Unmasking Fibonacci: Infinite Reversible Masks via Additive Shifts and the Period Cascade

This paper presents an experimental investigation into the robustness of the Fibonacci–Lucas period cascade under intentional disruption of formal modular signatures. Building on previous work by the author, which established the invariant chain of tail periods 4 -> 20 -> 4 -> 60 -> 12 -> 3 -> 3 -> 3 ..., we demonstrate that this cascade remains intact even when the sequence is subjected to an additive shift (adding a constant to every term) that destroys the obvious recursive property and alters the tail digits. Starting from a period-4 sequence (a Lucas mask), we added 113, applied the neighbour-sum transform T (summing terms one position apart), and obtained a degenerate period-1 sequence (all tails 6). Through successive divisions by 2, 5, and 3, and analysis of first differences, we recovered the pure Fibonacci sequence. We then show that this process is fully reversible, leading to a general construction of infinitely many masks for Fibonacci numbers, parameterised by arbitrary integer coefficients and initial values. The roles of powers of 2 and 5 are clarified: periods 4 and 20 are compressed forms of Lucas and Fibonacci obtained by division by powers of 2, while period 3 is a universal archive from which the classical sequences are extracted by division by powers of 5. These results confirm the universal and invertible nature of the period cascade.