子正则仿射胞腔与D型-1级顶点代数的Lean 4形式验证报告
A Lean 4 Verification Report for Subregular Affine Cells and the Level $-1$ Vertex Algebra of Type $D$
AI总结：
该研究完成了对D型-1级顶点代数相关证明架构的Lean 4形式验证，核验了证明的大量内部部分，将未构建基础架构的更高表示论结果分离为显式语义接口，而非从头形式化相关理论。
AI中文摘要：
我们报告了一篇伴随论文《子正则仿射胞腔与D型-1级顶点代数》（arXiv:2608.11997）的Lean 4形式验证工作。形式化内核对证明架构的大量内部部分进行了核验，包括第4节的成员/下降链、D型范数间隙论证、零轨道能量与符号置换刚性、节点权重与数值刚性计算、穷尽逻辑、候选商分类的推导过程、单对象计数、最终加法Grothendieck群比较，以及统一特征公式的系数替换层。那些基础架构尚未在文件中构建的更高表示论结果，被分离为显式语义接口，而非作为Lean公理引入。因此，该精确断言是从显式表示论边界输入出发的内核核验内部推导，而非在Mathlib内部从头形式化顶点代数、BRST约化、有限W代数或仿射Hecke理论。
英文摘要：
We report a Lean 4 formal verification accompanying the paper "Subregular Affine Cells and the Level -1 Vertex Algebra of Type D" (arXiv:2608.11997). The formalization kernel-checks substantial internal parts of the proof architecture, including the Section 4 membership/descent chain, the type-D norm-gap argument, zero-orbit energy and signed-permutation rigidity, node-weight and numerical rigidity calculations, the exhaustion logic, the passage to the candidate quotient classification, the simple-object count, the final additive Grothendieck-group comparison, and the coefficient-substitution layer of the uniform character formula. Higher representation-theoretic results whose foundational infrastructure is not presently constructed in the file are isolated as explicit semantic interfaces rather than introduced as Lean axioms. Thus the precise claim is a kernel-checked internal deduction from explicit representation-theoretic boundary inputs, not a from-scratch formalization of vertex algebras, BRST reduction, finite W-algebras, or affine Hecke theory inside Mathlib.
