YonedaAI Research Collective
An independent research initiative developing rigorous mathematical foundations for understanding and eliminating hallucination in large language models.
“Hallucination is not a bug — it’s a theorem.”
The Central Thesis
Every contemporary large language model implements a functor F: Sem → Vect_R from the semantic category (an ∞-groupoid with non-trivial homotopy groups) to finite-dimensional real vector spaces (a contractible space where all topological obstructions vanish). This functor necessarily collapses the higher categorical structure of meaning.
This collapse is not a flaw in any particular model — it is a structural inevitability. The Fundamental Obstruction Theorem establishes that no faithful functor exists from a semantically non-trivial category to a contractible codomain. Hallucination is a theorem.
The solution is not better statistics but richer mathematics: replacing the contractible codomain Vect with the ∞-topos Sem_∞, and replacing probabilistic generation with certified type inhabitation. The three papers develop this solution systematically.
The Trilogy
Author
Matthew Long is the founder of Magneton Labs LLC and lead researcher at the YonedaAI Research Collective, a Chicago-based independent research initiative focused on the mathematical foundations of artificial intelligence. His work applies tools from algebraic topology, category theory, and homotopy type theory to fundamental problems in machine learning.
The Homotopical Semantics Trilogy — three papers totalling 75 pages — represents the first complete mathematical framework proving that LLM hallucination is a structural theorem, not a statistical accident, and providing both a rigorous detection criterion and a certified generation alternative.
Organizations
An independent research collective dedicated to applying modern mathematics — category theory, homotopy theory, and type theory — to foundational questions in artificial intelligence. Named after the Yoneda Lemma, the fundamental result of category theory.
Research and development company focused on mathematically-grounded approaches to AI safety and reliability. Publisher of the Homotopical Semantics Trilogy.
Mathematical Background
The framework draws on three mathematical traditions:
Homotopy groups π_n, homology H_n, persistent homology, simplicial complexes, Vietoris–Rips filtrations. Used in Part I to classify the five hallucination types as topological invariants of the semantic knowledge complex.
Martin-Löf type theory, the Curry–Howard correspondence, proof search, type inhabitation, fibrations, dependent telescopes. Used in Part II to replace statistical sampling with certified proof construction.
∞-categories, ∞-toposes, adjoint functors, monads, sheaf cohomology, descent. Used in Part III to unify the topological and type-theoretic perspectives and prove the Completeness Theorem.