TITLE:
Matrix Boundary Functional for Recurrent Z-Class Transitions in Golden-Section Quasigeometric Sequences
AUTHORS:
Lovemore Mamombe
KEYWORDS:
Quasigeometric Sequences, Golden Section, Z-Class, Parent Numbers, Transition Matrix, Midway-Index Word, Boundary Functional, Fibonacci Recurrence
JOURNAL NAME:
Open Journal of Discrete Mathematics,
Vol.16 No.4,
October
9,
2026
ABSTRACT: Golden-section quasigeometric sequences possess a Z-class whose admissible same-parity pairs generate an ordered midway-index word. For the rth occurrence of the recurrent transition subword (8, 11, 9), let (ar, br), (cr, dr), and (er, fr) denote the three consecutive admissible same-parity Z-class pairs corresponding respectively to the indices 8, 11, and 9. Here ar, cr, and er are the lower endpoints, while br, dr, and fr are the corresponding upper endpoints. Their midpoints are
μ
r
=
(
a
r
+
b
r
)/2
,
ν
r
=
(
c
r
+
d
r
)/2
, and
ξ
r
=
(
e
r
+
f
r
)/2
. The transition matrix is
M
r
=(
(
a
r
,
b
r
)
(
c
r
,
d
r
)
(
e
r
,
f
r
)
μ
r
ν
r
ξ
r
)
. Thus each column is one symbol of the ordered transition, with its generating Z-class pair above and its midpoint below. The inaugural and first recurrent matrices are
M
1
=(
103/
117
137/
151
171/
185
110
144
178
)
,
M
2
=(
1090/
1104
1124/
1138
1158/
1172
1097
1131
1165
)
. The construction therefore produces a family of boundary pairs from the recurrent matrix structure.