Geometric consequences from local tilings
Can a corona teach a tile a larger marking?
Yes, in this Turtle experiment. Intersecting the glued markings of every 3-corona produces additional constraints that prune the growing search tree.
The common feature
Place the known marking on every tile in each corona and glue the overlapping values. Keep a value on the root tile only where every corona assigns that same value.
otherwise * (free).
With a complete enumeration, any global tiling compatible with the original marking must obey this extension. A finite patch can still fail at its boundary, where neighboring tiles need their own coronas.
Coronas here use positive point-support contact, with a viable outer frontier. Assigned zero is retained; missing values are free. This is an exact point-model experiment, with independent replay but no external exhaustive proof certificate.
What happened in the actual GCTS-I search?
Loading measurements…
| Seed | Known time / tiles | Extended time / tiles | Attempts: known → extended | Backtracks: known → extended |
|---|
These timings reuse the derived marking. The Turtle 3-corona enumeration took 84 seconds, plus 0.62 seconds to reduce its support. A single cold run has not demonstrated an end-to-end speedup.
Complete enumeration matters
| System | Corona | Coronas found | Original → common values | Status |
|---|
The partial Hat 3-corona intersection was not promoted. Compact markings preserve all additional pair conflicts, including contacts outside the tiles’ positive support. This is not a minimum-support proof.
On the Turtle index-3 lattice, all 3,005 original 3-coronas replay correctly. Decorating every tile with the extension leaves 2,996 internally compatible; 1,782 also retain an extended viable frontier. Thus the shared values also reveal boundary arrangements that need to be pruned.