A genus-two spectral curve of the Riemann Ξ-function: Real Structure and Certified Arithmetic Separation of Periods
Abstract
We associate a compact genus-two real spectral core with a transverse logarithmic-derivative discriminant of the Riemann Ξ-function. Directed interval arithmetic certifies its branch geometry, a real PIII(D6) scalar-normal-form realization, and associated Kummer and Shioda--Inose models. Zero-action resonance reduces to rationality of two real period ratios. Their certified heat motion yields generic local nonresonance and Diophantine separation, while explicit Farey windows impose denominator barriers of 1,036,003 and 140,769 throughout |κ|≤3×10−10 in the heat time κ. Fixed-point irrationality and exact Picard rank remain open. The heat tangent differs from the matched fixed-exponent isomonodromic direction, but admits a local moving-exponent curve lift. Separately, a reciprocal-head canopy certificate combined with published finite-RH and rectangular time-barrier verifications bounds the de Bruijn--Newman constant by Λ≤433/2500=0.1732 without a QRH assumption. These local arithmetic and global analytic results share a phase--transverse bridge; no equivalence between core nonresonance and RH is asserted.
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BibTeX
@misc{planat2026genustwo,
title = {{A genus-two spectral curve of the Riemann $Ξ$-function: Real Structure and Certified Arithmetic Separation of Periods}},
author = {Michel Planat},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32694},
eprint = {2610.32694},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32694},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A genus-two spectral curve of the Riemann Ξ-function: Real Structure and Certified Arithmetic Separation of Periods
AU - Michel Planat
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32694
UR - https://arxiv.org/abs/2610.32694
AB - We associate a compact genus-two real spectral core with a transverse logarithmic-derivative discriminant of the Riemann Ξ-function. Directed interval arithmetic certifies its branch geometry, a real P_III(D_6) scalar-normal-form realization, and associated Kummer and Shioda--Inose models. Zero-action resonance reduces to rationality of two real period ratios. Their certified heat motion yields generic local nonresonance and Diophantine separation, while explicit Farey windows impose denominator barriers of 1,036,003 and 140,769 throughout |κ|≤3×10^−10 in the heat time κ. Fixed-point irrationality and exact Picard rank remain open. The heat tangent differs from the matched fixed-exponent isomonodromic direction, but admits a local moving-exponent curve lift. Separately, a reciprocal-head canopy certificate combined with published finite-RH and rectangular time-barrier verifications bounds the de Bruijn--Newman constant by Λ≤433/2500=0.1732 without a QRH assumption. These local arithmetic and global analytic results share a phase--transverse bridge; no equivalence between core nonresonance and RH is asserted.
ER -