Six bent arms
The 25-cube shape shown in Sridhar Ramesh’s rotating “3d swastika” illustration: one central cube, six spokes, and one right-angle bend on each spoke. The model below is our reconstruction from the animation and the author’s direction-cycle definition.
Checked lattice result: \(H=0\). No complete first corona exists. Glucose supplied an UNSAT proof, independently accepted by DRAT-trim. This excludes tilings using integer translations and cubic-lattice rotations of this exact prototype.
Other bend arrangements: compare the three centrally symmetric variants and inspect all 72 non-overlapping shapes with these arm lengths.
Exact construction
Choose the cycle
\[e_x\longmapsto e_y\longmapsto e_z\longmapsto-e_x\longmapsto-e_y\longmapsto-e_z\longmapsto e_x.\]Writing this map as \(\sigma\), the voxel centers are
\[P=\{0\}\ \cup\!\!\bigcup_{d\in\{\pm e_x,\pm e_y,\pm e_z\}}\!\!\{d,\ 2d,\ 2d+\sigma(d),\ 2d+2\sigma(d)\}.\]This gives \(1+6\cdot4=25\) cubes. The cycle follows the author’s definition; choosing rotated coordinates does not change the tile. The arm lengths are read from the illustration. This report makes no claim that the source post discusses tiling.
What was checked
| Check | Result |
|---|---|
| First full touching corona | UNSAT |
| All possible root-neighbor placements | 4,396 |
| Variables / clauses | 111,770 / 319,778 |
| Glucose search, with proof recording | 17.34 seconds |
| Independent proof check | Verified; loading timing… |
Every voxel sector incident to a root-cube vertex must be covered, so face, edge, and vertex contacts are all included. Every rotated and translated tile that could cover such a voxel is eligible. Clauses require complete coverage and forbid overlap everywhere, including outside the root’s neighborhood. Outermost tiles have no further coverage requirement.
A separate verifier generates orientations by quarter turns, computes the boundary using cube-vertex incidence, and scans an explicit rectangular range of translations. It recovers the same complete placement universe. The CNF is regenerated byte for byte before DRAT-trim checks the proof. Thus the conclusion does not rely on failure to extend one chosen arrangement.
Scope: this is a specialized voxel-corona SAT calculation. It is not a GCTS marking benchmark. The times are individual runs, not a controlled solver comparison. Arbitrary rotations and noninteger translations are outside this model. Reflections add no new shapes here because the prototype has central inversion symmetry.
Reproduce the check
With Python, python-sat, and DRAT-trim installed, run python scripts/heesch-catalog/verify_bent_six_arm.py /path/to/drat-trim from the repository root. Add --solve to regenerate the Glucose run before verification. The receipt records the source and certificate hashes and the checker revision.