Selberg–Residue Analysis of the Equally Spaced Richardson–Gaudin Pairing Model

Abstract

Authorship, provenance, and verification. The scientific problem, original method, and research direction belong to W. V. Pogosov. This project builds on his published and unpublished work developed before the involvement of an advanced large language model (LLM). During the subsequent LLM-assisted phase, the LLM produced detailed derivations, calculations, and verification programs through an extended collaboration with the author. The author has not exhaustively verified every calculation and proof. The research developed through many iterations and branching lines of enquiry over a substantial period of time. The author posed and refined questions, assessed proposed directions, and guided the revision, rejection, and combination of intermediate approaches; the LLM supplied calculations and arguments that prompted further questions and revisions. The resulting manuscript is a record of this reciprocal, sustained research process. The manuscript develops a Selberg–residue approach to the equally spaced Richardson–Gaudin pairing model. Meromorphic contour periods are reduced to finite determinant expressions. Complementary kernels and sign-corrected minor expansions organize ground-state, marked, blocked, and nested-contour sectors. The text studies their thermodynamic asymptotics and candidate finite-size energy corrections while distinguishing exact period identities from physical eigenenergy interpretations. The method originated in W. V. Pogosov, “'Probabilistic' approach to Richardson equations,” Journal of Physics: Condensed Matter 24, 075701 (2012), doi:10.1088/0953-8984/24/7/075701 (https://doi.org/10.1088/0953-8984/24/7/075701), arXiv:1111.2907 (https://arxiv.org/abs/1111.2907). That work treated the ground state and obtained the required coupling dependence without a full direct evaluation of the determinant normalization. Within this line of work, it also first formulated explicitly the Hamiltonian origin of electron–hole symmetry in the exact Richardson analysis; its Appendix A acknowledges earlier observations of spectral duality. The subsequent paper, “Excited states in Richardson pairing model: 'probabilistic' approach,” Progress of Theoretical Physics 128, 1–14 (2012), doi:10.1143/PTP.128.1 (https://doi.org/10.1143/PTP.128.1), arXiv:1208.1088 (https://arxiv.org/abs/1208.1088), extended the approach to excited states, again leaving a common determinant normalization unevaluated directly. The author's 2014 doctoral dissertation, Superconductors and dilute superfluid Bose systems: From the microscopic to the macroscopic scale (in Russian), developed direct determinant evaluations by successive row differences and triangular reduction. See the Lebedev Physical Institute dissertation-council record (https://www.lebedev.ru/ru/fian-dissertation-councils/posts.html?id=188) and the complete thesis on ResearchGate (https://www.researchgate.net/publication/271840408_Sverhprovodniki_i_razrezennye_sverhtekucie_boze-sistemy_ot_mikro-_k_makrourovnu) (doi:10.13140/2.1.1066.5766 (https://doi.org/10.13140/2.1.1066.5766)). From the perspective of the broader program pursued here, these earlier results constituted a partial solution. The present project attempts to carry that program through; completion is not claimed. After these publications and the dissertation, the author developed further results and working materials that remained unpublished. Only then was an advanced LLM introduced to the accumulated corpus. The unpublished author-led stage therefore preceded the LLM-assisted development. Version 1.1 retains a repair of the confluent lattice-minor argument, using a saddle-localized continuation and an exponentially small error controlled on integer indices. Important open inputs remain: contour selection, uniform Gaussian energy control, the proposed Hessian-to-spectral-shift identification, and finite-defect comparisons. The correspondence between the period endpoint and the physical real-root density is also conjectural. The added Chapter 20 is restricted to the common all-pole Case A construction at doubled auxiliary inverse temperature: an explicit fourth-power Vandermonde moment Pfaffian, residue formulas, a positive quaternionic Grassmann representation proving nonvanishing of the full period, and fixed-spectrum coupling comparisons. No new Pfaffian results for mixed-contour Case B are included. The deposit contains the full twenty-chapter manuscript and fourteen appendices, LaTeX sources, bibliography, the retained scientific programs and historical reports, and a separate self-contained Case A Pfaffian library with two verification drivers and current JSON reports. The retained Revision 01 lattice report records 96 small exact-arithmetic comparisons and 24 larger minor and derivative checks. The Case A programs have been run for this release; the older suite has not been rerun in full. Independence here concerns implementations within the LLM-assisted workflow, not independent human verification. Finite tests do not prove the unresolved analytical assumptions.