Resurgence by Galois conjugation for a second-order operator over Q(√3): Stokes data and a Hankel integral

Abstract

We study a second-order linear differential operator over the field generated by the square root of three. At the origin it has a divergent power-series solution and a second formal solution with an irrational exponent. We prove that the series of the second solution is the Galois conjugate of the first; that the differential Galois group of the normal form is the special linear group; and that the Borel transform of the divergent series has one finite branch point on its principal sheet, whose amplitude determines the large-order amplitude and the Stokes constant. A Hankel integral of the Borel transform is shown to equal its leading-term value times an ordinary, not lateral, Borel sum of the conjugate series; this corrects the main identity of the author's earlier paper. The constants are evaluated numerically, and whether the operator governs the continued fraction from which it was first taken is left open. MSC2020: Primary 34M40; Secondary 34M25, 34M30, 34M35, 34M15 Preprint, not peer reviewed. This paper supersedes the main identity of the earlier record 10.5281/zenodo.20719042, which it corrects. AI Declaration Tools. AI assistance for this paper was obtained through GitHub Copilot CLI (versions 1.0.87-0, 1.0.88 and 1.0.97, and the client github/autopilot 0.0.0, as recorded in the session logs) and, in a supporting role, through the claude.ai chat interface (see Advisory model below). The models are named below by vendor label and by the model string recorded in those logs. The session records themselves are not part of the supplementary files; the checks that were run on them are (s14_sessions.out.txt). Anthropic Claude Opus 5 (claude-opus-5). Wrote the three drafts of the paper: the earlier draft, the second draft (GitHub Copilot CLI 1.0.87-0) and the third draft (client github/autopilot 0.0.0). Anthropic Claude Opus 5.5 (claude-opus-5.5; session metadata records claude-opus-5 for two of these sessions). Wrote the two blind reproductions, Report A (GitHub Copilot CLI 1.0.88) and Report B (client github/autopilot 0.0.0). In the assembly session (GitHub Copilot CLI 1.0.88) it carried out the adversarial referee reading of two of their proofs, with its test scripts, wrote the fixes that reading asked for, filled the gaps it found, assembled this paper and its audit, and reformatted the manuscript before this preprint version was prepared. OpenAI GPT-5.6 Sol (gpt-5.6-sol) and GPT-5.5 (gpt-5.5). Within the assembly session, GPT-5.6 Sol gave the second reading of fix A1, and GPT-5.5 a first comparison with the third draft, which was not used. Anthropic Claude Sonnet 5.5 (claude-sonnet-5.5). Prepared this preprint version from the earlier manuscript (GitHub Copilot CLI 1.0.97): it removed the venue-specific front matter and wording, wrote this declaration and the section Mathematical software, rebuilt the supplementary files and re-ran the verification scripts. It made no change to any statement or display of mathematics. Other models. Small models (gpt-4o-mini, gpt-5.4-nano) wrote only session titles and message classifications. Advisory model. Claude Opus 5.5 (Anthropic; model selection not logged), through the claude.ai chat interface, advised on the verification protocol (blind reproductions, adversarial referee reading, gap-filling), reviewed reports of the work, drafted many of the task instructions given to the CLI agents, and drafted correspondence. It did not construct the proofs. Extent of contribution. The mathematical arguments, calculations and the wording of the text were produced by the models above. Research directions were set by the author. The proofs were written by the models, not by the author. What was checked by whom. The proofs were checked by AI review only, including two blind reproductions by independent sessions and an adversarial referee reading; the author has not checked them by hand. AI review is not independent human validation and is not represented as such. The checks that were run, and the limits they state, are those of the verification appendix, Section 11, and of the Logs paragraph of Section 12. Responsibility. The author takes full responsibility for the content of this paper. Supplementary records. The supplementary files are deposited with this paper in the same record. Reproducibility. For this deposit the scripts were run again, and their printed output was compared with the recorded output. The original archive has 121 scripts; the deposit omits one of them (s16_operator_actions.py, a file of an editorial process), so 120 are deposited. Of the 121: 57 print output identical to a recorded output, apart from printed run times; 41 are not runnable as deposited, namely 24 that read intermediate files (.pkl, .json, .md, or the text of an earlier draft) that are not in the deposit, 11 that use absolute paths of the machine on which they were written, 4 that read private session files that are not in the deposit, and 2 that need a git worktree of Report B or a mode argument; 17 are library modules or have their recorded output under another name, and were not compared, and one further library module ran and has no recorded output; 1 timed out at 1500 seconds (s7_control.py); 1 was stopped after about 10.5 minutes of processor time (s10_minsearch.py, which has no recorded output); 1 differs by a parameter (s6_borel.py was recorded with coefficients to order 150 and run with 400, and one printed line differs for that reason only); and 1 has a record that predates a later file (s19b_pslq.py, whose record says that a file was not yet written; the lines before that are identical). The recorded output is reproduced only for the 57 scripts that passed; the others were not checked by this re-run, and their logs are as recorded. The audit’s own record of sources refers to a text of this paper that is not the present one.