# ⚛  L1 Principle — Industrial Computed Tomography (high-res x-ray CT for NDT)

**ID:** `L1-111` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Industrial Inspection — *Micro-/nano-CT non-destructive inspection*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D attenuation
> **📡 Carrier:** x_ray · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554199

---

## 🧠 1. Introduction

**Industrial Computed Tomography (high-res x-ray CT for NDT)** is a **linear inverse problem** whose unknown lives in **3D attenuation** space, within the **Micro-/nano-CT non-destructive inspection** sub-domain of **Industrial Inspection**.

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

The forward operator applies, in order: polyenergetic X-ray emission spectrum; rotates source / detector to acquire different projections; exponential attenuation along the propagation path; spreads measurements back along source rays (adjoint operator); integration over the solid angle of incidence/emission.

Observations are corrupted by Poisson shot noise from quantum-limited detection. Existence of the recovered 3D attenuation 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 moderately conditioned (kappa_eff ~= 18); beam_hardening dominates the stability cliff; scatter and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) sets the irreducible data-fidelity floor, while TV / wavelet-sparsity / deep priors stabilise recovery at the ill-conditioned end of Omega.

## ⚙ 2. Forward Model

Physical chain: **x** → X-ray source → Angular scan → Beer-Lambert attenuation → Angular integration → **y** (detector).

```
y = ∫dΩ exp(-∫µ dl) R(θ) I₀(E) x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.xray_source` | Polyenergetic x-ray emission spectrum |
| `S.scan.angular` | Rotates source / detector to acquire different projections |
| `L.beer_lambert` | Exponential attenuation along the propagation path |
| `int.angular` | Integration over the solid angle of incidence/emission |

**🛠 Solver components** _(used inside the solver, not in the forward equation)_:

| Primitive | What it does |
|---|---|
| `L.backproject` | Spreads measurements back along source rays (adjoint operator) |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Industrial Inspection |
| Sub domain | Micro-/nano-CT non-destructive inspection |
| Carrier | x_ray |
| Problem class | linear_inverse |
| Solution space | 3D_attenuation |
| Noise model | shot_poisson |
| Integration axis | angular |
| Difficulty delta | 5 |
| L dag | 3.8 |

## 📡 4. Measurement Model

Existence of the recovered 3D attenuation 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 moderately conditioned (kappa_eff ~= 18); beam_hardening dominates the stability cliff; scatter and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) sets the irreducible data-fidelity floor, while TV / wavelet-sparsity / deep priors stabilise recovery at the ill-conditioned end of Omega.

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

## 📏 5. Operating Range (Ω)

**Center problem class:** `industrial_ct` · **Forward operator:** `industrial_ct_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 2048 |
| W | px | 2048 |
| Z | — | 512 |
| Kv | — | 150 |
| Scatter | — | 0 |
| Pixel um | µm | 50 |
| Misalignment | — | 0 |
| N projections | — | 720 |
| Beam hardening | — | 0 |
| Photon count per ray | — | 10000 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 512 – 8192 |
| W | px | 512 – 8192 |
| Z | — | 64 – 4096 |
| Kv | — | 20 – 450 |
| Scatter | — | 0.0 – 0.4 |
| Pixel um | µm | 0.5 – 500 |
| Misalignment | — | 0.0 – 5.0 |
| N projections | — | 180 – 4320 |
| Ring artifact | — | 0.0 – 0.2 |
| Beam hardening | — | 0.0 – 0.3 |
| Photon count per ray | — | 100 – 100000 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 28.0

| Metric | Range |
|---|---|
| Psnr db | 5.0 – 40.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `360` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x7b559b400db1df155a398cb129f8e36a0278ccc22d8a27905ead9b4e0641ce27`
- **Chain tx hash:** `0x100aad798be7b7621886dff4347810c1c3dc6bf1934b65809a7b6f0378d8d6c0`
- **Chain block:** `41554199`

---

## File Mapping

This bundle consists of: `L1-111.md`, `L1-111.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-111.md` | Source of truth — edit this | Human or LLM |
| `L1-111.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 `L1-111`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._