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 $q/p < 1 + \alpha/n$ 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.
非均匀椭圆性前沿的神经符号学探索：通过拓扑斯理论推理模型形式化夏德理论的精确研究 / The Neurosymbolic Frontier of Nonuniform Ellipticity: Formalizing Sharp Schauder Theory via Topos-Theoretic Reasoning Models
这篇论文提出将数学中解决非均匀椭圆方程精确正则性问题的‘幽灵方程’方法，与基于拓扑斯理论的神经符号推理模型相结合，以自动生成机器可验证的证明，从而探索复杂物理系统分析的新范式。
源自 arXiv: 2602.10632
