A Calabi–Yau period in a four-electron correlation integral: modular physical fibre and a hypergeometric endpoint

Four electrons are joined by a ring of Coulomb links and one diagonal, with alternating Slater exponents. We extract the coefficient of one terminal dilogarithm in a fixed radial integration order and call it the sectorwise channel. We prove that its positive Horn series satisfies an inhomogeneous AESZ-330 equation. Exact certificates give homogeneous differential Galois group Sp₄ and a non-polylogarithm theorem for the channel; transfer to the full integral remains conditional. An all-prime denominator argument proves nonsplitting, with extension group 𝔾ₐ⁴ ⋊ Sp₄. The equal-exponent fibre is an identified rigid modular double octic with newform 32.4.a.a. There we prove the value (π²/4) ₄F₃(1/4, 1/4, 3/4, 3/4; 1, 1, 1; 1), identify it with an interpolated Apéry value, and certify 300 decimal places. Proved congruences are distinguished from conjectural homogeneous-period identities supported to 130 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. Earlier stages also used OpenAI GPT-5.6 Pro and Claude Fable 5.1. 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 14J32, 11F11, 33C20, 34M15, 11F67. Files: the compiled manuscript Four_Electron_Calabi_Yau_Period_v1.pdf (26 pages) and the source archive Four_Electron_Calabi_Yau_Period_v1_source.zip, which contains the LaTeX source main.tex with its bibliography (main.bbl, references.bib) and generated number macros (generated_numbers.tex), the build script build.py with its bibliography and metadata tools, and the ancillary directory anc/: exact data and certificates (telescoping coefficients, operator certificates, the integer endomorphism matrix, modular coefficients and the certified endpoint digits), verification scripts and libraries, regenerated outputs and logs, a README and PROVENANCE.json. Running python anc/run_all.py (Python 3.11 or later recommended; dependencies in anc/requirements.txt) reruns all 19 verification stages in fresh processes and regenerates generated_numbers.tex; python build.py rebuilds the PDF with pdfLaTeX, and pdflatex main.tex alone also works with the supplied main.bbl. The 300-place endpoint value is certified with exact rational arithmetic; other numerical agreement counts are observed, not interval bounds, and the conjectural period identities remain conjectures.