = Π_Dj ∘ T_ij ∘ Π_Di transport between charts
- Tree composition Ψ(ℓ) aggregates child adapters

This module implements the core SDTA infrastructure with:
1. State vectors over Q16_16
2. Degenerate charts with Sidon labels
3. Adapter morphisms with degeneracy preservation
4. Tree composition for hierarchical aggregation
5. Portability coefficient η computation

See 6-Documentation/docs/specs/sdta_spec.md for full specification.
-/

import Semantics.FixedPoint
import Mathlib.Data.Matrix.Basic

namespace Semantics.SDTA

open Semantics.FixedPoint

/-! ## §1: State Vectors -/

/-- A state vector of dimension n in Q16.16 fixed-point. -/
abbrev StateVec (n : Nat) := Fin n → Q16_16

/-- The zero state vector (origin). -/
def StateVec.zero (n : Nat) : StateVec n := fun _ => Q16_16.zero

/-- Pointwise addition of state vectors. -/
def StateVec.add (n : Nat) (x y : StateVec n) : StateVec n :=
  fun i => Q16_16.add (x i) (y i)

/-- Pointwise scalar multiplication. -/
def StateVec.smul (n : Nat) (c : Q16_16) (x : StateVec n) : StateVec n :=
  fun i => Q16_16.mul c (x i)

/-! ## §2: Degenerate Charts -/

/-- A degenerate semantic chart: a labeled subspace where phases are constant
    (the ZIM-style collapse manifold). The labels provide a Sidon-distinguishability
    structure within the degenerate sector. -/
structure DegenerateChart (n : Nat) where
  /-- Sidon labels for the chart modes (all pairwise sums unique) -/
  labels : List (Fin n)
  /-- Basis matrix for the chart (rows are basis vectors) -/
  basis : Fin n → Fin n → Q16_16
  /-- Dimension of the chart (≤ n) -/
  dim : Nat
  /-- Sidon property: all pairwise sums of labels are unique -/
  sidon : ∀ (a b c d : Fin n), a ∈ labels → b ∈ labels → c ∈ labels → d ∈ labels →
          a + b = c + d → a = c ∧ b = d ∨ a = d ∧ b = c

/-- The zero chart (all zeros, empty labels). -/
def DegenerateChart.zero (n : Nat) : DegenerateChart n where
  labels := []
  basis := fun _ _ => Q16_16.zero
  dim := 0
  sidon := by
    intros a b c d ha _ _ _ _
    cases ha

/-! ## §3: Degeneracy Projection Π_D -/

/-- Project a state vector into a degenerate chart. The projection collapses
    the dispersive component while preserving the chart's label structure.
    
    Implementation: Π_D(x) = B_D · B_D† · x
    where B_D is the chart basis and B_D† is its pseudoinverse. -/
def degeneracyProjection (D : DegenerateChart n) (x : StateVec n) : StateVec n :=
  -- NOTE(lean-port): implement per sdta_spec §3.3 as `Π_D(x) = B_D · B_D† · x`.
  --   Requires: (1) Q16_16 matrix-vector multiplication over `Fin n`,
  --             (2) Pseudoinverse `B_D†` (Jacobi iteration or QR-style, floor-bounded),
  --             (3) Hypothesis that `D.basis` is full-rank in its first `D.dim` rows.
  --   Until spec §7 item 1 (Q16_16 eigenstate decomposition) lands, keep zero placeholder.
  fun i => Q16_16.zero

/-- The projection is idempotent: Π_D(Π_D(x)) = Π_D(x). -/
theorem degeneracyProjection_idempotent (D : DegenerateChart n) (x : StateVec n) :
    degeneracyProjection D (degeneracyProjection D x) = degeneracyProjection D x :=
  -- NOTE(lean-port): currently `rfl` because both sides reduce to `fun _ => zero`.
  --   Once `degeneracyProjection` implements `B_D · B_D† · x`, this requires the
  --   Moore-Penrose idempotent law: `(B_D · B_D†)² = B_D · B_D†`, i.e.
  --   `Π_D ∘ Π_D = Π_D`.  Proof sketch post-impl:
  --     `unfold degeneracyProjection; rw [mat_mul_assoc, pseudoinverse_idempotent D.basis]`
  --   where `pseudoinverse_idempotent` must land in a future `Semantics.Q16Matrix`
  --   module (spec §7 item 1).
  rfl

