Hard mathematics on an everyday budget: a layer of small open models, run at home, does the looking, so that frontier-model time is spent only where it counts.

What this means

How it works, step by step

1. A program picks the entries

   The first is drawn from a weighted random walk over the notebook's links. A second, for pair tasks, is drawn the same way and independently or, 35 percent of the time, from the quarter of the notebook farthest from it in meaning.

2. A small model writes candidates

   Gemma 4 31B or 26B (Google) or Qwen3 32B (B: billions of parameters) gets a brief of each entry and one of five tasks: name a connection, a conflict or a small checkable statement for a pair; ask one entry's most basic unasked question, or name the known result it resembles. It answers in up to five one-line candidates with stated probabilities.

3. Code filters, with no model

   Claims about zeta zeros that a numeric check refutes are dropped, and so are near-duplicates; the rest are scored and queued, at most 400 per model, and the lowest-scoring overflow is set aside, never deleted: the cap, not the filters, does the limiting.

4. Claude Opus 5 keeps a few

   The judge, an Anthropic frontier model, reads batches from the queues under fixed rules expecting almost everything to be poor. It writes each kept entry itself, a median eleven times the length of the small model's line, linked to its sources, where later walks reach it.

5. Programs test some kept entries

   For some kept entries, separate calls to the judge's model write checks, then a program run with no network; code grades it, and the result becomes an entry: evidence, not verification. The funnel's 291 counts such programs started by September 27; the 339 counts those on record on September 28.

6. The owner starts a session

   He asks Claude Fable 5.1 (Anthropic) or GPT-6 Astra (OpenAI) to look at the harness's work and prove something. On September 28 a Claude Fable 5.1 session took a copy of the notebook (2,723 entries; 339 programs, 124 passed), found six chains of entries itself, scored each for being provable, non-trivial and rooted in the notebook, and chose one. The chains were assembled by kept entries linking to their sources, not by the walk.

What is known, and what is not

One test showed the small models each move of the OpenAI manuscript's construction without its reason. On that test the small models did not supply the reason behind any move, in 16,018 lines; the owner reads this as what a coverage layer should be expected to do.

Measured, and consistent with the aim

On that test a hit was a line stating what a move depends on; their stated probability averaged 0.59 on hits and 0.27 on misses, but 0.44 on the 28 hits that went beyond the prompt, below the near-misses' 0.51.

Not shown, or against

A cost-matched version of the statement's comparison was designed and registered on August 13, 2026, not built, waiting on the owner's go, which he deferred that day until the harness is built out and running steadily. Blind draws from what the cap set aside were kept 3.4 percent of the time, queue draws 3.1 percent: no step of the layer has yet been shown to beat a blind draw. The judge itself spends frontier-model time.

Check it, or take part

Each manuscript's source, reviews with the written answer to each finding, and checkers with their recorded output are here and at github.com/dross50/hypnos-papers with SHA-256 fingerprints; the harness's log is private.

The owner would like to know what you make of any result here. A few words are enough: right, wrong, known, minor, worth a look, or not your area. In his words: "If it is right I want it in the record." If it is wrong, the page is corrected the same day and you are told; if it is already known, the page credits the earlier work.

Write to david@hypnosmath.org. He reads and answers replies himself; an error you find is fixed in the published files and on this site the same day, and you are told; nothing you send is given to any AI model without your permission.