# ⚛  L1 Principle — Phylogenetic Likelihood Inference

**ID:** `L1-409` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Computational Biology — *Molecular evolution*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** phylogenetic tree topology
> **📡 Carrier:** N/A · **🌫 Noise:** multinomial
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41555219

---

## 🧠 1. Introduction

**Phylogenetic Likelihood Inference** is a **nonlinear inverse problem** whose unknown lives in **phylogenetic tree topology** space, within the **Molecular evolution** sub-domain of **Computational Biology**.

Measurements consist of N/A via a **molecular sequence alignment** sensing mechanism.

The forward operator applies, in order: applies a smooth nonlinear function element-wise; S · mcmc · tree space bayesian operator; O · atr · total tree likelihood operator.

Observations are corrupted by multinomial counting noise across categories. Existence of the recovered phylogenetic_tree_topology is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 2000); model_misspecification_substitution dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Multinomial sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → Pointwise nonlinearity → S · mcmc · tree space bayesian → O · atr · total tree likelihood → **y** (detector).

```
y = `O.atr.total_tree_likelihood` `S.mcmc.tree_space_bayesian` f(·) x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `N.pointwise` | Applies a smooth nonlinear function element-wise |
| `S.mcmc.tree_space_bayesian` | S · mcmc · tree space bayesian operator |
| `O.atr.total_tree_likelihood` | O · atr · total tree likelihood operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Computational Biology |
| Sub domain | Molecular evolution |
| Carrier | N/A |
| Problem class | nonlinear_inverse |
| Solution space | phylogenetic_tree_topology |
| Noise model | multinomial |
| Integration axis | phylogenetic_tree |
| Difficulty delta | 5 |
| L dag | 4.5 |

## 📡 4. Measurement Model

Existence of the recovered phylogenetic_tree_topology is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 2000); model_misspecification_substitution dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Multinomial sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | Robinson_Foulds_distance |
| Secondary | branch_length_RMSE |

## 📏 5. Operating Range (Ω)

**Center problem class:** `nonlinear_inverse` · **Forward operator:** `molecular_sequence_alignment`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N taxa | — | 50 |
| N sites | — | 5000 |
| Divergence time my | — | 100 |
| Mutation rate per site yr | — | 1e-09 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N taxa | — | 4 – 10000 |
| N sites | — | 100 – 100000 |
| Divergence time my | — | 1 – 1000 |
| Mutation rate per site yr | — | 1e-10 – 1e-07 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 5 Robinson_Foulds_distance

| Metric | Range |
|---|---|
| Robinson foulds distance | 0 – 20 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **Robinson_Foulds_distance**, with κ = `100000.0` and δ = `5`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0x89d3fb6cb7267412f3fd12ba45b4705b67200fc24e7b93d4c5cde61fbfa2b4e3`
- **Chain tx hash:** `0x820f0c2cd9fc9212901c8779f0a5ac0601dd802d9870bd446a567b786268b0f9`
- **Chain block:** `41555219`

---

## File Mapping

This bundle consists of: `L1-409.md`, `L1-409.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-409.md` | Source of truth — edit this | Human or LLM |
| `L1-409.json` | Structured metadata for the registry | LLM regenerates from the sections above |

**Prompt for your LLM after editing this Markdown:**

> Read the attached Markdown. Regenerate the sibling `.json` so every field matches.
> Preserve the schema documented in the rows above.
> Output each file in its own fenced code block tagged with the filename.
> Output only the JSON object.

_This Markdown was auto-synthesized from the catalog row for `L1-409`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._