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

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.

NOTE(lean-port): Connect to FPGA Warden Node AMMR accumulator
NOTE(lean-port): Integrate PhiRedundancy 3-stream scheme as erasure coding
NOTE(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