Problem 60 (no 5 points on a sphere): certificate-backed bounds through C(20) >= 50

I am reporting explicit integer-grid constructions establishing the following lower bounds:

- `C(14) >= 38`
- `C(15) >= 40`
- `C(16) >= 41`
- `C(17) >= 44`

The primary result is a 44-point subset of `{0,...,16}^3` with no five points on a common sphere or plane. It consists of 22 antipodal pairs about `(8,8,8)`. By monotonicity it also establishes `C(n) >= 44` for every `n >= 17`.

Repository and reproducibility package:
https://github.com/milesandmistakes/no-five-sphere-grid-certificates

Frozen v1.0.0 release:
https://github.com/milesandmistakes/no-five-sphere-grid-certificates/releases/tag/v1.0.0

For the 44-point certificate, verification exhaustively tests all `C(44,5) = 1,086,008` five-subsets. It finds zero forbidden determinants, with minimum absolute determinant 2 and maximum 776,880. The package includes two independent exact checks: vectorized int64 cofactor expansion and Python-integer Bareiss elimination. It also passes the published floating-point reference logic. SHA-256 hashes, complete verification output, the certificate, and a separately reproducible exact C search are included.

A clean rebuild and 20-second run of the frozen C source with seed 2 reproduced the same 44-point set at iteration 904, followed by exact verification with zero violations.

A pre-release public search found Numaro reporting `C(14) >= 34`, `C(15) >= 35`, `C(16) >= 39`, and `C(17) >= 40`, plus the existing `C(13) >= 36` issue here. I found no public `C(17) >= 44` or stronger certificate. I therefore describe these only as apparently new, certificate-backed lower bounds, not as optimal values or guaranteed priority over unpublished work.

Disclosure: the stepping-stone constructions were produced through Claude-driven OpenEvolve searches. After the n=17 OpenEvolve run stopped at 42 points, an OpenAI Codex continuation adapted and ran an independent exact C ruin-and-recreate search, finding the 44-point certificate. I directed the work and take responsibility for the claim. The mathematical claim rests on the published coordinates and reproducible exact checkers, not on trust in model output.

I am happy to prepare a CLA-signed pull request adding the certificates and bounds in whatever format the maintainers prefer.

Start by reviewing the frozen v1.0.0 release of the linked no-five-sphere-grid-certificates package, including its certificate, exact C and Python verification outputs, and reproducibility files. Check how this repository records problem results and ask maintainers which format and files should receive the reported bounds. Done means the certificates and bounds are integrated in the accepted format and their exact checks remain reproducible.

Written by the indexing model from the issue text.