/-- The projection preserves the chart subspace: Π_D(x) ∈ D for all x. -/
theorem degeneracyProjection_preserves_chart (D : DegenerateChart n) (x : StateVec n) :
    -- NOTE(lean-port): formalize "Π_D(x) ∈ D" via a chart-membership predicate
    --   `inChart (D : DegenerateChart n) (y : StateVec n) : Prop :=
    --      ∃ c : Fin D.dim → Q16_16, y = (D.basis)ᵀ · c`.
    --   Restated conclusion: `inChart D (degeneracyProjection D x)`, witnessed by
    --   `c := B_D† · x` (requires real `degeneracyProjection`, see TODO at line 76).
    --   Conclusion stays `True` until `inChart` predicate lands.
    True :=
  True.intro

/-! ## §4: Tree Transport T_ij -/

/-- Transport a state between two charts via tree structure. This is the
    inter-domain lift operation in the SDTA pipeline.
    
    T_ij: D_i → D_j
    where D_i and D_j are degenerate charts. -/
def treeTransport (D_i D_j : DegenerateChart n) (x : StateVec n) : StateVec n :=
  -- NOTE(lean-port): implement per sdta_spec §3.4 — tree-structured lift between charts.
  --   Target form: `T_ij(x) = B_Dj · M_ij · B_Di† · x` where `M_ij` is the
  --   inter-chart coupling matrix derived from the Sidon label overlap structure
  --   (spec §5.3 — Sidon label conservation).
  --   Requires: (1) Q16_16 matrix arithmetic, (2) Sidon overlap operator on
  --   `DegenerateChart.labels`, (3) `treeTransport_degenerate_linear` lemma
  --   (linear in degenerate sector, spec §3.4 caveat).
  --   Until spec §7 items 1–2 land, keep zero placeholder.
  fun i => Q16_16.zero

/-! ## §5: Adapter A_ij = Π_Dj ∘ T_ij ∘ Π_Di -/

/-- The SDTA adapter: collapse → transport → re-collapse. This is the
    fundamental morphism of the framework.
    
    Key property: degeneracy preservation
    Π_Dj ∘ A_ij ∘ Π_Di = A_ij
    
    This ensures the adapter never leaves the degenerate regime. -/
def adapter (D_i D_j : DegenerateChart n) (x : StateVec n) : StateVec n :=
  degeneracyProjection D_j (treeTransport D_i D_j (degeneracyProjection D_i x))

/-- Transport is natural with respect to projections:
    Π_Dj(T_ij(Π_Di(x))) = A_ij(x) -/
theorem treeTransport_natural (D_i D_j : DegenerateChart n) (x : StateVec n) :
    degeneracyProjection D_j (treeTransport D_i D_j (degeneracyProjection D_i x)) =
    adapter D_i D_j x :=
  rfl

/-- Adapter degeneracy preservation: applying projections before and after
    the adapter doesn't change it. -/
theorem adapter_degeneracy_preserved (D_i D_j : DegenerateChart n) (x : StateVec n) :
    degeneracyProjection D_j (adapter D_i D_j (degeneracyProjection D_i x)) =
    adapter D_i D_j x :=
  -- NOTE(lean-port): currently `rfl` because both sides reduce to `fun _ => zero`.
  --   Post-implementation this factors through two idempotence lemmas:
  --     (a) `degeneracyProjection_idempotent D_i`  — collapse the inner `Π_Di`,
  --     (b) `degeneracyProjection_idempotent D_j`  — collapse the outer `Π_Dj`.
  --   Proof sketch post-impl:
  --     `unfold adapter; rw [idempotent_Di, idempotent_Dj]`.
  --   Blocked on `degeneracyProjection_idempotent` post-stub (spec §7 item 1).
  rfl

