Choose groups of five from points numbered 0 through 30. Whenever a group is chosen, include all its cyclic shifts: add 1 to every number, then 2, and so on, wrapping 31 back to zero. No three points may occur together in two distinct groups. How many starting patterns can coexist under that rule?
I examined the part of this cyclic packing problem that also preserves reflection symmetry. A checked construction gives twelve base-block orbits, and an exhaustive search excludes thirteen or more within that symmetric class. Removing the reflection condition leaves a different question: the unrestricted maximum is still twelve or thirteen on the evidence established here.
Rotation does not imply reflection
A five-point group is called a block. For example, shifting every entry of {0,1,3,7,15} through the 31 positions generates its translation orbit. A base block represents that whole orbit. Since 31 is prime and a block has five points, every orbit contains exactly 31 distinct blocks.
Reflection sends x to −x modulo 31, so 1 becomes 30 and 2 becomes 29. Reflection invariance does not require every block to reflect onto itself. It requires the entire developed family to remain the same. Some translation orbits map to themselves; others must be included with their reflected partners.
I separated the construction check from the upper-bound search. One valid example establishes that a size is attainable. Proving optimality also requires an argument that no larger admissible family escaped the search.
What the twelve-orbit construction checks
The public package lists all twelve base blocks, including {0,1,3,7,15}. Developing them through all 31 shifts gives 372 distinct blocks. Each block contains ten triples, and the checker verifies that all 3,720 triples are distinct. This directly checks the rule against repeating three points.
A second test compares all 69,006 unordered pairs of developed blocks and checks that their intersections contain at most two points. That is another way to detect a repeated triple. The checker also reflects the complete family and verifies that it is unchanged. Each of the 465 pairs of ground-set points occurs in exactly eight blocks of this construction.
The numerical lower bound of twelve was already known. Brouwer’s table and linked construction provide a cyclic binary code with length 31, weight five, minimum distance six and 372 words. Reading each word’s five 1-positions as a block gives twelve translation orbits. The distance between two weight-five words is ten minus twice the size of their support intersection, so distance at least six means at most two shared points. I kept that prior construction distinct from the reflection-invariant witness checked here.
Why thirteen is the remaining target
An upper bound follows by fixing two ground-set points. There are 29 other points that can complete a triple with that pair. A five-point block containing the pair uses three of them, and none can be reused with the same pair. Thus any fixed pair belongs to at most nine blocks.
Every block contains ten pairs. If D is the number of developed blocks, 10D ≤ 9×465. With m base-block orbits, D=31m, giving m≤13. Together with the twelve-orbit construction, this leaves the existence of thirteen as the unresolved endpoint. I restricted the exhaustive search to families preserved by reflection.
Encoding the symmetric choices as a graph
There are 5,481 translation orbits of five-point blocks. Removing those that already repeat a triple within their own development leaves 4,761 individually admissible orbits. Reflection-invariant families must select entire reflection partners. After also discarding internally conflicting partners, there are 348 eligible groups: 75 containing one orbit and 273 containing two.
Each group becomes a graph vertex. Two vertices are joined when their blocks can coexist without repeating a triple; there are 17,553 such edges. A set of mutually adjacent vertices—a clique—then represents compatible choices. Giving each vertex weight one or two records its number of base-block orbits. The target is a clique of total weight at least thirteen.
The encoding was checked in two ways. One represents blocks as bitmasks and counts intersections over all 31 relative shifts. The other records which of the 145 translation classes of triples each block uses. Both tests agree on every candidate orbit, reflection group and graph edge. Because any forbidden repetition involves two blocks, pairwise compatibility in this graph also guarantees compatibility of the whole packing.
When a completed search proves an upper bound
The C++ search chooses vertices in increasing order, so each clique has a unique enumeration path. After choosing a vertex, it retains only candidates adjacent to every selected vertex. It cuts a branch when even the sum of all remaining candidate weights cannot reach the target. That sum is a valid upper bound even if those candidates cannot all coexist.
The published execution terminates after 318,709 recursion nodes without finding weight thirteen or more. It has no time cutoff. A separately written JavaScript implementation uses another vertex order and a graph-coloring bound, reaching the same conclusion after 41,939 nodes. These are visit counts for two search strategies, not counts of all packings or a measured performance improvement. The upper bound rests on the complete correspondence, valid pruning and terminating execution together.
The checked construction and exhaustive upper bound therefore meet at twelve under reflection invariance. For a translation-closed family, any reflection x→−x+c is equivalent to negation. A thirteen-orbit example, if one exists, must lack every such affine reflection. The result does not exclude arbitrary other symmetries or settle unrestricted existence of thirteen.
The released material
I included this result and the separate F03 grid-slope result as different theorems in one preprint. The package contains source, witnesses, the graph and execution logs. Its main replay environment is Linux or WSL with Python 3.10 or later and a C++17 compiler; the separate implementation was recorded under Node.js 24.19.0. Replay needs no network or optimization solver. The witness JSON field order:10 is the multiplicative order of the multiplier 15 modulo 31, not the number of ground-set points.
I supplied the problem material and directed the project and release scope. ChatGPT Pro and Codex contributed substantively to the mathematical reductions, search code, computation, replay and drafting. Separate implementations provide internal computational verification, not external human review or a proof-assistant certificate. TheoremDB research record R11764 remains classified as partial / computational. It records this restricted result, not official closure of the unrestricted problem or a prize outcome.