Supplementary Verification Code for the paper:
"Cofactor Dynamics and Pell Equations in GCD-Augmented Fibonacci Recurrences: The Algebraic Structure of Geometric Stabilization"
Mosab Hawarey
AIR Journal of Mathematics and Computational Sciences, Vol. 2026
Journal DOI: 10.65737/AIRMCS
Journal ISSN: 3142-7197
Publisher: Artificial Intelligence Review AIR Publishing House LLC
Article ID: AIRMCS2026571
Article DOI: https://doi.org/10.65737/AIRMCS2026571
https://doi.org/10.65737/AIRMCS2026571
https://airjournals.org/doi/10.65737.AIRMCS2026571.html
This repository contains a self-contained Python script that independently verifies all computational claims in the paper:
| Verification | Description |
|---|---|
| Table 1 | Pell equation solutions (x² − 5y² = 4) and cofactor orbit validity |
| Table 2 | Lock-in at n = 5 for odd seeds coprime to 3 (p = 1, 5, 7, …, 49) |
| Theorem 8 | Exhaustive uniqueness of the (3, 5) coprime period-2 orbit for all odd n ≤ 500 |
| Theorem 9 | Transient termination — cofactor pair (3, 5) reached by step n = 5 |
| Theorem 9, Step 3 | Explicit GCD: gcd(2p + 3, p + 2) = 1 for all valid seeds |
- Python 3.8+
- No external dependencies (uses only
math.gcdandmath.isqrt)
python verify_cofactor_dynamics.pyAll five verification suites run automatically and print ALL VERIFICATIONS COMPLETE on success.
We investigate the algebraic structure underlying geometric stabilization in GCD-augmented Fibonacci recurrences C_p(n) = C_p(n−1) + C_p(n−2) + gcd(C_p(n−1), C_p(n−2)), resolving Open Problem 4 from Hawarey (2026). The binary operation f(a, b) = a + b + gcd(a, b) admits the cofactor factorization f(a, b) = gcd(a, b) · (α + β + 1), decomposing the recurrence into scale factor dynamics and a cofactor map T(α, β) = (β, α + β + 1). We prove that the cofactor pair (3, 5) is the unique coprime period-2 orbit by reducing the existence question to the generalized Pell equation x² − 5y² = 4, whose solutions are parameterized by Lucas and Fibonacci numbers.
Hawarey, M. (2026). Cofactor Dynamics and Pell Equations in GCD-Augmented Fibonacci Recurrences: The Algebraic Structure of Geometric Stabilization. AIR Journal of Mathematics and Computational Sciences, Vol. 2026, AIRMCS2026571. https://doi.org/10.65737/AIRMCS2026571
Dr. Mosab Hawarey
PhD, Geodetic & Photogrammetric Engineering (ITU) | MSc, Geomatics (Purdue) | MBA (Wales) | BSc, MSc (METU)
- GitHub: https://github.com/mhawarey
- Personal: https://hawarey.org/mosab
- ORCID: https://orcid.org/0000-0001-7846-951X
MIT License