Real projective space 
is the space of one-dimensional linear subspaces
of 
.
Equivalently, it is the quotient of the sphere 
that identifies each point with its antipodal
point. Its first two positive-dimensional examples are the real projective line,
homeomorphic to a circle, and the real
projective plane.
Small triangulations provide finite combinatorial models of these spaces. Guyer et al. (2026) constructed a 24-vertex triangulation
of 
and a 45-vertex triangulation of 
. Chang (2026) proved that every finite simplicial triangulation
of 
has at least 24 vertices, establishing the minimality
of the former construction. The lower bound does not require
the triangulation to be combinatorial. Chang (2026)
reports Codex assistance with proof exploration and verification
code.
Kolosov (2026) subsequently reported a 44-vertex triangulation of 
with face vector
 |
Its alternating sum is 1, as required by the Euler characteristic of 
. The construction is an antipodal quotient of the boundary
of a centrally symmetric simplicial 7-polytope with 88 vertices.
Exact verification programs were replayed by VibeMathed (2026). ChatGPT assisted
the construction search and verification code, but independent expert review had
not been reported as of Sep. 7, 2026. The construction supplies an upper bound
of 44 vertices, not a proof
of minimality.
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18 Oct 2026. 
." 8 Mar 2026. 
." Sep. 6, 2026.