Matrix Boundary Functional for Recurrent Z-Class Transitions in Golden-Section Quasigeometric Sequences

Golden-section quasigeometric sequences possess a Z-class whose admissible same-parity pairs generate an ordered midway-index word. For the r^th occurrence of the recurrent transition subword (8, 11, 9), let (a_r, b_r), (c_r, d_r), and (e_r, f_r) denote the three consecutive admissible same-parity Z-class pairs corresponding respectively to the indices 8, 11, and 9. Here a_r, c_r, and e_r are the lower endpoints, while b_r, d_r, and f_r are the corresponding upper endpoints. Their midpoints are ${\mu }_r=\left(a_r+b_r\right)/2$ , ${\nu }_r=\left(c_r+d_r\right)/2$ , and ${\xi }_r=\left(e_r+f_r\right)/2$ . The transition matrix is $M_r=(\begin{array}{cc}\left(a_r,b_r\right)& ({}_{}\end{array}$