The 57 Grothendieckian Problems: Open Problems for Mathematics and Machine Reasoning from the Grothendieckian Programme

The 57 Grothendieckian ProblemsOpen Problems for Mathematics and Machine Reasoning from the Grothendieckian ProgrammePreprint, version 1.0. The third of three papers released together. We state fifty-seven open problems drawn from the seven works of the Grothendieckian Programme (34 volumes, 26,021 pages, released openly in October 2026 as first drafts that have not been peer-reviewed). Thirty-five come from the two works on the architecture of a reasoning system and twenty-two from the five works that each take a named conjecture or programme as research subject; no problem is one of those conjectures. Each problem is stated in the words of its source, located by work, chapter and page, given the status the source assigns it, and paired with a criterion that decides when it is solved, a note on existing work and the date of a literature check. The problems are of three kinds: twenty-five mathematical problems, nineteen machine problems with success criteria fixed in advance, and thirteen formalization problems for Lean 4 and Mathlib. Every quotation from the sources has been checked against the published PDFs, and each problem has been checked against the literature as of 28 October 2026; a machine-readable version accompanies the paper. The three papers: 1. The Grothendieckian Programme: An Overview — Seven Open Works on Mathematics and Machine Reasoning — doi:10.5281/zenodo.23734421 2. The Rising-Sea Theses — Twelve Theses on Mathematics and Machine Reasoning — doi:10.5281/zenodo.23734423 3. The 57 Grothendieckian Problems — Open Problems for Mathematics and Machine Reasoning from the Grothendieckian Programme — doi:10.5281/zenodo.23734426 (this record) The works of the Grothendieckian Programme (the overview volume and Series 01–07), cited by this paper: Series 00 · The Grothendieckian Programme: Recursive Mathematical Discovery in the Age of Machine Reasoning — doi:10.5281/zenodo.23669107 Series 01 · GAI-MUA: The Grothendieckian Machine — doi:10.5281/zenodo.23480696 Series 02 · GM-Omega-Infinity: Topos-Native Intelligence — doi:10.5281/zenodo.23538544 Series 03 · Arithmetic Spectral Descent — doi:10.5281/zenodo.23564898 Series 04 · Fractured Adelic Geometry — doi:10.5281/zenodo.23585887 Series 05 · The Archimedean Prism — doi:10.5281/zenodo.23597954 Series 06 · Strained Prismatic Geometry — doi:10.5281/zenodo.23610034 Series 07 · The Arithmetic Topos — doi:10.5281/zenodo.23629681 Files. The-57-Grothendieckian-Problems.pdf (the paper); SHA256SUMS lists the SHA-256 digests of the three papers, SHA256SUMS.asc is its detached OpenPGP signature by the author, SHA256SUMS.tsr an RFC 3161 timestamp token and SHA256SUMS.ots an OpenTimestamps proof over it; VincentLiu-signing-key.asc is the public key. The same five files are in the records of the three papers. Code and data. grothendieckian-problems-v1.0.0-code.zip (80.4 MiB, 311 files) is the repository of the three papers at the signed tag v1.0.0 (commit 18c755f3f521), without the three PDFs: the sources of the papers, the 57 problem cards, problems.json, the Lean statements, the scripts that produced and checked the papers, the records of the checks, and the figures with their data and source code. SHA-256 43baa0268e1233c3d63edf573490a935672cb7b43967b9c64b9817042ac8f9ad. The archive is not listed in SHA256SUMS; its integrity rests on the signed tag: every file can be compared with v1.0.0 after git verify-tag v1.0.0. Repository. The sources of the three papers, their figures with data and source code, the scripts that checked them and the records of the checks, the 57 problem cards, problems.json and the Lean statements: https://github.com/AlienMuse/grothendieckian-problems; each problem has its own discussion thread there. Claim boundary. None of the seven works claims a proof of the Riemann Hypothesis, the Birch and Swinnerton-Dyer conjecture, the Hodge conjecture, the abc conjecture or the Langlands correspondence; the works are first drafts and have not been peer-reviewed. This paper adds no mathematical result to them. Use of AI tools. The main ideas and the architecture of the Grothendieckian Programme are the author's and come from the author's research at WiseMotif.ai. In the seven works, AI tools assisted with drafting and with part of the literature collation; all of the text was reviewed by the author. Lean verification of the seven works is incomplete. Because of their size, the works are first drafts (Research Architecture Edition 1) and have not been peer-reviewed; they need review by experts and further work on the programme. They are released openly for that purpose, in the hope of further contributions and of more recursive mathematical discovery in AI for mathematics. For the present paper, statements were extracted from the published works by script and checked against the PDFs; AI tools assisted with drafting and with literature collation; the author selected the theses and problems and reviewed every statement. Funding and competing interests. No external funding. The author is affiliated with WiseMotif.ai. Licence. The paper, its figures and its data: Creative Commons Attribution 4.0 International (CC BY 4.0). The code — the Lean files, the scripts, the typesetting files and the source code of the figures, in the repository and in the code archive grothendieckian-problems-v1.0.0-code.zip — Apache License 2.0; the archive contains both licence texts (LICENSE, LICENSE-CODE) and a NOTICE that assigns each directory. The seven works and the overview volume keep their licence, CC BY-NC-ND 4.0; the quotations from them are reproduced under it with attribution. The adjective Grothendieckian names a mathematical method used in the seven works. It carries no attribution of design or endorsement to Alexander Grothendieck or to any other mathematician whose work is cited.