What this note offers
The note is intended for readers familiar with basic normed-space theory and convexity. It reorganizes the original argument and simplifies some steps; it is not an independent solution of Tingley’s problem.
Two points receive a detailed treatment: constructing a convex invariant set by successive convex enlargement, and using explicit support estimates before a direct comparison of the two chord coordinates. The full exposition, equations and references are in the PDF.
The underlying proof is attributed to OpenAI’s A positive solution to Tingley’s problem, dated 23 September 2026. The link above identifies the fixed manuscript used for this note.
A route through the note
Start with basic normed-space theory, convexity and the Hahn–Banach theorem.
- Sections 1–3: reduction and geometric preparation. Follow the zero-defect target, sphere geometry and ultrapower construction.
- Section 4: fixed-point background. Review the tools leading to Darbo’s theorem.
- Sections 5–6: the core chord argument. Follow the convex invariant-set construction, common supports and the final coordinate comparison.
If Darbo’s theorem is already familiar, use Theorem 4.3 and continue to Section 5, returning to Section 4 as needed. The PDF’s reading guide links to the relevant sections.
Materials and further reading
For complementary explanations, the following third-party resources may be useful. They are further reading, separate from the original proof source.
Version, responsibility and citation
Version 1. Prepared 13 October 2026. First published 14 October 2026; version publication date: 14 October 2026.
Publication responsibility: Ryotaro Tanaka. This role covers release, versioning and corrections. It does not claim authorship of the original proof. No author is assigned to this note’s bibliographic record.
The manuscript was drafted and revised using ChatGPT. The full use statement remains in the PDF. This note does not claim independent human proof review or Lean formalization.
Suggested reference
A reader’s reconstruction of the chord argument for Tingley’s problem. Exact Mathematics with AI — Expository Note, Version 1, published 14 October 2026, prepared 13 October 2026. Publication responsibility: Ryotaro Tanaka.
Version 1 PDF. See the existing content terms for explanatory text and its HTML/PDF presentations; third-party materials retain their own terms.
Corrections and questions
Please identify the note version and the relevant page, section or equation when reporting a correction.