Rank-two Nahm sums, modularity obstructions, and the McKay--Thompson series T31A
Abstract
We conjecture a positive four-shift identity for rank-two Nahm sums with denominator steps (1,31), giving a mixed-base realization of the Monster McKay--Thompson series T31A through the classical Rogers--Ramanujan function U(1,31). A theta dissection directly derives the modular target. For the underlying quadratic exponent 3r2+31rs+93s2, we prove nonmodularity of every individual sum with rational linear terms and normalization, together with an exclusion from finite Fricke systems with meromorphic finite-width expansions at infinity. Exact agreement through q2000 and six vanishing radial corrections support the proposed combination; a five-term dilogarithm certificate and an exact amplitude calculation prove its leading asymptotic behavior. We further propose a three-component Fricke law for a dual combination, with an explicit icosahedral matrix involving fifth-root constants, and a quadratic identity linking its components to T31A. The dual translation law and matrix algebra are proved; its Fricke transformation and moonshine identity remain conjectural, supported respectively by numerical evaluations and exact finite coefficient checks.
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BibTeX
@misc{hakimoglubrown2026ranktwo,
title = {{Rank-two Nahm sums, modularity obstructions, and the McKay--Thompson series $T_31A$}},
author = {Cetin Hakimoglu-Brown},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28509},
eprint = {2609.28509},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28509},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rank-two Nahm sums, modularity obstructions, and the McKay--Thompson series T_31A
AU - Cetin Hakimoglu-Brown
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.28509
UR - https://arxiv.org/abs/2609.28509
AB - We conjecture a positive four-shift identity for rank-two Nahm sums with denominator steps (1,31), giving a mixed-base realization of the Monster McKay--Thompson series T_31A through the classical Rogers--Ramanujan function U(1,31). A theta dissection directly derives the modular target. For the underlying quadratic exponent 3r^2+31rs+93s^2, we prove nonmodularity of every individual sum with rational linear terms and normalization, together with an exclusion from finite Fricke systems with meromorphic finite-width expansions at infinity. Exact agreement through q^2000 and six vanishing radial corrections support the proposed combination; a five-term dilogarithm certificate and an exact amplitude calculation prove its leading asymptotic behavior. We further propose a three-component Fricke law for a dual combination, with an explicit icosahedral matrix involving fifth-root constants, and a quadratic identity linking its components to T_31A. The dual translation law and matrix algebra are proved; its Fricke transformation and moonshine identity remain conjectural, supported respectively by numerical evaluations and exact finite coefficient checks.
ER -