Order-eight splitting methods for quartic potentials from a certified computer search, and why a smaller ℓ1 norm is not enough

We revisit time-symmetric splitting methods of order eight for y'' = g(y), g = -grad V, when V is a polynomial of degree at most four. For this class Escorihuela-Tomàs recently constructed the 19-stage method QA19, whose coefficient norm is l1 = 3.8225. A computer search by a team of AI agents produced twenty certificates, each accepted by two independently written interval-arithmetic checkers. They fall into twelve groups of numerically close 19-stage methods of order eight, all with l1 below that of QA19; the smallest is l1 = 3.0513898849, 20.2% below QA19. High-precision polishing gives numerical evidence that the four best groups contain strict local minima of l1. An independent numerical study shows, however, that a small l1 does not by itself make a method efficient. On three quartic test problems, the two methods with the smallest l1 use 7-20% more force evaluations than QA19 to reach a given energy accuracy. A third method, with l1 = 3.1264 and the smallest leading-error norm of the methods tested, uses 25-27% fewer force evaluations than QA19 on an FPU-beta chain and on a quartic oscillator, but 1-2% more on a phi^4 chain. We also show that, for quartic potentials in dimension d >= 2, the degree-nine component of the relevant Lie algebra has dimension 17, not 18 as tabulated, because modified potentials commute. Files: d1_note.pdf (the paper); d1_deposit.zip (the twenty certificates, the frozen checkers K1 and K2, a one-command re-check script reverify_d1.py, the catalogue of methods with 80-digit coefficients, the numerically polished local-minimum candidates, the efficiency study and the dimension computation, with SHA256SUMS). Use of AI and the author's role: the search was carried out by a multi-agent AI research system (Claude Opus 5.5, Anthropic) operated by the author. Claude also wrote checker K1 and the search and polishing code, re-proved the certificates and drafted the text. GPT-6 Astra (OpenAI) wrote checker K2 blind, carried out the efficiency study and the dimension computation, audited the literature, and fact-checked the manuscript. The author chose the project and its criteria, required cross-family review, chose the stopping rules and decided when to end the search, and re-ran both checkers on all certificates on the author's own computer (every check accepted). The written arguments were drafted with AI and have not yet been checked line by line by the author or by a mathematician. Anyone can check every certified method with one command (python reverify_d1.py). No AI system is an author; the author takes full responsibility for the content. Details are in the section "Use of artificial intelligence" of the paper.