The Riemann $\hat{I}ž$-function from primitive Markovian cycles II: Strip rigidity and divisor identification

Abstract

We compare the Riemann $\hat{I}ž$--function to a canonical real-entire reference family arising from the cycle Laplacian developed in Paper I. These spectral determinants have only real zeros by self-adjointness. Our main tool is a rigidity lemma for holomorphic functions on horizontal strips. Applied to a normalized seam ratio linking $\hat{I}ž(2\cdot )$ to the reference family, this lemma shows that, under explicit holomorphy and boundary nonvanishing hypotheses verified in the forthcoming Paper III, the seam ratio extends to a zero-free holomorphic function of bounded type on each overlap strip. It follows that, on every admissible overlap strip, $\hat{I}ž(2\cdot )$ and the reference family have the same zero divisor.

26 pages. Feedback welcome and appreciated