The Multi-Phase Period Pulsation of Fractional and Integer Golden Ratio Sequences under Body–Tail Encoding

We introduce a two-digit encoding system for integers, called body–tail encoding, where the tail is the last digit and the body is the digital root of the prefix (with 0 replaced by 9 for numbers >= 10). Applying this encoding to recurrent sequences of the form a_n = a_{n-1} + a_{n-2} (which include Fibonacci and Lucas sequences) reveals a rich periodic structure. We show that the periods of tails form a closed set {3,4,12,20,60}, all embedded in the fundamental 60-cycle. The periods of bodies are given by lcm(p, 24), where p is the tail period. A trigonometric projection of the codes onto the unit circle demonstrates a 120-step universal period and a full 360° closure. We also uncover a symmetry about position 60 and a cyclic transformation that interchanges Fibonacci and Lucas sequences. All results are verified by numerical experiments in Python and summarized in tables. This work provides a unified framework for understanding the hidden periodic structure of recurrent sequences with the golden ratio.