A semicontinuous relaxation of Saito's criterion and freeness as angular minimization

We introduce a nonnegative functional on the space of line arrangements in $\mathbb{P}^2$ that vanishes precisely on free arrangements, obtained as a semicontinuous relaxation of Saito's criterion for freeness. Given an arrangement $\mathcal{A}$ of $n$ lines with candidate exponents $(d_1, d_2)$, we parameterize the spaces of logarithmic derivations of degrees $d_1$ and $d_2$ via the null spaces of the associated derivation matrices and express the Saito determinant as a bilinear map into the space of degree $n$ polynomials. The functional then admits a natural geometric interpretation: it measures the squared sine of the angle between the image of this bilinear map and the direction of the defining polynomial $Q(\mathcal{A})$ in coefficient space, and equals zero if and only if its image contains the line spanned by $Q(\mathcal{A})$. This provides a computable measure of how far a given arrangement is from admitting a free basis of logarithmic derivations of the expected degrees. Using this functional as a reward signal, we develop a sequential construction procedure in which lines are added one at a time so as to minimize the angular distance to freeness, implemented via reinforcement learning with an adaptive curriculum over arrangement sizes and exponent types. Our results suggest that semicontinuous relaxation techniques, grounded in the geometry of polynomial coefficient spaces, offer a viable approach to the computational exploration of freeness in the theory of line arrangements.