/- 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