Structured Scaling of AI Discovery Across Diverse Scientific Domains — part 12/12

Cleaned-text code points [171691, 175457). Bibliography omitted here; source.md includes it.

Best results and discovered solutions. The n = 26 n=26 and n = 32 n=32 solutions recover the strongest known packing values under the same evaluator, but the two instances have visibly different structure. The n = 26 n=26 construction contains a dominant central circle, large boundary anchors, and a near-triangular ring of medium circles. The n = 32 n=32 construction is more homogeneous and resembles a fixed six-row quasi-hexagonal layout. These structures show that the search does not only adjust radii locally; it recovers global geometric organization suitable for each instance size ( Figure 10 g,h).

Case Analysis. The evolved n = 26 n=26 solver uses a coarse-to-fine strategy: broad center-set exploration first identifies promising layouts, and exact radius optimization then solves the radius subproblem under boundary and pairwise-distance constraints. The n = 32 n=32 solver simplifies to a more direct routine by reusing an incumbent or fixed quasi-hexagonal layout, then solving for radii through linear programming. The two programs therefore adapt the search granularity to the instance: broader geometric exploration for n = 26 n=26 and more efficient radius refinement for n = 32 n=32 .

##### Hadamard maximum-determinant problem of order 29 29 .

The Hadamard maximum-determinant problem of order 29 29 is a classical benchmark in extremal matrix theory and D D -optimal design.

Order 29 lies outside the Hadamard regime, so the best known solutions are near-extremal sign matrices rather than exact Hadamard matrices. The evaluator computes the determinant exactly using the Bareiss algorithm and reports both the raw determinant and a normalized score.

Best results and discovered solutions. The discovered sign matrices recover the long-standing classical lower-bound record. Their structure is best inspected through the Gram matrix rather than raw signs, since row and column permutations and sign flips can change the visible matrix while preserving equivalence. The high-determinant matrices exhibit near-orthogonal Gram structure, with off-diagonal interactions controlled so that rows remain as close to mutually orthogonal as possible under the order-29 constraint ( Figure 10 f).

Case Analysis. The strongest solver expands the initial search beyond a single quadratic-residue construction. It builds a diverse seed pool containing incumbent, quadratic-residue circulant, orthogonal-sign, and cropped Sylvester-style matrices, then alternates inverse-guided multi-flip hill climbing with simulated annealing. This combination allows the program to make coordinated sign changes rather than relying on isolated single-entry flips, which is essential for escaping low-determinant basins while preserving the global Gram structure needed for a near-extremal matrix.

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