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

AbstractThis 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.