Electron rings and the critical screening of the hydrogen atom: polylogarithms at every order and exact sum rules

A ring of N electrons has nuclear Slater spokes e⁻ʳ/r and neighbouring Coulomb links 1/rᵢⱼ; its integral divided by (4π)ᴺ is J_N. We prove 2ᴺJ_N = Σ_{n,ℓ} (2ℓ+1) λ_c(n,ℓ)ᴺ for N ≥ 2, where λ_c(n,ℓ) is the screened-hydrogen critical screening constant at which state (n,ℓ) reaches zero energy. The quadratic sum is 2; the cubic sum is 12 log 2 log(3/2) − 6 Li₂(1/4). Equal-spoke and symmetrized scalar rings with positive exponents and nonnegative integer radial powers have finite multiple-polylogarithm expressions at every order. Orders three and four are classical; order five has a motivic obstruction to classical tetralogarithms within rational {2,3,5}-unit arguments, modulo lower-weight products. For μ = λ_c(1s)/2, 0 ≤ J_N/μᴺ − 1 ≤ (0.261)ᴺ⁻³/20 for N ≥ 3. Exact-arithmetic certificates determine λ_c(1s) to at least 31 significant digits. 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 identification of the Yukawa–Birman–Schwinger connection 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 81V45; Secondary 33B30, 35P15, 11G55, 47B10. Files: the compiled manuscript Electron_Rings_Critical_Screening_v1.pdf (25 pages) and the source archive Electron_Rings_Critical_Screening_v1_source.zip, which contains the LaTeX source main.tex with its bibliography (main.bbl) and generated number macros (generated_numbers.tex); it compiles with pdfLaTeX.