# ⚛  L1 Principle — Merton Credit Risk Structural Model

**ID:** `L1-443` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Computational Finance — *Credit risk*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** firm value vol vector
> **📡 Carrier:** N/A · **🌫 Noise:** market gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41555259

---

## 🧠 1. Introduction

**Merton Credit Risk Structural Model** is a **nonlinear inverse problem** whose unknown lives in **firm value vol vector** space, within the **Credit risk** sub-domain of **Computational Finance**.

Measurements consist of N/A via a **equity option credit inversion** sensing mechanism.

The forward operator applies, in order: applies a smooth nonlinear function element-wise; O · newton · system 2eq operator; S · bs · equity as call operator.

Observations are corrupted by market gaussian. Existence of the recovered firm_value_vol_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 ~= 80); debt_complexity_multiple_tranches dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Market gaussian sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → Pointwise nonlinearity → O · newton · system 2eq → S · bs · equity as call → **y** (detector).

```
y = `S.bs.equity_as_call` `O.newton.system_2eq` f(·) x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `N.pointwise` | Applies a smooth nonlinear function element-wise |
| `O.newton.system_2eq` | O · newton · system 2eq operator |
| `S.bs.equity_as_call` | S · bs · equity as call operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Computational Finance |
| Sub domain | Credit risk |
| Carrier | N/A |
| Problem class | nonlinear_inverse |
| Solution space | firm_value_vol_vector |
| Noise model | market_gaussian |
| Integration axis | capital_structure |
| Difficulty delta | 5 |
| L dag | 2.5 |

## 📡 4. Measurement Model

Existence of the recovered firm_value_vol_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 ~= 80); debt_complexity_multiple_tranches dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Market gaussian sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | CDS_spread_prediction_RMSE_bp |
| Secondary | PD_calibration_error |

## 📏 5. Operating Range (Ω)

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

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| T maturity yr | — | 1 |
| R rate percent | — | 3 |
| Sigma e percent | — | 30 |
| Leverage ratio d v | — | 0.5 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| T maturity yr | — | 0.25 – 5.0 |
| R rate percent | — | 0.0 – 10.0 |
| Sigma e percent | — | 5 – 100 |
| Leverage ratio d v | — | 0.1 – 0.9 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 50 CDS_spread_prediction_RMSE_bp

| Metric | Range |
|---|---|
| Cds spread prediction rmse bp | 5 – 500 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **CDS_spread_prediction_RMSE_bp**, with κ = `2000` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x2efbf5051bb03d694a44a5906b3b18354043018770aedc0550d58bc5aec228af`
- **Chain tx hash:** `0x20a563530bc0c6f554d0d41d28074f54c77904246206bfe74a1de96a2afb6141`
- **Chain block:** `41555259`

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## File Mapping

This bundle consists of: `L1-443.md`, `L1-443.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-443.md` | Source of truth — edit this | Human or LLM |
| `L1-443.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 `L1-443`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._