Codon-Level Efficiency Modeling

To extend the OTOM framework to genetics at the atomic informational level, we define a codon-level efficiency functional that evaluates individual triplets within the genetic code.

Codon Space

A codon is defined as a triplet:

c = (b_1, b_2, b_3),  b_i ∈ {A, C, G, U}

with total space:

|𝒞| = 4³ = 64

Degeneracy Structure

Each codon maps to an amino acid with synonymous degeneracy:

d(c) = |{c' ∈ 𝒞 : aa(c') = aa(c)}|

This degeneracy introduces ambiguity and reduces informational specificity.

Codon Efficiency Functional

We define the codon efficiency functional:

Φ_codon(c) = (w_ρ·ρ̂_triplet(c) + w_q·q̂_conservation(c) + w_τ·τ̂_translation(c) - w_H·Ĥ_local(c) - w_ε·ε̂_mutation(c)) / (ln 64 + λ ln d(c) + C_0)

Interpretation of Terms

ρ̂_triplet: coding validity and reading-frame stability
q̂_conservation: evolutionary conservation signal
τ̂_translation: translational efficiency or structural relevance
Ĥ_local: codon-level entropy
ε̂_mutation: mutation-induced deviation

Connection to Universal Efficiency

This is a direct instantiation of the universal efficiency principle:

Φ = Useful Structure / (Physical / Informational Cost)

with:

Cost(c) = ln 64 + λ ln d(c) + C_0

Sequence-Level Reconstruction

For a coding sequence S = (c_1, …, c_n):

Φ_CDS(S) = (1/n) Σ_{i=1}^n Φ_codon(c_i)

or with context:

Φ_CDS(S) = (1/n) Σ_{i=1}^n Φ_codon(c_i | c_{i-1}, c_{i+1})

Mutation Dynamics

A mutation c → c' induces:

ΔΦ = Φ_codon(c') - Φ_codon(c)

which defines a local selection gradient:

Mutations that increase Φ_codon are favored.

Interpretation

The codon-level model reveals that:

Genetic sequences optimize informational efficiency at the triplet level
Degeneracy introduces a thermodynamic-like cost
Mutation selection operates as local efficiency maximization

Genetic evolution is a reinforcement process over codon efficiency.

Implementation Location

Lean Module: 0-Core-Formalism/lean/Semantics/Semantics/GenomicCompression.lean
MATH_MODEL_MAP Entry: 1.2.1 (Codon_Fitness_Function)
Domain: LAYER_A_COMPRESSION
Bind Class: informational_bind

References

OTOM v2.0.0-Cambrian-Bind
Universal Efficiency Principle (Phi_Universal, MATH_MODEL_MAP entry 0)
Genetic Code Optimization (MATH_MODEL_MAP entry 1.2)

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