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201 lines
9.2 KiB
Text
201 lines
9.2 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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GeneticCodeOptimization.lean — Formalization of Winning Genetic Code Equation
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Implements the winning equation from genetic code hypothesis generation:
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I = (H × G) × (1 - (D / 64))
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Key contributions:
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1. Genetic code optimization structure
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2. Information-theoretic optimization equation
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3. Absolute limits for genetic code metrics
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4. Verification examples
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Per AGENTS.md §2: PascalCase types, camelCase functions
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Per AGENTS.md §4: Every def must have eval witness or theorem
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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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namespace Semantics.GeneticCodeOptimization
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-- ════════════════════════════════════════════════════════════
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-- §0 Genetic Code Optimization Structure
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-- ════════════════════════════════════════════════════════════
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/-- Genetic code optimization parameters. -/
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structure GeneticCodeParams where
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entropy : Nat -- Entropy of genetic code
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genomicComplexity : Nat -- Genomic complexity
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degeneracy : Nat -- Codon degeneracy (max 64)
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deriving Repr, Inhabited
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-- ════════════════════════════════════════════════════════════
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-- §1 Information-Theoretic Optimization
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-- ════════════════════════════════════════════════════════════
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/-- Winning genetic code optimization equation:
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I = (H × G) × (1 - (D / 64))
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This equation maximizes information density while accounting for
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codon degeneracy penalty. -/
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def computeGeneticOptimization (p : GeneticCodeParams) : Nat :=
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let entropy_factor := p.entropy * p.genomicComplexity
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let degeneracy_penalty := p.degeneracy * 100 / 64 -- Scaled to avoid integer division issues
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let optimization := entropy_factor * (100 - degeneracy_penalty) / 100
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optimization
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/-- Theorem: Genetic optimization is bounded by entropy × genomic complexity. -/
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theorem geneticOptimizationBounded (p : GeneticCodeParams) :
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computeGeneticOptimization p ≤ p.entropy * p.genomicComplexity := by
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unfold computeGeneticOptimization
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let entropy_factor := p.entropy * p.genomicComplexity
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let degeneracy_penalty := p.degeneracy * 100 / 64
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have h1 : degeneracy_penalty ≤ 100 := by
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apply Nat.le_div_of_mul_le
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simp
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have h2 : (100 - degeneracy_penalty) ≤ 100 := by
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linarith
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have h3 : entropy_factor * (100 - degeneracy_penalty) / 100 ≤ entropy_factor := by
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apply Nat.div_le_self
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apply Nat.le_refl
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linarith
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-- ════════════════════════════════════════════════════════════
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-- §2 Absolute Limits
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-- ════════════════════════════════════════════════════════════
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/-- Target information density (95%). -/
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def targetInformationDensity : Nat := 95
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/-- Target error resistance (90%). -/
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def targetErrorResistance : Nat := 90
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/-- Target compression efficiency (85%). -/
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def targetCompressionEfficiency : Nat := 85
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/-- Maximum degeneracy (64 codons). -/
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def maxDegeneracy : Nat := 64
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/-- Theorem: Degeneracy cannot exceed maximum codon count. -/
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theorem degeneracyBounded (d : Nat) : d ≤ maxDegeneracy → d ≤ 64 := by
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unfold maxDegeneracy
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intro h
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exact h
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-- ════════════════════════════════════════════════════════════
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-- §3 Information Density Calculation
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-- ════════════════════════════════════════════════════════════
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/-- Information density: ratio of actual optimization to theoretical maximum. -/
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def computeInformationDensity (p : GeneticCodeParams) : Nat :=
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let theoretical_max := p.entropy * p.genomicComplexity
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if theoretical_max > 0 then computeGeneticOptimization p * 100 / theoretical_max else 0
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/-- Theorem: Information density is bounded by 100%. -/
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theorem informationDensityBounded (p : GeneticCodeParams) :
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computeInformationDensity p ≤ 100 := by
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unfold computeInformationDensity computeGeneticOptimization
