TITLE:
Foundations of Tribonacci Quasigeometric Sequences
AUTHORS:
Lovemore Mamombe
KEYWORDS:
Tribonacci Constant, Quasigeometric Sequence, Parent Number, Discrepancy, Barrenness, Primitive Barren Integer, Nearest-Integer Rounding, Invariant Defect, Barren Lineage
JOURNAL NAME:
Open Journal of Discrete Mathematics,
Vol.16 No.4,
October
29,
2026
ABSTRACT: A Tribonacci quasigeometric sequence (TQGS) is an integer sequence constrained simultaneously by the exact third-order recurrence and nearest-integer multiplication by the Tribonacci constant τ. The framework introduced here is defined by
h
n+3
=
h
n+2
+
h
n+1
+
h
n
,
h
n+1
=rnd(
τ
h
n
)
,
τ
3
=
τ
2
+τ+1
. Writing
δ
n
=
h
n+1
−τ
h
n
gives the exact discrepancy propagation law
δ
n+2
=(
1+τ−
τ
2
)
δ
n
+(
1−τ
)
δ
n+1
,
|
δ
n
|<
1
2
.These relations establish the basic TQGS compatibility dynamics, an explicit admissibility and non-degeneracy criterion for genuine Tribonacci parent numbers, and the complementary notion of global barrenness. For rounded orbits bn+1 = rnd(τbn), the integer Tribonacci defect
E
n
=
b
n
+
b
n+1
+
b
n+2
−
b
n+3
∈{
−1,0,+1 }
separates exact local TQGS compatibility (En = 0) from the two incompatible defect classes. A nonzero defect is invariant along a barren rounded orbit, so primitive barren roots generate infinite signed barren lineages. Thus the principal contribution is the introduction and development of the TQGS framework itself—its defining compatibility laws, discrepancy dynamics, parent-number criterion, and excluded-state dynamics—with barren integers forming one structural component of that broader theory.