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

/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Research Stack Team

CrossModalCompression.lean — Multi-Modal Biological Data Fusion via Field Theory

This module formalizes compression across multiple biological modalities:
- Sequence (DNA/RNA: 1D)
- Structure (Protein: 3D)  
- Function (Gene networks: graph)
- Expression (Transcriptomics: vector)

Key insight from MIRROR (2503.00374):
Multi-modal learning requires alignment between modalities, not just concatenation.

  REFERENCES:
    See 6-Documentation/docs/provenance/LANGUAGE_MATH_MODEL_SOURCES.cff
    for full DOIs. MIRROR: arxiv 2503.00374 (lookup current status at
    https://arxiv.org/abs/2503.00374).

The unified cross-modal field:
Φ_cross(x₁, x₂, ..., xₙ) = Σᵢ Φᵢ(xᵢ) + Σᵢ<ⱼ Φ_align(xᵢ, xⱼ)

Where:
- Φᵢ(xᵢ): Modality-specific field (sequence, structure, etc.)
- Φ_align(xᵢ, xⱼ): Alignment field between modalities i and j

Alignment field:
Φ_align(xᵢ, xⱼ) = -||projᵢ(xᵢ) - projⱼ(xⱼ)||²_κ / (1 + δ²)

Where:
- projᵢ: Projection to shared latent space
- ||·||²_κ: Geometry-aware distance (curvature κ)
- δ: Modality gap (how different the modalities are)

Per AGENTS.md §1.4: Q16_16 fixed-point for hardware extraction.
Per AGENTS.md §2: PascalCase types, camelCase functions.
Per AGENTS.md §4: Every def has eval witness or theorem.

NOTE(lean-port): Extract alignment formalism from MIRROR paper
NOTE(lean-port): Prove modality fusion improves compression
NOTE(lean-port): Connect to GenomicCompression for sequence-structure fusion
-/ 

import Mathlib.Data.Nat.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Matrix.Basic
import Mathlib.Tactic
import Semantics.FixedPoint

namespace Semantics.CrossModalCompression

open Semantics.Q16_16

-- ═══════════════════════════════════════════════════════════════════════════
-- §0  Modality Types
-- ═══════════════════════════════════════════════════════════════════════════

/-- Supported biological modalities. -/
inductive Modality
  | sequence    -- DNA/RNA sequence (1D)
  | structure   -- Protein 3D structure (coordinates)
  | function    -- Gene ontology / pathway (graph)
  | expression  -- Transcriptomics / proteomics (vector)
  | epigenetic  -- Methylation / chromatin state (tensor)
  deriving Repr, DecidableEq, Inhabited

namespace Modality

/-- Dimensionality of each modality. -/
def dimensionality : Modality → Nat
  | sequence => 1
  | structure => 3
  | function => 0  -- Graph: variable
  | expression => 1  -- Vector
  | epigenetic => 2  -- Tensor (position × modification)

/-- Human-readable names. -/
def name : Modality → String
  | sequence => "Sequence"
  | structure => "Structure"
  | function => "Function"
  | expression => "Expression"
  | epigenetic => "Epigenetic"

end Modality

/-- Generic modality data container (Q16.16). -/
structure ModalityData where
  modality : Modality
  data : List Q16_16  -- Flattened representation in Q16.16
  shape : List Nat   -- Original dimensions
  metadata : String  -- Additional info (e.g., gene ID)
  deriving Repr, Inhabited

-- ═══════════════════════════════════════════════════════════════════════════
-- §1  Modality-Specific Fields
-- ═══════════════════════════════════════════════════════════════════════════

/-- Parameters for sequence modality (from GenomicCompression) in Q16.16. -/
structure SequenceFieldParams where
  rhoAccuracy : Q16_16    -- Alignment accuracy
  vDynamics : Q16_16      -- Mutation/evolution rate
  sigmaDiversity : Q16_16 -- Nucleotide entropy
  deriving Repr, Inhabited

/-- Parameters for structure modality in Q16.16. -/
structure StructureFieldParams where
  rhoRMSD : Q16_16       -- Root-mean-square deviation
  tauTension : Q16_16    -- Structural strain
  kappaFold : Q16_16     -- Folding curvature
  deriving Repr, Inhabited

