Golden-section quasigeometric sequences (QGS) were introduced in the foundational work of Mamombe (2016) [1] and developed further within the Proportiones Perfectus formulation of Mamombe (2017) [2]. In this framework, the Fibonacci additive recurrence is considered together with a nearest-integer golden-section constraint, producing an arithmetic class of quasigeometric sequences with associated parent-number and indexed structures. The present work uses this QGS framework as its starting point.

Within the golden-section QGS, the Z-class is generated by the indexed transformation developed by Mamombe (2017) [2]. Admissible same-parity Z-class pairs determine ordered midpoints whose positions in their originating QGS form a midway-index word. The recurrent local word (8, 11, 9) is the object studied here. Each occurrence is represented by three consecutive admissible Z-class pairs and their three midpoints, naturally producing a 2 × 3 matrix.

Definition 1 (Z class).

A golden-section quasigeometric sequence H_nis a member of the Z class if10(h₅,h₆) = (p₁,p₂),where P_nis itself a golden-section quasigeometric sequence.

This is the defining membership condition introduced by Mamombe (2017). Within the Z class, when two successive sequences have the same arrangement of odd and even terms, they are symmetrical about a sequence Lₙ from the family generated by 7, 11, 18, 29, 47, … The same-parity Z-class pairs used in the present work are admissible pairs selected from this previously defined class; they do not define the Z class itself.

Fibonacci and Lucas recurrence theory is developed classically by Vajda [3] and Koshy [4]. Relations among Beatty sequences, Fibonacci numbers and the golden ratio are discussed by Russo and Schwiebert [5]. Falcón and Plaza [6]-[8] developed k-Fibonacci numbers, their Pascal 2-triangle and associated polynomial theory; Mamombe [9][10] extended this direction to k-Fibonacci/k-Lucas systems and generalized a:k:m-Fibonacci numbers. Recent work continues to study generalized Fibonacci and Lucas recurrences, arithmetic properties, return words, period structures, representation problems and intersections of recurrence sequences [11]-[26]. The QGS framework used here originates in Mamombe (2016) [1] and Mamombe (2017) [2].

Given

a golden-section QGS H= (h₁, h₂, …) satisfies

together with

The Fibonacci QGS indexing convention used here is

Let

denote the set of positive parent numbers, equivalently the positive integers omitted by the rounded golden map n ↦ rnd ( φ n ) . A parent-number obstruction centre C satisfies

,

,

.

Its associated parent-sandwich boundary is

Ordering admissible Z-class pairs by increasing lower endpoint and taking the midpoint of the parent numbers then computing the index of the midway in a valid QGS produces the midway-index word

The first appearances of successive new midway indices occur at positions 1, 2, 3, 5, 8, 13, 21, 34, … Moreover, the numbers of internal occurrences of the symbol 8 between the first admissions 10→11, 11→12, 12→13, 13→14 and 14→15 are 1, 1, 2, 3 and 5. The recurrent word (8, 11, 9) is therefore embedded in a Fibonacci-organized symbolic structure.

For the r^th occurrence of the ordered word (8, 11, 9), let qᵣ be the table position of its first symbol. Write the three consecutive admissible pairs as (a_r, b_r), (c_r, d_r), and (e_r, f_r), with associated midpoints μ r = ( a r + b r ) / 2 , ν r = ( c r + d r ) / 2 , and ξ r = ( e r + f r ) / 2 . The columns are fixed by the symbolic order 8, 11, 9.

Equivalently, if Table 1 (or its continuation) is denoted by T ( q ) = ( P ( q ) , m ( q ) , k ( q ) ) , then every occurrence q r satisfying ( k ( q r ) , k ( q r + 1 ) , k ( q r + 2 ) ) = ( 8 , 11 , 9 ) generates

This table-indexed form makes the matrix reproducible directly from the ordered Z-class data: locate an exact (8, 11, 9) block in the Index column, take the corresponding three admissible pairs as row 1, and take their midpoints as row 2.

Let the r^th occurrence of (8, 11, 9) be generated by the ordered admissible pairs P r ( 8 ) = ( a r ( 8 ) , b r ( 8 ) ) , P r ( 11 ) = ( a r ( 11 ) , b r ( 11 ) ) , and P r ( 9 ) = ( a r ( 9 ) , b r ( 9 ) ) . Define

The transition matrix is

The first row preserves the complete pair geometry, while the second row records the midpoint projection. Accordingly, an occurrence of the symbolic word is represented simultaneously in endpoint and midpoint coordinates.

Let

denote the positive parent-number line, i.e. the positive integers omitted by the nearest-integer golden map n ↦ rnd ( φ n ) . For the present argument it is useful to write the parent numbers in their explicit inhomogeneous Beatty form

This representation is complementary to rnd ( φ n ) = ⌊ φ n + 1 / 2 ⌋ : the slopes satisfy φ − 1 + φ − 2 = 1 , while the offsets satisfy ( 1 / 2 ) φ − 1 − ( φ / 2 ) φ − 2 = 0 . Hence the two sequences partition the positive integers. In particular,

.

