Learning without a known marking
From complete coronas to a family of markings
Learn several markings that accept every original one-corona. Extend each through the common values of its own larger coronas. The choice of marking changes what propagates.
The workflow tested
For each: complete marked 1-coronas → common marking → complete 2-coronas → common marking → …
All initial coronas supply agreement constraints. Individual components may be *; changing their support rebuilds the equalities. Each successive marking is the intersection of all glued sections at that stage. Only complete, nonempty intersections advance.
Both tiles use the index-3 sublattice, reflections and signed rank-3 components. Initial support stays within points covered by every original one-corona and at most one sublattice step from the tile. These are rooted-tile coronas, different from the demo’s pair-centered check.
How the alternatives developed
Loading measurements…
| Marking | Failed attachments blocked | Initial values | After 1-coronas | After 2-coronas | After 3-coronas |
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Enumeration evidence
| Marking / round | Coronas found | Common values | Status | Enumeration time |
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Values above are assigned components after reduction; “common values” here are before reduction. Every reduction preserves all pair conflicts, including marking-only overlaps. Unknown intersections are not used.
Using each marking to tile farther
| Marking | Seed 1: time / tiles | Attempts / backtracks | Seed 3: time / tiles | Attempts / backtracks |
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What this establishes
The workflow can produce stronger markings from scratch, and alternatives need not extend in the same way. A plateau at one radius need not persist at the next. It does not establish that the strongest initial classifier is the fastest tiler, or that all learned restrictions preserve unmarked global tilings.
Initial catalogues were independently enumerated again. Whole-corona agreement, completed intersections, support reduction and returned tiling patches were independently checked. The marking-space search is bounded; it is not an exhaustive search of all markings or an infinite-tiling proof.