# ⚛  L1 Principle — Transient Heat Conduction — inverse heat conduction (IHCP)

**ID:** `L1-195` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Heat Transfer — *Parabolic PDE inverse source/BC estimation*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D temperature field spacetime
> **📡 Carrier:** thermal · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554033

---

## 🧠 1. Introduction

**Transient Heat Conduction — inverse heat conduction (IHCP)** is a **linear inverse problem** whose unknown lives in **3D temperature field spacetime** space, within the **Parabolic PDE inverse source/BC estimation** sub-domain of **Heat Transfer**.

Measurements consist of thermal radiation / temperature measurements via a **thermocouples** sensing mechanism.

The forward operator applies, in order: L · initial condition operator; L · parabolic propagate operator; S · record · sensors operator; detector accumulates flux over the exposure window.

Observations are corrupted by additive Gaussian noise. IHCP is ill-posed (Hadamard): high-frequency BC content decays exponentially. Requires regularization (Tikhonov, future-time-stepping, PINN).

## ⚙ 2. Forward Model

Physical chain: **x** → L · initial condition → L · parabolic propagate → S · record · sensors → Temporal integration → **y** (detector).

```
y = ∫_t dt `S.record.sensors` `L.parabolic_propagate` `L.initial_condition` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.initial_condition` | L · initial condition operator |
| `L.parabolic_propagate` | L · parabolic propagate operator |
| `S.record.sensors` | S · record · sensors operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Heat Transfer |
| Sub domain | Parabolic PDE inverse source/BC estimation |
| Carrier | thermal |
| Problem class | linear_inverse |
| Solution space | 3D_temperature_field_spacetime |
| Noise model | gaussian |
| Integration axis | temporal |
| Difficulty delta | 5 |
| L dag | 3 |

## 📡 4. Measurement Model

IHCP is ill-posed (Hadamard): high-frequency BC content decays exponentially. Requires regularization (Tikhonov, future-time-stepping, PINN).

| Metric | Value |
|---|---|
| Metric | PSNR_dB |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `ihcp` · **Forward operator:** `ihcp_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 64 |
| W | px | 64 |
| Z | — | 64 |
| Fs hz | Hz | 10 |
| Snr db | dB | 30 |
| K error | — | 0 |
| Bc error | — | 0 |
| N samples | — | 200 |
| N sensors | — | 16 |
| Rho cp error | — | 0 |
| Sensor pos error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 16 – 512 |
| W | px | 16 – 512 |
| Z | — | 8 – 256 |
| Fs hz | Hz | 0.1 – 1000 |
| Snr db | dB | 0 – 40 |
| K error | — | 0.0 – 0.1 |
| Bc error | — | 0.0 – 0.5 |
| N samples | — | 20 – 10000 |
| N sensors | — | 4 – 256 |
| Rho cp error | — | 0.0 – 0.1 |
| Sensor pos error | — | 0.0 – 0.02 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 25.0

| Metric | Range |
|---|---|
| Psnr db | 10.0 – 45.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `800` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x366622440a8eb6dc6b7aadc3798ac1d2dc997b5aa608e00baf504c2d99c265eb`
- **Chain tx hash:** `0x0a179d9695a8fc81fd7610fafc8d16e723fac4c7e41bcf3ca5cac6074d737ae0`
- **Chain block:** `41554033`

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## File Mapping

This bundle consists of: `L1-195.md`, `L1-195.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-195.md` | Source of truth — edit this | Human or LLM |
| `L1-195.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 `L1-195`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._