# ⚛  L1 Principle — Cable Equation (Dendrite Propagation)

**ID:** `L1-398` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Computational Biology — *Computational neuroscience*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** dendritic conductance distribution
> **📡 Carrier:** N/A · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555217

---

## 🧠 1. Introduction

**Cable Equation (Dendrite Propagation)** is a **linear inverse problem** whose unknown lives in **dendritic conductance distribution** space, within the **Computational neuroscience** sub-domain of **Computational Biology**.

Measurements consist of N/A via a **intracellular voltage recording** sensing mechanism.

The forward operator applies, in order: gradient / divergence with respect to position; S · green function · cable operator; O · tikhonov · cable inversion operator.

Observations are corrupted by additive Gaussian noise. Existence of the recovered dendritic_conductance_distribution 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 ~= 80); dendritic_branching_approximation 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** → Spatial derivative → S · green function · cable → O · tikhonov · cable inversion → **y** (detector).

```
y = `O.tikhonov.cable_inversion` `S.green_function.cable` ∇ x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `D.space` | Gradient / divergence with respect to position |
| `S.green_function.cable` | S · green function · cable operator |
| `O.tikhonov.cable_inversion` | O · tikhonov · cable inversion operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Computational Biology |
| Sub domain | Computational neuroscience |
| Carrier | N/A |
| Problem class | linear_inverse |
| Solution space | dendritic_conductance_distribution |
| Noise model | gaussian |
| Integration axis | spatial_dendritic |
| Difficulty delta | 3 |
| L dag | 3 |

## 📡 4. Measurement Model

Existence of the recovered dendritic_conductance_distribution 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 ~= 80); dendritic_branching_approximation dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | conductance_distribution_RMSE |
| Secondary | voltage_reconstruction_RMSE_mV |

## 📏 5. Operating Range (Ω)

**Center problem class:** `linear_inverse` · **Forward operator:** `intracellular_voltage_recording`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Noise sigma mv | — | 0.1 |
| Cable length um | µm | 500 |
| N recording sites | — | 3 |
| Lambda space constant um | µm | 200 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Noise sigma mv | — | 0.01 – 2.0 |
| Cable length um | µm | 50 – 5000 |
| N recording sites | — | 1 – 20 |
| Lambda space constant um | µm | 50 – 1000 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.15 conductance_distribution_RMSE

| Metric | Range |
|---|---|
| Conductance distribution rmse | 0.02 – 0.5 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **conductance_distribution_RMSE**, with κ = `2000` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xbf5701b997d296c4255b5f3a22c2099911540aae48b30b204735225b387a10d4`
- **Chain tx hash:** `0xc4273774dc450ea251277e9f87bdd83363eec2be99447d5a90379260fd24f78d`
- **Chain block:** `41555217`

---

## File Mapping

This bundle consists of: `L1-398.md`, `L1-398.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-398.md` | Source of truth — edit this | Human or LLM |
| `L1-398.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 `L1-398`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._