# ⚛  L1 Principle — Oxygen Transport (Krogh Cylinder)

**ID:** `L1-411` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Computational Biology — *Physiology*
> **🎯 Problem class:** parameter estimation · **🧮 Solution space:** oxygen distribution field
> **📡 Carrier:** N/A · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555219

---

## 🧠 1. Introduction

**Oxygen Transport (Krogh Cylinder)** is a **parameter-estimation problem** whose unknown lives in **oxygen distribution field** space, within the **Physiology** sub-domain of **Computational Biology**.

Measurements consist of N/A via a **po2 measurement fiber optic** sensing mechanism.

The forward operator applies, in order: gradient / divergence with respect to position; S · fdm · oxygen diffusion operator; O · chi2 · po2 profile operator.

Observations are corrupted by additive Gaussian noise. Existence of the recovered oxygen_distribution_field 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); non_uniform_capillary_spacing 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 · fdm · oxygen diffusion → O · chi2 · po2 profile → **y** (detector).

```
y = `O.chi2.po2_profile` `S.fdm.oxygen_diffusion` ∇ x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `D.space` | Gradient / divergence with respect to position |
| `S.fdm.oxygen_diffusion` | S · fdm · oxygen diffusion operator |
| `O.chi2.po2_profile` | O · chi2 · po2 profile operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Computational Biology |
| Sub domain | Physiology |
| Carrier | N/A |
| Problem class | parameter_estimation |
| Solution space | oxygen_distribution_field |
| Noise model | gaussian |
| Integration axis | radial_tissue |
| Difficulty delta | 3 |
| L dag | 2.8 |

## 📡 4. Measurement Model

Existence of the recovered oxygen_distribution_field 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); non_uniform_capillary_spacing dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | pO2_profile_RMSE_mmHg |
| Secondary | anoxic_fraction_error |

## 📏 5. Operating Range (Ω)

**Center problem class:** `parameter_estimation` · **Forward operator:** `po2_measurement_fiber_optic`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| R t tissue um | µm | 50 |
| Q max mlo2 ml s | s | 0.0005 |
| Arterial po2 mmhg | — | 100 |
| Capillary radius um | µm | 5 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| R t tissue um | µm | 10 – 200 |
| Q max mlo2 ml s | s | 0.0001 – 0.005 |
| Arterial po2 mmhg | — | 50 – 150 |
| Capillary radius um | µm | 2 – 20 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 5 pO2_profile_RMSE_mmHg

| Metric | Range |
|---|---|
| Po2 profile rmse mmhg | 0.5 – 30 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **pO2_profile_RMSE_mmHg**, with κ = `1000` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x04892ea35760c3889fa934f728182804e3f053623ababf6b45d343a17885cbca`
- **Chain tx hash:** `0x5e5eb665acbb6f52a4a75a7ab90a69309f68eafff1619f8289bc502746e887ac`
- **Chain block:** `41555219`

---

## File Mapping

This bundle consists of: `L1-411.md`, `L1-411.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-411.md` | Source of truth — edit this | Human or LLM |
| `L1-411.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 `L1-411`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._