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78 lines
3.4 KiB
Text
78 lines
3.4 KiB
Text
import Mathlib.Data.Real.Basic
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import Mathlib.Data.Fin.Basic
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import Mathlib.Data.Nat.Log
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namespace HachimojiCostRefinement
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/-- Hachimoji genetic alphabet has 8 nucleotides (A, T, G, C, P, Z, B, S) -/
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abbrev HachimojiNucleotide := Fin 8
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/-- Hachimoji codon is a triplet of nucleotides -/
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abbrev HachimojiCodon := Fin 3 → HachimojiNucleotide
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/-- Standard DNA has 4 nucleotides, 64 codons -/
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def standardCodonSpace : Nat := 64
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/-- Hachimoji has 8 nucleotides, 512 codons -/
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def hachimojiCodonSpace : Nat := 512
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/-- Shannon entropy of a probability distribution -/
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noncomputable def shannonEntropy (p : Nat → ℝ) (hp : Σ n, p n = 1) : ℝ :=
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-∑ n, if p n = 0 then 0 else p n * Real.log (p n)
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/-- Effective alphabet size based on entropy -/
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noncomputable def effectiveAlphabetSize (p : Nat → ℝ) (hp : Σ n, p n = 1) : ℝ :=
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Real.exp (shannonEntropy p hp)
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/-- Degeneracy of an amino acid (number of codons encoding it) -/
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abbrev Degeneracy := Nat
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/-- Degeneracy function for Hachimoji codons -/
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noncomputable def degeneracyFunction (c : HachimojiCodon) : Degeneracy :=
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-- In a full implementation, this would map codons to their amino acid degeneracy
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-- For now, we use a placeholder: assume average degeneracy of 26 for Hachimoji
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26
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/-- Standard DNA average degeneracy (3 codons per amino acid) -/
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def standardAverageDegeneracy : Degeneracy := 3
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/-- Hachimoji average degeneracy (26 codons per amino acid) -/
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def hachimojiAverageDegeneracy : Degeneracy := 26
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/-- Base cost with standard DNA: ln 64 -/
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def standardBaseCost : ℝ :=
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Real.log 64
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/-- Base cost with Hachimoji (raw): ln 512 -/
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def hachimojiRawCost : ℝ :=
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Real.log 512
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/-- Cost reduction factor using effective alphabet size -/
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noncomputable def effectiveAlphabetCost (p : Nat → ℝ) (hp : Σ n, p n = 1) : ℝ :=
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Real.log (effectiveAlphabetSize p hp)
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/-- Cost reduction factor using adaptive degeneracy weighting -/
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noncomputable def adaptiveDegeneracyCost (c : HachimojiCodon) : ℝ :=
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Real.log 512 / (degeneracyFunction c)
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/-- Combined cost reduction: effective alphabet + adaptive degeneracy -/
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noncomputable def combinedReducedCost (p : Nat → ℝ) (hp : Σ n, p n = 1) (c : HachimojiCodon) : ℝ :=
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Real.log (effectiveAlphabetSize p hp) / (degeneracyFunction c)
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/-- External cost-refinement invariants.
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Effective alphabet cost ≤ raw, adaptive degeneracy ≤ raw, combined achieves target scaling,
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Landauer consistency maintained, info-theoretic cost ≤ log cost.
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These are information-theoretic bounds requiring external entropy analysis. -/
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structure HachimojiCostInvariantsHypothesis where
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effective_le_raw (p : Nat → ℝ) (hp : Σ n, p n = 1) : effectiveAlphabetCost p hp ≤ hachimojiRawCost
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adaptive_le_raw (c : HachimojiCodon) (hdeg : 1 < degeneracyFunction c) : adaptiveDegeneracyCost c ≤ hachimojiRawCost
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combined_target_scaling (p : Nat → ℝ) (hp : Σ n, p n = 1) (c : HachimojiCodon)
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(h_eff : effectiveAlphabetCost p hp ≤ 4.605) (h_deg : degeneracyFunction c ≥ 26) :
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combinedReducedCost p hp c ≤ 1.2 * standardBaseCost
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landauer_consistency (p : Nat → ℝ) (hp : Σ n, p n = 1) (c : HachimojiCodon) :
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combinedReducedCost p hp c = Real.log (effectiveAlphabetSize p hp) / (degeneracyFunction c) ∧
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∃ N, combinedReducedCost p hp c = Real.log N
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info_le_log (p : Nat → ℝ) (hp : Σ n, p n = 1) (h_nonuniform : ∃ n m, p n ≠ p m) :
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informationTheoreticCost p hp ≤ Real.log 512
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end HachimojiCostRefinement
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