One entry, followed from August 12 to September 29, 2026
How an entry changes
Hypnos is a research harness, a program whose loop 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, read a growing notebook of mathematics and write one-line candidate ideas about its entries. Claude Opus 5, a frontier model made by Anthropic (the judge), reads samples of those lines, keeps about 3 in 100 of those it reads, and writes each kept idea into the notebook as an entry; in sessions he starts, other frontier models read chains of those entries and try to prove something. How it works explains the loop.
The entry followed here is a question the judge wrote on August 12, 2026, the harness's second day, from ten words one of the three small models had written: "The Kadets 1/4-theorem regarding the stability of bases of exponentials."
What changes about an entry, and what never does
August 12, 2026
Written by the judge, from a ten-word line
A program handed one of the three small models an entry of the notebook and asked which known result it resembles; one of its one-line answers was the ten words (a small model's line). They name Kadec's 1964 theorem, a sufficient condition: if the n-th point of a sequence lies within less than 1/4 of n for every n, the exponentials built on the sequence form a Riesz basis, a system that behaves like an orthonormal basis up to constants, on the interval from minus pi to pi.
The judge kept the line. What went into the notebook (the judge's entry, Claude Opus 5) is 21.7 times as long: a question, whether the unfolded zeros of the Riemann zeta function (their heights, rescaled to the density of the integers) stay within 1/4 of the integers; a test, the first index where 1/4 is crossed among the first 10,000 zeros; and a prediction that it fails early. It went in at the lowest trust level, with one link, to the entry the small model had been shown, and with the ten words stored beside it.
August 12 to September 18, 2026
Linked to its source, where later walks can reach it
Its one link records an event, a derivation, and the random walk over the notebook's links, by which a program picks what the small models read, follows only links of that kind. The entry joined the walk at full weight for recency, which halves every 72 hours unless the entry is touched again. On September 27, 2026, 3,968 of the notebook's 10,708 links (37 percent) were code's comparisons of a mechanism with a problem statement (both explained below), pass or fail; the walk stopped following them on September 19, because a failed comparison had pulled it as hard as a derivation, so more than a third of the notebook's links no longer steer the walk.
Over the next five weeks more lines on the same question followed, each in two steps (a small model's line, then the judge's entry, Claude Opus 5). On August 14 a second recall of Kadec's theorem, from the same source entry, was kept. On August 16 a small model, shown that entry and another, claimed falsely that the first N zeros satisfy Kadec's condition; the judge rewrote the claim as the statement that the condition is exactly a uniform bound on the argument of zeta at the zeros, and named Avdonin's averaged condition and Pavlov's criterion as the live question. Later came a third recall (August 20), a conjecture about the first 50 rescaled zeros and a recall of Pavlov's results that the eventual paper does not cite (both September 2), and a request to compute Riesz constants (September 18).
August 19, 2026
Measured by a program, which leaves trust where it was
On August 19, 2026 a program (written by the judge's model and graded by code) ran on the August 16 entry. With the chain's later programs it measured what the August 12 entry had asked: the first zero to stray 1/4 or more from its integer is the ninth, at height 48.005, and the largest distance among the first 100,000 zeros is 1.129. The prediction held. Whether the Riesz property is lost, or only unproved by Kadec's test, became the question of two later entries (September 2 and 18).
Each result is filed as an entry of its own, linked to the idea it tested, and no result moves trust: a single result is evidence, not verification (how the programs are written and graded).
Unchanged since August 12, 2026
Trust: the level it entered at, and why nothing raised it
Every kept entry enters at the lowest of four trust levels (speculative, numerically supported, proved informally, proved formally in Lean), and nothing in the running harness raises it; the code that could, on a recorded check, is called by nothing but a test. A review of the whole notebook on August 17, 2026 found every entry at the lowest level, so recency, not trust, sets an entry's weight in the walk. A separate usability tier, described below, says whether a mechanism can be used; the August 12 entry, a question, has none.
The theorem that answered the chain's question lives in a paper, not in the entry. On September 29, 2026 Claude Fable 5.1, a frontier model made by Anthropic, proved that the unfolded zeros, counted with multiplicity, form no Riesz basis of exponentials on any bounded interval; the paper says this answers the question of those two later entries (the property is lost), and its theorem that Kadec's condition fails for every centering also answers the August 12 question.
