Many engineering failures (thermal hotspot concentration, Hertz contact fatigue localization, boundary-layer loss, mixing dead zones) are geometric failure modes: changing the material delays the failure; changing the geometry eliminates it. Despite this, no formal metric exists for evaluating how uniformly a convex body distributes surface contact during rolling, with direct engineering implications. We introduce the Contact Distribution Score (CDS), a scalar metric defined as the area-weighted variance of contact time over a rolling surface, and its stress-domain counterpart the Stress Distribution Score (SDS), the area-weighted variance of accumulated Hertz contact pressure. CDS -> 0 indicates uniform contact; SDS -> 0 indicates uniform stress. We implement a three-layer oracle architecture (approximate oracle for search, rigid-body oracle for validation, Hertz contact pressure oracle for SDS). A parametric search over 45 members of the developable roller family identifies the oloid (Schatz, 1929) at CDS = 8.2 x 10^-7, with the conventional cylinder baseline at 4.75 x 10^-5: a 58x discrimination. Independent curvature-driven analysis under uniform contact yields a geometry-only SDS of 4.8 x 10^-8, indicating the oloid's surface curvature contributes minimal additional stress non-uniformity beyond the contact distribution. We extend the analysis to fatigue (FDS), thermal (TDS), and wear (WDS) scores, finding the oloid's 58x advantage transfers consistently across linear and multiplicative metrics in a 46-68x range. The nonlinear fatigue metric diverges due to Basquin S-N amplification but still shows oloid superiority over all tested alternatives. This work establishes the formal vocabulary and computational infrastructure for substrate geometry: the study of geometric forms as engineering substrates classified by their operational invariants.

翻译：许多工程失效（如热热点集中、赫兹接触疲劳局部化、边界层损失、混合死区）本质上是几何失效模式：更换材料只能延缓失效，而改变几何形状则可消除失效。然而，目前尚无正式指标评估凸体滚动过程中表面接触的均匀性，这直接影响工程应用。我们引入接触分布得分（CDS）——定义为滚动表面接触时间的面积加权方差，及其应力域对应量——应力分布得分（SDS）——定义为累积赫兹接触压力的面积加权方差。CDS趋近于0表示接触均匀；SDS趋近于0表示应力均匀。我们构建了三层预言机架构（近似预言机用于搜索、刚体预言机用于验证、赫兹接触压力预言机用于计算SDS）。对可展滚子族45个成员进行参数搜索，发现Oloid体（Schatz，1929）的CDS为8.2×10⁻⁷，而传统圆柱基线的CDS为4.75×10⁻⁵：区分度达58倍。在均匀接触条件下，基于曲率的独立分析得到仅含几何因素的SDS为4.8×10⁻⁸，表明Oloid表面曲率除接触分布外仅贡献极小额外应力不均匀性。我们将分析扩展到疲劳（FDS）、热（TDS）和磨损（WDS）得分，发现Oloid的58倍优势在线性和乘法指标中一致传递，范围达46–68倍。非线性疲劳指标因Basquin S-N曲线放大效应出现发散，但Oloid仍优于所有测试替代方案。本研究为基板几何学建立了正式词汇与计算基础设施：该学科将几何形态视为工程基板，以其操作不变量进行分类。