/- 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 StreamCompression.lean — DSP-Aware Stream Compression with Spectral Analysis This module formalizes DSP-aware compression for streaming data: - Sample-block processing (DSP standard) - Spectral redundancy detection (FFT-based) - Q16_16 fixed-point for hardware extraction - FPGA DSP slice integration (TSM opcodes 0x14, 0x42) - Energy-aware decompression scheduling Key insight: DSP compression treats data as continuous signals, not discrete bytes. Spectral analysis identifies redundancy in frequency domain, not just spatial. 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): Connect to FPGA Warden Node AMMR accumulator TODO(lean-port): Integrate PhiRedundancy 3-stream scheme as erasure coding TODO(lean-port): Integrate swarm design review -/ import Mathlib.Data.Nat.Basic import Mathlib.Data.Real.Basic import Mathlib.Tactic import Semantics.FixedPoint import Semantics.SwarmDesignReview import Semantics.Timing namespace Semantics.StreamCompression open Semantics.Q16_16 open Semantics.SwarmDesignReview open Semantics.Timing -- ═══════════════════════════════════════════════════════════════════════════ -- §0 DSP Sample Types -- ═══════════════════════════════════════════════════════════════════════════ /-- DSP sample in Q16.16 fixed-point format. -/ abbrev DspSample := Q16_16 /-- Sample block (DSP standard processing unit). -/ structure SampleBlock where samples : Array DspSample sampleRate : Nat -- Hz deriving Repr, Inhabited /-- Frequency domain representation (FFT output). -/ structure FrequencyDomain where real : Array DspSample -- Real components imag : Array DspSample -- Imaginary components deriving Repr, Inhabited /-- Spectral band for compression decisions. -/ structure SpectralBand where lowFreq : Nat -- Lower bound (Hz) highFreq : Nat -- Upper bound (Hz) energy : Q16_16 -- Band energy (Q16.16) redundancy : Q16_16 -- Redundancy factor (0 = unique, 1 = fully redundant) deriving Repr, Inhabited -- ═══════════════════════════════════════════════════════════════════════════ -- §1 DSP Primitives (Fixed-Point) -- ═══════════════════════════════════════════════════════════════════════════ /-- Compute energy of sample block: E = Σ x². -/ def computeEnergy (block : SampleBlock) : Q16_16 := block.samples.foldl (fun acc s => acc + s * s) zero /-- Compute mean of sample block. -/ def computeMean (block : SampleBlock) : Q16_16 := if block.samples.isEmpty then zero else let sum := block.samples.foldl (fun acc s => acc + s) zero div sum (ofNat block.samples.size) /-- Compute variance: Var = E[x²] - E[x]². -/ def computeVariance (block : SampleBlock) : Q16_16 := let mean := computeMean block let meanSq := mean * mean let energy := computeEnergy block let energyPerSample := div energy (ofNat block.samples.size) sub energyPerSample meanSq /-- Simple FIR filter: y[n] = Σ h[k] * x[n-k]. Stub: returns block unchanged. -/ def firFilter (coeffs : Array Q16_16) (block : SampleBlock) : SampleBlock := block -- ═══════════════════════════════════════════════════════════════════════════ -- §2 Spectral Analysis (FFT Approximation) -- ═══════════════════════════════════════════════════════════════════════════ /-- Power-of-2 check for FFT. -/ def isPowerOfTwo (n : Nat) : Bool := n > 0 && (n &&& (n - 1)) = 0 /-- Next power of 2 ≥ n (iterative, no recursion). -/ def nextPowerOfTwo (n : Nat) : Nat := if n = 0 then 1 else let n1 := n - 1 let n2 := n1 ||| (n1 >>> 1) let n3 := n2 ||| (n2 >>> 2) let n4 := n3 ||| (n3 >>> 4) let n5 := n4 ||| (n4 >>> 8) let n6 := n5 ||| (n5 >>> 16) n6 + 1 /-- Cooley-Tukey FFT (simplified, fixed-point). Note: Full FFT implementation requires complex arithmetic. This is a placeholder for spectral energy estimation. -/ def computeFFT (block : SampleBlock) : FrequencyDomain := let n := block.samples.size if ¬isPowerOfTwo n then -- Zero-pad to next power of 2 let paddedSize := nextPowerOfTwo n let padded := Array.mk (List.replicate paddedSize zero) let filled := padded.foldl (fun arr _ => arr.push zero) block.samples { real := filled, imag := Array.mk (List.replicate paddedSize zero) } else { real := block.samples, imag := Array.mk (List.replicate n zero) } /-- Compute spectral energy in frequency band. -/ def bandEnergy (freq : FrequencyDomain) (low high : Nat) (sampleRate : Nat) : Q16_16 := let n := freq.real.size if n = 0 then zero else let binWidth := sampleRate / n let lowBin := low / binWidth let highBin := high / binWidth let indices := List.range n |>.filter (fun i => i ≥ lowBin && i ≤ highBin) indices.foldl (fun acc i => if i < freq.real.size && i < freq.imag.size then let magSq := freq.real[i]! * freq.real[i]! + freq.imag[i]! * freq.imag[i]! acc + magSq else acc) zero /-- Detect spectral redundancy: compare band energy to total. -/ def spectralRedundancy (bandEnergy totalEnergy : Q16_16) : Q16_16 := if totalEnergy = zero then zero else div bandEnergy totalEnergy -- ═══════════════════════════════════════════════════════════════════════════ -- §3 DSP-Aware Compression -- ═══════════════════════════════════════════════════════════════════════════ /-- Compression decision based on spectral analysis. -/ inductive CompressionMode where | lossless -- No compression (unique spectral content) | lossy8 -- 8-bit quantization | lossy4 -- 4-bit quantization | spectral -- Spectral redundancy coding | genetic -- Genetic compression (DNA/Protein field-guided encoding) deriving Repr, DecidableEq /-- DSP compression parameters with curvature coupling from self-compression and genomic field. -/ structure DspCompressionParams where sampleRate : Nat blocksize : Nat quantizationBits : Nat -- 8, 4, or 1 spectralThreshold : Q16_16 -- Redundancy threshold kappaSquared : Q16_16 -- Curvature coupling κ² from self-compression (arXiv:2301.13142) -- Genomic field parameters for genetic compression 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 /-- Analyze spectral bands and select compression mode with curvature and genomic awareness. κ² modulates the spectral threshold: higher curvature = more aggressive compression. Genomic field parameters enable genetic compression when data shows sequence-like structure. This connects self-compression geometric structure and genomic field to DSP spectral decisions. -/ def selectCompressionMode (block : SampleBlock) (params : DspCompressionParams) : CompressionMode := let freq := computeFFT block let totalEnergy := bandEnergy freq 0 (params.sampleRate / 2) params.sampleRate let lowBandEnergy := bandEnergy freq 0 (params.sampleRate / 4) params.sampleRate let redundancy := spectralRedundancy lowBandEnergy totalEnergy -- Curvature-aware threshold: κ² increases effective threshold (more compression) let curvatureFactor := Q16_16.one + params.kappaSquared let adjustedThreshold := mul params.spectralThreshold curvatureFactor -- Genomic field strength: sum of genomic parameters let genomicStrength := params.rhoSeq + params.vEpigenetic + params.tauStructure + params.sigmaEntropy + params.qConservation let hierarchyFactor := Q16_16.one + params.kappaHierarchy let genomicWeight := mul genomicStrength hierarchyFactor -- Genetic compression enabled when genomic field is strong enough if genomicWeight > (ofNat 100) then CompressionMode.genetic else if redundancy > adjustedThreshold then CompressionMode.spectral else if totalEnergy < (ofNat 100) then -- Low energy = simple signal CompressionMode.lossy4 else CompressionMode.lossless /-- Compress sample block using selected mode with curvature and genomic-aware compression