A penalised Saito functional for heuristic search of free line arrangements

We introduce the penalised Saito functional 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 infinity 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 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) at least2, 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.