Single-centre expansions of three-centre 1s Slater integrals: convergence rates and certified error bounds

Single-centre expansions of Slater-orbital integrals converge only algebraically when the two distances from the expansion centre are equal. We determine their convergence rates and derive rigorous remainder bounds for a three-centre electron-repulsion integral with four normalized 1s orbitals, its two-centre exchange limit, a nuclear-attraction integral with two exponential factors, and a Yukawa prototype. Near coincidence of the displaced centres, each integral has a convergent expansion consisting of an analytic part and a half-integer power times an analytic function, without logarithms. These expansions determine the partial-wave coefficients. Unequal distances produce geometric decay; equal distances give inverse powers, including (ℓ + ½)⁻⁸ for the four-orbital integral. We explain why cancellation makes this rate faster than the rates of separately estimated components. Explicit second and fourth coefficients, a uniform transition formula, and two-sided remainder bounds yield certified numerical intervals. At equal distances, relative widths reach 4.17 × 10⁻²⁰ and 4.26 × 10⁻²⁷ for the one- and two-electron integrals. All sixty test geometries are enclosed; fifty meet a 10⁻¹² width target. Generative-AI disclosure and author responsibility. OpenAI ChatGPT-6 Pro was used for derivations, computations, and drafting; Claude Opus 5.5 (Anthropic) was used for independent re-derivation and audit. The closed form of the Class II coefficient d₂, the centred transition majorant, the reverse-Bessel form of the tail kernels and the 7/5 overestimate of the ζ₆ tail correction arose during this independent audit. The author directed and audited the work throughout and takes full responsibility for the mathematics and the final text. 2020 Mathematics Subject Classification: Primary 81V55; Secondary 41A60, 33C45, 65G20, 65G30, 40A25. Files: the compiled manuscript Slater_v1_FINAL.pdf (39 pages) and the verification package CLASSII_PAPER_VERIFICATION.zip, which contains the exact interval endpoints, per-channel enclosures, parameter sets, proof records and theorem statements, the literature-comparison matrix, a claims map linking each theorem and table to its sources, provenance records with SHA-256 hashes, the LaTeX sources with their table generators, and the code that generates and replays the certificates (Python, and the original C++ interval engine using MPFR/GMP). Running python -B verify.py --quick (Python 3.12; SymPy 1.14.0 and mpmath 1.3.0 pinned in requirements.txt; see README.md) checks the payload hashes and replays the 17 exact intervals, the sixty-geometry summaries and the publication tables; adding --proof-checks --self-checks runs the symbolic, model-function and high-precision checks. code/regenerate_certificate.py recompiles the native engine (C++17 with MPFR/GMP) and regenerates one degree-192 four-orbital head and three tail records, which the recorded run reproduced byte-identically. Replay and numerical comparisons establish reproducibility and consistency; the mathematical arguments and their hypotheses determine the claims.