# ⚛  L1 Principle — Wright-Fisher Population Genetics

**ID:** `L1-408` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Computational Biology — *Population genetics*
> **🎯 Problem class:** parameter estimation · **🧮 Solution space:** population genetic parameter vector
> **📡 Carrier:** N/A · **🌫 Noise:** multinomial
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41555219

---

## 🧠 1. Introduction

**Wright-Fisher Population Genetics** is a **parameter-estimation problem** whose unknown lives in **population genetic parameter vector** space, within the **Population genetics** sub-domain of **Computational Biology**.

Measurements consist of N/A via a **allele frequency sequencing** sensing mechanism.

The forward operator applies, in order: applies a smooth nonlinear function element-wise; O · likelihood · allele frequency operator; S · mcmc · popgen posterior operator.

Observations are corrupted by multinomial counting noise across categories. Existence of the recovered population_genetic_parameter_vector 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 ~= 100); population_structure_subdivision 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 → O · likelihood · allele frequency → S · mcmc · popgen posterior → **y** (detector).

```
y = `S.mcmc.popgen_posterior` `O.likelihood.allele_frequency` f(·) x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `N.pointwise` | Applies a smooth nonlinear function element-wise |
| `O.likelihood.allele_frequency` | O · likelihood · allele frequency operator |
| `S.mcmc.popgen_posterior` | S · mcmc · popgen posterior operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Computational Biology |
| Sub domain | Population genetics |
| Carrier | N/A |
| Problem class | parameter_estimation |
| Solution space | population_genetic_parameter_vector |
| Noise model | multinomial |
| Integration axis | generations |
| Difficulty delta | 5 |
| L dag | 3 |

## 📡 4. Measurement Model

Existence of the recovered population_genetic_parameter_vector 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 ~= 100); population_structure_subdivision dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Multinomial sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | Ne_estimation_RMSE_log |
| Secondary | selection_coefficient_RMSE |

## 📏 5. Operating Range (Ω)

**Center problem class:** `parameter_estimation` · **Forward operator:** `allele_frequency_sequencing`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| S selection | — | 0 |
| Ne effective | — | 1000 |
| N generations | — | 20 |
| N sampled alleles | — | 100 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| S selection | — | -0.1 – 0.1 |
| Ne effective | — | 10 – 100000 |
| N generations | — | 5 – 100 |
| N sampled alleles | — | 10 – 1000 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.3 Ne_estimation_RMSE_log

| Metric | Range |
|---|---|
| Ne estimation rmse log | 0.05 – 1.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **Ne_estimation_RMSE_log**, with κ = `2000` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x4032c97662301a11eb728fee50ab6352f9031854e796853bc3533d2002b8f0c4`
- **Chain tx hash:** `0xa78c1cdf96ae810ee096d0a93e8f4d4ee0ea9c94742642b5f2e428fe3ba97876`
- **Chain block:** `41555219`

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## File Mapping

This bundle consists of: `L1-408.md`, `L1-408.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-408.md` | Source of truth — edit this | Human or LLM |
| `L1-408.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 `L1-408`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._