A penalised Saito functional for heuristic search of free line arrangements

We introduce the penalised Saito functional \mathfrak S_{λ,β}(A;d_1,d_2) for a reduced arrangement A of n lines and a prescribed pair d_1+d_2=n-1. It measures the alignment of a candidate Saito determinant with the defining polynomial while penalising the failure of the candidate derivations to be logarithmic. We prove that the functional takes values in [0,1], vanishes exactly when A is free with exponents (1,d_1,d_2), and lies strictly between 0 and 1 otherwise. For fixed (d_1,d_2), it is upper semicontinuous on the reduced configuration space, continuous at arrangements free with the prescribed pair, and converges as λ\to\infty to the corresponding binary freeness test. We use a numerical approximation of this functional, together with a small b_2-shell term, to guide fixed-cardinality line-replacement searches over \mathbb{Q} and selected quadratic extensions. Numerical values are used only to select candidates; every reported arrangement is certified in exact arithmetic using Saito's criterion. At the current snapshot, the certified database contains 6{,}146 representatives with distinct Weisfeiler--Leman fingerprints and cardinalities up to n=28. Among them, 3{,}012 have multiplicity gap ε(A)=d_1-m(A)\geq2, including lower-bound-extremal examples with ε=7. These non-supersolvable arrangements provide test cases for studying realisation spaces and the persistence of freeness among realisations of the same intersection lattice, in connection with Terao's conjecture.