/-- Adapter composition law: A_jk ∘ A_ij = A_ik when charts are compatible. -/
theorem adapter_composition (D_i D_j D_k : DegenerateChart n) (x : StateVec n) :
    adapter D_j D_k (adapter D_i D_j x) = adapter D_i D_k x :=
  -- NOTE(lean-port): currently `rfl` because both sides reduce to `fun _ => zero`.
  --   WARNING: per sdta_spec §5.1 the real statement requires a compatibility
  --   hypothesis `Compatible D_i D_k` — without it, the equation is FALSE for
  --   real adapters between incompatible charts (spec §5.1 defines the
  --   associator `obs(i,j,k) := A_jk ∘ A_ij − A_ik` as the obstruction).
  --   The theorem statement is preserved (per "do not delete/rename" rule);
  --   when real `adapter` lands, EITHER:
  --     (a) add `(hcompat : Compatible D_i D_k)` to the hypothesis list, OR
  --     (b) restate the goal as `adapter D_j D_k (adapter D_i D_j x) − adapter D_i D_k x
  --         = obstruction(D_i, D_j, D_k, x)`.
  rfl

/-! ## §6: Semantic Mass Weighting -/

/-- Semantic mass between two charts. High mass means the charts are tightly
    coupled (low portability); low mass means loosely coupled (high portability).
    
    Computed as the overlap integral of the chart bases. -/
def semanticMass (D_i D_j : DegenerateChart n) : Q16_16 :=
  -- NOTE(lean-port): implement per sdta_spec §3.6 — basis overlap integral
  --   `SM(D_i, D_j) = Σ_{a ∈ D_i.labels, b ∈ D_j.labels} inner(D_i.basis a, D_j.basis b)`
  --   where `inner : (Fin n → Q16_16) → (Fin n → Q16_16) → Q16_16` is the Q16_16 dot
  --   product.  Requires `Finset.sum` over Q16_16 plus a `Q16Matrix.inner` primitive.
  --   Until spec §7 item 2 (SMN semantic definition) lands, keep zero placeholder.
  Q16_16.zero

/-- Semantic mass is symmetric: m_s(D_i, D_j) = m_s(D_j, D_i). -/
theorem semanticMass_symmetric (D_i D_j : DegenerateChart n) :
    semanticMass D_i D_j = semanticMass D_j D_i :=
  -- NOTE(lean-port): currently `rfl` because `semanticMass` returns `zero` on both sides.
  --   Post-implementation requires:
  --     (a) `Q16_16.mul_comm`   — NOT yet in `FixedPoint.lean` (would follow from
  --        `Int.mul_comm` modulo saturating `ofRawInt`; needs an LSB-preserving proof),
  --     (b) `Finset.sum_comm`   — to swap the inner-product factors across `a,b`.
  --   Blocked on spec §7 item 2 (SMN semantic definition).  See
  --   `FixedPoint.mul_self_nonneg` for an existing Q16_16 dot-product lemma.
  rfl

/-- Semantic mass is non-negative: m_s(D_i, D_j) ≥ 0. -/
theorem semanticMass_nonneg (D_i D_j : DegenerateChart n) :
    Q16_16.zero ≤ semanticMass D_i D_j :=
  -- NOTE(lean-port): currently `by rfl` because `semanticMass = zero` and `0 ≤ 0` (Int).
  --   Post-implementation requires `Finset.sum_nonneg` over a family of
  --   `mul_self_nonneg (inner ...)` terms — i.e. each summand is a square of
  --   a Q16_16 dot product and hence non-negative (see `FixedPoint.mul_self_nonneg`,
  --   proved).  Blocked on spec §7 item 2 (SMN semantic definition).
  by rfl

/-! ## §7: Tree Composition -/

/-- A node in the SDTA composition tree. -/
structure TreeNode (n : Nat) where
  /-- The chart at this node -/
  chart : DegenerateChart n
  /-- Semantic mass weight for this node -/
  mass : Q16_16
  /-- Child nodes (leaves have empty list) -/
  children : List (TreeNode n)

/-- Tree composition: aggregate child adapters bottom-up, weighted by
    semantic mass, plus the residual at the current node.
    
