A 44-vertex triangulation of
Does real projective 6-space admit a simplicial triangulation with fewer
than 45 vertices? This is Question 3.2 of Guyer, Steinerberger and Yang,
“An Efficient Triangulation of RP^5”, arXiv:2603.07808 (2026).
The question asks for an improvement on their 45-vertex construction,
not for the exact minimum number of vertices.
- Result
- Proved(see note)
- Status
- Resolved
- AI contribution
- AI-assisted
- Method
- Construction
- Field
- Combinatorial topology; triangulations of manifolds; convex polytopes
- Posed by
- Dan Guyer, Stefan Steinerberger, Yirong Yang — Question 3.2, arXiv:2603.07808
- Year posed
- 2026
- Years open
- 0y
- Solved
- 2026-09-06
- Model
- ChatGPT (OpenAI, model version unstated)
- Vendor
- OpenAI
- Collaborators
- Alexey M. Kolosov
- Verification
- Site-confirmed
- Publication
- Announced
- Significance
- 10 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
The manuscript gives an explicit centrally symmetric simplicial
7-polytope with 88 integer vertices. Its antipodal boundary quotient
is a 44-vertex triangulation of real projective 6-space, with f-vector
(44, 938, 7024, 22555, 34936, 25914, 7404).
This improves the 45-vertex construction of Guyer, Steinerberger and
Yang and gives an affirmative answer to their Question 3.2. Integer
coordinates, facet lists and exact-arithmetic verification code
accompany the manuscript. No claim that 44 is vertex-minimal is made.
What the AI did
The manuscript's Computational assistance section: "OpenAI's ChatGPT was used in the computational exploration, in the development of the verification programs, and in preparing this manuscript. The two implementations were developed within the same AI-assisted workflow; their agreement is not external independent validation." Search assistance, code and prose: the assisted tier.
Verification
Site-confirmed: both of the author's exact-arithmetic verifiers (anc/verify.py with integer Bareiss elimination and anc/verify_rational.py with rational Gaussian elimination, Python standard library) were replayed here on 6 September 2026 from a clone of cheptil/44-vertex-triangulation at 2c62b31. Both pass: every facet's supporting hyperplane, completeness of the facet list, the antipodal disjoint-star condition, f-vector (44, 938, 7024, 22555, 34936, 25914, 7404), quotient Euler characteristic 1, in about 3.5 seconds; the negative controls (perturbed coordinate, missing facet) fail as they should. Question 3.2 confirmed verbatim in arXiv 2603.07808. The manuscript's "Computational assistance" section discloses ChatGPT's role and says the two verifiers share one workflow, which is why an independent replay was worth doing.
Sources
Submitted by cheptil on