# 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)