ratio. κ² increases compression ratio for curved manifolds (quantization structure). Genomic field parameters enable DNA/Protein-like compression with hierarchy-aware encoding. -/ def compressBlock (block : SampleBlock) (params : DspCompressionParams) (mode : CompressionMode) : Nat × Q16_16 := let originalSize := block.samples.size * 2 -- 2 bytes per Q16_16 -- Curvature increases compression ratio for spectral mode let curvatureBoost := Q16_16.one + params.kappaSquared -- Genomic field denominator for genetic compression let kappaSq := params.kappaHierarchy * params.kappaHierarchy let geomTerm := Q16_16.one + kappaSq let mutTerm := Q16_16.one + params.epsilonMutation let genomicDenom := mul geomTerm mutTerm let genomicNumerator := params.rhoSeq + params.vEpigenetic + params.tauStructure + params.sigmaEntropy + params.qConservation let genomicWeight := div genomicNumerator genomicDenom match mode with | CompressionMode.lossless => (originalSize, Q16_16.one) | CompressionMode.lossy8 => (originalSize / 2, ofNat 2) | CompressionMode.lossy4 => (originalSize / 4, ofNat 4) | CompressionMode.spectral => let baseRatio := ofNat 8 let adjustedRatio := mul baseRatio curvatureBoost (originalSize / 8, adjustedRatio) | CompressionMode.genetic => let baseRatio := ofNat 12 -- Higher base ratio for genetic compression let genomicBoost := Q16_16.one + genomicWeight let adjustedRatio := mul baseRatio genomicBoost (originalSize / 12, adjustedRatio) -- ═══════════════════════════════════════════════════════════════════════════ -- §4 FPGA DSP Slice Integration -- ═══════════════════════════════════════════════════════════════════════════ /-- FPGA DSP opcode mapping (from substrate_isa_spec.md). -/ inductive FpgaDspOpcode where | resonate -- 0x14: TSM_RESONATE / PHONON_LOCK (Phi=1.618) | mergeModes -- 0x42: TSM_MERGE_MODES | ingestVib -- 0x47: TSM_INGEST_VIBRATION deriving Repr, DecidableEq /-- DSP slice configuration for compression. -/ structure DspSliceConfig where opcode : FpgaDspOpcode phi : Q16_16 -- Resonance parameter (1.618 for golden ratio) clockCycles : Nat -- Estimated cycles deriving Repr /-- Map compression mode to FPGA DSP opcode. -/ def modeToOpcode (mode : CompressionMode) : FpgaDspOpcode := match mode with | CompressionMode.spectral => FpgaDspOpcode.resonate | CompressionMode.genetic => FpgaDspOpcode.resonate -- Genetic compression uses resonance (phi-encoding) | CompressionMode.lossless => FpgaDspOpcode.mergeModes | _ => FpgaDspOpcode.ingestVib /-- Estimate DSP slice energy cost (cycles * power). -/ def dspEnergyCost (config : DspSliceConfig) : Q16_16 := let baseCost := ofNat config.clockCycles let phiWeight := config.phi mul baseCost phiWeight -- ═══════════════════════════════════════════════════════════════════════════ -- §5 Energy-Aware Decompression Scheduling -- ═══════════════════════════════════════════════════════════════════════════ /-- Decompression task with energy cost and self-compression curvature. -/ structure DecompressionTask where blockId : Nat mode : CompressionMode energyCost : Q16_16 -- DSP energy cost kappaSquared : Q16_16 -- Curvature coupling from self-compression (arXiv:2301.13142) priority : Nat -- Lower = higher priority deriving Repr /-- Combined cost: DSP energy + curvature penalty from self-compression. Higher κ² = higher structural complexity = higher scheduling priority. -/ def combinedCost (task : DecompressionTask) : Q16_16 := let curvaturePenalty := mul task.kappaSquared (ofNat 1000) -- Scale κ² to energy units task.energyCost + curvaturePenalty /-- Energy-aware scheduler: high-cost (energy + curvature) blocks first. This integrates self-compression geometric structure into scheduling decisions. -/ def scheduleDecompression (tasks : Array DecompressionTask) : Array DecompressionTask := tasks.qsort (fun t1 t2 => combinedCost t1 > combinedCost t2) -- ═══════════════════════════════════════════════════════════════════════════ -- §6 Theorems -- ═══════════════════════════════════════════════════════════════════════════ /-- Claim boundary: computed sample energy should be non-negative. -/ def energyNonneg (block : SampleBlock) : Prop := computeEnergy block ≥ zero /-- Claim boundary: computed sample variance should be non-negative. -/ def varianceNonneg (block : SampleBlock) : Prop := computeVariance block ≥ zero /-- Claim boundary: spectral redundancy should stay in [0, 1]. -/ def redundancyBounded (band total : Q16_16) (_hPos : total > zero) : Prop := zero ≤ spectralRedundancy band total ∧ spectralRedundancy band total ≤ Q16_16.one /-- Claim boundary: compression ratio should be ≥ 1 (no expansion). -/ def compressionRatioAtLeastOne (block : SampleBlock) (params : DspCompressionParams) (mode : CompressionMode) : Prop := Q16_16.one ≤ (compressBlock block params mode).2 /-- Claim boundary: combined cost should be non-negative (energy + curvature penalty). -/ def combinedCostNonneg (task : DecompressionTask) : Prop := combinedCost task ≥ zero /-- Claim boundary: higher κ² should increase scheduling priority (combined cost). -/ def curvatureIncreasesPriority (task : DecompressionTask) (kappa1 kappa2 : Q16_16) (_h : kappa1 > kappa2) : Prop := combinedCost { task with kappaSquared := kappa1 } > combinedCost { task with kappaSquared := kappa2 } -- ═══════════════════════════════════════════════════════════════════════════ -- §8 Swarm Design Review Integration -- ═══════════════════════════════════════════════════════════════════════════ /-- Extract geometric parameters from DSP compression params for swarm review. -/ def extractGeometricParams (params : DspCompressionParams) : GeometricParameters := { kappaSquared := params.kappaSquared, rhoSeq := params.rhoSeq, vEpigenetic := params.vEpigenetic, tauStructure := params.tauStructure, sigmaEntropy := params.sigmaEntropy, qConservation := params.qConservation, kappaHierarchy := params.kappaHierarchy, epsilonMutation := params.epsilonMutation } /-- Run swarm design review on compression parameters. Returns swarm analysis with consensus and recommendations for improvement. -/ def runSwarmDesignReview (params : DspCompressionParams) : SwarmState := let geomParams := extractGeometricParams params let swarm := initializeSwarm runSwarmAnalysis swarm geomParams /-- Apply swarm recommendations to improve compression parameters. This function interprets swarm consensus and adjusts parameters accordingly. -/ def applySwarmRecommendations (params : DspCompressionParams) (swarm : SwarmState) : DspCompressionParams := -- If consensus is low (< 0.5), increase geometric parameters if swarm.consensus < (ofNat 32768) then -- 0.5 in Q16.16 -- Boost all geometric parameters to improve utilization { params with kappaSquared := mul params.kappaSquared (ofNat 150) -- 1.5x boost kappaHierarchy := mul params.kappaHierarchy (ofNat 150) epsilonMutation := mul params.epsilonMutation (ofNat 150) rhoSeq := mul params.rhoSeq (ofNat 120) vEpigenetic := mul params.vEpigenetic (ofNat 120) tauStructure := mul params.tauStructure (ofNat 120) sigmaEntropy := mul params.sigmaEntropy (ofNat 120) qConservation := mul params.qConservation (ofNat 120) } else -- Parameters are well-tuned, keep as-is params -- ═══════════════════════════════════════════════════════════════════════════ -- §9 FAMM Integration (Frustration-Aware Manifold Memory) -- ═══════════════════════════════════════════════════════════════════════════ /-- FAMM-aware compression parameters with frustration-based timing. -/ structure FammCompressionParams where baseParams : DspCompressionParams -- FAMM timing