# ⚛  L1 Principle — Pharmacodynamics Dose-Response Inversion

**ID:** `L1-404` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Computational Biology — *Pharmacodynamics*
> **🎯 Problem class:** parameter estimation · **🧮 Solution space:** PD parameter vector
> **📡 Carrier:** N/A · **🌫 Noise:** lognormal
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555218

---

## 🧠 1. Introduction

**Pharmacodynamics Dose-Response Inversion** is a **parameter-estimation problem** whose unknown lives in **PD parameter vector** space, within the **Pharmacodynamics** sub-domain of **Computational Biology**.

Measurements consist of N/A via a **effect concentration measurement** sensing mechanism.

The forward operator applies, in order: applies a smooth nonlinear function element-wise; O · nls · pd fit operator; S · bayesian · pd posterior operator.

Observations are corrupted by log-normal multiplicative noise. Existence of the recovered PD_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 ~= 20); inter_individual_variability_PD dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Lognormal sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → Pointwise nonlinearity → O · nls · pd fit → S · bayesian · pd posterior → **y** (detector).

```
y = `S.bayesian.pd_posterior` `O.nls.pd_fit` f(·) x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `N.pointwise` | Applies a smooth nonlinear function element-wise |
| `O.nls.pd_fit` | O · nls · pd fit operator |
| `S.bayesian.pd_posterior` | S · bayesian · pd posterior operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Computational Biology |
| Sub domain | Pharmacodynamics |
| Carrier | N/A |
| Problem class | parameter_estimation |
| Solution space | PD_parameter_vector |
| Noise model | lognormal |
| Integration axis | concentration_space |
| Difficulty delta | 3 |
| L dag | 2 |

## 📡 4. Measurement Model

Existence of the recovered PD_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 ~= 20); inter_individual_variability_PD dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Lognormal sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | EC50_RMSE_fold |
| Secondary | dose_response_fit_RMSE |

## 📏 5. Operating Range (Ω)

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

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N hill | — | 1.5 |
| Ec50 nm | nm | 100 |
| N dose groups | — | 6 |
| Noise cv percent | — | 20 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N hill | — | 0.5 – 5.0 |
| Ec50 nm | nm | 0.1 – 1000000.0 |
| N dose groups | — | 3 – 20 |
| Noise cv percent | — | 5 – 100 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 1.5 EC50_RMSE_fold

| Metric | Range |
|---|---|
| Ec50 rmse fold | 1.05 – 5.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **EC50_RMSE_fold**, with κ = `500` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xee72c89da66e97ecd73154e158d753de9735dce17b199c8d1c2705bf9f7b2306`
- **Chain tx hash:** `0xf1404d5483048628ad972f47ef4f4a0777d7ae50c5149a0db8d6a701d545f40d`
- **Chain block:** `41555218`

---

## File Mapping

This bundle consists of: `L1-404.md`, `L1-404.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-404.md` | Source of truth — edit this | Human or LLM |
| `L1-404.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 `L1-404`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._