The Rising-Sea Theses: Twelve Theses on Mathematics and Machine Reasoning

The Rising-Sea ThesesTwelve Theses on Mathematics and Machine ReasoningPreprint, version 1.0. The second of three papers released together. The Grothendieckian Programme is a set of seven works on machine reasoning about mathematics (34 volumes, 26,021 pages), released openly in October 2026 as first drafts that have not been peer-reviewed. This paper draws from them twelve theses on how mathematics and machine reasoning should be combined. Each thesis is stated in one sentence, argued, tied to the passages of the seven works on which it rests, related to existing work, and paired with an observation that would refute it. The theses concern the unit of mathematical knowledge for a reasoning system, the place of epistemic status, the latent space of mathematical reasoning, attention, failure, authority, and the method of discovery. The paper adds no mathematical result to the seven works; none of them claims a proof of the Riemann Hypothesis, the Birch and Swinnerton-Dyer conjecture, the Hodge conjecture, the abc conjecture or the Langlands correspondence. 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 (this record) 3. The 57 Grothendieckian Problems — Open Problems for Mathematics and Machine Reasoning from the Grothendieckian Programme — doi:10.5281/zenodo.23734426 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-Rising-Sea-Theses.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. An archive of the repository of the three papers at the signed tag v1.0.0 — sources, problem cards, Lean statements, scripts, records of the checks and figure sources — is in the record of the third paper (doi:10.5281/zenodo.23734426). 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.