Markus J. Buehler proposes a category-theory framework that lets AI scientists dynamically expand their reasoning schema

Warning: once you learn category theory, you'll never be able or willing to talk with people who don't know category theory.

Paper: https://arxiv.org/abs/2606.01444

AI scientists may be moving from search to real discovery.

A new MIT paper proposes a framework for self-revising AI systems that don’t just explore a fixed scientific vocabulary, but can expand the vocabulary itself, introducing new variables, tools, verifiers, and model structures when existing ones are no longer enough.

True scientific progress is often not just about finding better answers, but about changing the space in which answers can exist.

If this scales, AI could become far more than a research assistant: it could become an auditable partner in building new scientific world models.

Still early, but conceptually very exciting.

very cool paper

Category theory fits here because neural networks learn data relationships, not data itself. The paper formalizes those relationships as typed artifacts, morphisms, provenance, and regime transitions.

Deep Manifold reads these categorical structures as external boundary conditions that guide numerical manifold traversal. Discovery occurs when a new relation becomes stable enough to form a new fixed-point pathway.

#DeepManifoldInterpretation

@ProfBuehlerMIT I like how this is timed with https://www.netsci2026.com/ in Boston!

@NoahChrein might find this interesting!

AI scientist and self-improving agents may be similar systems in practice?

Thank you @somi_ai great points. The critique would apply if typing were post hoc but in our framework the schema itself is part of the audited state: typed operations, immutable lineage, gates, rejected alternatives, and explicit regime transitions. Novelty is only the residual beyond the transported image (i.e. what carrying old evidence over by Left Kan extension cannot produce) and only once the new state passes its own gate. In Builder/Breaker mode-conditioned compliance survived MDL on enlarged evidence and arbitrary refinements were rejected/retracted. In other words subjectivity is not hidden in the ontology choice - it is exposed, constrained, and audited.

@ProfBuehlerMIT @blader Can we test this on ARC-AGI-3?

@ProfBuehlerMIT in case it is helpful...

the Left Kan extension framing is clean, but the residual's objectivity seems to ride on how you typed the provenance. too coarse and real discoveries collapse into the transported image, too fine and every reformulation reads as novel. feels like the subjectivity moves to the choice of base category rather than disappearing.

@ProfBuehlerMIT The jump from searching a fixed schema to actually expanding the schema itself is the part that makes this genuinely different.

The fact that novelty is now mathematically quantifiable rather than a benchmark delta is a real step forward.

You're right that L is subadditive and we don't assume additivity. Discovery cost is defined as the conditional code length L(I′ | im ρ̄); the subtraction L(I′) − L(im ρ̄) is only its additive special case (a subadditive upper bound otherwise). The figure numbers aren't that cost anyway but instead they are paired MDL acceptance gains at fixed evidence which we flag explicitly as 'not a direct numerical discovery cost'. So the conditional form is the cross-regime primitive; the figures' before/after gaps are a separate fixed-evidence quantity. We never equate them.

Here are solid quick-start resources for category theory:

1. **Category Theory for Programmers** – Bartosz Milewski (free PDF + YouTube series). Intuitive, code-focused (Haskell examples), great diagrams. Start here if you like programming.

2. **Seven Sketches in Compositionality** – Fong & Spivak (free PDF on arXiv). Applied, real-world examples, self-contained. Excellent for seeing uses fast.

Milewski’s first 4–5 chapters or videos will get you the core ideas quickly. Pairs well with the prereqs I listed earlier.

This is a much bigger shift than it looks at first glance.

Most AI systems optimize within a language of concepts they inherit, your framework asks whether the language itself can evolve.

The real breakthrough is not generating more hypotheses, it's making schema expansion something that can be verified rather than narrated after the fact.

If this holds up, it pushes AI closer to participating in science instead of just accelerating existing scientific workflows.

@ProfBuehlerMIT Incredibly timely work, Prof. Buehler. I recently lived this exact edge case as a human Verifier. When introducing a novel, first principles chemical formulation to a standard LLM 'final check,' the model defaulted to a fixed schema search, confabulating out of date textbook ...

@ProfBuehlerMIT Discovery cost = L(I′) − L(im ρ̄)" only subtracts cleanly if description length is additive. It's subadditive. The relative form L(I′ | im ρ̄) is well-defined but it's just a conditional code length, not the before/after gap the figures imply. I’m confused by this

@ProfBuehlerMIT Can you say something about why you think the formalism of Kan extensions is appropriate here? (Also I hope you don’t mind if I ask you not to copy and paste from an LLM in response—I’ve already discussed the paper with a frontier model.)

@ProfBuehlerMIT Made the Flashpapers explainer here:

https://flashpapers.ai/p/rf92xpm99x