# ⚛  L1 Principle — 3D Gaussian Splatting (3DGS) — explicit real-time rendering

**ID:** `L1-083` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Computational Optics — *Explicit primitive-based 3D reconstruction*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 3D explicit primitives
> **📡 Carrier:** photon · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554183

---

## 🧠 1. Introduction

**3D Gaussian Splatting (3DGS) — explicit real-time rendering** is a **nonlinear inverse problem** whose unknown lives in **3D explicit primitives** space, within the **Explicit primitive-based 3D reconstruction** sub-domain of **Computational Optics**.

Measurements consist of photons collected by an optical detector via a **multi view photogrammetry** sensing mechanism.

The forward operator applies, in order: S · scan · view operator; L · project · gaussian operator; L · alpha composite operator; int · depth operator.

Observations are corrupted by additive Gaussian noise. Non-convex parameter fitting. Unique 3D Gaussian cloud only up to primitive-reparameterization (splitting one Gaussian into two overlapping ones yields identical rendering). Stability risks: pose error introduces baked-in floaters; aggressive densification produces over-opaque pancake Gaussians (mitigated by Mip-Splatting / 2DGS).

## ⚙ 2. Forward Model

Physical chain: **x** → S · scan · view → L · project · gaussian → L · alpha composite → int · depth → **y** (detector).

```
y = `int.depth` `L.alpha_composite` `L.project.gaussian` `S.scan.view` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `S.scan.view` | S · scan · view operator |
| `L.project.gaussian` | L · project · gaussian operator |
| `L.alpha_composite` | L · alpha composite operator |
| `int.depth` | Int · depth operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Computational Optics |
| Sub domain | Explicit primitive-based 3D reconstruction |
| Carrier | photon |
| Problem class | nonlinear_inverse |
| Solution space | 3D_explicit_primitives |
| Noise model | gaussian |
| Integration axis | angular |
| Difficulty delta | 5 |
| L dag | 3.5 |

## 📡 4. Measurement Model

Non-convex parameter fitting. Unique 3D Gaussian cloud only up to primitive-reparameterization (splitting one Gaussian into two overlapping ones yields identical rendering). Stability risks: pose error introduces baked-in floaters; aggressive densification produces over-opaque pancake Gaussians (mitigated by Mip-Splatting / 2DGS).

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

## 📏 5. Operating Range (Ω)

**Center problem class:** `gaussian_splatting_novel_view` · **Forward operator:** `gaussian_splat_rasterize`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 1080 |
| W | px | 1920 |
| L sh | — | 3 |
| K views | — | 200 |
| N gaussians | — | 1000000 |
| Photon count | — | 2000 |
| Non lambertian | — | 0 |
| Pose error rad | rad | 0 |
| Per frame exposure variation | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 400 – 4000 |
| W | px | 400 – 4000 |
| L sh | — | 0 – 4 |
| K views | — | 10 – 1000 |
| N gaussians | — | 100000 – 10000000 |
| Photon count | — | 100 – 10000 |
| Non lambertian | — | 0.0 – 0.5 |
| Pose error rad | rad | 0.0 – 0.1 |
| N budget too small | — | 0.0 – 0.8 |
| Motion blur per frame | — | 0.0 – 0.1 |
| Per frame exposure variation | — | 0.0 – 0.5 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 31.0 dB PSNR

| Metric | Range |
|---|---|
| Psnr db | 18.0 – 42.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `4000` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x129e3a3a772e37f0a0a226bb79566e255da24be9cc182ba6c339adf3894b0ad7`
- **Chain tx hash:** `0xdd1929f81145c7971c11e75b579dc2ab281677a4da64a899749fe9aa1c081723`
- **Chain block:** `41554183`

---

## File Mapping

This bundle consists of: `L1-083.md`, `L1-083.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-083.md` | Source of truth — edit this | Human or LLM |
| `L1-083.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 `L1-083`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._