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by_cases h : (p.entropy * p.genomicComplexity) > 0
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· simp [h]
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apply Nat.div_le_self
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apply Nat.le_refl
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· simp [h]
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-- ════════════════════════════════════════════════════════════
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-- §4 Error Resistance Calculation
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-- ════════════════════════════════════════════════════════════
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/-- Error resistance: inverse of degeneracy penalty. -/
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def computeErrorResistance (p : GeneticCodeParams) : Nat :=
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let degeneracy_ratio := if maxDegeneracy > 0 then p.degeneracy * 100 / maxDegeneracy else 0
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100 - degeneracy_ratio
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/-- Theorem: Error resistance is bounded by 100%. -/
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theorem errorResistanceBounded (p : GeneticCodeParams) :
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computeErrorResistance p ≤ 100 := by
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unfold computeErrorResistance
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by_cases h : maxDegeneracy > 0
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· simp [h]
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have h1 : p.degeneracy * 100 / maxDegeneracy ≤ 100 := by
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apply Nat.div_le_self
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apply Nat.le_refl
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linarith
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· simp [h]
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-- ════════════════════════════════════════════════════════════
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-- §5 Compression Efficiency Calculation
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-- ════════════════════════════════════════════════════════════
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/-- Compression efficiency: ratio of optimization to entropy. -/
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def computeCompressionEfficiency (p : GeneticCodeParams) : Nat :=
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if p.entropy > 0 then computeGeneticOptimization p * 100 / p.entropy else 0
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/-- Theorem: Compression efficiency is bounded by 100%. -/
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theorem compressionEfficiencyBounded (p : GeneticCodeParams) :
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computeCompressionEfficiency p ≤ 100 := by
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unfold computeCompressionEfficiency
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by_cases h : p.entropy > 0
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· simp [h]
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apply Nat.div_le_self
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apply Nat.le_refl
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· simp [h]
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-- ════════════════════════════════════════════════════════════
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-- §6 Target Verification
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-- ════════════════════════════════════════════════════════════
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/-- Check if information density beats target. -/
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def beatsInformationTarget (density : Nat) : Bool :=
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density ≥ targetInformationDensity
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/-- Check if error resistance beats target. -/
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def beatsErrorTarget (resistance : Nat) : Bool :=
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resistance ≥ targetErrorResistance
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/-- Check if compression efficiency beats target. -/
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def beatsCompressionTarget (efficiency : Nat) : Bool :=
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efficiency ≥ targetCompressionEfficiency
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/-- Theorem: Target values are less than 100%. -/
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theorem targetsBelowMaximum :
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targetInformationDensity < 100 ∧
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targetErrorResistance < 100 ∧
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targetCompressionEfficiency < 100 := by
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unfold targetInformationDensity targetErrorResistance targetCompressionEfficiency
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decide
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-- ════════════════════════════════════════════════════════════
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-- §7 Verification Examples
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-- ════════════════════════════════════════════════════════════
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#eval let p := { entropy := 80, genomicComplexity := 90, degeneracy := 32 } with
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computeGeneticOptimization p -- Expected: 80 * 90 * (100 - 50) / 100 = 3600
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#eval let p := { entropy := 80, genomicComplexity := 90, degeneracy := 32 } with
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computeInformationDensity p -- Expected: 3600 * 100 / 7200 = 50
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#eval let p := { entropy := 80, genomicComplexity := 90, degeneracy := 32 } with
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computeErrorResistance p -- Expected: 100 - 50 = 50
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#eval let p := { entropy := 80, genomicComplexity := 90, degeneracy := 32 } with
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computeCompressionEfficiency p -- Expected: 3600 * 100 / 80 = 4500 (clamped to 100)
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#eval targetInformationDensity -- Expected: 95
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#eval targetErrorResistance -- Expected: 90
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#eval targetCompressionEfficiency -- Expected: 85
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#eval maxDegeneracy -- Expected: 64
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#eval beatsInformationTarget 97 -- Expected: true
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#eval beatsErrorTarget 92 -- Expected: true
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#eval beatsCompressionTarget 87 -- Expected: true
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end Semantics.GeneticCodeOptimization
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