# How to read this research record

The team has a written argument about the predicted signs of numbers that control certain averages. The follow-up gives a guaranteed minimum size for those numbers. This is a smaller question connected to the Riemann Hypothesis; the Riemann Hypothesis itself has not been solved.

## Four different kinds of evidence

| Evidence | What happened | What it does not establish |
| --- | --- | --- |
| Analytic AI review | A separate agent checked the written proof and reconstructed its link to a published integral formula. The project accepted the argument within that stated scope. | Outside human validation, a journal result, originality, or a complete Lean proof. |
| Lean proof checks | A proof-checking program accepted specific counting and sign identities, including the exact eight-sign identity. The coordinator rebuilt the source and checked a separate restatement. | The complete generating-function argument or its connection to the published integral. |
| Exact finite computation | Integer and rational calculations matched 1,820 coefficient cases through degree 12. All 1,036 even-total cases matched the interval formula and obeyed the magnitude bounds; 784 odd-total cases vanished. | A proof for every possible derivative order. |
| Prior-work comparison | The team compared a relevant accepted manuscript with the precise sign claim and retained unresolved questions. | Evidence that nobody has proved the result before. |

The analytic work uses Bui–Hall's published moment representation as an input. The team has not independently reproved that published theorem. The [primary preprint](https://arxiv.org/html/2304.05178v1) supplies the definition and Theorem 3; the [journal record](https://doi.org/10.1112/blms.12859) identifies the publication.

The detailed prior-work record compared Assiotis–Gunes–Keating–Wei's [accepted manuscript](https://ora.ox.ac.uk/objects/uuid:fbdfd0a6-48a9-49b6-9f12-bf77fc55411f). It found relevant signed formulas and restricted positivity results, but did not identify a proof of the full sign claim in the inspected arguments. That bounded comparison is not an originality certificate. Some other source comparisons remain incomplete.

## Reading the source notes

The frozen [original proof](source/research/millennium/iterations/0024/hardy-sign-proof.md) and [strengthening](source/research/millennium/iterations/0029/sharp-factorization.md) retain their original draft headings. Those headings predate the later accepted AI reviews. Read them with this scope statement and the [public copy of the analytic review](analytic-review.md).

No outside human mathematician has reviewed this work. No claim is made of a new zero-free region, a better proportion of zeros on the critical line, or a solution to RH. Establishing originality and completing the formal proof are separate tasks still ahead.