Your AI Doesn't Discover Anything. Here's the Math That Proves It.

Your model is a formidable recombiner. Calling it anything else is a category error.

Hand it the whole corpus of structural biology and it will produce fluent, plausible, occasionally correct recombinations of everything already written down. What it won't do, on its own, is notice that the vocabulary it's reasoning in is the thing that's wrong, tear it up, and prove the replacement earned its place. That gap finally has a name. It also has, for once, a piece of real mathematics behind it.

A new MIT preprint draws the line cleanly. Fiona Y. Wang and Markus J. Buehler give discovery a precise shape in category theory: a structural fact you can record, verify, and price in bits, in place of "a sufficiently surprising output." The paper is Self-Revising Discovery Systems for Science, posted to arXiv on 31 May 2026. It's a preprint, not peer-reviewed, and the demonstrations are narrow materials-science cases. Keep that firmly in mind before anyone sells you a "self-evolving AI scientist" on the strength of it.

But the spine of the argument is the most useful thing I've read on agentic systems in months. Once you can say formally what it means to change the representational regime, you can also say how a verifier should be built, what provenance must be recorded, how much was actually discovered, and why scaling a fixed model is categorically different from building a system that can mint new commitments.

Why a builder should care

I build agent systems that have to survive an audit. Typed artifacts, provenance, gates in the path, and the unglamorous one nobody funds: knowing whether a step actually changed the model or just overwrote a number. This paper is a formal account of exactly that distinction, which is why I'm writing about it instead of the week's model releases.

The authors separate three operations that practitioners constantly blur into one undifferentiated "the agent did stuff" log. The difference is structural, and has nothing to do with how impressive the output looks.

The first two never alter what kinds of things can exist. Only the third does. That gives you a definition of discovery with no appeal to subjective novelty: you don't ask whether the output feels new, you ask whether the space of admissible artifacts changed. That's a structural fact about the system, and structural facts can be audited. The lineage is Popper, Kuhn, Lakatos: science as the revision of frameworks rather than the accumulation of answers. The contribution is making "revision of framework" precise enough to check.

01 · A discovery system's state is a copresheaf

A chat transcript, a hidden vector, a single checkpoint: none of those is the persistent state of an agentic discovery system. That state is a growing, typed record of artifacts and how each one was produced.

Formalise it. A regime carries a schema category $\mathcal{S}_b$: its objects are artifact types (a sequence, a structure, a contact graph, a symbolic model, a measurement, a report) and its morphisms are the operations allowed between them. The system's state at time $t$ is then a covariant, $\mathbf{Set}$-valued functor: $I_t : \mathcal{S}_b \longrightarrow \mathbf{Set}.$

For each type $A$, the set $I_t(A)$ holds the actual artifacts of that type the system currently has; for each operation $f : A \to B$, the function $I_t(f) : I_t(A) \to I_t(B)$ records how applying it turns an $A$-artifact into a $B$-artifact. The functor cleanly separates the regime (the schema, which can stay fixed) from its contents (the population, which grows).

Now take the category of elements of that functor, written $\int_{\mathcal{S}_b} I_t$. Its objects are pairs $(A,x)$, a type and an actual artifact of it, and a morphism $(A,x)\to(B,y)$ exists exactly when some operation $f$ satisfies $I_t(f)(x)=y$. There's a provenance edge from $x$ to $y$ precisely when a legal operation actually produced $y$ from $x$. Nothing here depicts a provenance graph. The category of elements is the typed provenance DAG: every accepted artifact, its parents, and the operation that made it. Vendors gloss this. Everyone ships a "lineage view." Almost no one has lineage as the substrate the reasoning actually runs on.

Bundle the regime as a tuple $b = (\mathcal{S}_b, \Gamma_b, V_b, L_b)$, meaning schema, generators, a verifier, and an optional description-length functional. The full live state then becomes a single object the paper calls a knowledge-computation graph: $\mathfrak{K}_t^{\,b} = \bigl(\mathcal{S}_b,\ \Gamma_b,\ I_t,\ \mathsf{Prov}_t,\ V_b,\ L_b,\ \mathsf{D}_t,\ \pi_t\bigr).$

Forget everything but $\mathcal{S}_b$ and a flattened $\mathsf{Prov}_t$ and you recover an ordinary knowledge graph; keep the production edges but drop the gates and you recover a workflow-provenance graph. The claim is that you want all of it at once, in one executable, verifier-aware object.

