# ⚛  L1 Principle — Self-Consistent Field Theory (SCFT)

**ID:** `L1-475` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Polymer Physics — *Block copolymer microphase separation*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 3D composition field
> **📡 Carrier:** none · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41555300

---

## 🧠 1. Introduction

**Self-Consistent Field Theory (SCFT)** is a **nonlinear inverse problem** whose unknown lives in **3D composition field** space, within the **Block copolymer microphase separation** sub-domain of **Polymer Physics**.

Measurements consist of none via a **scattering forward** sensing mechanism.

The forward operator applies, in order: E · propagator diffusion operator; a fixed-point or gradient iteration on the unknown; O · density field operator; O · scattering operator.

Observations are corrupted by additive Gaussian noise. Multiple local minima (lamellae, gyroid, hexagonal) → non-unique inversion; phase diagram navigation requires morphology seeding.

## ⚙ 2. Forward Model

Physical chain: **x** → E · propagator diffusion → O · density field → O · scattering → **y** (detector).

```
y = `O.scattering` `O.density_field` `E.propagator_diffusion` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.propagator_diffusion` | E · propagator diffusion operator |
| `O.density_field` | O · density field operator |
| `O.scattering` | O · scattering operator |

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

| Primitive | What it does |
|---|---|
| `O.iter` | A fixed-point or gradient iteration on the unknown |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Polymer Physics |
| Sub domain | Block copolymer microphase separation |
| Carrier | none |
| Problem class | nonlinear_inverse |
| Solution space | 3D_composition_field |
| Noise model | gaussian |
| Integration axis | chain_contour |
| Difficulty delta | 5 |
| L dag | 3.8 |

## 📡 4. Measurement Model

Multiple local minima (lamellae, gyroid, hexagonal) → non-unique inversion; phase diagram navigation requires morphology seeding.

| Metric | Value |
|---|---|
| Metric | field_RMSE |
| Secondary | S_q_peak_position_error |

## 📏 5. Operating Range (Ω)

**Center problem class:** `scft_morphology_inversion` · **Forward operator:** `scft_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| F | — | 0.5 |
| N ds | — | 100 |
| Chi n | N | 15 |
| Grid n | N | 64 |
| L box r g | g | 4 |
| Mixing alpha | — | 0.05 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| F | — | 0.2 – 0.8 |
| N ds | — | 50 – 400 |
| Chi n | N | 10 – 40 |
| Grid n | N | 32 – 128 |
| L box r g | g | 2.0 – 10.0 |
| Mixing alpha | — | 0.01 – 0.2 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** phi_A RMSE <= 0.025

| Metric | Range |
|---|---|
| Field rmse | 0.01 – 0.2 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **field_RMSE**, with κ = `1200` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xc56bbf2b12bad462c8eb1df69631cf5ddf19be662a2224471a85d2cf5c85a6cd`
- **Chain tx hash:** `0x6cb7d134443dc1a082d20e89d16c4df0be62d806a2efcda2597e62905da09678`
- **Chain block:** `41555300`

---

## File Mapping

This bundle consists of: `L1-475.md`, `L1-475.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-475.md` | Source of truth — edit this | Human or LLM |
| `L1-475.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 `L1-475`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._