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78 lines
3.4 KiB
Text
78 lines
3.4 KiB
Text
import Semantics.FixedPoint
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import Semantics.GeneticCode
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namespace Semantics.Spectrum
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/-! # Spectral Encoding
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Derived from the Erdős #1196 solution via piecewise eigenvector construction.
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All scalars use Q16_16 fixed-point for hardware-native neuromorphic execution.
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-/
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/-- Default number of spectral bins. -/
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def binCount : Nat := 8
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/-- A spectral signature is a finite vector of amplitudes. -/
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structure SpectralSignature where
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bins : List Q16_16
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deriving Repr, BEq, DecidableEq
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namespace SpectralSignature
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def empty : SpectralSignature := ⟨List.replicate binCount Q16_16.zero⟩
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def activeBins (sig : SpectralSignature) : List (Nat × Q16_16) :=
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(List.zip (List.range sig.bins.length) sig.bins).filter (λ p => p.2 != Q16_16.zero)
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/-- Peak distance in bin index space. -/
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def peakDistance (i j : Nat) : Nat :=
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if i > j then i - j else j - i
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/-- Erdős-Hooley constant δ ≈ 0.08607 as Q16_16.
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Computed as 5643 / 65536 ≈ 0.08609 (within 0.02% of true value). -/
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def erdosHooleyDelta : Q16_16 := ⟨5643⟩ -- 5643/65536 ≈ 0.08609 (within 0.02% of true δ ≈ 0.08607)
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#eval erdosHooleyDelta -- Expected: ⟨5643⟩
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/-- Verify no two active peaks are adjacent (minimum separation = 1 bin). -/
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def verifySpectralGap (sig : SpectralSignature) : Bool :=
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let active := sig.activeBins.map (λ p => p.1)
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active.all (λ i => active.all (λ j => i == j || peakDistance i j > 1))
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/-- Map an event to a discrete spectral signature (one peak per base).
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Each base type (a, t, g, c) gets a unique spectral peak position.
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This creates a spectral barcode for genetic event encoding. -/
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def eventSpectrum : Semantics.GeneticCode.EventType → SpectralSignature
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| Semantics.GeneticCode.EventType.a => { bins := [Q16_16.one, Q16_16.zero, Q16_16.zero,
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Q16_16.zero, Q16_16.zero, Q16_16.zero,
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Q16_16.zero, Q16_16.zero] }
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| Semantics.GeneticCode.EventType.t => { bins := [Q16_16.zero, Q16_16.one, Q16_16.zero,
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Q16_16.zero, Q16_16.zero, Q16_16.zero,
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Q16_16.zero, Q16_16.zero] }
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| Semantics.GeneticCode.EventType.g => { bins := [Q16_16.zero, Q16_16.zero, Q16_16.one,
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Q16_16.zero, Q16_16.zero, Q16_16.zero,
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Q16_16.zero, Q16_16.zero] }
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| Semantics.GeneticCode.EventType.c => { bins := [Q16_16.zero, Q16_16.zero, Q16_16.zero,
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Q16_16.one, Q16_16.zero, Q16_16.zero,
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Q16_16.zero, Q16_16.zero] }
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/-- Compute spectral overlap (inner product) between two signatures. -/
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def spectralOverlap (sig1 sig2 : SpectralSignature) : Q16_16 :=
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List.zipWith (λ a b => Q16_16.mul a b) sig1.bins sig2.bins
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|>.foldl (λ acc x => Q16_16.add acc x) Q16_16.zero
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/-- Piecewise eigenvector merge: superposition with saturation. -/
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def piecewiseMerge (left right : SpectralSignature) : SpectralSignature :=
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let merged := List.zipWith (λ a b => Q16_16.min Q16_16.one (Q16_16.add a b)) left.bins right.bins
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⟨merged⟩
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/-- Count resonance degeneracy (overlapping non-zero bins). -/
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def resonanceDegeneracy (left right : SpectralSignature) : Nat :=
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List.zipWith (λ a b => if a != Q16_16.zero && b != Q16_16.zero then 1 else 0) left.bins right.bins
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|>.foldl Nat.add 0
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/-- Density bound predicate: active bins must not exceed threshold. -/
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def withinDensityBound (sig : SpectralSignature) (maxActive : Nat) : Bool :=
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sig.activeBins.length ≤ maxActive
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end SpectralSignature
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end Semantics.Spectrum
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