# ⚛  L1 Principle — Digital Mammography

**ID:** `L1-036` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Medical Imaging — *Low-energy X-ray breast imaging*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 2D breast projection
> **📡 Carrier:** x_ray · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41553385

---

## 🧠 1. Introduction

**Digital Mammography** is a **linear inverse problem** whose unknown lives in **2D breast projection** space, within the **Low-energy X-ray breast imaging** sub-domain of **Medical Imaging**.

Measurements consist of X-ray photons transmitted through (or scattered by) the sample via a **mammography xray** sensing mechanism.

The forward operator applies, in order: polyenergetic X-ray emission spectrum; L · compression operator; exponential attenuation along the propagation path; pixel-level spatial averaging on the detector.

Observations are corrupted by Poisson shot noise from quantum-limited detection. Existence of the recovered 2D breast projection is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by the declared priors. Stability is well-conditioned (kappa_eff ~= 9); scatter dominates the stability cliff; compression_variation and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) sets the irreducible data-fidelity floor, while mild Tikhonov or analytic inversion is sufficient at the nominal Omega point.

## ⚙ 2. Forward Model

Physical chain: **x** → X-ray source → L · compression → Beer-Lambert attenuation → Spatial integration → **y** (detector).

```
y = ∫_A dA exp(-∫µ dl) `L.compression` I₀(E) x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.xray_source` | Polyenergetic x-ray emission spectrum |
| `L.compression` | L · compression operator |
| `L.beer_lambert` | Exponential attenuation along the propagation path |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Medical Imaging |
| Sub domain | Low-energy X-ray breast imaging |
| Carrier | x_ray |
| Problem class | linear_inverse |
| Solution space | 2D_breast_projection |
| Noise model | shot_poisson |
| Integration axis | spatial |
| Difficulty delta | 3 |
| L dag | 2.8 |

## 📡 4. Measurement Model

Existence of the recovered 2D breast projection is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by the declared priors. Stability is well-conditioned (kappa_eff ~= 9); scatter dominates the stability cliff; compression_variation and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) sets the irreducible data-fidelity floor, while mild Tikhonov or analytic inversion is sufficient at the nominal Omega point.

| Metric | Value |
|---|---|
| Metric | PSNR_dB |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `mammography` · **Forward operator:** `mammography_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 4096 |
| W | px | 3328 |
| Kvp | — | 28 |
| Mas | — | 75 |
| Scatter | — | 0.1 |
| Pixel um | µm | 50 |
| Dense tissue overlap | — | 0 |
| Compression variation | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 2048 |
| W | px | 1664 |
| Kvp | — | 22 – 34 |
| Mas | — | 10 – 400 |
| Scatter | — | 0.0 – 0.3 |
| Pixel um | µm | 25 – 100 |
| Dense tissue overlap | — | 0.0 – 0.5 |
| Compression variation | — | 0.0 – 0.2 |
| Geometric magnification | — | 1.0 – 2.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 35.0

| Metric | Range |
|---|---|
| Psnr db | 5.0 – 45.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `180` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xeff332ec150d64d8ce07d27d7f3d4370132bc437f35c6726f04a0a9fb59d816d`
- **Chain tx hash:** `0x1cb31a095e9536b02a750de30141eb5f77f2e96740c572822bb3960a1c2d5bba`
- **Chain block:** `41553385`

---

## File Mapping

This bundle consists of: `L1-036.md`, `L1-036.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-036.md` | Source of truth — edit this | Human or LLM |
| `L1-036.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 `L1-036`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._