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# 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

Paper 1 — An Elementary Proof of a Universal Approximation Theorem

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 · Point-Set Separator · Set-Set Separator · UAT Contradiction Machine

Connection: Complements Chapter 19 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