Other bends, the same arm lengths

Each of the six arms takes two steps out from the central cube and then two perpendicular steps. Each bend has four possible directions. We enumerate every assignment, exclude arms that share a cube, and identify shapes related by proper cubic rotation. Face and edge contacts between different arms are allowed.

There are three centrally symmetric shapes: opposite arms bend oppositely. Without that requirement there are 72 shapes, with mirror images kept distinct unless a proper rotation identifies them. All 72 shapes have now received first-corona searches. The search status below distinguishes checked certificates from inconclusive runs.

First-corona search

The 70 variants beyond the two original checked exclusions each received four-second Glucose and CaDiCaL probes, plus at least one native-cardinality probe with a 20-second budget. Selected cases received 120-second probes; Kissat was also tried with a 200,000-conflict limit. Solved cases can terminate before their budget. No complete first-corona witness was found for these bends. As a positive control, both native-cardinality solvers did find independently verified first coronas for the ring octocube (coordinates and checks).

The table lists certified results and the three centrally symmetric cases. Select any of the 72 shapes below to inspect its individual search status.

A checked UNSAT first-corona proof establishes \(H=0\). A time limit establishes no upper bound. These individual runs are not controlled timing comparisons. Placement searches use proper cubic rotations and integer translations, with full face/edge/vertex touching coronas; they are specialized SAT controls, not GCTS marking benchmarks.

Inspect the alternatives

What the enumeration covers

The full assignment set has \(4^6=4096\) choices. Of these, 2,600 make two arms share at least one cube; they are excluded because they change the 25-cube construction. The remaining 1,496 assignments give 72 proper-rotation classes. Exactly 40 assignments are centrally symmetric, making three classes.

Two of those three arrangements use every bend direction once: the original single six-cycle, and two separate three-cycles. The third repeats some bend directions, while still making opposite arms bend oppositely. If “variation” also requires the author’s single-cycle rule, the original is the only centrally symmetric proper-rotation class in this fixed-length family.

The enumeration independently checks its classes using voxel shapes and quarter-turn rotations. First-corona exclusions use the same encoding as the original tile; separate rectangular placement scans and DRAT-trim proof checks audit the certified cases. The additional shapes shown by “All bend arrangements” have bounded searches recorded individually. An undecided run does not establish either the existence or the impossibility of a first corona.

Arm lengths, additional bends, and configurations with shared cubes are outside this census. All claims concern the stated lattice placement model.

Reproduction

python scripts/heesch-catalog/bent_arm_variants.py --enumerate rebuilds the census. The same script’s --verify /path/to/drat-trim --ids bend_018 checks the new certified three-cycle variant. Each result links its run record and available certificate. The original six-cycle keeps its previously checked certificate. Add --solve --ids bend_018 to regenerate the Glucose run; python scripts/heesch-catalog/bent_arm_cadical.py repeats the second solver’s probe of the unresolved case. Regeneration overwrites the corresponding run artifacts.

python scripts/heesch-catalog/search_bent_arm_coronas.py --solver minicard --seconds 20 searches the census using native nonoverlap constraints. Solver choices also include Glucose, CaDiCaL and Gluecard. Kissat uses a conflict budget through --conflicts, rather than a time budget. The optional --incremental mode for Minicard or Gluecard adds missing coverage requirements in batches while retaining learned constraints. Every positive result must pass a separate coordinate-based verifier. These are specialized SAT experiments, not the reference GCTS schedule. See the encoding and conformance notes.