Fibonacci Meets Triangular Numbers: Uncovering Both Lucas and Fibonacci via the Period Cascade

This paper presents an experimental investigation into the robustness of the Fibonacci–Lucas period cascade under a non-linear additive perturbation. Building on previous work by the author, which established the invariant chain of tail periods 4 -> 20 -> 4 -> 60 -> 12 -> 3 -> 3 -> 3 ..., we add the sequence of triangular numbers T_n = n(n+1)/2 to the Fibonacci 60 sequence. This creates a more complex mask, as triangular numbers possess their own tail period (20). We then apply a sequence of simple operations — two applications of the neighbour-sum transform T, differencing, subtraction with a shift of 3, and division by 2 or 10 — and show that the system yields the pure Lucas sequence (period 12) from one mask and the pure Fibonacci sequence (period 60) from another. This demonstrates that the cascade is universal and remains intact even when the added perturbation is non-linear. The results confirm that masks are merely intermediate layers that can be removed by elementary arithmetic operations, and that period 4 is a compressed form of Lucas accessible by division by 2. The newly observed period 15 is case-specific and illustrates the richness of possible masks when combining Fibonacci with other sequences. All operations are reversible, confirming the bidirectional nature of the period cascade.