Reptend Cycles as Finite Recurrence Operators: A Mathematical Core with a VEF Interpretation Appendix

Reptend Cycles as Finite Recurrence Operators: A Mathematical Core with a VEF Interpretation Appendix presents a rigorous analysis of cyclic numbers arising from full reptend primes (e.g., 7 and 17) and formalizes their behavior using finite permutation operators. The paper establishes that the repeating decimal blocks of reciprocals of full reptend primes are invariant under modular multiplication, producing cyclic permutations that can be represented by a linear shift operator. The resulting operator has a discrete Fourier spectrum consisting of roots of unity, providing a precise model of finite recurrence under closed modular constraints. All mathematical claims are confined to number theory and linear algebra, with explicit limits and falsifiers identified. An appendix clearly separates a non-causal, analogical interpretation within the Virtual Ego Framework (VEF). This interpretive layer treats reptend cycles as symbolic models of endurance, phase rotation, and resolution in closed systems, without asserting physical, metaphysical, or numerological causation. The mathematical core stands independently of this appendix. The work is intended for researchers in number theory, mathematical physics, and systems modeling, as well as interdisciplinary readers interested in how finite cyclic structures can model recurrence and invariance without invoking infinite dynamics or unbounded growth.Keywords:reptend primes, cyclic numbers, finite recurrence, modular arithmetic, permutation operators, shift operators, roots of unity, number theory, linear algebra, spectral analysis, discrete Fourier transform, closed systems, invariants, finite dynamical systems, Virtual Ego Framework, VEF, structural models