October Research Notes: Graphs, Proofs, and Experiments
Recent work on graph saturation, formal mathematics, and conversational memory, with a few experiments that changed what I thought I was measuring.
Alex Towell's research notebook
Mathematics, code, and ideas worth testing
I'm Alex Towell. I explore AI, mathematics, and privacy through code, experiments, and writing. This is where the pieces come together.
From the notebook
Recent writing
Finding Every Smallest C6-Saturated Graph
An eleven-vertex witness, 205 minimum graphs on fourteen vertices, and the difference between finding an example and proving a minimum.
Watch: Calculus, The Big Ideas
Calculus as one idea examined from every side: a seven-episode animated playlist running from limits through the derivative, the integral, the fundamental theorem, gradients, and optimization.
Watch: Quantum Computing from Scratch, in Python
The quantum-from-scratch series is now a ten-episode animated playlist: a NumPy simulator built from nothing, amplitude bars you can watch interfere, and a classical on-ramp with the coin under the cup.
Watch: Next-Token Prediction, from Solomonoff to Transformers
The sequential-prediction series is now an eight-episode animated playlist: what-comes-next as the foundation of the language-model era, traced from Solomonoff induction to the transformer.
Watch: Inductive Biases, What Your Architecture Assumes
The inductive-biases series is now a ten-episode animated playlist: every architecture is a bet about the world, and the bet is the bias. No free lunch first, then each architecture read as its assumptions, graded on one …
Open source, in practice
Build with the code
arkiv
Universal personal data format. JSONL in, SQL out, SQL back to JSONL. One format, one database, one query interface.
memex
Conversation knowledge base with enrichments, provenance tracking, and MCP server
chartfold
Patient-facing tool for consolidating personal health data from multiple EHR systems into a single SQLite database
Questions, methods, evidence
Explore the research
Cognitive MRI of AI Conversations: Analyzing AI Interactions through Semantic Embedding Networks
Through a single-user case study of 449 ChatGPT conversations, we introduce a cognitive MRI applying network analysis to reveal thought topology hidden in linear conversation logs. We construct semantic similarity …
Reliability Estimation in Series Systems Maximum Likelihood Techniques for Right-Censored and Masked Failure Data
This paper investigates maximum likelihood techniques to estimate component reliability from masked failure data in series systems. A likelihood model accounts for right-censoring and candidate sets indicative of masked …
Encrypted Search - Enabling Standard Information Retrieval Techniques for Several New Secure Index Types While Preserving Confidentiality Against an Adversary With Access to Query Histories and Secure Index Contents
Encrypted Search, a technique for securely searching documents on untrusted systems. The author designs and compares several secure index types to determine their performance in terms of time complexity, space …