02 · The audit contract most stacks quietly fail

Inside a fixed regime, an agent just updates the state, which is an endofunctor on the category of copresheaves: $\Phi_b : [\mathcal{S}_b,\mathbf{Set}] \longrightarrow [\mathcal{S}_b,\mathbf{Set}].$

Saying $\Phi_b$ is an endofunctor is a substantive claim about the implementation. A raw program that maps one JSON ledger to another is merely an endomap. It becomes a functor only when it preserves refinement: if a state $J$ extends a state $I$ by adding verified artifacts without overwriting prior provenance, then $\Phi_b(J)$ must extend $\Phi_b(I)$ the same way.

I'll put it more bluntly than the paper does. If your agent framework mutates records in place, reuses IDs, or drops failed calls on the floor, what you have is a confident text generator with a database bolted to it, and none of the mathematics below applies to you. Fix the contract first. And before anyone objects that reality is stochastic (agents sample, tools fail, schedulers branch), the framework absorbs that without flinching: read $\Phi_b$ as a stochastic kernel, a morphism in the Kleisli category of a probability monad. The structural claims survive.

03 · Discovery = transport + residual

Search is iterating $\Phi_b$ inside one regime; discovery is moving to a new one. A genuine regime change is a schema map $u : \mathcal{S}_b \to \mathcal{S}_{b'}$, a translation from the old vocabulary into a richer one. The old evidence is carried into the new schema by a left Kan extension $\operatorname{Lan}_u I_t$: the least committal way to reinterpret old data in the new vocabulary, adding nothing it isn't forced to. Then a verified transition supplies a comparison map

$\bar\rho : \operatorname{Lan}_u I_t \longrightarrow I'_{t+1}.$

The image of $\bar\rho$ is everything the new world can explain by merely reinterpreting the old. Everything outside that image is genuinely new content. Because this is all set-valued, you can price it in bits.

The sharpest line in the paper is the Kan obstruction. If the new regime introduces a type $A'$ that no operation reaches from the old world, the comma category indexing the colimit is empty, so transport hands it the empty set: $(\operatorname{Lan}_u I_t)(A') = \varnothing.$

An isolated new type, a quantity nothing old can produce, starts life genuinely empty. The only way to populate it is to go get new evidence, run a new tool, admit a new construction. Discovery is transport plus residual, and transport alone never gets you there.

04 · The protein example, end to end

Abstraction is cheap; the paper's most convincing section is a worked run. A Builder/Breaker loop tries to learn a symbolic law for how flexible each residue in a protein is. A Breaker picks new proteins designed to expose the current model's failures; a Builder edits a symbolic computation graph; a gate decides what survives. The physics base is the Gaussian Network Model, where a protein's slow collective motions and per-residue mobility fall out of its contact topology alone.

Start from a chain $p$ with residues $i = 1,\dots,N_p$ and Cα coordinates $\mathbf{r}_{pi}\in\mathbb{R}^3$. Build the contact graph by thresholding distance, then form the GNM Kirchhoff (graph-Laplacian) matrix: $A^{(p)}_{ij} = \mathbf{1}\{\, i \neq j,\ \|\mathbf{r}_{pi}-\mathbf{r}_{pj}\| < r_c \,\}, \qquad r_c = 10\,\text{Å},$

$\Gamma^{(p)}_{ij} = \begin{cases} -A^{(p)}_{ij}, & i \neq j, \\ \sum_{k\neq i} A^{(p)}_{ik}, & i = j. \end{cases}$

Diagonalise it, $\Gamma_p\,\mathbf{u}_{pk} = \lambda_{pk}\,\mathbf{u}_{pk}$, and the all-mode compliance of a residue, its softness summed over every nonzero mode, is the diagonal of the pseudoinverse: $C_{pi} = (\Gamma_p^{+})_{ii} = \sum_{\lambda_{pk}>0} \frac{u_{pik}^{2}}{\lambda_{pk}}.$

