Learning to Complete Extremal Mathematical Structures

Extremal combinatorics seeks rare mathematical structures that satisfy hard constraints while maximizing a global objective. Most existing neural systems for mathematical discovery train a model to generate complete configurations from scratch. We argue that this framing mismatches the dynamics of effective search: high-quality solutions are more effectively found by preserving a feasible partial core and conditionally regenerating the remaining uncertain objects. We formulate extremal structure discovery as object-level active-core completion: the model is trained to conditionally complete masked objects given a feasible partial core, capturing inter-object relations rather than fitting a distribution over complete configurations. We instantiate this with a full-attention Transformer over object-aligned slots, and propose Inpainting PatternBoost, a variant of PatternBoost. In each iteration, elite constructions from a top pool are partially destroyed by random or structured remasking; the model learns to reconstruct the missing objects, and their completions are reinserted into the pool. We evaluate on the no-three-in-line problem and maximum sum-of-radii circle packing in the unit square, under a single shared abstraction. On no-three-in-line, our framework recovers the optimal $2n$-point configuration for $n$ up to $18$, and reaches $2n-1$ points for $n=19,20,25$, outperforming prior neural baselines. On circle packing, where FlowBoost combines neural generation with gradient-based local search, our approach matches or exceeds FlowBoost when paired with the same local search; even without local search, it reaches an average of 88.1% of FlowBoost's reported score across instances.