# ⚛  L1 Principle — de Gennes-Doi-Edwards Reptation Theory

**ID:** `L1-476` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Polymer Physics — *Entangled melt dynamics*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** relaxation spectrum H tau
> **📡 Carrier:** none · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41555300

---

## 🧠 1. Introduction

**de Gennes-Doi-Edwards Reptation Theory** is a **linear inverse problem** whose unknown lives in **relaxation spectrum H tau** space, within the **Entangled melt dynamics** sub-domain of **Polymer Physics**.

Measurements consist of none via a **oscillatory rheometry** sensing mechanism.

The forward operator applies, in order: E · doi edwards tube operator; O · composite method operator; int · rheology operator; O · viscoelastic spectrum operator.

Observations are corrupted by additive Gaussian noise. Well-posed for monodisperse melts; polydisperse inversion is ill-conditioned in the transition zone between Rouse and terminal regimes.

## ⚙ 2. Forward Model

Physical chain: **x** → E · doi edwards tube → O · composite method → int · rheology → O · viscoelastic spectrum → **y** (detector).

```
y = `O.viscoelastic_spectrum` `int.rheology` `O.composite_method` `E.doi_edwards_tube` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.doi_edwards_tube` | E · doi edwards tube operator |
| `O.composite_method` | O · composite method operator |
| `int.rheology` | Int · rheology operator |
| `O.viscoelastic_spectrum` | O · viscoelastic spectrum operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Polymer Physics |
| Sub domain | Entangled melt dynamics |
| Carrier | none |
| Problem class | linear_inverse |
| Solution space | relaxation_spectrum_H_tau |
| Noise model | gaussian |
| Integration axis | time |
| Difficulty delta | 5 |
| L dag | 3.5 |

## 📡 4. Measurement Model

Well-posed for monodisperse melts; polydisperse inversion is ill-conditioned in the transition zone between Rouse and terminal regimes.

| Metric | Value |
|---|---|
| Metric | log_G_prime_RMSE |
| Secondary | eta_0_relative_error |

## 📏 5. Operating Range (Ω)

**Center problem class:** `rheological_spectrum_inversion` · **Forward operator:** `reptation_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Pdi | — | 1.05 |
| T c | — | 25 |
| M e kda | — | 5 |
| N over ne | — | 20 |
| Strain pct | — | 1 |
| Omega range decades | — | 7 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Pdi | — | 1.01 – 3.0 |
| T c | — | -50 – 200 |
| M e kda | — | 0.5 – 50.0 |
| N over ne | — | 5 – 200 |
| Strain pct | — | 0.1 – 10.0 |
| Omega range decades | — | 3 – 10 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** log10 G'(omega) RMSE <= 0.05

| Metric | Range |
|---|---|
| Log g prime rmse | 0.03 – 0.3 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **log_G_prime_RMSE**, with κ = `300` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x8ce98f5a72d471ae0540b0ff01c8ef6e4304310fcd72872cd73c28baf8f7e308`
- **Chain tx hash:** `0x9872e5ee09c179fb14c5db94f09e9a5aacc41d20ac4e270ea9ee5d77f1b86250`
- **Chain block:** `41555300`

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## File Mapping

This bundle consists of: `L1-476.md`, `L1-476.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-476.md` | Source of truth — edit this | Human or LLM |
| `L1-476.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 `L1-476`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._