/-- Parameters for function modality (graph) in Q16.16. -/
structure FunctionFieldParams where
  rhoConnectivity : Q16_16  -- Network density
  qFlow : Q16_16          -- Information flow (PageRank-like)
  kappaTopology : Q16_16  -- Graph curvature
  deriving Repr, Inhabited

/-- Parameters for expression modality in Q16.16. -/
structure ExpressionFieldParams where
  rhoMean : Q16_16       -- Mean expression level
  sigmaVariance : Q16_16 -- Expression variance
  vTemporal : Q16_16     -- Temporal dynamics
  deriving Repr, Inhabited

/-- Unified modality field parameters with genomic compression support. -/
structure ModalityFieldParams where
  sequence : SequenceFieldParams
  structure : StructureFieldParams
  function : FunctionFieldParams
  expression : ExpressionFieldParams
  -- Genomic field parameters for genetic compression (from GenomicCompression.lean)
  rhoSeq : Q16_16      -- ρ_seq²: sequence alignment accuracy
  vEpigenetic : Q16_16 -- v_epigenetic²: methylation dynamics
  tauStructure : Q16_16 -- τ_structure²: 3D folding tension
  sigmaEntropy : Q16_16 -- σ_entropy²: nucleotide diversity
  qConservation : Q16_16 -- q_conservation²: evolutionary constraint
  kappaHierarchy : Q16_16 -- κ_hierarchy²: chromatin levels
  epsilonMutation : Q16_16 -- ε_mutation: mutation rate
  deriving Repr, Inhabited

namespace ModalityFieldParams

/-- Default parameters for sequence-structure fusion (Q16.16). -/
def sequenceStructureFusion : ModalityFieldParams :=
  { sequence := { rhoAccuracy := one, vDynamics := ofNat 20, sigmaDiversity := ofNat 30 }
    structure := { rhoRMSD := ofNat 50, tauTension := ofNat 40, kappaFold := ofNat 30 }
    function := { rhoConnectivity := zero, qFlow := zero, kappaTopology := zero }
    expression := { rhoMean := zero, sigmaVariance := zero, vTemporal := zero }
    -- Genomic parameters for DNA/protein fusion
    rhoSeq := ofNat 80
    vEpigenetic := ofNat 30
    tauStructure := ofNat 50
    sigmaEntropy := ofNat 20
    qConservation := ofNat 25
    kappaHierarchy := ofNat 30
    epsilonMutation := ofNat 10 }

/-- Default parameters for multi-omics (sequence + expression + epigenetic) in Q16.16. -/
def multiOmicsFusion : ModalityFieldParams :=
  { sequence := { rhoAccuracy := ofNat 80, vDynamics := ofNat 30, sigmaDiversity := ofNat 20 }
    structure := { rhoRMSD := zero, tauTension := zero, kappaFold := zero }
    function := { rhoConnectivity := zero, qFlow := zero, kappaTopology := zero }
    expression := { rhoMean := ofNat 90, sigmaVariance := ofNat 50, vTemporal := ofNat 40 }
    -- Genomic parameters for multi-omics
    rhoSeq := ofNat 90
    vEpigenetic := ofNat 50
    tauStructure := ofNat 10
    sigmaEntropy := ofNat 30
    qConservation := ofNat 20
    kappaHierarchy := ofNat 25
    epsilonMutation := ofNat 15 }

end ModalityFieldParams

-- ═══════════════════════════════════════════════════════════════════════════
-- §2  Cross-Modal Alignment Field
-- ═══════════════════════════════════════════════════════════════════════════

/-- Alignment field parameters between two modalities (Q16.16). -/
structure AlignmentParams where
  kappa : Q16_16     -- Curvature of shared latent space
  delta : Q16_16     -- Modality gap (intrinsic difference)
  weight : Q16_16    -- Importance of this alignment
  
  wf_kappa_nonneg : kappa ≥ zero
  wf_delta_pos : delta ≥ zero
  wf_weight_pos : weight > zero
  deriving Repr

/-- Compute geometry-aware distance in curved space (Q16.16).
    Simplified: Euclidean distance with curvature correction. -/
def curvedDistance (x y : List Q16_16) (kappa : Q16_16) : Q16_16 :=
  -- Flatten to same length
  let n := min x.length y.length
  let xTrunc := x.take n
  let yTrunc := y.take n
  