Theorem 1 (Parent-Number Production Theorem).

Let (a, b) be any admissible Z-class pair occurring in a transition matrix Mᵣ,with

and b–a = 14,and let m = (a + b)/2 = a + 7.Then the same pair geometry necessarily produces the additional parent numbers

while a − 1,m = a + 7,and b + 1 = a + 15 are not parent numbers.Consequently every admissible matrix column produces the three parent-sandwich boundaries

Proof

Write a = p k and θ = { φ 2 k − φ / 2 } .Since b = a + 14 is also a parent number,the monotonic parent sequence gives b = p k + 5 .Using φ 2 = 2 + 1 / φ ,the condition p k + 5 − p k = 14 is equivalent to ⌊ θ + 5 / φ ⌋ = 4 ,hence θ ≥ 4 − 5 / φ .It follows successively that p k − 1 = a − 2 , p k + 2 = a + 6 , p k + 3 = a + 8 , p k + 5 = a + 14 = b ,and p k + 6 = a + 16 = b + 2 .Because the parent sequence is strictly increasing,the integers lying between each indicated consecutive parent pair—namely a − 1,a + 7 = m,and a + 15 = b+ 1—are omitted from

Thus the three displayed parent-sandwich boundaries are produced by the admissible pair itself,without a separate parent-number test.

The inaugural transition matrix is presented in Table 3.

Table 3. Matrix M₁ for the inaugural occurrence of (8, 11, 9).

By Theorem 1, each of the three admissible columns of M₁ produces three parent-sandwich boundaries. The inaugural matrix therefore produces

The first recurrent transition matrix is presented in Table 4.

Table 4. Matrix M₂ for the first recurrent occurrence of (8, 11, 9).

By Theorem 1, the three columns of M₂ produce the boundary family

Two distinct displacement variables are required. The symbolic displacement measures movement in the ordered Z-class table, whereas the arithmetic displacement measures movement of the corresponding matrix coordinates. Keeping these quantities separate makes the construction reproducible without assuming that a translated matrix automatically generates a parent-number boundary.

Lemma 1 (Return-Position Lemma).

Let k(q) denote the midway index in row q of the ordered Z-class table.Define q₁as the first table position for which ( k ( q 1 ) , k ( q 1 + 1 ) , k ( q 1 + 2 ) ) = ( 8 , 11 , 9 ) ,and recursively define

Define the symbolic return displacement by

Thenq_ris uniquely determined by the ordered midway-index word whenever the r-th occurrence exists.

Proof

The set in the recursive definition contains exactly the later starting positions at which the three consecutive midway indices are8, 11, 9.Taking its minimum selects the next occurrence and excludes intervening permutations such as (9, 11, 8).Thus the recursion enumerates the occurrences in their table order.

Lemma 2 (Matrix Reconstruction Lemma).

Let T ( q ) = ( P ( q ) , m ( q ) , k ( q ) ) denote the admissible pair,midpoint and midway index at table position q.For an occurrence beginning atq_r,the transition matrix is

For the Z-class local geometry displayed by the computed(8, 11, 9)occurrences,let ν r = m ( q r + 1 ) be the central midpoint.Then the matrix takes the one-coordinate form

Proof

Each admissible pair has endpoint separation14,hence each pair is seven units on either side of its midpoint.In a(8, 11, 9)block the three consecutive midpoints are separated by34,so they areν_r − 34,νᵣandν_r + 34.Substituting±7about each midpoint gives the six displayed endpoints.Thusq_rfixes the three table rows,and the central midpoint νᵣfixes every integer coordinate of M_r.

Definition 4 (Arithmetic matrix displacement).

For successive occurrences define

This identity is the exact bridge between symbolic return positions and arithmetic matrix translations.

Corollary 3.1 (Transition-Matrix Translation Law).

For two successive occurrences in the same local Z-class geometry,every endpoint and midpoint is translated byΔν(r).Hence

where scalar addition to a pair entry translates both of its endpoints.Equivalently,

A 2 × 3 matrix representation of the recurrent Z-class word (8, 11, 9) yields a reproducible parent-number production and boundary framework for golden-section quasigeometric sequences. The Return-Position Lemma determines the starting table position q_r of each occurrence; the Matrix Reconstruction Lemma determines M_r; and the Parent-Number Production Theorem shows that every admissible separation-14 column necessarily generates additional parent numbers and three parent-sandwich boundaries. Accordingly, each transition matrix produces a nine-member boundary family without a separate parent-number test. The inaugural matrix reproduces 117/119 among its generated boundaries, while the first recurrent reproduces 1130/1132, which boundaries have been used in physcical correspondences [2]. On the computed domain, the arithmetic displacement spectrum is { 377 , 610 , 987 } = { f 13 , f 14 , f 15 } . The resulting formulation links symbolic recurrence, Z-class matrix geometry, parent-number production, and boundary formation through explicit reproducible maps.