TITLE:
Golden Discrepancy and Binary Stability in Quasigeometric Sequences
AUTHORS:
Lovemore Mamombe
KEYWORDS:
Quasigeometric Sequence, Golden Ratio, Golden-Section Discrepancy, Discrepancy Decay, Binary Stability, Parent Numbers
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.10,
October
28,
2026
ABSTRACT: The golden section
φ
is the defining irrational constant in golden section quasigeometric integer sequences generated by nearest-integer iteration. Such sequences do not reproduce exact geometric progression; instead, rounding introduces a structured discrepancy whose magnitude decays geometrically while its sign oscillates. The first term
h
1
of a quasigeometric sequence
H
n
is called a parent number
p
and it satisfies strict computational constraints. Parent numbers can also be generated by perturbing terms of an established quasigeometric sequence
H
n
by unity, so that
p=
h
n
+ϵ
,
ϵ∈{
−1,+1 }
. The analysis of this perturbation shows that for
2≤n≤4
, only the positive or negative branch is admissible, and a transition occurs at
n=5
when binary stability is attained. This paper introduces the golden discrepancy decay variable and examines its significance at the critical fifth stage of quasigeometric evolution. Starting from the nearest-integer bound, the analysis establishes the characteristic discrepancy scale 1/(2φ4) ≈ 0.072949 at
n=5
, the very onset of the stable binary perturbation regime.