# ⚛  L1 Principle — Cardiac Electrophysiology (Monodomain)

**ID:** `L1-399` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Computational Biology — *Cardiac modeling*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 3D transmembrane potential
> **📡 Carrier:** N/A · **🌫 Noise:** observation gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41555217

---

## 🧠 1. Introduction

**Cardiac Electrophysiology (Monodomain)** is a **nonlinear inverse problem** whose unknown lives in **3D transmembrane potential** space, within the **Cardiac modeling** sub-domain of **Computational Biology**.

Measurements consist of N/A via a **ecg body surface potential** sensing mechanism.

The forward operator applies, in order: gradient / divergence with respect to position; S · eikonal · fast marching operator; O · ecg · inverse body surface operator.

Observations are corrupted by observation gaussian. Existence of the recovered 3D_transmembrane_potential 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); torso_conductivity_uncertainty dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Observation gaussian sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → Spatial derivative → S · eikonal · fast marching → O · ecg · inverse body surface → **y** (detector).

```
y = `O.ecg.inverse_body_surface` `S.eikonal.fast_marching` ∇ x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `D.space` | Gradient / divergence with respect to position |
| `S.eikonal.fast_marching` | S · eikonal · fast marching operator |
| `O.ecg.inverse_body_surface` | O · ecg · inverse body surface operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Computational Biology |
| Sub domain | Cardiac modeling |
| Carrier | N/A |
| Problem class | nonlinear_inverse |
| Solution space | 3D_transmembrane_potential |
| Noise model | observation_gaussian |
| Integration axis | 3d_cardiac_volume |
| Difficulty delta | 5 |
| L dag | 4.5 |

## 📡 4. Measurement Model

Existence of the recovered 3D_transmembrane_potential 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); torso_conductivity_uncertainty dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Observation gaussian sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | activation_time_error_ms |
| Secondary | body_surface_potential_RMSE_mV |

## 📏 5. Operating Range (Ω)

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

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N mesh nodes | — | 10000 |
| Noise snr db | dB | 30 |
| N electrodes bsp | — | 128 |
| Heart torso distance mm | mm | 50 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N mesh nodes | — | 1000 – 100000 |
| Noise snr db | dB | 10 – 60 |
| N electrodes bsp | — | 32 – 512 |
| Heart torso distance mm | mm | 20 – 100 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 5 activation_time_error_ms

| Metric | Range |
|---|---|
| Activation time error ms | 1 – 30 |

## ⚖ 7. Hardness Function

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

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x781340fe947f4a5e7eb4920a752ea2c505bdbcedb5928b5df5e8afd99b236717`
- **Chain tx hash:** `0x8d057901026adf6d57bcb86b8bc050e92aeaed8b462e04aef8991d80536894e2`
- **Chain block:** `41555217`

---

## File Mapping

This bundle consists of: `L1-399.md`, `L1-399.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-399.md` | Source of truth — edit this | Human or LLM |
| `L1-399.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 `L1-399`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._