Algebra

Reverse-Process Synthetic Data Generation for Math Reasoning

Training LLMs on mathematical reasoning by inverting easy-to-solve problems: generate derivatives, reverse them into integration exercises with full step-by-step solutions.

Watch: Algorithms Arise from Algebraic Structure

The Stepanov series is now a nine-episode animated playlist: declare the algebraic structure, and the algorithms come for free. From the monoid to lattices, in teacher and student dialogue.

Synthesis: Codecs as Structure

The series closes by restating the codes-as-priors thesis across all twelve instances and connecting the wire-format side to the Stepanov type-algebra side.

A Book of Abstract Algebra

Abstract Algebra

Language Calculus: An Algebraic Framework for LLM Composition

A mathematical framework that treats language models as algebraic objects with compositional structure.

Algebraic Hashing A Modern C++20 Library for Composable Hash Functions Version 2.0

Algebraic Hashing: Composable Hash Functions Through XOR

A C++ library for composable hash functions using algebraic structure over XOR, with template metaprogramming.

algebraic.mle: MLEs as Algebraic Objects

An R package that treats MLEs as algebraic objects. They carry Fisher information, compose through independent likelihoods, and propagate uncertainty correctly.

algebraic.dist: Distributions as Algebraic Objects in R

An R package that treats probability distributions as algebraic objects. They compose through standard operations. The algebra preserves distributional structure.

Polynomials as Euclidean Domains

The same GCD algorithm works for integers and polynomials because both are Euclidean domains. One structure, many types, same algorithms.

Modular Arithmetic as Rings

Integers modulo N form a ring, an algebraic structure that determines which algorithms apply. Understanding this structure unlocks algorithms from cryptography to competitive programming.