parameters torsionalStress : Q16_16 -- Σ²: torsional stress from manifold state interlockingEnergy : Q16_16 -- I_lock: interlocking energy laplacianEnergy : Q16_16 -- Δϕ: Hodge-Laplacian vibration energy deriving Repr /-- Derive FAMM timing from compression geometric parameters. Maps κ² to torsional stress and κ_hierarchy² to interlocking energy. -/ def deriveFammTiming (params : DspCompressionParams) : FammCompressionParams := -- Map curvature κ² to torsional stress (higher curvature = more stress) let torsionalStress := params.kappaSquared -- Map hierarchy κ² to interlocking energy (hierarchy depth = lock strength) let kappaSq := params.kappaHierarchy * params.kappaHierarchy let interlockingEnergy := div kappaSq (Q16_16.one + kappaSq) -- Map spectral redundancy to laplacian energy (redundancy = neighbor vibration) let laplacianEnergy := params.spectralThreshold { baseParams := params, torsionalStress := torsionalStress, interlockingEnergy := interlockingEnergy, laplacianEnergy := laplacianEnergy } /-- Compress with FAMM-aware timing adjustments. Uses FAMM timing to modulate compression mode selection and ratios. -/ def compressBlockFamm (block : SampleBlock) (fammParams : FammCompressionParams) : Nat × Q16_16 × ManifoldTiming := let timing := deriveTiming { phi := zero, x_pos := { x := fammParams.interlockingEnergy, y := zero }, x0_pos := { x := zero, y := zero }, g := { xx := zero, xy := zero, yy := zero }, t := { t1_12 := fammParams.torsionalStress, t2_12 := zero }, a := { xx := Q16_16.one, xy := zero, yy := zero } } fammParams.laplacianEnergy let params := fammParams.baseParams let originalSize := block.samples.size * 2 -- FAMM-aware compression: adjust parameters based on timing let adjustedParams := -- If tTCL is high (high stress), use more aggressive compression if timing.tcl > (ofNat 22) then -- Above baseline { params with kappaSquared := mul params.kappaSquared (ofNat 120), -- Boost κ² spectralThreshold := mul params.spectralThreshold (ofNat 110) -- Tighten threshold } -- If tMRE is high (slipping manifold), refresh more aggressively else if timing.mre > (ofNat 131072) then -- Above baseline { params with kappaHierarchy := mul params.kappaHierarchy (ofNat 120) -- Boost hierarchy } else params let mode := selectCompressionMode block adjustedParams let (compressedSize, ratio) := compressBlock block adjustedParams mode (compressedSize, ratio, timing) -- ═══════════════════════════════════════════════════════════════════════════ -- §7 Verification Examples -- ═══════════════════════════════════════════════════════════════════════════ def exampleBlock : SampleBlock := { samples := #[Q16_16.one, two, Q16_16.one, two, Q16_16.one, two, Q16_16.one, two], sampleRate := 48000 } def exampleParams : DspCompressionParams := { sampleRate := 48000, blocksize := 8, quantizationBits := 8, spectralThreshold := ofNat 50, -- 50% redundancy threshold kappaSquared := zero, rhoSeq := zero, vEpigenetic := zero, tauStructure := zero, sigmaEntropy := zero, qConservation := zero, kappaHierarchy := zero, epsilonMutation := zero } #eval computeEnergy exampleBlock -- Expected: 8 * (1² + 2²) = 8 * 5 = 40 (in Q16.16) #eval! computeMean exampleBlock -- Expected: (1+2+1+2+1+2+1+2)/8 = 12/8 = 1.5 #eval! computeVariance exampleBlock -- Expected: E[x²] - E[x]² = 5 - 2.25 = 2.75 #eval! selectCompressionMode exampleBlock exampleParams -- Expected: spectral (high redundancy in repeating pattern) #eval! compressBlock exampleBlock exampleParams CompressionMode.lossy4 -- Expected: (16/4 = 4 bytes, 4.0 ratio) #eval modeToOpcode CompressionMode.spectral -- Expected: resonate #eval dspEnergyCost { opcode := FpgaDspOpcode.resonate, phi := ofNat 16180, clockCycles := 100 } -- Expected: 100 * 1.618 ≈ 161.8 end Semantics.StreamCompression