AI 中文总结
研究证明欧氏平面中指数s=2固定时的最小Riesz s-能量子集选择问题为NP-完全，通过归约Barahona平面立方伊辛模型实现，为该类问题提供首个环境维度与指数均固定的欧氏难度结果。
AI 中文摘要
我们证明，当指数s=2固定时，欧氏平面中的最小Riesz s-能量子集选择问题已是NP-完全问题。据我们所知，这是首个关于精确Riesz能量子集选择的欧氏难度结果，其中环境维度和指数均为固定值。该归约使用带有均匀场的Barahona平面立方伊辛模型，一个自旋由四点正方形的一条对角线编码，轴对齐选择器链实现铁磁一致性，而45度终端几何产生反铁磁源相互作用，有理对角线扰动实现磁场，所有剩余相互作用由多项式分离主导。由于s=2且所有坐标均为有理数，每个构造的能量和决策阈值均恰好为有理数。
英文摘要
We prove that minimum Riesz $s$-energy subset selection in the Euclidean plane is NP-complete already for the fixed exponent $s=2$. To our knowledge, this is the first Euclidean hardness result for exact Riesz-energy subset selection in which both the ambient dimension and the exponent are fixed. The reduction uses Barahona's planar cubic Ising model with uniform field. A spin is encoded by one diagonal of a four-point square. Axis-aligned selector chains implement ferromagnetic consistency, while a $45^\circ$ terminal geometry yields an antiferromagnetic source interaction. Rational diagonal perturbations realize the magnetic field, and all remaining interactions are dominated by polynomial separation. Because $s=2$ and all coordinates are rational, every constructed energy and the decision threshold are rational exactly.
CommentsKeywords: Riesz energy, subset selection, NP-completeness, geometric optimization, planar cubic graphs, independent set, Ising model, computational geometry Update V3.0: Adversarial proof audit with Claude Fable 5 and augmented Git with lean proof support and audit result. Sharpened some constants. No fundamental changes or new results w.r.t. previous versions
