— and the wisdom to know the difference.
Eduardo Dueñez · Mathematics, UTSA
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The object is so complex that no finite description pins it down. In the combinatorial form: every pattern of answers is realized — the independence property, infinite VC dimension.
Then no procedure learns it, and none ever will, at any sample size.
Nothing is tangled at all — the answer simply is not determined. The next coin flip. When this atom decays. Thursday's weather.
Here the concept can be perfectly learnable while each instance stays unpredictable.
My work is about where that line falls — and it turns out the line is not a line. It is a ladder.
Joint with J. Iovino (UTSA) and collaborators in Toronto; and, on the applied side, with the research team at Gibran.AI in London.
Newton's method on f(z) = z³ − 1. Colour each starting point by where it stands after n = 1 steps.
Drag the white dot: its orbit is drawn step by step.
Scroll to zoom anywhere in the picture; 0 returns to this view.
Let gm(x) = 1 exactly when m!·x is an integer. Each gm is a limit of continuous functions. Their limit is Dirichlet's function — nowhere continuous, and not Baire-1.
Fix an irrational x∞ and take any rationals xn → x∞. Grothendieck's criterion asks whether two iterated limits agree — one index over functions, one over points.
Computed live in exact integer arithmetic — never floating point.
n — the point xn →
0 ≠ 1 — the exchange fails
When it does not fail, the deep computation is explicitly computable. That is the theorem.
tame = Baire-1 = NIP = finite VC = PAC-learnable
Topology, model theory and statistical learning drew the same line independently. But above that line there is structure: four levels, distinguished by how the closure sits topologically.
Underneath sits Todorčević's trichotomy for Rosenthal compacta — proved with forcing and infinite Ramsey theory; decades on, still no elementary proof. Argyros–Dodos–Kanellopoulos refine it to seven minimal prototypes.
Defined for countable families already known to be NIP. The levels are, so far, purely topological: whether a combinatorial characterization exists is open.
A biased coin, p unknown. Two different questions: what is p, and what is the next flip.
The first is learnable to any precision you like. On the second you will be wrong min(p, 1−p) of the time — and no quantity of data moves that, ever.
So "not computable" and "not predictable" are different failures. The theory extends: if the family is NIP, every deep computation is universally Monte Carlo computable — measurable against every reasonable measure. Whether the converse holds is open.
Quantum computation is, loosely, statistical classical computation — which is where this theory is headed next. (That framing is mine; it is not yet in the papers.)
Both plotted as error. Cyan: how far the estimate of p is from the truth — it goes to zero. Magenta: how often the next flip is called wrong — it stops at min(p, 1−p) and stays there forever.
Thirteen published decoders for quantum error correction, each hand-designed by an expert over a paper's worth of work. We expressed the primitives they share as 38 typed operations — then evolved new programs in that language. Mutation and crossover. No language model anywhere.
R5-E058 — more accurate than BP+OSD-CS at 1/22 of its cost.
R5-E059 — grown from random initialization, no expert seeds: more accurate than the strongest decoder on the panel, at indistinguishable cost, while enumerating zero candidate corrections.
12 hours of wall-clock on a 185-node cloud CPU cluster — about 6×10⁷ decoder evaluations. Twelve of sixteen finalists cleared a pre-registered bar on a 200,000-shot hold-out set. Submitted to EQTC 2026; work with Gibran.AI.
Selected rows from the EQTC 2026 abstract. Cost is a machine-independent work proxy, internal to this panel — the margins over BP+AC are inside its resolution.
The group J. Iovino and I run has been mathematical home to some of our most talented students. Topology and continuous logic, aimed at what "computable" even means.
Newly organized by UTSA Math faculty, and looking for its regulars. Coding theory, belief propagation, and algorithms that write themselves.
Kimberly Roe is asking which structural features of an optimization problem let any optimizer beat the others on average — dependence, NIP, the same dividing line as before. Early days. Room for company.
Everything here is yours.
Slides, runnable notebooks, papers —
supernumero.us/files/talks/talk-2026-what-we-can-compute/
eduardo.duenez@utsa.edu · and the Problem-Solving Club meets weekly, with pizza
Four points, sixteen sign patterns; a line realizes —. The ones it misses are what VC dimension measures. Drag the points.
Thomae's function (1/q at p/q in lowest terms, 0 at irrationals) is Baire-1: continuous at every irrational, discontinuous exactly on a countable — hence meager — set. Dirichlet's indicator of ℚ is nowhere continuous, so by Baire's theorem it cannot be Baire-1. That pair is the tame/wild boundary on the real line.
R5-E059's advantage does not scale indefinitely: past n = 144 its logical error rate per round turns upward, 2.2 → 3.8 × 10⁻⁴, while the Ambiguity-Clustering family keeps descending. Transfer results are labelled preliminary.
Deep Equilibria — accepted, to appear in Math. Structures in Computer Science. Complexity of Deep Computations — arXiv:2601.00528, in preparation. EQTC and the two NeurIPS workshop papers — submitted, non-archival.
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