A Three-Round Peeling Bound for Internal Partitions of 5- and 8-Regular Graphs

Lin and Zeng proved that every 5-regular or 8-regular graph with at least 10000000 vertices has an internal partition. We refine their argument to the explicit threshold 500000. A third round of degree-p deletion in their weighted local-density argument bounds the number of degree-p vertices by N/p^3 + (2 + 2/p + 2/p^2)|X| under a local inequality on sets of at most 53 vertices. Explicit integer choices in the final small-surplus-set argument yield the improved threshold. More precisely, each side of every minimum bisection contains a ceil(r/2)-cohesive set. The existence results for these two degrees, the bisection strategy and the small-surplus-set lemma are prior work. This is a quantitative refinement, not a solution for arbitrary regular degree or an optimal-threshold claim. Exact finite computations supplement the written proof. This is an unrefereed, AI-assisted preprint; no independent review, formal verification or absolute priority certification is claimed.