Fibonacci Harmony: A Reflection-Parity Representation of Bilateral Fibonacci Magnitudes

This mathematical note formalizes the structure represented in the accompanying Fibonacci Harmony illustration. Starting from the ordinary Fibonacci sequence, each nonnegative magnitude F_n is assigned to two oppositely oriented components, −F_n and +F_n. Their signed balance vanishes identically, while their total unsigned magnitude remains 2F_n. Directional cancellation is therefore separated from disappearance of magnitude. The bilateral pair is represented by v_n = (−F_n, +F_n)^T. Under the exchange involution J, satisfying J² = I, the state obeys Jv_n = −v_n and lies exactly in the odd-parity eigenspace. Fibonacci scaling and reflection parity consequently separate: F_n controls magnitude, whereas the normalized bilateral state is independent of n. The work also distinguishes directional sign reflection from negative-index Fibonacci numbers and golden-ratio conjugation, and defines a falsifiable geometric interpretation in which bilateral equality and Fibonacci scaling are tested as separate hypotheses. Natural forms appearing in the accompanying visualization are illustrative motifs rather than evidence of exact Fibonacci scaling.