Set aside, never removed
Parking: half weight in the walk, and a way back
Nothing in the notebook is deleted, but an entry can be parked. Code parks entries under the rules of consolidation, the pass that groups recent entries into themes: only under capacity pressure, and then only the least recently touched ones that the small models have not drawn lately and no kept entry cites. A parked entry stays in the walk at half weight. A new kept entry that cites it wakes it, and so can a stored condition checked by code; that second kind of wake also queues a review that nothing handles yet.
Parked or not, the August 12 entry stayed in the notebook; it is one of the 17 entries the paper cites.
Step 1 of 5
The design behind the notebook
Entries as mechanisms with interfaces, and how much of that the code carries out
The notebook follows a design compiled on August 11, 2026 from the owner's own documented way of working through problems. Its two phases, in his words:
Phase one: "Concepts enter as 'hollow shells' (labeled, unusable). They gain usability only through enforced interaction. Everything is stored as a typed mechanism, not a fact."
Phase two: "Problems are compiled to domain-free mechanism signatures. Inventory is test-fitted against signatures by structure, not vocabulary: candidate mechanisms are connected as nodes on a graph, and most fail on connection compatibility. Before rejection, near-misses whose failure is localized to the interface boundary trigger a bounded adapter search. Wrong-specific/right-general matches are kept as pointers. Failed fits refine the problem signature. Partial fits combine into modules; modules re-enter the inventory through the same gate."
In the code, a mechanism is an idea stored as a tool with an interface, what it accepts and what it emits, written in a fixed vocabulary of 43 terms such as self-adjoint operator, and with one of six shapes, so that tools from different fields compare as the same kind of thing. The vocabulary is fixed so that a fit is a symbolic difference computed by code, with no language model involved. Problems are written in the same form: the notebook started with four ways of looking at the Riemann hypothesis written as problem statements with interfaces, and four classical bridges between fields stored as mechanisms.
A mechanism normally enters as a hollow shell and becomes usable only through a gate that code computes over recorded artifacts: at least three worked examples, at least two links from executed programs to mechanisms already usable, and one known consequence re-derived. Six mechanisms were usable by August 12, 2026: four built on computational instruments checked against published values, which entered usable with those checks as evidence, and two starting bridges that passed the gate, Weil's explicit formula and the Montgomery-Dyson bridge to random matrices. No mechanism is recorded passing the gate since, so after the harness's second day this machinery made nothing newly usable; and its comparison never reaches what the loop adds, since every kept idea, program result, theme and question, the August 12 entry among them, carries no interface.
Phase two, step by step:
Built. Problems compiled to interfaces, as the four starting ways of looking were.
Built. Fitting by structure: code compares interface, shape and invariants and records the whole difference as a link, pass or fail, with one of three verdicts: fits, fails only at the interface, fails in structure. When the walk hands a small model a mechanism and a problem statement together, the judge sees the fit, and it counts in the score that code gives the line.
Built in part. The adapter search on a near miss: code looks for a usable mechanism, or a chain of two, whose outputs cover the missing piece; the bounded research task for when none exists is not built.
Not built. Failed fits refining the problem statement, and partial fits combining into modules that re-enter through the gate.
What the sessions do with entries
A session reads a chain whole, and chooses
Nothing in the loop hands an entry to the paper-writing models: the owner starts a session with Claude Fable 5.1 (Anthropic) or GPT-6 Astra (OpenAI) and asks it to read the harness's work and prove something it finds there. On September 28, 2026 such a Claude Fable 5.1 session read a copy of the notebook, scored six chains of entries for being provable, non-trivial and rooted in the harness's work, and chose the chain around the August 12 entry, 17 entries and 10 programs. Its proof joins Pavlov's criterion with Selberg's theorem on the growth of the argument of zeta; whether that route came from the notebook, where the August 16 entry and the September 2 recall had named Pavlov, or from what the model already knew is not recorded. The walk had never drawn the chain's own nine pairs of entries together in the small models' 117,339 steps through September 29, and handed those pairs that day, no small model wrote a line containing the proof's mechanism; the session that ran the test concluded that proof-level moves need several ingredients held together, which two entries, each shown cut to 500 characters, never carry. How often later sessions use kept entries has not been measured.