Understanding train_gpt.py: A Complete Mental Model of the Parameter Golf Pipeline

A deep breakdown of the full training, evaluation, and compression pipeline — and where the real optimization levers are.

When the Heart Speaks Math: AI-Controlled Coupled Oscillators in the Browser

Modeling respiratory sinus arrhythmia with two coupled Van der Pol oscillators and a PyTorch controller deployed via ONNX and Next.js. The full train → export → serve pipeline, running live in the browser.

The Future Programmer Won't Just Code — They'll Choose a Model of Reality

Programming is fracturing into layers — each with its own model of what 'correct' means. Formal proofs, simulations, probabilistic thinking. The future programmer won't just write code, they'll choose an epistemology.

FirstProof — The First Real Research Math Test for AI

A new experiment testing whether AI can autonomously solve real research mathematics — and what it reveals about the future of reasoning.

Basis as a Coordinate System (Not Just a Set of Vectors)

Why coordinates depend on the basis, and why a basis is really a measuring grid.

Why a Basis Needs Both Spanning and Independence

Coordinates must exist and must be unique. That’s why bases require spanning and linear independence.

Putnam 2016 A1: Where Insight Lives (and Why Formal Proofs Expose It)

A mathematical look at Putnam 2016 A1, focusing on where the real difficulty lies—and what formalization makes explicit.

Putnam, Proof Assistants, and the Moving Boundary of Intelligence

What PUTNAMBENCH tells us about AI, formal mathematics, and why intelligence keeps shifting as tools improve.

Formalizing Compactness in Lean: From Paper to Proof Assistant

Translating the continuous image of compact sets theorem into Lean 4

The Image of Compact Sets: A Love Letter to Continuity

Exploring how continuous functions preserve compactness in topological spaces

Why Dependent Types Actually Make Sense: A Mathematician's Journey into Lean

Dependent type theory sounds abstract, but it's actually the natural way to express mathematical ideas. A hands-on exploration using examples from topology, analysis, and abstract algebra—with Mermaid diagrams to visualize how types depend on values.

Discovering Lean: Teaching Computers to Understand Mathematical Proof

What if you could write proofs where a computer checks every step, catches mistakes immediately, and helps you figure out what to do next? Welcome to Lean, the proof assistant that's changing how we think about mathematics.

Hamiltonian Systems: The Mathematics of Conservation

Why some motions persist forever — energy, phase space, and structure

How Dustin Used Algebra to Discover the Upside Down’s Circular Wall

Stranger Things meets geometry: how three measurements and perpendicular bisectors reveal a circular boundary around the Upside Down.

Dialogue on Brains, Robots, and the Equation We Can’t Write

A personal dialogue on robots, brains, Gödel, and why I still believe in chasing the missing equation—even if we may never write it.

Subset sums, generating functions, and roots of unity

From a small example to a slick root-of-unity trick.

Van der Pol Oscillator in the Browser

A WASM-powered phase portrait demo.

Putnam 2017 A1

Our exploration of the 2017 Putnam A1 problem with Dr. Pragel — and how a simple modular idea connects everything beautifully.

Characters of a Group and What They Remember

From traces of representations to character tables for Z₄, S₃, and D₄.

Putnam 2016 A1

My handwritten exploration, Dr. Pragel’s explanation, and the reasoning behind why the smallest integer j is 8.

From Lines to Planes: Understanding Two-Dimensional Linear Systems

A geometric journey through two-dimensional linear systems — centers, spirals, saddles, and the beauty of phase portraits.

3I/ATLAS: The Interstellar Visitor Older Than the Sun

For the third time in human history, astronomers have spotted an object that doesn’t belong to our Solar System. Meet 3I/ATLAS — an icy traveler from beyond the stars.

From Pendulums to Planets: How Perturbation Theory Explains Perihelion Precession

From small-angle pendulums to Mercury’s orbit, perturbation theory reveals how tiny nonlinearities create measurable cosmic effects.

Perturbation Theory

A clear introduction to perturbation theory — how small parameters help us approximate and understand complex systems.

The Pigeonhole Principle

Pigeonhole Principle explanation

One-Dimensional Dynamical Systems

An intuitive guide to understanding one-dimensional dynamical systems through geometric and physical perspectives.

Five Points on a Sphere

Five Points on a Sphere

Building a Local AI-Powered Shell Assistant (and Making It Global!)

How I created my own custom ChatGPT-powered terminal assistant that understands natural language, runs locally with Ollama + Mistral, and can be launched from anywhere as a global CLI.

Verifying the GVZ Property in GAP: A Computational Approach

We explore how to check whether a finite group is a GVZ-group using GAP, character theory, and code. Includes visual flow and example analysis.

How I Learned to Break Down Proofs (and Think Like a Mathematician)

Dr. Harmon’s advice helped me move beyond memorization and learn how to understand the structure of mathematical proofs.

Elysium Isn’t Just Sci-Fi — It’s a Warning About Tech Inequality

Tech Inequality

Astro Crash Course: Build Lightning-Fast Sites With Ease

A quick guide for learning Astro the practical way, perfect for small business sites, blogs, and component-driven projects.

Conic sections with eccentricity

Interactive demo of how e changes the shape

GRE Group Theory Problem

Step-by-step solution and visual explanation of a basic GRE series problem

Lessons Learned: Making Animated Math Work in Astro + React + MDX

A behind-the-scenes look at how I combined Framer Motion animations, MathJax rendering, and Astro content to create smooth visual math walkthroughs.

Minimum of a Parabola

Step-by-step solution and visual explanation of a basic GRE algebra problem

Extracting Topology from Data: A Persistent Homology Study with the Iris Dataset

Exploring topological data analysis using persistent homology and the Iris dataset

Reflections on the Cantor Set and My Journey Through Topology

How the Cantor set and compactness helped reshape my intuition for infinity, structure, and rigor in topology.