# ⚛ L1 Principle — Density Matrix Renormalization Group (DMRG)
**ID:** `L1-322` · **Status:** ⊙ Testnet (genesis catalog)
> **🌐 Domain:** Quantum Mechanics — *Matrix-product-state ground states*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** matrix product state
> **📡 Carrier:** electron · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 10 · **⛓ Block:** 41554081
---
## 🧠 1. Introduction
**Density Matrix Renormalization Group (DMRG)** is a **nonlinear inverse problem** whose unknown lives in **matrix product state** space, within the **Matrix-product-state ground states** sub-domain of **Quantum Mechanics**.
Measurements consist of electrons collected by an electron detector via a **energy correlation benchmark** sensing mechanism.
The forward operator applies, in order: E · MPO hamiltonian operator; adds a prior term that biases the solution toward smoothness/sparsity; O · gs energy operator.
Observations are corrupted by additive Gaussian noise. Exponential precision with bond dimension M in 1D; 2D DMRG limited by transverse correlation length.
## ⚙ 2. Forward Model
Physical chain: **x** → E · MPO hamiltonian → O · gs energy → **y** (detector).
```
y = `O.gs_energy` `E.MPO_hamiltonian` x + n, n ~ 𝒩(0, σ²)
```
**Measurement DAG:**
| Primitive | What it does |
|---|---|
| `E.MPO_hamiltonian` | E · mpo hamiltonian operator |
| `O.gs_energy` | O · gs energy operator |
**🛠 Solver components** _(used inside the solver, not in the forward equation)_:
| Primitive | What it does |
|---|---|
| `O.regularize` | Adds a prior term that biases the solution toward smoothness/sparsity |
## 🔬 3. Physics Fingerprint
| Property | Value |
|---|---|
| Domain | Quantum Mechanics |
| Sub domain | Matrix-product-state ground states |
| Carrier | electron |
| Problem class | nonlinear_inverse |
| Solution space | matrix_product_state |
| Noise model | gaussian |
| Integration axis | site_index |
| Difficulty delta | 10 |
| L dag | 3.8 |
## 📡 4. Measurement Model
Exponential precision with bond dimension M in 1D; 2D DMRG limited by transverse correlation length.
| Metric | Value |
|---|---|
| Metric | gs_energy_error_mHa_per_site |
| Secondary | entanglement_entropy_error |
## 📏 5. Operating Range (Ω)
**Center problem class:** `dmrg` · **Forward operator:** `dmrg_forward`
**Center point:**
| Parameter | Unit | Value |
|---|---|---|
| Model | — | Heisenberg_spin_half |
| L sites | — | 100 |
| N sweeps | — | 20 |
| Bond dim m | m | 200 |
**Allowed bounds:**
| Parameter | Unit | Range |
|---|---|---|
| L sites | — | 10 – 10000 |
| N sweeps | — | 5 – 200 |
| Bond dim m | m | 20 – 10000 |
## 🎯 6. Tolerance (ε)
**Center tolerance:** E/site error <= 0.1 mHa
| Metric | Range |
|---|---|
| Gs energy error mha per site | 0.01 – 10 |
## ⚖ 7. Hardness Function
Hardness scales as **`epsilon_fn`** on **gs_energy_error_mHa_per_site**, with κ = `2000` and δ = `10`.
## 💾 8. Reference Dataset
- **primary** · weight 1.0 · IPFS _(not pinned yet)_
## 9. On-chain Registration
- **Chain hash:** `0xbc5d5c6cd3a3ecd4c21449e1696807dbe2d7d4a9eb7dfdc3cecb2049cc1f0131`
- **Chain tx hash:** `0x8bdc1ba462a809c8da5d0beb1dc469cb04ac5311b487f0907277292e32cfa2fc`
- **Chain block:** `41554081`
---
## File Mapping
This bundle consists of: `L1-322.md`, `L1-322.json`.
| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-322.md` | Source of truth — edit this | Human or LLM |
| `L1-322.json` | Structured metadata for the registry | LLM regenerates from the sections above |
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> Read the attached Markdown. Regenerate the sibling `.json` so every field matches.
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