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320 lines
14 KiB
Text
320 lines
14 KiB
Text
/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Research Stack Team
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DspErasureCoding.lean — DSP-Aware 3-Stream Erasure Coding
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This module formalizes DSP-aware erasure coding for streaming data:
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- 3-stream redundancy scheme (inspired by PhiRedundancy)
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- Sample-block based erasure recovery
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- Q16_16 fixed-point for hardware extraction
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- FPGA DSP slice integration
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- Spectral-aware erasure detection
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Key insight:
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DSP erasure coding treats streams as continuous signals, not discrete bytes.
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Spectral analysis identifies erasures in frequency domain, not just bit errors.
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Per AGENTS.md §1.4: Q16_16 fixed-point for hardware extraction.
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Per AGENTS.md §2: PascalCase types, camelCase functions.
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Per AGENTS.md §4: Every def has eval witness or theorem.
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TODO(lean-port): Connect to FPGA Warden Node AMMR accumulator
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TODO(lean-port): Integrate with StreamCompression spectral analysis
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-/
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import Mathlib.Data.Nat.Basic
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import Mathlib.Data.Fin.Basic
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import Mathlib.Tactic
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import Semantics.FixedPoint
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namespace Semantics.DspErasureCoding
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open Semantics.Q16_16
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §0 DSP Stream Types
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- DSP sample in Q16.16 fixed-point format. -/
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abbrev DspSample := Q16_16
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/-- Sample block (DSP standard processing unit). -/
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structure SampleBlock where
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samples : Array DspSample
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blockId : Nat
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deriving Repr, Inhabited
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/-- Stream identifier for 3-stream redundancy. -/
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inductive StreamId where
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| primary -- Stream 0: original data
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| recovery1 -- Stream 1: first permutation
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| recovery2 -- Stream 2: second permutation
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deriving Repr, DecidableEq, BEq
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/-- Erasure marker for damaged samples. -/
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structure ErasureMarker where
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isErased : Bool
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confidence : Q16_16 -- Confidence that this is truly an erasure (Q16.16)
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deriving Repr, Inhabited
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 3-Stream Redundancy Scheme
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Redundancy scheme parameters with genomic compression support (Q16.16). -/
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structure RedundancyScheme where
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blockSize : Nat
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step1 : Nat -- Coprime to blockSize for permutation 1
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step2 : Nat -- Coprime to blockSize for permutation 2
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offset1 : Nat
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offset2 : Nat
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-- Genomic field parameters for genetic compression
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rhoSeq : Q16_16 -- ρ_seq²: sequence alignment accuracy
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vEpigenetic : Q16_16 -- v_epigenetic²: methylation dynamics
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tauStructure : Q16_16 -- τ_structure²: 3D folding tension
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sigmaEntropy : Q16_16 -- σ_entropy²: nucleotide diversity
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qConservation : Q16_16 -- q_conservation²: evolutionary constraint
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kappaHierarchy : Q16_16 -- κ_hierarchy²: chromatin levels
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epsilonMutation : Q16_16 -- ε_mutation: mutation rate
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deriving Repr
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/-- Check if step is coprime to n (gcd = 1). -/
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def isCoprime (n step : Nat) : Bool :=
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let rec gcd (a b : Nat) : Nat :=
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if b = 0 then a else gcd b (a % b)
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gcd n step = 1
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/-- Valid scheme requires coprime steps. -/
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def isValidScheme (sch : RedundancyScheme) : Bool :=
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isCoprime sch.blockSize sch.step1 ∧ isCoprime sch.blockSize sch.step2
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/-- Affine permutation: π(i) = (offset + step * i) mod n. -/
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def affinePerm (n step offset i : Nat) : Nat :=
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(offset + step * i) % n
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/-- Inverse lookup for affine permutation. -/
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def affinePermInv? (n step offset target : Nat) : Option Nat :=
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let rec go (j : Nat) : Option Nat :=
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if j < n then
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if affinePerm n step offset j = target then some j else go (j + 1)
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else
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none
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go 0
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Stream Construction
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Build primary stream (identity permutation). -/
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def buildPrimaryStream (sch : RedundancyScheme) (block : SampleBlock) : SampleBlock :=
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{ samples := block.samples, blockId := block.blockId }
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/-- Build recovery stream 1 (affine permutation). -/
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def buildRecoveryStream1 (sch : RedundancyScheme) (block : SampleBlock) : SampleBlock :=
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let permuted := (Array.range sch.blockSize).map (fun j =>
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block.samples[affinePerm sch.blockSize sch.step1 sch.offset1 j]!
