# ⚛  L1 Principle — Nuclear Magnetic Resonance (NMR) Spectroscopy

**ID:** `L1-146` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Spectroscopy — *Zeeman-split spin-state spectroscopy*
> **🎯 Problem class:** parametric spectral estimation · **🧮 Solution space:** chemical shift spectrum
> **📡 Carrier:** RF · **🌫 Noise:** complex gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554228

---

## 🧠 1. Introduction

**Nuclear Magnetic Resonance (NMR) Spectroscopy** is a **parametric spectral estimation** whose unknown lives in **chemical shift spectrum** space, within the **Zeeman-split spin-state spectroscopy** sub-domain of **Spectroscopy**.

Measurements consist of RF coil signals (typical of MRI) via a **zeeman precession** sensing mechanism.

The forward operator applies, in order: L · precess · zeeman operator; L · synthesis · harmonic operator; detector accumulates flux over the exposure window.

Observations are corrupted by complex-valued Gaussian noise on real and imaginary parts. Linear inverse via FFT; peak overlap in crowded spectra is dominant ill-posedness source. Mismatch: B0 shim drift, B1 miscalibration, phase drift.

## ⚙ 2. Forward Model

Physical chain: **x** → L · precess · zeeman → L · synthesis · harmonic → Temporal integration → **y** (detector).

```
y = ∫_t dt `L.synthesis.harmonic` `L.precess.zeeman` x + n,    n ~ 𝒞𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.precess.zeeman` | L · precess · zeeman operator |
| `L.synthesis.harmonic` | L · synthesis · harmonic operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Spectroscopy |
| Sub domain | Zeeman-split spin-state spectroscopy |
| Carrier | RF |
| Problem class | parametric_spectral_estimation |
| Solution space | chemical_shift_spectrum |
| Noise model | complex_gaussian |
| Integration axis | temporal |
| Difficulty delta | 3 |
| L dag | 3 |

## 📡 4. Measurement Model

Linear inverse via FFT; peak overlap in crowded spectra is dominant ill-posedness source. Mismatch: B0 shim drift, B1 miscalibration, phase drift.

| Metric | Value |
|---|---|
| Metric | chemical_shift_RMSE_ppm |
| Secondary | integral_NMSE |

## 📏 5. Operating Range (Ω)

**Center problem class:** `1d_1h_nmr_peak_picking` · **Forward operator:** `nmr_fid_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| K | — | 10 |
| N | — | 32768 |
| B0 mhz | MHz | 500 |
| Snr db | dB | 30 |
| Phase distortion | — | 0 |
| B0 inhomogeneity hz | Hz | 1 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| K | — | 1 – 200 |
| N | — | 1024 – 262144 |
| B0 mhz | MHz | 60 – 1200 |
| Snr db | dB | -5 – 50 |
| Baseline roll | — | 0.0 – 0.2 |
| Phase distortion | — | 0.0 – 0.5 |
| B1 miscalibration | — | 0.0 – 0.3 |
| B0 inhomogeneity hz | Hz | 0.0 – 20 |
| Solvent suppression residual | — | 0.0 – 0.1 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.002 ppm RMSE

| Metric | Range |
|---|---|
| Rmse ppm | 0.0001 – 0.1 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **chemical_shift_RMSE_ppm**, with κ = `200` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x552e4cb298b8331e88586fbcf99d52bb96e987da73ac46998ba6417c61cbc624`
- **Chain tx hash:** `0x67005d87bff2b33db657b876350aabc7167ca0809dec97bd58ccf713d15badda`
- **Chain block:** `41554228`

---

## File Mapping

This bundle consists of: `L1-146.md`, `L1-146.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-146.md` | Source of truth — edit this | Human or LLM |
| `L1-146.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 `L1-146`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._