Read the overview (21.6 MB) · The theses (36.1 MB) · The problems (32.2 MB) · Problem table · Take part · Website

The 57 Grothendieckian Problems

An overview, twelve theses and fifty-seven open problems on mathematics and machine reasoning, drawn from the seven works of the Grothendieckian Programme.

“All right, take 57.” — as told in A. Jackson, Notices of the AMS 51 (2004), p. 1196. Fifty-seven is 3 × 19.

This repository holds three papers and their data, released together:

1. The Grothendieckian Programme: An Overview — Seven Open Works on Mathematics and Machine Reasoning (PDF, doi:10.5281/zenodo.23734421): the architecture, the five research subjects, the method, the series and its records, and the research plan, in ten pages.
2. The Rising-Sea Theses — Twelve Theses on Mathematics and Machine Reasoning (PDF, doi:10.5281/zenodo.23734423, text).
3. The 57 Grothendieckian Problems — Open Problems for Mathematics and Machine Reasoning from the Grothendieckian Programme (PDF, doi:10.5281/zenodo.23734426, cards, problems.json, Lean statements).

The thirteen figures of the three papers, with their sources and data, are in figures/. The PDFs are signed: papers/SHA256SUMS with its OpenPGP signature and timestamps; see papers/VERIFY.md.

_{Figure 1 of The 57 Grothendieckian Problems (600 ppi; the 13 figures of the three papers are in figures/).}

The seven source works ([S01; S02; S03; S04; S05; S06; S07]; overview volume [S00]) comprise 34 volumes and 26,021 pages, released on Zenodo in October 2026 as first drafts that have not been peer-reviewed. 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, and none of the problems below is one of those conjectures.

The twelve theses

1. Worlds, not isolated theorems. For a reasoning system, the unit of mathematical knowledge is a typed, verifiable world, not an isolated theorem.
2. Status is part of the type. The epistemic status of a statement belongs to its type.
3. A structured latent space. Mathematical reasoning needs a structured, non-vectorial latent space.
4. Attention as descent. Attention can be rebuilt as descent.
5. Report failure; abstain without evidence. A reasoning system should report where local information fails to glue, and abstain when independent overlap data are absent.
6. A chain of non-implications. Operational completion does not imply mathematical validity, which does not imply epistemic promotion, which does not imply publication.
7. Machines propose; humans authorize. Machines may propose; humans authorize publication.
8. The rising sea. Hard problems are better approached by raising the surrounding theory than by direct attack.
9. State what is not established. A result is not complete until it states what it does not establish.
10. Discovery is recursive. Mathematical discovery is recursive: propose, test against what is known, abstract what survives, ask again.
11. Creating language. A mathematical reasoning system must be able to create notation and language, not only use them.
12. Judge a method by its open ledger. A method should be judged by an open ledger of what it establishes, what it reduces to stated hypotheses, and what it leaves open.

The fifty-seven problems

Problems 1–35 come from the two works on the architecture of a reasoning system for mathematics (Series 01 and 02); Problems 36–57 from the five works that each take a named conjecture or programme as research subject (Series 03–07). Every problem states what counts as a solution. Every quotation has been checked against the published PDFs, and every problem against the literature (28 October 2026; the records are in the paper and in problems.json).

#	Problem	Source	Kind	Level	Thesis
1	The first rising-sea event	S01	machine	programme	8
2	A bounded fixed point of the coupled world–language cycle	S01	formalization	programme	10
3	From obstruction to conjecture	S01	machine	research	10
4	Minimal extension questions	S01	machine	research	1
5	Target fidelity	S01	machine	research	6
6	Verified encodings of universal properties	S01	formalization	research	3
7	The AURORA-K0 kernel, I: syntax and checking	S01	formalization	research	2
8	The AURORA-K0 kernel, II: a model and relative consistency	S01	formalization	research	2
9	The seven obligations of descent execution	S01	formalization	research	5
10	Non-self-authorization	S01	formalization	research	7
11	Evolution without authority	S01	formalization	research	7
12	Deterministic identity across implementations	S01	machine	entry	6
13	A frozen benchmark for counterexample and proof discovery	S01	machine	entry	12
14	The language forge, end to end	S01	machine	programme	11
15	A preregistered test of typed base change	S01	machine	entry	1
16	A preregistered test of the four judgments of gluing	S01	machine	entry	5
17	A preregistered test of obstruction records	S01	machine	entry	5
18	Discovering the site	S02	machine	programme	3
19	The capability wager	S02	machine	research	4
20	Nerve baselines between GRPO and RLOO	S02	machine	research	4
21	Relative obstruction as a predictor of unsupported answers	S02	machine	research	5
22	Topos-MoE against modern sparse mixture-of-experts	S02	machine	research	4
23	A defence against a globally coherent falsehood	S02	machine	programme	5
24	Compiled-to-streaming descent stability	S02	mathematical	research	4
25	The Beck–Chevalley square 12.6.3	S02	mathematical	research	4
26	Sharp accumulation bounds for equivariance defects	S02	mathematical	research	6
27	Local dynamics of the restriction-frame locus	S02	mathematical	research	4
28	A base-change bound for the descent solve	S02	mathematical	research	4
29	The training-to-serving deviation	S02	mathematical	research	6
30	Monotone refinement budgets	S02	mathematical	research	5
31	How many ambiguity axes are there?	S02	mathematical	research	2
32	Second-order disagreement without engineering	S02	mathematical	research	5
33	Formalizing uniform circuit classes	S02	formalization	research	9
34	A second kernel	S02	formalization	entry	6
35	The computational-univalence bridge	S02	formalization	programme	4
36	Polarization of a cyclotomic spectrum	S03	mathematical	research	8
37	Construction Problem A1 on the syntomic stack	S03	mathematical	research	8
38	Archimedean Frobenius from quaternionic THH	S03	mathematical	research	8
39	A certified inertia for the pole-removed Weil form	S03	machine	entry	9
40	The Tamagawa factor as a local term	S04	mathematical	research	9
41	The arithmetic flow	S04	mathematical	programme	8
42	Sub-conductor rank detection	S04	mathematical	research	12
43	Murmurations from the flow: a falsification test	S04	machine	entry	9
44	The Hodge decomposition in Lean	S05	formalization	programme	9
45	No δ-structure on the archimedean prism	S05	mathematical	research	9
46	Independence of the twelve prism axioms	S05	mathematical	entry	12
47	Descent beyond the polarized case	S05	mathematical	research	5
48	A prism reading of Voisin’s counterexample	S05	mathematical	entry	9
49	Close the formal algebra leaf	S06	formalization	entry	9
50	Toward an algorithm for prismatic cohomology	S06	machine	research	12
51	The real-place gauge	S06	mathematical	research	8
52	The gluing problem for the global substrate	S06	mathematical	programme	8
53	The multiplicative-group gate	S07	mathematical	research	9
54	The first falsification test for GL$_2$	S07	mathematical	research	9
55	Ambidexterity over the arithmetic base	S07	mathematical	programme	8
56	Construct the global arithmetic curve	S07	mathematical	programme	8
57	Coherence for arithmetic modes	S07	formalization	research	11

