The Hardy Reality Net
Year: 2026
Type: preprint
Abstract: We isolate the analytically productive core of the critical-line material around a single real entire function, the Hardy form of the completed zeta, Z(t)=ξ(½+it). Off the critical line the completed ξ is exactly the analytic continuation of Z: ξ(½+u+iv)=Z(v−iu). This identity gives exact parity laws, identifies the off-line stiffness as a(v)=−Z′(v), and shows that the reality locus of ξ is a net — the critical-line spine plus parabolic rungs through the extrema of Z. A high-precision scan verifies the local rung law and the rung count. Two controls bound the picture: random-phase fields with the critical n^(−½) spectrum have Kac–Rice zero density 1/√3 of the band reference and extremum-to-zero ratio 3/√5; and the de Bruijn–Newman zero flow possesses a finite quadratic symmetry-defect functional with an exact monotone dissipation law. This edition adds an independent recomputation of every result, a local off-line criterion u²=2Z(v⁎)/Z″(v⁎) that unifies the reality net, the scan, and the defect, a measured asymptotic for the rung distribution, and an explicit integration with the Spectral Ledger. Nothing here proves the Riemann Hypothesis; the controls and finite-system results are labelled as such.
Author: Desmond, Timothy
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212304507
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