# ⚛  L1 Principle — Mantle Convection — high-Ra thermal convection

**ID:** `L1-286` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Geophysics — *Extended Boussinesq mantle flow*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** mantle flow and thermal structure
> **📡 Carrier:** thermal · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 10 · **⛓ Block:** 41554065

---

## 🧠 1. Introduction

**Mantle Convection — high-Ra thermal convection** is a **nonlinear inverse problem** whose unknown lives in **mantle flow and thermal structure** space, within the **Extended Boussinesq mantle flow** sub-domain of **Geophysics**.

Measurements consist of thermal radiation / temperature measurements via a **seismic tomography constraint** sensing mechanism.

The forward operator applies, in order: L · stokes infinite prandtl operator; L · energy operator; L · rheology operator; detector accumulates flux over the exposure window.

Observations are corrupted by additive Gaussian noise. Chaotic for Ra > 10^7; initial condition of mantle is unknown.

## ⚙ 2. Forward Model

Physical chain: **x** → L · stokes infinite prandtl → L · energy → L · rheology → Temporal integration → **y** (detector).

```
y = ∫_t dt `L.rheology` `L.energy` `L.stokes_infinite_prandtl` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.stokes_infinite_prandtl` | L · stokes infinite prandtl operator |
| `L.energy` | L · energy operator |
| `L.rheology` | L · rheology operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Geophysics |
| Sub domain | Extended Boussinesq mantle flow |
| Carrier | thermal |
| Problem class | nonlinear_inverse |
| Solution space | mantle_flow_and_thermal_structure |
| Noise model | gaussian |
| Integration axis | temporal |
| Difficulty delta | 10 |
| L dag | 4 |

## 📡 4. Measurement Model

Chaotic for Ra > 10^7; initial condition of mantle is unknown.

| Metric | Value |
|---|---|
| Metric | flow_rel_error_percent |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `mantle_convection` · **Forward operator:** `mantle_convection_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 128 |
| W | px | 128 |
| Z | — | 128 |
| Ra | — | 1e+07 |
| Snr db | dB | 25 |
| N times | — | 100 |
| Bc error | — | 0 |
| Rheology error | — | 0.1 |
| Time history unknown | — | 0.3 |
| Viscosity contrast error | — | 0.2 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 32 – 1024 |
| W | px | 32 – 1024 |
| Z | — | 32 – 1024 |
| Ra | — | 100000.0 – 1000000000.0 |
| Snr db | dB | 0 – 40 |
| N times | — | 10 – 10000 |
| Bc error | — | 0.0 – 0.3 |
| Rheology error | — | 0.0 – 0.3 |
| Time history unknown | — | 0.0 – 0.5 |
| Viscosity contrast error | — | 0.0 – 0.5 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 15.0

| Metric | Range |
|---|---|
| Flow rel | 1.0 – 60.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **flow_rel_error_percent**, with κ = `1600` and δ = `10`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x6c41c889e917cbf9bf0f5f86ccdb1ce1fa181f681d8df7288a2eec67eea3d4ba`
- **Chain tx hash:** `0x7aa1fc0f0655ea4954c7121ad8f04879dad9246b8e7371551f4678dc225ffd1f`
- **Chain block:** `41554065`

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## File Mapping

This bundle consists of: `L1-286.md`, `L1-286.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-286.md` | Source of truth — edit this | Human or LLM |
| `L1-286.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 `L1-286`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._