  -- Euclidean distance
  let euclidean := (xTrunc.zip yTrunc).foldl (fun acc (xi, yi) => 
    acc + (xi - yi) * (xi - yi)
  ) zero
  
  -- Curvature correction: sin(√κ · d) / √κ ≈ d - κ·d³/6
  if kappa > ofNat 1 then
    let sqrtK := sqrt kappa
    let kd := sqrtK * sqrt euclidean
    div (sin kd) sqrtK
  else
    sqrt euclidean

/-- Alignment field between two modalities (Q16.16).
    Φ_align = -||projᵢ(xᵢ) - projⱼ(xⱼ)||²_κ / (1 + δ²) -/
def alignmentField (data1 data2 : ModalityData) (params : AlignmentParams) : Q16_16 :=
  let d := curvedDistance data1.data data2.data params.kappa
  let d2 := d * d
  let denominator := one + params.delta * params.delta
  neg (div (d2 * params.weight) denominator)

-- ═══════════════════════════════════════════════════════════════════════════
-- §3  Unified Cross-Modal Field
-- ═══════════════════════════════════════════════════════════════════════════

/-- Compute modality-specific field value (Q16.16). -/
def modalityField (data : ModalityData) (params : ModalityFieldParams) : Q16_16 :=
  match data.modality with
  | Modality.sequence =>
      let p := params.sequence
      p.rhoAccuracy + p.vDynamics + p.sigmaDiversity
  | Modality.structure =>
      let p := params.structure
      p.rhoRMSD + p.tauTension + p.kappaFold
  | Modality.function =>
      let p := params.function
      p.rhoConnectivity + p.qFlow + p.kappaTopology
  | Modality.expression =>
      let p := params.expression
      p.rhoMean + p.sigmaVariance + p.vTemporal
  | Modality.epigenetic =>
      -- Epigenetic uses expression params as approximation
      let p := params.expression
      p.rhoMean + p.sigmaVariance

/-- Cross-modal field: sum of individual fields + alignment terms (Q16.16). -/
def crossModalField 
    (modalities : List ModalityData)
    (modalityParams : ModalityFieldParams)
    (alignmentParams : List (Nat × Nat × AlignmentParams))  -- (i, j, params)
    : Q16_16 :=
  -- Sum of individual modality fields
  let individualSum := modalities.foldl (fun acc m => 
    acc + modalityField m modalityParams
  ) zero
  
  -- Sum of alignment fields
  let alignmentSum := alignmentParams.foldl (fun acc (i, j, params) =>
    if i < modalities.length && j < modalities.length then
      let mi := modalities.get! i
      let mj := modalities.get! j
      acc + alignmentField mi mj params
    else
      acc
  ) zero
  
  individualSum + alignmentSum

/-- Cross-modal compression loss: L = -Φ in Q16.16. -/
def crossModalLoss 
    (modalities : List ModalityData)
    (modalityParams : ModalityFieldParams)
    (alignmentParams : List (Nat × Nat × AlignmentParams)) : Q16_16 :=
  neg (crossModalField modalities modalityParams alignmentParams)

-- ═══════════════════════════════════════════════════════════════════════════
-- §4  Compression with Cross-Modal Fusion
-- ═══════════════════════════════════════════════════════════════════════════

/-- Compress multi-modal data using fused field with genetic compression (Q16.16). -/
def compressMultiModal
    (modalities : List ModalityData)
    (modalityParams : ModalityFieldParams)
    (alignmentParams : List (Nat × Nat × AlignmentParams)) : Q16_16 × Q16_16 :=
  let totalSize := ofNat (modalities.foldl (fun acc m => 
    acc + m.data.length
  ) 0)
  
  let fieldValue := crossModalField modalities modalityParams alignmentParams
  
  -- Genomic field strength for genetic compression
  let genomicNumerator := modalityParams.rhoSeq + modalityParams.vEpigenetic + 
                      modalityParams.tauStructure + modalityParams.sigmaEntropy + 
                      modalityParams.qConservation
  let kappaSq := modalityParams.kappaHierarchy * modalityParams.kappaHierarchy
  let geomTerm := one + kappaSq
  let mutTerm := one + modalityParams.epsilonMutation
  let genomicDenom := mul geomTerm mutTerm
  let genomicWeight := div genomicNumerator genomicDenom
  
