The Infinite Two-Way Fibonacci–Lagrange–Lucas Period Cascade via 4–20–4–20–4–60–12–3–3–3… in Tail-Terminology

We present a formal analysis of the period cascade generated by the tail sequences (last digits) of recurrent sequences governed by an=an−1+an−2an=an−1+an−2. By applying the iterative transform bn=an−1+an+1bn=an−1+an+1, we uncover a rigid, self-similar trajectory of tail periods: 4→20→4→60→12→3→3→⋯4→20→4→60→12→3→3→⋯. This cascade is not closed but opens into an infinite stationary attractor at period 3, which acts as a fractal archive containing alternating Fibonacci (period 60) and Lucas (period 12) macro-structures. Sequential divisions by powers of 5 progressively extract these embedded layers. All results are verified by algorithmic experiments and presented in tables. The cascade is shown to be infinite in both directions, with the number of preliminary 4–20 phases determined by the power of 2 dividing the initial conditions.