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)
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{ samples := permuted, blockId := block.blockId }
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/-- Build recovery stream 2 (second affine permutation). -/
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def buildRecoveryStream2 (sch : RedundancyScheme) (block : SampleBlock) : SampleBlock :=
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let permuted := (Array.range sch.blockSize).map (fun j =>
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block.samples[affinePerm sch.blockSize sch.step2 sch.offset2 j]!
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)
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{ samples := permuted, blockId := block.blockId }
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/-- Complete 3-stream redundancy bundle. -/
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structure StreamBundle where
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primary : SampleBlock
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recovery1 : SampleBlock
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recovery2 : SampleBlock
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deriving Repr, Inhabited
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/-- Build complete stream bundle. -/
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def buildStreamBundle (sch : RedundancyScheme) (block : SampleBlock) : StreamBundle :=
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{
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primary := buildPrimaryStream sch block,
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recovery1 := buildRecoveryStream1 sch block,
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recovery2 := buildRecoveryStream2 sch block
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}
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Erasure Detection (Spectral Analysis)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Detect erasures using spectral anomaly detection with genomic compression (Q16.16). -/
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def detectErasureSpectral (block : SampleBlock) (threshold : Q16_16) (sch : RedundancyScheme) : Array ErasureMarker :=
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-- Compute energy of each sample
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let energies := block.samples.map (fun s => s * s)
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let meanEnergy := div (energies.foldl (fun acc e => acc + e) zero) (ofNat energies.length)
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-- Genomic field strength for adaptive threshold
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let genomicNumerator := sch.rhoSeq + sch.vEpigenetic + sch.tauStructure +
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sch.sigmaEntropy + sch.qConservation
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let kappaSq := sch.kappaHierarchy * sch.kappaHierarchy
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let geomTerm := one + kappaSq
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let mutTerm := one + sch.epsilonMutation
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let genomicDenom := mul geomTerm mutTerm
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let genomicWeight := div genomicNumerator genomicDenom
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let adaptiveThreshold := mul threshold (one + genomicWeight)
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-- Mark samples with energy deviation > adaptive threshold as potential erasures
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block.samples.mapIdx (fun i s =>
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let deviation := abs (s * s - meanEnergy)
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{
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isErased := deviation > adaptiveThreshold,
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confidence := if deviation > adaptiveThreshold then div deviation adaptiveThreshold else zero
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}
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)
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/-- Detect erasures using simple threshold with genomic compression (Q16.16). -/
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def detectErasureThreshold (block : SampleBlock) (threshold : Q16_16) (sch : RedundancyScheme) : Array ErasureMarker :=
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-- Genomic field strength for adaptive threshold
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let genomicNumerator := sch.rhoSeq + sch.vEpigenetic + sch.tauStructure +
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sch.sigmaEntropy + sch.qConservation
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let kappaSq := sch.kappaHierarchy * sch.kappaHierarchy
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let geomTerm := one + kappaSq
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let mutTerm := one + sch.epsilonMutation
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let genomicDenom := mul geomTerm mutTerm
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let genomicWeight := div genomicNumerator genomicDenom
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let adaptiveThreshold := mul threshold (one + genomicWeight)
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block.samples.map (fun s =>
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{
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isErased := abs s > adaptiveThreshold,
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confidence := if abs s > adaptiveThreshold then div (abs s) adaptiveThreshold else zero
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}
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)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 Erasure Recovery
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Sample with optional erasure marker. -/
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abbrev MarkedSample := Option DspSample
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/-- Fetch sample from stream with inverse permutation. -/
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def fetchSample
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(stream : SampleBlock)
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(inv? : Nat → Option Nat)
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(logicalIdx : Nat) : MarkedSample :=
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match inv? logicalIdx with
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| some j =>
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if j < stream.samples.size then
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some stream.samples[j]!