No learning produced this feature. It is the harmonic mobility implied by the contacts, and standard GNM ties it straight to the crystallographic B-factor. Because the target is normalised within each chain, the global constants drop out. What remains to explain is the pattern of flexibility, with absolute scale set aside. The two features the run lands on are a compressed log-compliance coordinate and a clipped slow-mode participation weight: $\phi_{pi} = z_p\!\bigl(\log(C_{pi}+\epsilon)\bigr), \qquad \psi_{pi} = \bigl[\, z_p(|u_{pi2}|) + \theta \,\bigr]_{+},$ where $u_{p2}$ is the first nonzero mode, the dominant collective deformation. After paired refitting under the gate, the surviving law is strikingly compact: $\widehat{B}^{(z)}_{pi} = \alpha + \beta\,\phi_{pi}\,\psi_{pi}, \qquad \alpha=-0.1332,\ \ \beta=0.2239,\ \ \theta=2.2678.$

The mechanical reading is clean: experimental flexibility is best compressed as local compliance expressed through participation in the dominant collective mode. And the physics lives entirely in a typed, interpretable pipeline, $\{\mathbf{r}_{pi}\} \mapsto A_p \mapsto \Gamma_p \mapsto \{(\lambda_{pk},\mathbf{u}_{pk})\}_k \mapsto (C_{pi},|u_{pi2}|) \mapsto \widehat{B}^{(z)}_{pi}$, with no opaque regressor anywhere in it.

Modes were in the schema from the start, so the discovery is not "the system used a normal mode." What changed is that the regime came to admit a new multi-input morphism: $\texttt{LogNormCompliance} \times \texttt{ReLUModeAmpl} \longrightarrow \texttt{ModeConditionedCompliance}.$

Run a Kan-transport audit over each accepted transition and every new type sorts into three buckets: generator-reachable (an old type hands it an immediate unary morphism), composite-reachable (it only appears once a new multi-input composition is admitted), or isolated (even composites can't reach it). The two ingredient features are generator-reachable. But ModeConditionedCompliance is composite-reachable only: it cannot exist until the regime admits the product that multiplies the two. That product is the scientific commitment, the discovery located exactly.

The last two columns are what a naive "accuracy went up" story misses entirely: the later accepted transitions shrink the model code while delivering the largest compression gains. The law gets simpler even as it explains more. Discovery here includes retraction and compression as well as accretion. The descriptive fit even wanders the wrong way, with $R^2$ going $0.48 \to 0.68 \to 0.54 \to 0.41$, because each score is on a harder, larger evidence set as the Breaker adds adversarial proteins (open and closed adenylate kinase, PDB 4AKE and 1AKE). Across the whole run the gate admits 25 of 388 proposed edits, about 6.4%.

05 · The gates: MDL and AIC

Everything above hinges on the verifier $V_b$, the thing that refuses to commit an artifact just because an agent proposed it. The Builder/Breaker loop runs on Minimum Description Length: score a model by the total bits to state the model and the data given it, $L(M, D) = L_{\text{model}}(M) + L_{\text{data}}(D \mid M),$ and admit a revision $M'$ only if it pays for itself on the enlarged, shared evidence after both are refit: $L(M', D \cup E) < L(M, D \cup E)$. That's the formal sense in which a productive failure becomes structure, and it is exactly why a monotone single-number score is the wrong thing to track. The second case study swaps in Akaike, $\mathrm{AIC} = 2k - 2\ln\hat{L}$ (lower wins), to select an anisotropic orientation-tensor stiffness surrogate over a simpler isotropic fibre-count descriptor. The rejected descriptor is kept as typed provenance instead of deleted, which is what the contract requires.

06 · What this means if you ship agents

The mathematics does two jobs at once. As a language, it gives discovery a definition with no subjective novelty in it: a verified change of regime with non-trivial residual content, priced in bits. As a specification, it hands you an unforgiving checklist.

Searching harder inside a fixed regime, however vast, is a categorically different operation from minting a new representational commitment and proving, against preserved old evidence, that it added something transport couldn't. A bigger model gets superhuman at recombination and may never, once, change the vocabulary. You cannot spend your way out of that. It is a structure problem, and the empty set $(\operatorname{Lan}_u I_t)(A') = \varnothing$ is the proof.

So read the paper, then check whether your stack could even pass the audit contract. Most can't. Mine has work to do too. That's the useful kind of paper: the one that hands you the test you've been failing without a name for it.

---

Tarry Singh is the founder and CEO of Real AI (realai.eu), an enterprise AI advisory and deployment firm working with global enterprises on production agent systems, model risk, and AI sovereignty strategy. He also leads Earthscan (earthscan.io) for Energy AI, and is a founding contributor to the EU-funded HCAIM and PANORAIMA programmes for responsible AI education across European universities. He writes at tarrysingh.com.