Elliptic structure of the singular three-electron Hylleraas integrals

We give an elliptic-polylogarithmic form of the singular three-electron Hylleraas master h(a,b,c) for positive exponents. Pachucki and Puchalski (2005, 2008) represented this inverse-square interelectronic integral as a one-dimensional elliptic quadrature. Its curve is Eₓ: Y² = 2X³ + 4xX² − 2X − 4x, with letter set E[4] ∪ (±Q′ + E[2]). The six-block expression reduces to classical weight-three functions on both degenerate planes. At equal exponents the value is a finite Chen combination on X₀(24). The master is not a finite sum of products of multiple polylogarithms with algebraic arguments and coefficients. The specified iterated monodromy of h(t,1,1) is 8π²K(1 − t²/4)/t; a regulator lattice and rational enclosure fix its coefficient. Closed forms for the nuclear inverse-square and inverse-cube masters and the interelectronic inverse-cube master correct five entries of Table I of Pachucki and Puchalski (2008). Green-function quadrature and integrated moment series verify the equal-exponent constant to 420 significant digits; the classical formula and full Green moment verify the degenerate value to 200. Reference values have 60 or 100 digits with stated precision. Reduction of the equal-exponent constant to modular L-values remains open. All the work was done by AI. Generative-AI disclosure and author responsibility. ChatGPT-6 Pro was used for derivations, computations, and drafting; Claude Opus 5.5 (rounds 1–19) and Claude Fable 5.1 (round 20) were 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 33E05, 33B30, 11G55, 11F67, 65D30. Files: the compiled manuscript Matuzas_2026_Elliptic_structure_singular_three_electron_Hylleraas_integrals_v8.pdf (60 pages) and the verification package LITHIUM_QED_MASTER_P1_v8.zip. The package contains paper/ with the clean LaTeX source main.tex, its sections, tables and bibliography, the same PDF, the separately retained annotated source and the build checks; PAPER1_VERIFICATION/ with README.md, the entry point run_checks.py, MANUSCRIPT_MAP.json mapping the manuscript's results, equations and tables to source files, SHA-256 pins and check IDs, the frozen source files, requirements.txt and manifest.sha256; the claims ledger CLAIMS_LEDGER.json; and the provenance and revision records. From PAPER1_VERIFICATION/, sha256sum -c manifest.sha256 or python3 run_checks.py --mode hashes checks the package hashes; python3 run_checks.py --mode smoke --work-dir DIR runs selected fresh checks, and --mode full --work-dir DIR replays all 29 printed-formula groups (695 checks), where DIR is a new directory outside the package (recorded environment: Python 3.12; mpmath 1.3.0, SymPy 1.14.0 and NumPy 2.3.5 pinned in requirements.txt). Running latexmk -pdf main.tex in paper/ rebuilds the manuscript; two clean builds with TeX Live 2023 reproduced the deposited PDF byte for byte. Multiprecision comparisons verify the arithmetic of the printed expressions and are not interval proofs; the formal proofs are in the manuscript. Preprint. External peer review is not claimed. Website companion, full text and citation files: https://jonas-matuzas-research.matuzas77.chatgpt.site/papers/2026-elliptic-structure-singular-three-electron-hylleraas-integrals-23858519/ ResearchGate preprint: https://www.researchgate.net/publication/417295358_Elliptic_structure_of_the_singular_three-electron_Hylleraas_integrals