    Ψ(ℓ)(x) = Σ_j w_j · A_{parent, child_j}(x) + R_parent(x)
    
    where w_j are semantic mass weights and R is the residual. -/
def treeComposition (root : TreeNode n) (x : StateVec n) : StateVec n :=
  -- NOTE(lean-port): implement per sdta_spec §3.7 — recursive bottom-up aggregation.
  --     Leaf:        `Ψ(leaf)(x)              = Π_D(leaf.chart) x`
  --     Internal:    `Ψ(node)(x)              = Σ_j node.children[j].mass · A_{node, child_j}(x)
  --                                            + R_node(x)`  (residual at parent chart)
  --   Requires: (1) `adapter` real implementation (TODO at line 113),
  --             (2) Q16_16 weighted-sum primitive (`StateVec.smul` + `StateVec.add`,
  --                 already defined above, suffice), (3) structural recursion proof
  --                 (well-founded on `TreeNode.children` acyclicity invariant).
  --   Until spec §7 item 3 (SDTA theorems) lands, keep zero placeholder.
  fun i => Q16_16.zero

/-- Tree composition preserves degeneracy: the result is in the root's chart. -/
theorem treeComposition_preserves_chart (root : TreeNode n) (x : StateVec n) :
    -- NOTE(lean-port): formalize "Ψ(root)(x) ∈ root.chart" using the same
    --   `inChart` predicate as `degeneracyProjection_preserves_chart` (TODO at
    --   line 87).  Proof by structural induction on `root.children`:
    --     base case (empty children): `Ψ(leaf)(x) = Π_D(x) ∈ D` (degeneracy_idempotent
    --       + preserves_chart).
    --     inductive case: weighted sum of `adapter` outputs in `root.chart`;
    --       each `adapter` lands in `root.chart` by an `adapter_target_in_chart`
    --       lemma (also TBD), closed under `StateVec.add` / `StateVec.smul`.
    --   Conclusion stays `True` until `inChart` predicate lands.
    True :=
  True.intro

/-! ## §8: Portability Coefficient η -/

/-- The portability coefficient measures how much of the problem structure
    is captured in the degenerate subspace.
    
    η(A, k) = ||Π_k A Π_k†||_F / ||A||_F
    
    where Π_k is the projection onto the top-k singular vectors.
    
    High η (≈1) means the problem is essentially flat in the degenerate sector.
    Low η (≈0) means high semantic mass and low portability. -/
def portabilityCoefficient (n : Nat) (A : Matrix (Fin n) (Fin n) Q16_16) (k : Nat) : Q16_16 :=
  -- NOTE(lean-port): implement per sdta_spec §4.1 — SVD truncation energy ratio
  --   `η(A, k) = ‖Π_k A Π_k†‖_F / ‖A‖_F` where `Π_k` projects onto the top-`k`
  --   singular vectors of `A`.  Requires:
  --     (1) Q16_16 Frobenius norm (Σ diagonal via `mul_self_nonneg`, proved),
  --     (2) Q16_16 eigenstate decomposition — Jacobi iteration with floor-bounded
  --         error (spec §4.2 explicitly calls out "production Lean must use Q16_16
  --         eigenstate decomposition, not NumPy SVD"),
  --     (3) floor-truncated division `η = num_MSBs / den_MSBs` (Q16_16 `div` is
  --         available; needs `den ≠ 0` hypothesis for nonzero `A`).
  --   Until spec §7 item 1 (Q16_16 eigenstate decomposition) lands, keep zero
  --   placeholder.  Note: zero placeholder makes `portabilityCoefficient_bounded`
  --   trivially true (`0 ≤ 0` ∧ `0 ≤ 65536`) — re-verify the upper bound (`≤ one`)
  --   once the real ρ-normalized impl lands.
  Q16_16.zero

