# ⚛  L1 Principle — Optimal Experimental Design

**ID:** `L1-500` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Optimization — *Information-theoretic design*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** experiment design vector
> **📡 Carrier:** N/A · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41555318

---

## 🧠 1. Introduction

**Optimal Experimental Design** is a **nonlinear inverse problem** whose unknown lives in **experiment design vector** space, within the **Information-theoretic design** sub-domain of **Optimization**.

Measurements consist of N/A via a **fisher information optimization** sensing mechanism.

The forward operator applies, in order: adds a prior term that biases the solution toward smoothness/sparsity; O · d optimality · criterion operator; S · gradient · update operator.

Observations are corrupted by additive Gaussian noise. Existence of the recovered experiment_design_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 ~= 50); prior_parameter_uncertainty dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → O · d optimality · criterion → S · gradient · update → **y** (detector).

```
y = `S.gradient.update` `O.d_optimality.criterion` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `O.d_optimality.criterion` | O · d optimality · criterion operator |
| `S.gradient.update` | S · gradient · update operator |

**🛠 Solver components** _(used inside the solver, not in the forward equation)_:

| Primitive | What it does |
|---|---|
| `O.regularize` | Adds a prior term that biases the solution toward smoothness/sparsity |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Optimization |
| Sub domain | Information-theoretic design |
| Carrier | N/A |
| Problem class | nonlinear_inverse |
| Solution space | experiment_design_vector |
| Noise model | gaussian |
| Integration axis | parameter_space |
| Difficulty delta | 5 |
| L dag | 2.8 |

## 📡 4. Measurement Model

Existence of the recovered experiment_design_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 ~= 50); prior_parameter_uncertainty dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | D_optimality_criterion |
| Secondary | parameter_estimation_RMSE |

## 📏 5. Operating Range (Ω)

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

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| P parameters | — | 5 |
| Sigma2 noise | — | 0.1 |
| N experiments | — | 20 |
| Nonlinearity level | — | 0.5 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| P parameters | — | 2 – 20 |
| Sigma2 noise | — | 0.01 – 1.0 |
| N experiments | — | 5 – 100 |
| Nonlinearity level | — | 0.1 – 2.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.80 D_optimality_criterion

| Metric | Range |
|---|---|
| D optimality criterion | 0.3 – 1.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **D_optimality_criterion**, with κ = `1000.0` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xe56535577bd0cb33edad3f534ebd44c2540e17ec7040920c32b7456bfe695242`
- **Chain tx hash:** `0xd0e0e30d2fee837aedd2225cb2239be38be2f079c300dec27c8e15a0eec5a7f0`
- **Chain block:** `41555318`

---

## File Mapping

This bundle consists of: `L1-500.md`, `L1-500.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-500.md` | Source of truth — edit this | Human or LLM |
| `L1-500.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 `L1-500`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._