A genus-two spectral curve of the Riemann $Ξ$-function: Real Structure and Certified Arithmetic Separation of Periods

Source 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 $P_{\mathrm{III}}(D_6)$ 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 $|κ|\le3\times10^{-10}$ 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 $Λ\le433/2500=0.1732$ 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.