Plain language

What this result means

The feasibility rule is crisp, but the search space mixes positions with rotations and many possible contact graphs. Escaping the established eight-corner topology exposed a packing family that two strong prior methods had missed.

  • All 55 cube pairs have a certified separating axis; the smallest certified gap is 4.69×10^-7.
  • The C₃ structure contains two body-diagonal cubes, three near face centers, three edge-shifted cubes, and three tilted cubes.
  • The result was a best-known improvement when produced. Friedman's live table now lists a smaller 2.88295+ construction by Haowei Lin.

Visual notes

How to read the result

Comparison of container side lengths for eleven-cube packings from Friedman, AlphaEvolve, Berthold, Numaro, and the current public table.
The benchmark ladderThe Numaro packing cleared the published 2026 benchmark when found; the live table has since moved lower.
Schematic projection of a threefold-symmetric eleven-cube arrangement.
A different symmetry basinThe schematic shows the four C₃ orbits that replace the conventional eight-corner family.

Result table

A new C₃-symmetric packing beat both AlphaEvolve and the published 2026 benchmark.

CellBaselineNumaroDeltaNote
Friedman 19982.9120962.891999838-0.020096certified beat
AlphaEvolve 20252.8945312.891999838-0.002531same objective reproduced
Berthold et al. 20262.8944467812.891999838-0.002447published benchmark at discovery
Current public table2.88295+2.891999838supersededHaowei Lin, July 2026

Method

How it was found

Batched penalty optimization searched cube centers and unit quaternions under containment and separating-axis penalties, with diverse structural seeds rather than a fixed eight-corner template.

  • Calibrated the objective on the published AlphaEvolve coordinates.
  • Ran GPU multistart search across structurally different seeds.
  • Refined the new C₃-symmetric basin near its feasibility wall.
  • Certified containment and pair separation with interval arithmetic and two independent geometric checks.

Verification

How it was checked

The standalone verifier uses 200-bit interval separating-axis tests, convex-QP minimum distances, and point-in-cube sampling. It certifies all 55 pairs as separated and the bounding side at 2.891999838304098.

Scope

What is not being claimed

This is a feasible packing, not an optimality proof, and it is no longer the smallest public value. Its durable contribution is the independently certified C₃-symmetric construction that beat the published benchmark at discovery.

References

Baseline sources

Citation

How to cite

Numaro AI Autoresearch. "Eleven freely rotated cubes in a smaller containing cube." Numaro Research Report NUMARO-2026-019, 2026.

@techreport{numaro2026Packing11Unit,
  title = {Eleven freely rotated cubes in a smaller containing cube},
  author = {Numaro AI Autoresearch},
  institution = {Numaro},
  number = {NUMARO-2026-019},
  year = {2026},
  url = {https://numaro.tech/research/packing-11-unit-cubes-2026/}
}