Taking part

Each problem has an issue: post a solution, a partial result, a counterexample to the problem as stated, a reference showing that it is already solved, or a correction. A solution that meets the criterion of its card is recorded, under the solver's name, in the next version of the papers, in the next edition of the source work, and in SOLVED.md. No prize is attached. Corrections to the papers go to ERRATA.md. See CONTRIBUTING.md.

For AI laboratories and tool builders: problems.json lists every problem with its source, status, kind and decision criterion, and formal/ contains Lean 4 statements against a pinned Mathlib; both may be used as a benchmark under the licences below.

Licence and name

Text, data and figures: CC BY 4.0. Code — the Lean files (formal/), the scripts (tools/), the typesetting files (styles/, filters/) and the source code of the figures (figures/web/): Apache-2.0. The fonts and three.js in figures/web/ keep their own licences. The seven works keep their own licence (CC BY-NC-ND 4.0). The adjective Grothendieckian names a method used in the works and implies no attribution of design or endorsement to Alexander Grothendieck; see NOTICE. Author: Vincent Liu, WiseMotif.ai, ORCID 0009-0000-8718-222X, vincent.liu@wisemotif.ai.

References

[S00] Vincent Liu, The Grothendieckian Programme: Recursive Mathematical Discovery in the Age of Machine Reasoning, Zenodo, 2026, An overview of Series 01–07; WiseMotif Monograph of Mathematical Intelligence, Series 00, doi:10.5281/zenodo.23669107.

[S01] Vincent Liu, GAI-MUA: The Grothendieckian Machine. A Mathematical Universe Architecture for Verifiable and Self-Evolving Artificial Intelligence, Zenodo, 2026, WiseMotif Monograph of Mathematical Intelligence, Series 01; 6 volumes, doi:10.5281/zenodo.23480696.

[S02] Vincent Liu, GM-Omega-Infinity: Topos-Native Intelligence. A Sheaf-Theoretic Architecture for Verifiable Machine Intelligence, Zenodo, 2026, Series 02; 5 volumes, doi:10.5281/zenodo.23538544.

[S03] Vincent Liu, Arithmetic Spectral Descent: Obstruction Theory, Hodge–Krein Realization, and a Falsifiable Architecture for the Riemann Hypothesis, Zenodo, 2026, Series 03; 4 volumes, doi:10.5281/zenodo.23564898.

[S04] Vincent Liu, Fractured Adelic Geometry: A Foundational Framework for the Birch and Swinnerton-Dyer Conjecture, Zenodo, 2026, Series 04; 4 volumes, doi:10.5281/zenodo.23585887.

[S05] Vincent Liu, The Archimedean Prism: Tame Period Geometry and the Hodge Conjecture, Zenodo, 2026, Series 05; 5 volumes, doi:10.5281/zenodo.23597954.

[S06] Vincent Liu, Strained Prismatic Geometry: Deformation over the Absolute Base and the abc Conjecture, Zenodo, 2026, Series 06; 5 volumes, doi:10.5281/zenodo.23610034.

[S07] Vincent Liu, The Arithmetic Topos: Foundations of a Trace-Theoretic Approach to the Langlands Programme, Zenodo, 2026, Series 07; 5 volumes, doi:10.5281/zenodo.23629681.

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_{WiseMotif Monograph of Mathematical Intelligence · Three papers on the Grothendieckian Programme
Series 00: The Grothendieckian Programme · Series 01: GAI-MUA: The Grothendieckian Machine · Series 02: GM-Omega-Infinity: Topos-Native Intelligence · Series 03: Arithmetic Spectral Descent · Series 04: Fractured Adelic Geometry · Series 05: The Archimedean Prism · Series 06: Strained Prismatic Geometry · Series 07: The Arithmetic Topos
© 2026 Vincent Liu · ORCID 0009-0000-8718-222X · grothendiecklab.com}