Worked case: From a small model's line to a theorem
From ten words by a small model to a theorem about the zeros of zeta
On August 12, 2026, one of Hypnos's three small open AI models wrote ten words naming Kadec's 1/4 theorem. Claude Opus 5 (Anthropic), the frontier model that reads samples of their lines, kept the line and wrote a question from it. More lines, sharper questions and ten programs followed, and on September 29 a Claude Fable 5.1 (Anthropic) session proved that the unfolded zeros of the Riemann zeta function, counted with multiplicity, form no Riesz basis of exponentials on any bounded interval.
Hypnos is a research harness with two parts. In the loop, which has run since August 11, 2026, with pauses between runs, three small open AI models on the graphics cards of the owner, the person who runs Hypnos, write one-line ideas about entries in a notebook of mathematics; Claude Opus 5 (Anthropic), the judge, reads samples and writes the few worth a closer look into the notebook as entries. In sessions the owner starts by hand, Claude Fable 5.1 (Anthropic) and GPT-6 Astra (OpenAI) read chains of entries and do the deriving and the checking; whether the small models save them any effort is untested. How it works, in full
The paper at the end of this case was written by Claude Fable 5.1 (Anthropic). As of October 3, 2026, it has not been peer reviewed, and no human mathematician has read it. This case runs from August 12 to October 2, 2026.
Six steps and the five links between them
Each step is one piece of the work; each link between steps says how one step's output became the next one's input. Every step names who did the work, by model and maker, as far as the published sources record it; they do not say which small model wrote a line. Each panel links to the published page that reports the case.
Six steps and five links, in order
Step 1 · August 12, 2026
A small model names Kadec's theorem
A program, not a model, chose what the small model read: a random walk over the notebook's links drew one entry, shown to the model as a short brief (the sources do not name the entry). The task was recall: name the known result, technique or paper the entry resembles. The model wrote up to four one-line answers, each with a probability it gave itself; one read, in full, "The Kadets 1/4-theorem regarding the stability of bases of exponentials." Kadec's theorem (1964) is about Riesz bases, the images of orthonormal bases under bounded invertible maps: if every point of a sequence stays within a fixed distance less than 1/4 of its integer, the exponentials with those frequencies form a Riesz basis of the square-integrable functions on (−π, π). The condition is sufficient, not necessary.
Started from
One entry drawn by the walk, and the recall task.
Done by
One of the three small models (which one is not recorded).
Produced
One line with a stated probability, filtered and queued by code.
Code handles every line first, with no model involved: a numerical check of statements about zeta zeros, removal of near-duplicates, a score, and a queue of at most 400 lines per small model, the rest set aside and never deleted. The judge reads batches from the queues under rules that expect almost everything to be "slop"; from August 11 to September 27, 2026 it kept about 3 in 100 of the lines it read. Code files each kept entry; the small models cannot write to the notebook.
Claude Opus 5 kept the line and wrote the entry itself, 21.7 times as long as the line. Reading the zeros as sampling points for band-limited functions, it asked whether the unfolded ordinates (the heights of the zeros, rescaled to sit about one apart) are close enough to the integers for Kadec's theorem to apply. It set a test, how far each of the first 10,000 rescaled zeros sits from its integer and where 1/4 is first crossed, and predicted that the condition would fail early, because the argument of zeta is unbounded.
Started from
The line, in a batch drawn from the queues.
Done by
Claude Opus 5 (Anthropic), the judge.
Produced
A question with a test, linked to its source, at the lowest trust level.
A kept entry is linked to the entry it came from and carries full weight as a recent entry in the random walk that picks what the small models read, so walks near its source reach it and it becomes material for later pairs. The August 14 recall was kept as an entry; two days later the program drew that entry with another, and the false statement of August 16 came from that pair.
August 14 and 20 (small models' lines, kept by the judge): two more recalls of Kadec's theorem. August 16 (a small model's line, then the judge's entry): a false statement that the first N rescaled zeros all meet Kadec's condition, which the judge rewrote: the condition comes down to a bound on the argument of zeta at the zeros, and the live question is Avdonin's averaged version of it, with Pavlov's criterion as the sharp test. September 2 (a small model's line, kept by the judge): a recall of Pavlov's stability results, as an entry the paper does not cite. September 2 (a small model's line, then the judge's entry): a conjecture about the exponentials on the first 50 rescaled zeros, from which the judge wrote that Kadec's condition is only sufficient, so its failure decides nothing about whether the rescaled zeros still act as a sampling set, and asked for a measurement from a Gram matrix, with a random-jitter control, instead. September 18 (a small model's line, then the judge's entry): a request to compute the Riesz constants of the first 1,000 rescaled zeros, from which the judge asked whether the lower Riesz constant stays away from zero as N grows or falls toward it, so that the Riesz property is lost.
