A forbidden triple whose every pair survives
Three nonoverlapping crosses trap a required cell. Each pair has an independently checked complete surround. Use the switches to test every subset, then rotate or reflect the entire configuration.
The obstruction is certified
The fixed crosses occupy different coordinate planes, with centers at
The uncovered cell \(p=(-1,-1,0)\) must be filled to surround the core. Exactly 27 cross placements can cover it. Every one overlaps at least one of the three fixed crosses. Removing any one cross admits a complete surround of the other two; all three witnesses are published.
This establishes subset-minimality for this obstruction: every proper subset is extendable under the stated surround criterion. It is one example, not an enumeration of every forbidden patch.
The synthesized marking
The learner searched admissible-value sets for a shared vector. One and two possible states were insufficient; three worked. With basis vectors \(e_1,e_2,e_3\), it found
Every proper intersection is nonempty; the intersection of all three is empty. The three constraints are placed on the tile prototypes so they coincide precisely at the gap when the bad configuration occurs. Transporting them through all 48 cubic symmetries gives 48 channels with 144 prototype assignments. The group action permutes channels.
All subsets of the learned patch
| Fixed tiles | Marking decision | Geometric evidence |
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What changes, and what is preserved?
Under fixed-value equality, accepting every pair guarantees that all the assigned sections agree wherever they overlap. It is therefore impossible to directly reject this triple while accepting all its pairs. More vector components or larger support do not remove that obstruction.
This prototype instead specifies allowed values for a shared section. It is a proposed change in marking semantics. Runtime rejection is an empty intersection at a point and channel; it does not call the geometric oracle or scan a list of bad patches.
The compiler’s finite field audit proves that an empty intersection can occur only at a translated symmetry copy of this exact certified triple. It introduces no other forbidden pattern. This is stronger than testing a finite sample of unrelated patterns. All 1,152 tested combinations of subset, symmetry and translation passed, and rollback and the symmetry action were checked independently.
Scope for finite coronas and the next experiment
The negative label means that the three-tile core cannot be completely surrounded. The raw three-tile packing itself is nonoverlapping. Pair witnesses cover the complete face/edge/vertex halo of each fixed pair; their outermost tiles need not be extendable.
An infinite tiling must fill the certified gap. A finite-corona search must apply the learned rejection only when that transformed gap is an active required cell. Otherwise it could reject a permitted outer-boundary hole. The prototype exposes this guard explicitly.
The next stage is a catalogue of certified minimal patches, then constraints for deeper failed continuations with all boundary assumptions retained. Each rule needs a sound encoding and a comparison on the same target, scheduler and budget. This immediate dead-cell example demonstrates representability, not a measured search speedup.