  -- Combined coherence: cross-modal alignment + genomic field
  let coherence := expNeg (neg fieldValue)
  let genomicBoost := one + genomicWeight
  let combinedCoherence := mul coherence genomicBoost
  
  -- Compression ratio with genetic compression enabled
  let compressedSize := div totalSize (one + combinedCoherence)
  let ratio := div totalSize compressedSize
  
  (compressedSize, ratio)

-- ═══════════════════════════════════════════════════════════════════════════
-- §5  Theorems: Fusion Benefits
-- ═══════════════════════════════════════════════════════════════════════════

/-- Theorem: Cross-modal compression includes all modality-specific fields (Q16.16).
    Individual compression is a special case (no alignment terms). -/
theorem crossModalGeneralizesSingleModal
    (modalities : List ModalityData)
    (modalityParams : ModalityFieldParams)
    (hSingle : modalities.length = 1) :
    let noAlignment : List (Nat × Nat × AlignmentParams) := []
    crossModalField modalities modalityParams noAlignment =
    modalities.foldl (fun acc m => acc + modalityField m modalityParams) zero := by
  -- No alignment terms for single modality
  unfold crossModalField
  simp [noAlignment]
  -- alignmentSum is 0 for empty list
  have hAlignZero := List.foldl (fun acc (i, j, params) => 
    if i < modalities.length && j < modalities.length then
      let mi := modalities.get! i
      let mj := modalities.get! j
      acc + alignmentField mi mj params
    else
      acc
  ) zero [] = zero
  exact hAlignZero

/-- Theorem: Alignment improves compression when modalities are coherent (Q16.16).
    If modalities are related (small δ), alignment field is less negative. -/
theorem alignmentHelpsWhenCoherent
    (d1 d2 : ModalityData)
    (p1 p2 : AlignmentParams)
    (hCoherent : p1.delta < p2.delta)
    (hSameKappa : p1.kappa = p2.kappa)
    (hSameWeight : p1.weight = p2.weight)
    (hSameData : d1 = d2) :
    alignmentField d1 d2 p1 > alignmentField d1 d2 p2 := by
  -- Unfold alignmentField definition
  unfold alignmentField
  -- Since data and kappa are same, curvedDistance is equal
  have hDistEq : curvedDistance d1.data d2.data p1.kappa = curvedDistance d1.data d2.data p2.kappa := by
    rw [hSameKappa]
  
  -- Let d = curvedDistance, w = weight
  let d := curvedDistance d1.data d2.data p1.kappa
  let w := p1.weight
  
  -- Compare: -d²/(1+δ₁²) * w > -d²/(1+δ₂²) * w
  -- Since w > 0 and d² ≥ 0, we can divide both sides
  have hWPos : w > zero := by exact p1.wf_weight_pos
  have hD2Nonneg : d * d ≥ zero := by exact mul_self_nonneg d
  
  -- Multiply both sides by -1 (flips inequality)
  suffices hDenomLt : one + p2.delta * p2.delta < one + p1.delta * p1.delta from
    have hFinal := neg (div (d * d * w) (one + p1.delta * p1.delta)) > neg (div (d * d * w) (one + p2.delta * p2.delta)) := by
      have hNumNonneg := neg (d * d * w) ≤ zero := by
        exact mul_nonpos (neg_nonneg hD2Nonneg) (by simp [zero, le])
      exact (div_lt_div_iff hWPos hDenomLt).mp (by rfl)
    exact hFinal
  
  -- Since δ₁ < δ₂ and both ≥ 0, δ₁² < δ₂²
  have hDeltaSqLt : p1.delta * p1.delta < p2.delta * p2.delta := by
    apply mul_lt_mul_of_pos_left hCoherent p1.wf_delta_pos
  
  -- Add 1 to both sides preserves inequality
  exact add_lt_add_left hDeltaSqLt one

-- TODO(lean-port): Add crossModalRatioAtLeastOne theorem after proving exp positivity
-- Theorem: Cross-modal compression ratio ≥ 1.0 (no expansion)

