/- 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. 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. TODO(lean-port): Extract alignment formalism from MIRROR paper TODO(lean-port): Prove modality fusion improves compression TODO(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