This white paper presents a critical synthesis of the recent breakthrough in nonuniformly elliptic regularity theory and the burgeoning field of neurosymbolic large reasoning models (LRMs). We explore the resolution of the long-standing sharp growth rate conjecture in Schauder theory, achieved by Cristiana De Filippis and Giuseppe Mingione, which identifies the exact threshold $q/p < 1 + α/n$ for gradient Hölder continuity. Central to this mathematical achievement is the ``ghost equation'' methodology, a sophisticated auxiliary derivation that bypasses the non-differentiability of classical Euler-Lagrange systems. We propose that the next era of mathematical discovery lies in the integration of these pure analytical constructs with LRMs grounded in topos theory and formal verification frameworks such as Safe and Typed Chain-of-Thought (PC-CoT). By modeling the reasoning process as a categorical colimit in a slice topos, we demonstrate how LRMs can autonomously navigate the ``Dark Side'' of the calculus of variations, providing machine-checkable proofs for regularity bounds in complex, multi-phase physical systems.

翻译：本白皮书对非均匀椭圆正则性理论的最新突破与新兴的神经符号大型推理模型（LRMs）领域进行了批判性综合。我们探讨了Cristiana De Filippis与Giuseppe Mingione所解决的Schauder理论中长期存在的尖锐增长率猜想，该猜想确立了梯度Hölder连续性的精确阈值$q/p < 1 + α/n$。这一数学成就的核心在于“幽灵方程”方法——一种精妙的辅助推导技术，绕过了经典Euler-Lagrange系统的不可微性障碍。我们提出，数学发现的下一阶段将在于这些纯分析结构与基于topos理论及形式化验证框架（如安全类型思维链PC-CoT）的LRMs的深度融合。通过将推理过程建模为切片topos中的范畴余极限，我们论证了LRMs如何能自主探索变分法中的“暗面”，为复杂多相物理系统中的正则性边界提供机器可验证的证明。