# Exact certificate: a 44-vertex triangulation of RP^6

This directory accompanies Alexey M. Kolosov, "A 44-vertex triangulation of
real projective 6-space" (manuscript dated September 6, 2026).

## The object

Let a_1,...,a_44 be the rows of coordinates44.txt. Define

    P = conv(a_1,...,a_44,-a_1,...,-a_44).

The certificate proves that P is a centrally symmetric simplicial 7-polytope,
and that its antipodal boundary quotient is a triangulation of RP^6.
No normalization of the integer coordinates is to be performed.
No assertion of vertex-minimality is made.

## Files and conventions

- coordinates44.txt: 44 rows of 7 integers. Rows are numbered 1 through 44.
- facets88_half.txt: 7404 antipodal facet representatives. Each row has 7
  zero-based labels. Labels i=0,...,43 mean a_(i+1); i=44,...,87 mean
  -a_(i-43). The antipode of i is (i+44) modulo 88. Adjoin all antipodes
  to obtain 14808 facets. The input file is not trusted as a convex-hull
  output: supporting hyperplanes and completeness are checked.
- facets_RP6.txt: the 7404 quotient facets with one-based labels 1,...,44.
  Both programs reconstruct the quotient and compare it with this file.
- verify.py: exact integer verifier using fraction-free Bareiss elimination.
- verify_rational.py: separate implementation using rational Gaussian
  elimination; it does not import verify.py.
- diagnostics.py: exact first-seven-row determinant, missing edges of the
  quotient graph, and inner-product statistics. It uses no hull software.
- test_verifiers.py: accepts the valid data and rejects three types of
  deliberately corrupted data, using temporary directories.
- verification_result.json and verification_rational.json: reports of runs.
- diagnostics.json: auxiliary computed invariants.
- test_results.json: results of the rejection tests.
- SHA256SUMS: SHA-256 digests of the stable data and program files.

The names in this directory are local file names. The programs determine
its location from __file__, so the directory itself may be renamed.

## Run

Keep the files together. Python 3.9 or newer, standard library only:

    python verify.py
    python verify_rational.py
    python diagnostics.py
    python test_verifiers.py

The programs have been run on the supplied data. They use no network,
third-party modules, cached hull, or floating-point geometric decisions.
Timing fields in their reports are not part of the mathematical tests.
Reports are overwritten when rerun; checksums intentionally cover the
stable inputs and programs rather than reports with changing timings.

Expected principal outputs:

    vertices of P: 88
    edges of P: 1876
    ridges of P: 51828
    facets of P: 14808
    antipodal common-neighbor violations: 0
    f-vector of the quotient:
        (44, 938, 7024, 22555, 34936, 25914, 7404)
    Euler characteristic: 1
    quotient facet file matches the computed quotient: true

## Why the finite tests prove the result

Each proposed facet has a nonsingular 7 by 7 vertex matrix. The checker
solves B*u=1 exactly, clears denominators, and tests an integer support
pair (c,b), b>0. Equality holds precisely at its seven listed points, and
strict inequality at the other 81 points. The opposite facet follows by
changing c to -c. Every input point appears in a certified simplex facet.

Each six-element subset of a certified facet is a genuine ridge. A ridge
of a convex polytope belongs to exactly two facets. The ridge-incidence
check therefore proves closure under every adjacency step in the full
facet graph. This graph is connected, so a nonempty closed list is the
full list. Consequently the entire polytope is simplicial.

The graph test rejects an edge between antipodes and any common neighbor
of a pair of antipodal vertices. This makes the quotient an actual
simplicial complex with unique simplex lifts up to simultaneous sign.
Its realization is the antipodal quotient of the polytopal sphere, and
radial projection identifies this quotient with RP^6. The manuscript
contains full proofs of these implications.

## Provenance and limitations

The integer data are unchanged from the coordinate and facet certificate
produced in the computational exploration on September 6, 2026.
The proof improves the 45-vertex construction in:

D. Guyer, S. Steinerberger, Y. Yang, An Efficient Triangulation of RP^5,
arXiv:2603.07808v1 (2026), Section 2.3 and Question 3.2.

The numerical exploration used that 45-pair construction as a starting
point, removed one pair, and varied the remaining points with antipodal
symmetry. These files reproduce the exact verification of the final
object, not the complete search trajectory.

ChatGPT assisted the exploration, programming, and manuscript preparation.
The two verifier implementations were developed within the same workflow;
they are not an external independent review. No global priority claim or
claim that 44 is the minimum is established by this certificate.