The third-order term of the Lee–Yang expansion in closed form, and the fourth-order ring coefficient of the dilute Fermi gas

Abstract. In many-body perturbation theory, the ground-state energy of a balanced dilute Fermi gas has the formal expansion E/N = (kF²/2M)[3/5 + (g − 1)(C₁ + C₂ + C₃ + ⋯)]. The first two coefficients have been known in closed form since 1929 and 1957. We evaluate the two constants previously known only numerically in C₃ = [A + (g − 3)B](kFa)³ + ⋯: B = (17π² + 64L − 52)/(90π³), A = 3/8 − 19/(90π) + L²/(2π²) + [62 + 316L − 360GL − 720 Im Li₃((1 + i)/2)]/(45π³), where L = log 2 and G is Catalan’s constant. This gives the complete third-order term for every spin degeneracy g. For hard spheres at g = 2, the coefficient of (kFa)³ in the bracket of E/N is A − B + 2/(3π) = 0.2303919405…; the Wellenhofer–Drischler–Schwenk (WDS) coefficient ε₃ is 5/3 times this value only because WDS normalize the energy to that of the free Fermi gas. The closed forms agree with the numerical values of WDS (for A, on truncation): they are exact evaluations of the same quantities, not corrections. We also evaluate the fourth-order particle–hole ring contribution, the only fourth-order class quadratic in g after removal of the common factor (g − 1). Its coefficient of (g − 2)²(kFa)⁴ is P₄ = −16/1575 + 8(1 + 3L)/(945π²) + [14/45 + (32/9)L − (12/5)ζ(3)]/π⁴ = −0.0086394470…, independent of three-body forces and the renormalization scale. Contour deformations reduce the ring integrals to finite logarithmic moments; a divergence identity reduces the ladder integral to the Fermi-surface arc. An exact identity between the exterior fourth-order ring region and a ladder triangle moment is the fourth-order member of a family that follows from analyticity of the in-medium loop; for the physical weight the same argument gives exact vanishing relations at every higher order. Independent quadratures of the defining integrals agree with A, B and P₄ to more than 80 significant digits. The theorems are exact statements about these integrals; they bear on the ground-state energy only through the formal diagrammatic expansion. All the work was done by AI. Generative-AI disclosure and author responsibility. OpenAI ChatGPT-6 Pro was used for derivations, computations, and drafting; Claude Opus 5.5 and Claude Fable 5.1 (Anthropic) were used for the integral representations, the numerical identifications, and 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. This record holds the manuscript (39 pages) and its verification package Matuzas_LeeYang_Third_Order_Verification_v1.0.1.zip (version 1.0.1): the complete LaTeX source with bibliography, figures and symbol registry; exact symbolic checks of every derivative certificate, with machine-readable rational coefficient data; independent quadratures of the defining integrals; programs that generate all tables and displayed decimals; the recorded fast run; and SHA-256 manifests. README.md in the package is the entry point: python3 replay.py --fast runs 34 stages in about two minutes, and python3 replay.py --long runs 46 stages, evaluating the high-precision defining integrals afresh, in about 20–30 minutes with three workers. The package was checked with Python 3.12.14 and 3.13.18, mpmath 1.3.0 and SymPy 1.14.0. 2020 Mathematics Subject Classification: Primary 81V74; Secondary 81V70, 81T18, 82D05, 33B30, 65D30. License scope: the manuscript, data files and documentation are licensed under CC BY 4.0; the verification code is licensed under MIT. Complete license texts and scope are included in the package. Preprint. External peer review is not claimed. Companion links: Website paper and reading guide · ResearchGate preprint. Version 1.0.1. The abstract and the introduction now state that the theorems are exact statements about explicitly defined integrals, which bear on the ground-state energy only through the formal diagrammatic expansion. The abstract also says that the factor 5/3 relative to the published hard-sphere coefficient is a matter of normalization. The paper names the authors of the rigorous results and adds six references. Two statements that rest on the diagrammatic expansion are no longer stated inside a theorem or corollary. The verification package also runs with its workspace inside the paper's source folder. No result or number has changed.