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math.HO — History and Overview

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

Contributed by Vincent Liu

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reasoning about mathematics (34 volumes, 26,021 pages), released openly in October 2026 as first drafts that have not been peer-reviewed.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.

Author comments: 26 pages, 5 figures. Version 1.0 of 28 October 2026; the same text is deposited at Zenodo, DOI 10.5281/zenodo.23734424 (all versions: 10.5281/zenodo.23734423). This copy was typeset with pdfLaTeX for Hexagon: its text is identical to version 1.0 at Zenodo, whose sealed PDF was typeset with LuaLaTeX, except that a broken cross-reference on page 3 ("igref{works}") is corrected to "Figure 1"; a few line and page breaks differ. The twelve theses are also set out at https://github.com/AlienMuse/grothendieckian-problems/blob/main/THESES.md; the seven works on which they rest, each with its DOI, are listed at https://github.com/AlienMuse/grothendieckian-programme. Corrections recorded since version 1.0 to the seven works and to the papers: https://github.com/AlienMuse/grothendieckian-problems/blob/main/ERRATA.md. Companion papers: The Grothendieckian Programme: An Overview (DOI 10.5281/zenodo.23734421) and The 57 Grothendieckian Problems (DOI 10.5281/zenodo.23734426).

A mix of human-written and AI-generated textHuman understanding: all partsGrothendieckian Programmedescentepistemic statusformal verificationmachine reasoningrecursive mathematical discovery

Nov 10, 202626 pages

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hexagon:2611.00237v1

math.HO — History and Overview

The Grothendieckian Programme: An Overview — Seven Open Works on Mathematics and Machine Reasoning

Contributed by Vincent Liu

The Grothendieckian Programme is a series of seven works on machine reasoning about mathematics (34 volumes, 26,021 pages), released openly on Zenodo in October 2026 with an overview volume. Two of the works specify a reasoning architecture — a core that holds mathematical worlds as typed, versioned and verifiable objects, and a kernel that reasons inside such worlds by sites, sheaves and descent. Five apply one method, recursive mathematical discovery, to the Riemann Hypothesis, the conjecture of Birch and Swinnerton-Dyer, the Hodge conjecture, the abc conjecture and the Langlands programme, and record what it proves, what it reduces to stated hypotheses and what it leaves open. This paper describes the architecture, the five research subjects, the method, the published corpus and the open process by which the works are to be reviewed and continued. None of the works claims a proof of any of these conjectures; they are first drafts and have not been peer-reviewed.

Author comments: 10 pages, 3 figures. Version 1.0 of 28 October 2026; the same text is deposited at Zenodo, DOI 10.5281/zenodo.23734422 (all versions: 10.5281/zenodo.23734421). This copy was typeset with pdfLaTeX for Hexagon: its text, and its line and page breaks, are the same as in version 1.0 at Zenodo, whose sealed PDF was typeset with LuaLaTeX. The seven works it describes, each with its DOI, are listed at https://github.com/AlienMuse/grothendieckian-programme. Corrections recorded since version 1.0 to the seven works and to the papers: https://github.com/AlienMuse/grothendieckian-problems/blob/main/ERRATA.md. Companion papers: The Rising-Sea Theses (DOI 10.5281/zenodo.23734423) and The 57 Grothendieckian Problems (DOI 10.5281/zenodo.23734426).

A mix of human-written and AI-generated textHuman understanding: all partsGrothendieckian Programmedescentmachine reasoningmathematical worldsopen sciencerecursive mathematical discovery

Nov 10, 202610 pages

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hexagon:2611.00208v1

math.HO — History and Overview

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

Contributed by Vincent Liu

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.

Author comments: 54 pages, 5 figures. Version 1.0 of 28 October 2026; the same text is deposited at Zenodo, DOI 10.5281/zenodo.23734427 (all versions: 10.5281/zenodo.23734426). This copy was typeset with pdfLaTeX for Hexagon: its text is identical to version 1.0 at Zenodo, whose sealed PDF was typeset with LuaLaTeX, and a few line and page breaks differ. Each problem has a public issue at https://github.com/AlienMuse/grothendieckian-problems, which also holds the machine-readable data, the literature-check records and the Lean statement of Problem 49. Corrections recorded since version 1.0, including literature missed by the check for Problem 18: https://github.com/AlienMuse/grothendieckian-problems/blob/main/ERRATA.md. Companion papers: The Grothendieckian Programme: An Overview (DOI 10.5281/zenodo.23734421) and The Rising-Sea Theses (DOI 10.5281/zenodo.23734423).

A mix of human-written and AI-generated textHuman understanding: all partsGrothendieckian ProgrammeLeandescentformal verificationmachine reasoningopen problems

Nov 10, 202654 pages

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