An explicit 36-point configuration raises the recorded lower bound for the 13 × 13 × 13 grid.

This project establishes C(13) ≥ 36 with a centrally symmetric subset of {0,…,12}³ containing no five points on a common sphere or plane. The certificate improves the inherited bound C(13) ≥ 33 and supplies a directly checkable target for Problem 6.60 in the AlphaEvolve repository; it does not claim that 36 is optimal.

The configuration consists of 18 antipodal pairs on pairwise distinct central shells. It is accepted by three independent exact routes — Python cofactor expansion, Python Bareiss elimination, and Rust — and by a fourth checker written from scratch by an independent model. A seven-mutant suite exercises the verification boundary.

The frozen problem

The official problem page states: “For n a natural number, let C(n) denote the size of the largest subset of [n]^3 = {1,…,n}^3 such that no 5 points lie on a sphere or a plane. Obtain upper and lower bounds for C(n) that are as strong as possible.”

The result

For the exhibited set, each of the C(36,5) = 376,992 lifted 5×5 determinants with columns [x, y, z, x²+y²+z², 1] is a nonzero exact integer. The minimum absolute determinant is 2, which certifies the lower bound C(13) ≥ 36.

The same checkers accept all six official AlphaEvolve record sets for n = 7,…,12, so the comparison uses the convention that produced the prior records.

Prior record

The paper and official repository list lower bounds only through n = 12, including C_6.60(12) ≥ 33 in November 2025. This gives the inherited baseline C(13) ≥ C(12) ≥ 33. To our knowledge, documented searches on 12 June 2026 found no public claim of C(13) ≥ 34.

Artifacts

Limitations

This is a lower bound for n = 13 only. C(13) = 36 is not claimed, 36 is not claimed maximal even empirically, and no claim is made for other values of n; the best upper bound remains the trivial 52. The priority statement is limited to the documented public search and does not exclude unposted concurrent work.

The result has not been externally refereed. The authors were notified privately. Corrections will be logged, dated, and never silent.