The Riemann $\mathrm{\Xi }$-function from primitive Markovian cycles II: Strip rigidity and divisor identification

We compare the Riemann $\mathrm{\Xi }$–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 $\mathrm{\Xi }(2\cdot )$ 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, $\mathrm{\Xi }(2\cdot )$ and the reference family have the same zero divisor.