Eigenvector zeros of a linear shear between the zeta and Möbius matrices of a divisor lattice

On the divisor lattice of a positive integer let be the zeta matrix and the Möbius matrix. We consider the one-parameter family , which interpolates () between the divisor-sum operator and its Möbius transform. The spectrum of is frozen at the divisors of for every ; only the eigenvectors move. We show that each eigenvector component depends only on the ratio of the two divisors indexing it, defining a family of polynomials , and we study their zeros in the open interval . Our main results are: (i) closed endpoint values (for squarefree ) and ; (ii) a pinching: every zero of lies in , where is the least prime factor of , with the upper endpoint attained exactly when is a prime power; and (iii) a parity law: has exactly one zero in when is odd and none when is even. We prove the parity law for all prime powers, for squarefree with , and for the families ; the general case is stated as a conjecture with a single remaining count bound.