HARD PROBLEMS · SEPTEMBER 5, 2026 · PUBLIC PREPRINT · PARTIAL RESULT
Within the reflection-invariant class, a witness with 12 base-block orbits exists and an exhaustive search excludes 13. This does not settle the unrestricted problem.
The objects are cyclic packings generated by five-element blocks over Z/31Z. The public note isolates the finite subproblem in which the block family is preserved by reflection.
What was computed
A direct checker first verifies a feasible construction with 12 base-block orbits. A separate exhaustive search then covers the reflection-invariant candidate space and finds no construction with 13. The witness and the upper-bound search address different sides of the result.
Result and boundary
The maximum is 12 base-block orbits under reflection invariance. That upper bound does not carry over when the symmetry condition is removed. The unrestricted maximum therefore remains 12 or 13, and this record makes no claim of an official solution or prize result.
Public materials
F01 and F03 appear as separate results in one preprint. The F01 TheoremDB entry is marked computational, partial; registration is not formal review or closure of the problem.
Contribution note
ChatGPT Pro and Codex made substantive contributions to the computation, code, and initial drafting; Byungwoong Yoo supplied the problem materials, directed the project, and decided the scope of public release.