Three-electron Hylleraas integrals with two inverse-square distances: exact reduction and high-precision evaluation for the fine structure of lithium

Abstract. Two Coulomb fields produce singular three-electron integrals that limited Hylleraas evaluations of lithium fine-structure operators. Two integrations by parts reduce the eight relevant operators to two inverse-square families and previously treated integrals. We give one-dimensional representations for their masters T₂, T₃, and alternative cube–square anchors C₂, C₃; C₃ is expressed in classical polylogarithms of weight at most three. Parity separates decoupled sectors from genuinely three-electron sectors. Exact rational reductions through index sum 32 use twelve sufficient masters at the tested exponent sets. Double screening and Gaussian coordinate integration determine these masters to sixty significant digits. Exact collection of the source functions controls cancellation near equal exponents at four tested separations; equal exponents use a separate exact reduction. Numerical comparisons cover 104 pre-specified recursive members and an independent direct sample of 136 decoupled members through index sum 32. Production-size basis matrix elements of P₄ and one component of P₁ agree by two methods to 41–45 digits. On one Intel Xeon Silver 4410Y machine, both families require 5.85–6.99 CPU hours per exponent set at the stated 64-digit output precision. Uniform error bounds, minimal master sets, and the uncovered lower integrals in some class-2 matrix elements remain open. This record holds the manuscript (version 11, revision 4; 38 pages), the verification bundle LITHIUM_QED_MASTER_P2_v11.zip (LaTeX source and PDF, the supplementary verification package with reference values, kernels, exact identities, the map from every statement to its source and the replay script run_checks.py, the claims ledger and the provenance of the manuscript; 3,050 files, checked by manifest.sha256), the clean arXiv LaTeX source, and SHA256SUMS.txt. README.md explains how to verify the files and run the checks; PAPER2_VERIFICATION/README.md inside the bundle is the entry point for the replay. Companion article: J. Matuzas, Elliptic structure of the singular three-electron Hylleraas integrals, doi:10.5281/zenodo.23858519. Generative-AI disclosure and author responsibility. ChatGPT-6 Pro (ChatGPT-5.6 Pro in parts of rounds 10 and 12; GPT-6 Astra Pro in writing the manuscript) was used for derivations, computations, and drafting; OpenAI Codex ran the production computations on a local server; Claude Opus 5.5 proposed several of the reduction steps and was used for independent re-derivation and audit of every increment. The author directed and audited the work throughout and takes full responsibility for the mathematics and the final text. 2020 Mathematics Subject Classification. Primary 81V45; Secondary 65D30, 65F35, 33B30, 33E05, 81-08. All the work was done by AI. Preprint. External peer review is not claimed. Website companion, full text, reading guide and citation files: https://jonas-matuzas-research.matuzas77.chatgpt.site/papers/2026-three-electron-hylleraas-integrals-two-inverse-square-distances-23977551/ ResearchGate preprint: ResearchGate publication 417503397.