Bilateral Fibonacci Symmetry: A Signed-Magnitude Construction for Mirrored Recursive Growth

Abstract

This preprint introduces Bilateral Fibonacci Symmetry, a signed-magnitude construction in which equal Fibonacci magnitudes are assigned to opposite spatial directions. For each Fibonacci magnitude F_n, the construction defines a_n^- = -F_n and a_n^+ = +F_n. Their signed balance is B_n = a_n^- + a_n^+ = 0, while their total unsigned magnitude is S_n = |a_n^-| + |a_n^+| = 2F_n. The construction separates directional cancellation from disappearance of magnitude and provides a mirrored geometric realization while preserving the Fibonacci recurrence and its asymptotic golden-ratio behavior. Two independent geometric quantities are introduced: A, measuring bilateral asymmetry, and E_F, measuring relative deviation from Fibonacci scaling. A six-scenario synthetic test battery demonstrates that bilateral symmetry and Fibonacci scaling can fail independently. The archived version also includes a reproducibility appendix and reference implementation for regenerating the synthetic tests. The scope is deliberately limited. This work does not propose a new Fibonacci sequence, does not claim that natural trees exhibit exact Fibonacci bilateral symmetry, and does not interpret B_n = 0 as a physical conservation law. The tree illustration is a conceptual geometric representation rather than botanical evidence. Version 1.0 is frozen as the first archival version of the construction and its reproducibility protocol.