A remark on first spacing in Gauss's circle problem and Dirichlet's divisor problem

For every ε > 0, we prove that the error terms in Gauss's circle problem and Dirichlet's divisor problem are O_ε(X^(573/1828+ε)). Here X denotes the squared radius in the circle problem and the summation cutoff in the divisor problem. The main idea is a refined use of Guth and Maldague's amplitude-dependent wave-envelope theorem in the first-spacing analysis, retaining the amplitude information throughout the argument. 33 pages, 2 figures. Comments are welcome. Lean 4 formalization project: https://github.com/hxypqr/gauss-circle-divisor-energy 2020 Mathematics Subject Classification: Primary 11P21; Secondary 11L07, 11N37, 42B10, 42B25.