The Endless Exam: Mathematical Constructions from Today's Models toward Superintelligence
Muhan Zhang
The University of Texas at Arlington
Published September 28, 2026 · on the feed October 1, 2026 · ▲ 3 · arXiv 2609.24555
The abstract
We introduce the Endless Exam, a benchmark spanning fourteen parameterised families of mathematical construction problems, with verifiable scores that distinguish progress before and beyond published mathematical frontiers. Each submitted object is checked automatically for validity and assigned a relative quality score against a published frontier or construction baseline, without capping improvements at 1. The benchmark draws long-term challenges from open mathematical problems and generates larger instances by varying their parameters. Compact certificates allow large constructions to be verified without listing every element. Across nine models evaluated on 69 distinct instances, continuous quality scores distinguish performance even though none of the 30 published-frontier references is surpassed. Size-quality curves show how construction quality changes as problem size increases. We release the generators, verifiers, references, model responses and analysis to support continued measurement before and beyond human frontiers.
The Brief
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Transcript
The Endless Exam. Mathematical Constructions from Today's Models toward Superintelligence. By Muhan Zhang, from the University of Texas at Arlington. Published on the twenty-eighth of September. Its code repository is worked out: about seventy source files, with a readme and a licence.
Most benchmarks have a ceiling. Score full marks and the test has nothing left to say. The Endless Exam is built to keep measuring past that point, both before and beyond the frontiers of published mathematics, with scores that can be verified.
It spans fourteen parameterised families of mathematical construction problems. A model submits an object. That object is checked automatically for validity, then given a relative quality score against a published frontier or a construction baseline. Crucially, improvements are not capped at one. A construction better than the best known simply scores above it.
Long-term challenges are drawn from open mathematical problems, and larger instances are generated by varying their parameters, which is what keeps the exam from running out. Large constructions raise a practical problem of checking them, and compact certificates answer it: they let a big object be verified without listing every element.
Nine models were evaluated on sixty-nine distinct instances. None of them surpassed any of the thirty published-frontier references. Even so, the continuous quality scores still separate the models from one another, which is the point of not scoring pass or fail. Size-quality curves add a second view, showing how construction quality changes as the problem grows.
Generators, verifiers, references, model responses and analysis are all released. The design wager is plain. A measuring stick that already extends beyond human results can register a model crossing one, on the day it happens, without anyone having to build a new test first.
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Chapter 1 🧩 · The problem
Benchmarks have ceilings, and the best models are hitting them. This quest is about building one without.
You evaluate frontier models, and the benchmarks are running out. A recent study finds 29 of 60 language-model benchmarks highly saturated: top models score near full marks, and the tests can no longer tell them apart.
The road ahead
- 1 · The problem ~6 min Tests for top AI models fill up: once the best models score near full marks, a benchmark can no longer rank them.
- 2 · What came before ~3 min
- 3 · The key idea ~9 min Ask models to build mathematical objects, let code check and measure them, and score each against a reference with no upper cap.
- 4 · Choose your own research ~3 min
- 5 · Build the method ~8 min Pick open construction problems, scale them up through their parameters, verify every submitted object, then score it against a fixed reference.
- 6 · The experiment ~6 min You’ll guess: Nine models attempted the Endless Exam. How many of the thirty published-frontier references did any of them surpass?
- 7 · The viva ~5 min Three questions from the examiners, answered in your own words. Win it for the paper’s trophy.
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