The Arithmetic Topos — Research Architecture Edition 1 (v1.0.0)

The Arithmetic Topos

Foundations of a Trace-Theoretic Approach to the Langlands Programme

Vincent Liu · Research Architecture Edition 1 · version 1.0.0 · first public edition

A monograph in five volumes bound as one — 7 parts, 40 chapters, 2,001 pages, 120 vector plates and 280 references — and the seventh work of the series: a self-contained research subject of the Grothendieckian Programme, following Strained Prismatic Geometry (doi:10.5281/zenodo.23610034). It asks why the Langlands correspondence can be proved over function fields and remains open over number fields in general, and what object would let the function-field proofs, which take traces of Frobenius over a curve, be imitated over ℤ. Four obstructions are argued to be one; a geometry is defined as a six-functor formalism with duality and a primitive trace, by twenty-eight axioms; the global arithmetic curve 𝔛_ℤ is specified, not built, and its existence is a conjecture (GC.1). The Main Conjecture (Conjecture 19.5) states three equivalences, each a separate conjecture, from which eight corollaries are derived with the hypotheses each consumes. Every one of the 1,609 numbered statements carries one of five status marks; the ledger of 111 items holds 53 conjectures.

This work does not prove the Langlands correspondence, and no case of its Main Conjecture is established unconditionally in it (Proposition 40.4).

New here? Start with the preview (72.9 MB): the Series Note, the Abstract, the Preface, How to Read This Book, notation, the contents, the list of plates and the whole of Part I.

	File	Pages	Size
Preview	The-Arithmetic-Topos-v1.0.0-Preview.pdf	283	72.9 MB
Complete monograph (Volumes I–V and Appendices A–G)	The-Arithmetic-Topos-v1.0.0-Complete.pdf	2,001	85.5 MB
Companion code (Apache-2.0)	The-Arithmetic-Topos-v1.0.0-code.zip	—	93.7 KB

Permanent archive and citation — Zenodo, concept DOI 10.5281/zenodo.23629681 (this edition: 10.5281/zenodo.23629682). The files here are byte-identical to the Zenodo record.

Authenticity — SHA256SUMS is signed with OpenPGP key DA1A 1AEF D8B2 C17C FF3F 1D1B 3905 0F50 7A4F EC17, timestamped by two RFC 3161 authorities (FreeTSA, DigiCert) and anchored in Bitcoin (OpenTimestamps); the key is the same as for Series 01–06. This release is immutable. See VERIFY.md.

sha256sum -c SHA256SUMS --ignore-missing
gpg --import VincentLiu-signing-key.asc && gpg --verify SHA256SUMS.asc SHA256SUMS

License — CC BY-NC-ND 4.0 for the work; the code listings, aletheia and the Lean 4 statement skeleton are additionally available under the Apache License 2.0, as the copyright page states (NOTICE.md). Text and data mining rights are reserved: model training requires written permission, individual reading with AI tools is welcome — TDM-AND-AI-POLICY.md.

README · 中文说明 · Errata · Discussions