-- ═══════════════════════════════════════════════════════════════════════════
-- §6  Integration with OTOM
-- ═══════════════════════════════════════════════════════════════════════════

/-! ## Connections to Other Modules

### GenomicCompression.lean
- Sequence modality parameters exported from GenomicCompression
- Alignment field connects sequence ↔ structure (protein folding)

### ResearchAgent.lean
- Cross-modal fusion guides multi-source literature synthesis
- Alignment field models: paper A + paper B → unified insight

### SSMS.lean (State Machine)
- Multi-modal data as MLGRU state vectors
- Alignment as phantom coupling between modalities

### BettiSwoosh.lean (Topology)
- κ² alignment curvature relates to simplicial complex geometry
- Cross-modal graph as filtered simplicial complex

## Biological Applications

1. **Structure Prediction**: Sequence → 3D structure (AlphaFold-style)
   - Φ_seq(x) + Φ_struct(y) + Φ_align(seq, struct)

2. **Multi-Omics**: DNA + RNA + Protein + Methylation
   - 4-modality fusion with 6 alignment terms

3. **Pathway Analysis**: Function + Expression
   - Graph + vector alignment for active pathway detection
-/ 

-- ═══════════════════════════════════════════════════════════════════════════
-- §7  Verification Examples
-- ═══════════════════════════════════════════════════════════════════════════

#eval let seqData := { modality := Modality.sequence, data := [one, zero, one, zero], 
                       shape := [4], metadata := "ATCG" : ModalityData }
      let structData := { modality := Modality.structure, data := [zero, one, zero, one],
                          shape := [4], metadata := "folded" : ModalityData }
      let params := ModalityFieldParams.sequenceStructureFusion
      let align := [(0, 1, { kappa := ofNat 10, delta := ofNat 50, weight := one, 
                             wf_kappa_nonneg := by simp [zero, le_refl], 
                             wf_delta_pos := by simp [zero, le_refl],
                             wf_weight_pos := by simp [zero, lt] } : AlignmentParams)]
      crossModalField [seqData, structData] params align
-- Expected: Individual sums + alignment (negative if dissimilar) in Q16.16

#eval compressMultiModal 
  [ { modality := Modality.sequence, data := [one, ofNat 20, ofNat 30], shape := [3], metadata := "test" }
  , { modality := Modality.expression, data := [one, ofNat 20, ofNat 30], shape := [3], metadata := "test" } ]
  ModalityFieldParams.multiOmicsFusion
  [(0, 1, { kappa := ofNat 10, delta := ofNat 10, weight := one,
            wf_kappa_nonneg := by simp [zero, le_refl],
            wf_delta_pos := by simp [zero, le_refl], 
            wf_weight_pos := by simp [zero, lt] })]
-- Expected: High compression ratio (similar data) in Q16.16

-- ═══════════════════════════════════════════════════════════════════════════
-- §8  Future Work
-- ═══════════════════════════════════════════════════════════════════════════

/-! ## Research Directions

### Immediate (This Week)
- [ ] Connect to GenomicCompression for sequence parameters
- [ ] Implement Python shim for modality data loading
- [ ] Test on AlphaFold structures + sequences

### Short-term (Next 2 Weeks)
- [ ] Multi-omics fusion: ENCODE + GTEx data
- [ ] Prove crossModalAtLeastBestSingle theorem
- [ ] Benchmark vs single-modal baselines

### Medium-term (Next Month)
- [ ] Full 5-modality fusion (sequence + structure + function + expression + epigenetic)
- [ ] Application: Cancer subtype classification
- [ ] Paper: "Unified Field Theory for Multi-Omics Integration"

## References

- MIRROR (2503.00374): Multi-modal pathological learning
- AlphaFold: Structure prediction from sequence
- ENCODE: Encyclopedia of DNA Elements (multi-modal data)
-/ 

-- TODO(lean-port):
-- 1. Complete alignmentHelpsWhenCoherent proof
-- 2. Complete crossModalAtLeastBestSingle proof
-- 3. Add projection functions (proj_i: modality → shared latent)
-- 4. Connect to BettiSwoosh for topological alignment
-- 5. Implement Python data loaders (h5ad, FASTA, PDB)

end Semantics.CrossModalCompression