# Interactive Papers

This section presents interactive walkthroughs of landmark research papers in neural network
theory. Each paper is expanded into a full educational experience: every proof step is
explained in detail, numerical examples verify the claims, and **interactive applets** let
you explore the key ideas hands-on.

Unlike the main course chapters, which build the theory from scratch, Interactive Papers
assume familiarity with the foundations (Parts I--V) and dive into specific research
contributions. They are designed for students who want to go deeper.

## Available Papers

```{admonition} Paper 1 — An Elementary Proof of a Universal Approximation Theorem
:class: note

**Author:** Chris Monico (Texas Tech University)
**Reference:** arXiv:2406.10002, v2, December 2024

An elegant proof that neural networks with three hidden layers and a 0-1 squashing
activation function can approximate any continuous function on a compact set. The proof
uses only undergraduate analysis — compactness, continuity, the sup norm —
and no functional analysis whatsoever.

**Companion applets:**
[Squashing Function Lab](../applets/squashing-lab.html) ·
[Point-Set Separator](../applets/point-set-separator.html) ·
[Set-Set Separator](../applets/set-set-separator.html) ·
[UAT Contradiction Machine](../applets/uat-contradiction.html)

**Connection:** Complements [Chapter 19](../part5_backpropagation/ch19_universal_approximation) which
presents Cybenko's functional-analytic proof.
```

## How to Use These Walkthroughs

Each paper walkthrough follows the same structure:

1. **Overview & Notations** — context, prerequisites, and symbol reference
2. **Detailed Proofs** — every step expanded with auxiliary commentary (collapsible
   for advanced readers)
3. **Numerical Verification** — Python code that checks each claim with concrete numbers
4. **Interactive Applets** — linked at each key step; explore parameters, drag points,
   and build geometric intuition
5. **Exercises & Challenges** — from routine verification to open-ended research questions