Started from
Each time, one or two entries drawn by the walk, and one of five tasks.
Done by
The lines: small models (which ones is not recorded). The entries: Claude Opus 5 (Anthropic).
An entry that carries a runnable check is eligible for the execution step. Separate calls to the judge's model write the checks, then the program, which is shown the frozen checks; one check must reproduce a known value. The program runs with no network access, and code grades the result and files it back as an entry linked to the question.
Ten programs ran on the chain by September 28. The ninth rescaled zero, at height 48.005, is already at least 1/4 from its integer, and over the first 100,000 the largest distance is 1.129, so Kadec's condition certifies nothing here. The lower Riesz constant of the first N exponentials (the smallest eigenvalue of their Gram matrix, scaled) fell from 0.1115 at N = 100 to 0.0159 at N = 1,000. The paper reports these numbers apart from its proofs, as observations, not evidence. Its checkers (programs that recompute its numbers, published beside the output they recorded) reproduce the distances but read a private table of zeros, so they cannot be rerun from the published files alone; the Riesz constants are quoted from the harness's private log.
Started from
The chain's questions, and the harness's table of 100,000 zeros.
Done by
Programs written by Claude Opus 5 (Anthropic), the checks first, in a separate call; code graded them.
Produced
Ten results, each filed beside the question it answered.
Nothing in the loop hands a problem to a paper-writing model. On September 28, 2026, the owner started a Claude Fable 5.1 session and asked it to read the harness's work and prove something it found there. It took a copy of the notebook (2,723 entries, 339 programs run), scored six chains of entries, with their programs' verdicts, on whether each was provable, non-trivial and rooted in the notebook, and chose this one. It read the chain whole, where a small model sees one or two short briefs at a time.
Claude Fable 5.1 proves the Riesz property is lost
The paper's main theorem: take the positive heights of the zeros of the Riemann zeta function, counted with multiplicity (a height counts twice if two zeros share it), rescale them to unit density and add their mirror images. Then for every choice of the rescaling's additive constant, every shift, every finite change (points added or removed) and every bounded interval, the exponentials with those frequencies are not a Riesz basis of the square-integrable functions on that interval. Nothing about the zeros is assumed. A Riesz basis cannot repeat an element, so the real content is the set of distinct frequencies, and for that set the paper proves the theorem only when all but finitely many heights are simple; otherwise it says nothing about the set. The proof joins Pavlov's criterion, in the form of Hruščev, Nikol'skii and Pavlov (a Riesz basis forces the counting function's deviation from a straight line to have bounded mean oscillation), with Selberg's theorem that the variance of the argument of zeta grows without bound, which bounded mean oscillation forbids. The paper says this answers the judge's questions of September 2 and 18: the Riesz property is lost, not merely uncertified.
Started from
The chain's questions and numbers, and the classical results the paper cites.
Done by
Claude Fable 5.1 (Anthropic), in a session the owner started; he read it for red flags and did not check the proofs.
Produced
An 18-page paper dated September 29, 2026, published here with its checkers, their outputs, four AI reviews and every answer.
The chain shows the division of labor running once: a small model's line, the judge's entries, programs' numbers, a session's proof. It does not show that the small models saved that session any effort: the comparison that would test it, the same model on the same problem with and without the notebook's material, has not been run, and a cost-matched version registered on August 13, 2026 has not been built. Frontier models also worked inside the loop at every step after the first.
On September 29 the chain's own nine pairs of entries were handed back to the small models: no line contained the proof's mechanism, its two halves (Pavlov's condition, Selberg's variance growth) appeared in lines from different small models and were never joined, and the program that draws pairs had never drawn those together in 117,339 steps of normal operation. Where Claude Fable 5.1 found its route is not recorded: the judge's entry of August 16 had named Pavlov's criterion, and a small model had recalled Pavlov's results on September 2, but the sources do not say whether the session drew on those entries or on what the model already knew, or whether it read beyond the 17 entries it cites. The paper calls its own literature search bounded and says that the search does not establish priority.
Started from
The chain's pairs, handed back to the small models, and the paper's account of its sources.
Done by
The replayed lines: small models. The paper's account: Claude Fable 5.1 (Anthropic).
Produced
One limit: the harness's log shows where the questions and numbers came from, not where the proof's route came from.
A small model's line started this chain; its questions were written by Claude Opus 5, its measurements by programs, and its theorem by Claude Fable 5.1.