AI 中文总结
该研究提出一种几何框架，通过双二次曲面上的有限网格构造科亨-斯佩克构型，可得到无限族构型，其典范完备化能恢复F₄、H₄等著名四维语境构型，揭示了它们的统一射影几何来源。
AI 中文摘要
我们开发了一个几何框架，用于构造和组织四维实空间中的科亨-斯佩克（Kochen-Specker）构型。该构造利用了通过欧几里得配极相关的一对光滑二次曲面上的有限网格，其关联几何直接产生正交测量语境和量子语境性的奇偶性证明，从而得到无限族构型，以及对先前已知的循环构造的几何解释和扩展。对于一个特殊的子族，相同的几何允许由割线确定的典范完备化，该完备化的前两个实例恢复了F₄和H₄型的例外根构型，而最小的情况还恢复了Cabello构型及其在Peres构型中的嵌入。因此，几个此前通过不同构造得到的著名四维语境构型，均源自单一的射影几何机制。
英文摘要
We develop a geometric framework for constructing and organizing Kochen--Specker configurations in real four-dimensional space. The construction uses finite grids on pairs of smooth quadrics related by Euclidean polarity. Their incidence geometry directly produces orthogonal measurement contexts and parity proofs of quantum contextuality, yielding infinite families of configurations and a geometric interpretation and extension of a previously known cyclic construction. For a distinguished subfamily, the same geometry admits a canonical completion determined by secant lines. The first two instances of this completion recover the exceptional root configurations of types $F_4$ and $H_4$, while the smallest case also recovers the Cabello configuration and its embedding in the Peres configuration. Thus several prominent four-dimensional contextual configurations, previously obtained from different constructions, arise from a single projective-geometric mechanism.
