# ⚛  L1 Principle — PillCam-SPECTRA Bleeding-Region Detection (PWDR)

**ID:** `L1-512` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Medical Imaging — *Capsule-endoscopic bleeding / polyp / mucosal-anomaly classification via thresholded chromophore recovery*
> **🎯 Problem class:** linear inverse with categorical readout · **🧮 Solution space:** 2D categorical mucosal label map
> **📡 Carrier:** photon · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41553373

---

## 🧠 1. Introduction

**PillCam-SPECTRA Bleeding-Region Detection (PWDR)** is a **linear inverse with categorical readout** whose unknown lives in **2D categorical mucosal label map** space, within the **Capsule-endoscopic bleeding / polyp / mucosal-anomaly classification via thresholded chromophore recovery** sub-domain of **Medical Imaging**.

Measurements consist of photons collected by an optical detector via a **diffuse reflectance capsule with threshold classifier** sensing mechanism.

The forward operator applies, in order: L · optical source operator; L · diffuse reflectance operator; exponential attenuation along the propagation path; L · detector response operator; L · chromophore unmixing operator; L · threshold classifier operator; pixel-level spatial averaging on the detector; detector sums all spectral bands.

Observations are corrupted by additive Gaussian noise. Existence and uniqueness inherited from L1-511 analytical core; threshold function is piecewise-constant and Lipschitz-continuous on the underlying chromophore-concentration state, so existence and uniqueness lift from continuous to discrete readout under the standard categorical-readout argument (PWM_GATE_CLASSES_AND_SCALING.md Case A). Stability inherits L1-511's kappa_eff ~ 15 plus a small additive contribution C_threshold * |threshold_calibration_uncertainty| from per-class threshold-value uncertainty. The threshold-continuity_proof in discrete_readout demonstrates that boundaries between class regions form measure-zero hyperplanes in chromophore space, so misclassification probability scales linearly with chromophore-recovery error away from the boundary.

## ⚙ 2. Forward Model

Physical chain: **x** → L · optical source → L · diffuse reflectance → Beer-Lambert attenuation → L · detector response → L · chromophore unmixing → L · threshold classifier → Spatial integration → Spectral integration → **y** (detector).

```
y = Σ_λ ∫_A dA `L.threshold_classifier` `L.chromophore_unmixing` `L.detector_response` exp(-∫µ dl) `L.diffuse_reflectance` `L.optical_source` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.optical_source` | L · optical source operator |
| `L.diffuse_reflectance` | L · diffuse reflectance operator |
| `L.beer_lambert` | Exponential attenuation along the propagation path |
| `L.detector_response` | L · detector response operator |
| `L.chromophore_unmixing` | L · chromophore unmixing operator |
| `L.threshold_classifier` | L · threshold classifier operator |
| `int.spatial` | Pixel-level spatial averaging on the detector |
| `int.spectral` | Detector sums all spectral bands |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Medical Imaging |
| Sub domain | Capsule-endoscopic bleeding / polyp / mucosal-anomaly classification via thresholded chromophore recovery |
| Carrier | photon |
| Problem class | linear_inverse_with_categorical_readout |
| Solution space | 2D_categorical_mucosal_label_map |
| Noise model | gaussian |
| Integration axis | spectral_spatial |
| Difficulty delta | 3 |
| L dag | 6.9 |

## 📡 4. Measurement Model

Existence and uniqueness inherited from L1-511 analytical core; threshold function is piecewise-constant and Lipschitz-continuous on the underlying chromophore-concentration state, so existence and uniqueness lift from continuous to discrete readout under the standard categorical-readout argument (PWM_GATE_CLASSES_AND_SCALING.md Case A). Stability inherits L1-511's kappa_eff ~ 15 plus a small additive contribution C_threshold * |threshold_calibration_uncertainty| from per-class threshold-value uncertainty. The threshold-continuity_proof in discrete_readout demonstrates that boundaries between class regions form measure-zero hyperplanes in chromophore space, so misclassification probability scales linearly with chromophore-recovery error away from the boundary.

| Metric | Value |
|---|---|
| Metric | categorical_accuracy |
| Secondary | mIoU_per_class |

## 📏 5. Operating Range (Ω)

**Center problem class:** `pillcam_spectra_bleeding_pwdr` · **Forward operator:** `pillcam_spectra_pwdr_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 256 |
| W | px | 256 |
| Snr db | dB | 26 |
| Tau bvf | — | 0.04 |
| Tau hbo2 | — | 0.5 |
| Tau edema | — | 0.45 |
| N wavelengths | — | 8 |
| Lambda range nm | nm | 400 – 950 |
| Tau lipid polyp | — | 0.15 |
| Tau total blood | — | 0.06 |
| Tau scatter neop | — | 22 |
| Tau scatter polyp | — | 18 |
| Tau oxy ratio neop | — | 4 |
| Working distance mm | mm | 2 |
| Scattering anisotropy g | g | 0.85 |
| Source detector distance mm | mm | 5 |
| Motion induced label flicker | — | 0 |
| Between region smoothing artifact | — | 0 |
| Threshold calibration uncertainty | — | 0 |
| Capsule orientation mucosal assignment | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 128 – 1024 |
| W | px | 128 – 1024 |
| Snr db | dB | 10.0 – 38.0 |
| N wavelengths | — | 3 – 16 |
| Motion induced label flicker | — | 0.0 – 0.4 |
| Between region smoothing artifact | — | 0.0 – 0.3 |
| Threshold calibration uncertainty | — | 0.0 – 0.3 |
| Capsule orientation mucosal assignment | — | 0.0 – 0.4 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.85_accuracy

| Metric | Range |
|---|---|
| Categorical accuracy | 0.5 – 0.99 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **categorical_accuracy**, with κ = `80` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xeb1e83e6df8f26f674a4f3c080ee808696aa30e1cba1fad3f8e0044f7f4d9b85`
- **Chain tx hash:** `0x3504adc2e3ab9fe2688f49a503724aa99f6d3ac7813dbb3d95122cb52d8f5485`
- **Chain block:** `41553373`

---

## File Mapping

This bundle consists of: `L1-512.md`, `L1-512.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-512.md` | Source of truth — edit this | Human or LLM |
| `L1-512.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 `L1-512`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._