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Symmetric Sparse Recurrences with Central Inversion: Factorization Cycles and k-nacci Attractors

Year: 2026

Type: article

Abstract: This paper introduces a class of symmetric sparse recurrence relations generating infinitefamilies of Salem numbers. By implementing a central negative inversion and symmetricallyexpanding the recurrence using a sparsity parameter i, we derive palindromic characteristicpolynomials Pk,i(x). We identify a phenomenon of “algebraic memory”: at periodic intervalsL = 2(k + 1), the polynomials become reducible and explicitly contain the base kernel as afactor. A systematic 30-step analysis for orders k = 2 to k = 8 confirms the resonance cyclesand the rapid convergence of dominant roots to k-nacci constants, which act as attractorsfor the system.

Source: Zenodo (CERN European Organization for Nuclear Research)

DOI: 10.5281/zenodo.19204943

Authors: Helmdach, Emma, Helmdach, Emma

Language: English

Topic: Advanced Mathematical Theories and Applications

Subfield: Statistical and Nonlinear Physics

Field: Physics and Astronomy

Domain: Physical Sciences

Open Access status: green

OpenAlex ID: W7140179871

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