/-- Portability coefficient is bounded: 0 ≤ η ≤ 1. -/
theorem portabilityCoefficient_bounded (n : Nat) (A : Matrix (Fin n) (Fin n) Q16_16) (k : Nat) :
    Q16_16.zero ≤ portabilityCoefficient n A k ∧
    portabilityCoefficient n A k ≤ Q16_16.one :=
  by
    constructor
    · unfold portabilityCoefficient; rfl
    · unfold portabilityCoefficient; decide

/-- High portability implies low semantic mass. -/
theorem portability_high_semantic_mass_low (n : Nat) (A : Matrix (Fin n) (Fin n) Q16_16) (k : Nat)
    (hη : Q16_16.one ≤ portabilityCoefficient n A k) :
    -- NOTE(lean-port): formalize the inverse relationship
    --   `η(A, k) ≥ 1  ⟹  SMN(A) ≤ ε` (spec §4.1 + §4.3).
    --   Currently vacuously true: hypothesis `Q16_16.one ≤ portabilityCoefficient n A k`
    --   is FALSE against the zero stub (`toInt one = 65536` vs `toInt zero = 0`,
    --   ∀ A k).  Post-impl, needs:
    --     (a) a `semanticMassOfMatrix (A : Matrix (Fin n) (Fin n) Q16_16) : Q16_16`
    --         predicate (spec §4.3 — SMN is `Σ_{i<j} |A[i,j] * A[j,i]| / C(n,2)`),
    --     (b) the inequality chain tying the truncated-Frobenius ratio to the
    --         SMN coupling strength (spec §4.1 interpretation).
    --   Blocked on spec §7 item 2 (SMN semantic definition).
    True :=
  True.intro

/-! ## §9: Type Checks -/

#check @StateVec
#check @StateVec.zero
#check @StateVec.add
#check @StateVec.smul
#check @DegenerateChart
#check @DegenerateChart.zero
#check @degeneracyProjection
#check @degeneracyProjection_idempotent
#check @treeTransport
#check @treeTransport_natural
#check @adapter
#check @adapter_degeneracy_preserved
#check @adapter_composition
#check @semanticMass
#check @semanticMass_symmetric
#check @semanticMass_nonneg
#check @TreeNode
#check @treeComposition
#check @treeComposition_preserves_chart
#check @portabilityCoefficient
#check @portabilityCoefficient_bounded
#check @portability_high_semantic_mass_low

/-! ## §10: Category-Theoretic Structure -/

/-- The category of degenerate charts with adapters as morphisms.
    
    Objects: DegenerateChart n
    Morphisms: A_ij : D_i → D_j
    Composition: adapter_composition
    Identity: adapter_degeneracy_preserved (identity case) -/
structure DegenerateChartCategory (n : Nat) where
  /-- Objects are degenerate charts -/
  obj : Type
  /-- Morphisms between objects -/
  hom (D_i D_j : obj) : Type
  /-- Identity morphism -/
  id (D : obj) : hom D D
  /-- Composition of morphisms -/
  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
  /-- Category laws -/
  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) :
      comp (comp f g) h = comp f (comp g h)
  left_id {D_i D_j : obj} (f : hom D_i D_j) : comp (id D_j) f = f
  right_id {D_i D_j : obj} (f : hom D_i D_j) : comp f (id D_i) = f

/-- The SDTA forms a category where adapters are morphisms.
    Proved constructively from function composition. -/
def SDTA_is_category (n : Nat) : DegenerateChartCategory n where
  obj := DegenerateChart n
  hom _ _ := StateVec n → StateVec n
  id _ := id
  comp f g := f ∘ g
  assoc _ _ _ := rfl
  left_id _ := rfl
  right_id _ := rfl

end Semantics.SDTA