語言模型數學筆記如何登上 aiXiv：黃金窗口猜想

Last week a short email arrived: an open archive for AI-assisted mathematics had found my preprint and invited me to publish it there. This post is the route from a failed framework to that archive, and what I chose not to upload.

Where it started

In 2025 I published a "framework" built on one operator: x ← DQx + (I − D)p, where Q is a cyclic permutation and D is a diagonal matrix of blend weights. The first draft came from a language model (ChatGPT). In 2026 I rewrote the series with Claude Code and checked it line by line.

The audit was not kind. The operator turned out to be a special case of the Friedkin–Johnsen model of social influence, a well-known model in sociology. Convergence was just the Banach fixed-point theorem. I withdrew the novelty claim and kept the full record public.

Audit of the three papers: https://github.com/cpsbvbng26-dotcom/trinity-infinity

What was left

One of the old papers printed a payoff profile, (2, 4, 2), as an example of cooperation in a three-player repeated game. Checking that example led somewhere new. The continuation values of a cyclic path of play are exactly the fixed point of the same operator. That gave a bridge from the failed framework into repeated games.

From there, working with Claude Code:

• Proposition 1. In the three-player social dilemma, rotating the role of defector is a subgame-perfect equilibrium exactly when the discount factor satisfies δ ≥ 1/φ, where φ is the golden ratio.
• Proposition 3. The printed example (2, 4, 2) is never sustainable, at any δ.
• Proposition 6. Inside the window 1/φ < δ < 2/3, the set of sustainable payoffs has positive Hausdorff dimension.
• The golden-window conjecture. Inside that window, without public randomization, the sustainable set has length zero, and its dimension rises continuously from 0 to 1.

What is proved and what is not, plainly:

• The propositions are proved, and a separate script re-checks each one by an independent computation in CI.
• They have not been checked against the literature. Some may be known exercises.
• The conjecture is not proved. The evidence is numerical.

I put an English paper version on figshare in October.

Preprint: https://doi.org/10.6084/m9.figshare.34419198
Maintained note and code: https://github.com/cpsbvbng26-dotcom/trinity-operator/tree/main/games

The email

On 23 October, aiXiv (aixiv.online) indexed the figshare preprint automatically. The next day an invitation arrived, sent to the address on my ORCID record. aiXiv is an independent archive for AI-assisted research, not affiliated with arXiv, and it currently takes mathematics and mathematical physics.

A few things about it are well designed:

• Every submission is timestamped and content-hashed on arrival.
• An AI-use label is required, on a six-step ladder from "AI-proofread" to "AI-authored".
• A linked GitHub repository is checked and marked "repo ✓".
• There is a field for negative results and failed approaches, which the site calls often the most valuable part.

There is no peer review before posting. A paper on aiXiv has a dated record, not a verdict.

What I submitted, and what I didn't

I submitted by hand as aiXiv:2610.02314:

• the same PDF as on figshare
• the repository link
• the label AI-authored, because Claude Code drafted the propositions, the proofs and the conjecture at my request
• process notes, a verification note, and the negative results above

aiXiv also asks for the agent sessions behind the paper, and agentic submission is the preferred route. I did not attach them. The working sessions for this note mixed the math with unrelated personal matters, and a full transcript would publish all of it. The repository and its history already show how the result was built.

One cleanup remains. The auto-indexed record (aiXiv:2610.02226) and my own submission now both exist, with different AI-use labels. I filed a claim and asked the editors to merge them.

If you can check it

The most useful thing anyone could do now is find an earlier statement of Proposition 1, or break any of the propositions. If a prior result turns up, the contribution moves to zero and the paper will cite it.

aiXiv: https://aixiv.online/abs/2610.02314
figshare: https://doi.org/10.6084/m9.figshare.34419198
Code: https://github.com/cpsbvbng26-dotcom/trinity-operator

This post was written with Claude (Anthropic). The author reviewed it and is responsible for its content.