提出新函数检测线排列是否自由，可高效筛选非超可解的自由排列。
A penalised Saito functional for heuristic search of free line arrangements
- 设计惩罚型Saito泛函，结合对齐度与对数导子约束判断自由性。
- 在6146个已验证排列中发现3012个多重性间隙≥2的非超可解例。
- 适合研究交集格的自由性保持性与Terao猜想的反例构造。
我们为含n条线的约化排列𝒜及给定对(d₁,d₂)（满足d₁+d₂=n−1）引入带惩罚的Saito泛函ℑ_{λ,β}(𝒜; d₁,d₂)，用于衡量候选Saito行列式与定义多项式的对齐程度，并惩罚候选导子不满足对数条件的情况。证明该泛函取值于[0,1]，仅当𝒜以指数(1, d₁, d₂)自由时为零，否则严格介于0与1之间。对固定(d₁,d₂)，其在约化配置空间上上半连续，且在自由排列处连续，当λ→∞时收敛至对应二元自由性判别。结合数值近似与小的b₂-壳项，指导在ℚ及特定二次扩张上的固定基数线替换搜索。数值仅用于候选筛选；所有报告排列均通过Saito准则在精确算术下验证。当前已验证数据库包含6146个具有不同Weisfeiler–Leman指纹的排列，基数最大达n=28。其中3012个具有多重性间隙ε(𝒜)=d₁−m(𝒜)≥2，包括ε=7的极小例子。这些非超可解排列为研究实现实空间及同交集格下自由性保持性提供了测试案例，与Terao猜想相关。
原文摘要 · Abstract (English)
We introduce the penalised Saito functional $\mathfrak S_{λ,β}(\mathcal{A};d_1,d_2)$ for a reduced arrangement $\mathcal{A}$ 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 $\mathcal{A}$ 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 $ε(\mathcal{A})=d_1-m(\mathcal{A})\geq2$, 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.
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