# ⚛  L1 Principle — Flory-Huggins Polymer Solution Theory

**ID:** `L1-335` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Materials Science — *Polymer solution thermodynamics*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** phi T phase boundary
> **📡 Carrier:** none · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554128

---

## 🧠 1. Introduction

**Flory-Huggins Polymer Solution Theory** is a **linear inverse problem** whose unknown lives in **phi T phase boundary** space, within the **Polymer solution thermodynamics** sub-domain of **Materials Science**.

Measurements consist of none via a **cloud point turbidimetry** sensing mechanism.

The forward operator applies, in order: E · mixing entropy operator; E · interaction chi operator; computes eigen-pairs of a linear operator; O · phase diagram operator.

Observations are corrupted by additive Gaussian noise. Well-posed; chi(T) inversion from cloud-point curves is mildly ill-conditioned (entropy/enthalpy trade-off).

## ⚙ 2. Forward Model

Physical chain: **x** → E · mixing entropy → E · interaction chi → O · phase diagram → **y** (detector).

```
y = `O.phase_diagram` `E.interaction_chi` `E.mixing_entropy` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.mixing_entropy` | E · mixing entropy operator |
| `E.interaction_chi` | E · interaction chi operator |
| `O.phase_diagram` | O · phase diagram operator |

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

| Primitive | What it does |
|---|---|
| `E.eigensolve` | Computes eigen-pairs of a linear operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Materials Science |
| Sub domain | Polymer solution thermodynamics |
| Carrier | none |
| Problem class | linear_inverse |
| Solution space | phi_T_phase_boundary |
| Noise model | gaussian |
| Integration axis | composition |
| Difficulty delta | 3 |
| L dag | 3 |

## 📡 4. Measurement Model

Well-posed; chi(T) inversion from cloud-point curves is mildly ill-conditioned (entropy/enthalpy trade-off).

| Metric | Value |
|---|---|
| Metric | phase_boundary_K_error |
| Secondary | chi_AB_relative_error |

## 📏 5. Operating Range (Ω)

**Center problem class:** `flory_huggins` · **Forward operator:** `flory_huggins_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N a | — | 100 |
| N b | — | 100 |
| T k | — | 300 |
| Chi 0 | — | 0.6 |
| Chi slope | — | 0.001 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N a | — | 10 – 10000 |
| N b | — | 10 – 10000 |
| Chi 0 | — | 0.1 – 2.0 |
| T k range | — | 200 – 500 |
| Chi slope | — | 0 – 0.01 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** phase boundary error <= 2 K

| Metric | Range |
|---|---|
| Phase boundary k error | 0.5 – 30.0 |

## ⚖ 7. Hardness Function

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

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x1569fda13b503d650bebc7e018da659b862499b84b07411484cfb74366e6391b`
- **Chain tx hash:** `0x16d9b8ac2a5b8ac2c0e90b98feb2fb80c4087d5c57cef9eecf232a021e87baef`
- **Chain block:** `41554128`

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## File Mapping

This bundle consists of: `L1-335.md`, `L1-335.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-335.md` | Source of truth — edit this | Human or LLM |
| `L1-335.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 `L1-335`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._