How many 8-regular graphs exist on n vertices? The generating series that encodes this answer satisfies a differential equation — but until now, that differential equation could not be computed. The mathematical object existed in principle. No algorithm could reach it.
Brochet, Chyzak, and Lairez extend the Griffiths-Dwork reduction method to holonomic systems, creating a computational pathway that reaches farther into the space of parameterized integrals than creative telescoping — the previous state of the art for D-finite cases. Their method does not beat creative telescoping on its home territory. Instead, it reaches territory that creative telescoping cannot access at all.
The differential equation for 8-regular graphs is the proof of concept. Previous methods could handle lower regularity — 4-regular, 6-regular — but 8-regular was beyond reach. The computational complexity scaled past what existing algorithms could handle. The new method, implemented in Julia, navigates the holonomic landscape differently, finding paths through the reduction that avoid the combinatorial explosion.
The structural insight: computational reach is not a smooth function of problem size. The jump from 6-regular to 8-regular graphs crosses a boundary where one method fails and another is needed. The new method extends the frontier — but somewhere beyond 8-regular, another boundary will appear, requiring yet another technique. The space of computable mathematical objects is not expanding uniformly. It advances in discrete jumps, each unlocked by a specific algorithmic insight.