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else
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none
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| none => none
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/-- Vote-based recovery from up to 3 candidates (Q16.16). -/
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def recoverSample (c1 c2 c3 : MarkedSample) : DspSample :=
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let candidates := [c1, c2, c3].filterMap id
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if candidates.isEmpty then zero
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else
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let sum := candidates.foldl (fun acc s => acc + s) zero
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div sum (ofNat candidates.length)
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/-- Recover complete block from 3 streams with erasure markers. -/
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def recoverBlock
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(sch : RedundancyScheme)
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(primary recovery1 recovery2 : SampleBlock)
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(primaryMarkers recovery1Markers recovery2Markers : Array ErasureMarker)
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: SampleBlock :=
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let recovered := (Array.range sch.blockSize).map (fun i =>
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let c1 := if primaryMarkers[i]!.isErased then none else some primary.samples[i]!
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let c2 := if recovery1Markers[i]!.isErased then none
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else fetchSample recovery1 (affinePermInv? sch.blockSize sch.step1 sch.offset1) i
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let c3 := if recovery2Markers[i]!.isErased then none
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else fetchSample recovery2 (affinePermInv? sch.blockSize sch.step2 sch.offset2) i
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recoverSample c1 c2 c3
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)
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{ samples := recovered, blockId := primary.blockId }
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 FPGA DSP Integration
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- FPGA DSP opcode for erasure coding. -/
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inductive ErasureOpcode where
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| resonate -- 0x14: TSM_RESONATE / PHONON_LOCK
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| mergeModes -- 0x42: TSM_MERGE_MODES
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deriving Repr, DecidableEq
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/-- DSP erasure coding configuration. -/
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structure ErasureConfig where
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opcode : ErasureOpcode
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phi : Q16_16 -- Resonance parameter (1.618 for golden ratio)
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clockCycles : Nat
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deriving Repr
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/-- Map erasure mode to FPGA opcode. -/
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def modeToOpcode (mode : StreamId) : ErasureOpcode :=
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match mode with
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| StreamId.primary => ErasureOpcode.mergeModes
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| StreamId.recovery1 => ErasureOpcode.resonate
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| StreamId.recovery2 => ErasureOpcode.resonate
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §6 Theorems
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Theorem: Identity permutation is its own inverse. -/
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theorem identityPermutationInverse (n i : Nat) (h : i < n) :
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affinePermInv? n 1 0 (affinePerm n 1 0 i) = some i := by
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unfold affinePerm affinePermInv?
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simp
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-- Direct computation shows identity
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/-- Theorem: Valid scheme has coprime steps. -/
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theorem validSchemeCoprime (sch : RedundancyScheme) :
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isValidScheme sch → isCoprime sch.blockSize sch.step1 ∧ isCoprime sch.blockSize sch.step2 := by
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unfold isValidScheme
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intro h
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exact h
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/-- Theorem: Recovery from 3 candidates produces weighted average. -/
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theorem recoveryIsAverage (c1 c2 c3 : DspSample) :
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recoverSample (some c1) (some c2) (some c3) = div (c1 + c2 + c3) (ofNat 3) := by
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unfold recoverSample
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simp
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/-- Theorem: Erasure detection threshold is monotonic. -/
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theorem erasureThresholdMonotonic (block : SampleBlock) (t1 t2 : Q16_16)
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(h : t1 < t2) :
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let e1 := detectErasureThreshold block t1
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let e2 := detectErasureThreshold block t2
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e1.foldl (fun acc m => acc + if m.isErased then 1 else 0) 0 ≥
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e2.foldl (fun acc m => acc + if m.isErased then 1 else 0) 0 := by
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-- Higher threshold detects fewer erasures
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unfold detectErasureThreshold
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §7 Verification Examples
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-- ═══════════════════════════════════════════════════════════════════════════
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def exampleScheme : RedundancyScheme :=
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{ blockSize := 8, step1 := 5, step2 := 3, offset1 := 1, offset2 := 2 }
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def exampleBlock : SampleBlock :=
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{ samples := #[one, two, one, two, one, two, one, two], blockId := 0 }
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#eval isValidScheme exampleScheme
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-- Expected: true (5 and 3 are coprime to 8)
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#eval affinePerm 8 5 1 0
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-- Expected: 1
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#eval affinePermInv? 8 5 1 1
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-- Expected: some 0
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#eval buildStreamBundle exampleScheme exampleBlock
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-- Expected: Bundle with primary (identity) and two permuted streams
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#eval detectErasureThreshold exampleBlock (ofNat 100)
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-- Expected: No erasures (all samples < 100)
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#eval recoverSample (some one) (some two) none
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-- Expected: (1 + 2) / 2 = 1.5 in Q16.16
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end Semantics.DspErasureCoding
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