Two Fibonacci Butterflies: A Reflection-Parity / Stokes Dictionary for the Fibonacci–Mellin Two-Sheet Transform

This paper develops a controlled mathematical interface between two independently defined Fibonacci constructions: Ali Alhawarat’s reflection-parity geometry, in which opposite directional components carry equal Fibonacci magnitude and opposite sign, and Pasquale Camelia’s Fibonacci–Mellin Transform (FMT), a ten-node Vandermonde frame on a golden multiplicative lattice with LOW and HIGH output sheets. For a two-component state Ψ = (L,H)ᵀ, the exchange involution is diagonalized by the normalized Hadamard matrix, connecting the parity-channel energies directly to the Stokes combinations of the FMT readout. The paper carefully distinguishes fixed-gauge statistics from gauge-invariant coherence and states the conditions required to define a phase-referenced odd-parity fidelity. A general second-order recurrence theorem shows that an asymptotically uniform log-amplitude lattice is not uniquely Fibonacci but occurs whenever one eigenvalue dominates. The Fibonacci specialization enters through the spectral pair (φ,−φ⁻¹). Independent numerical tests recover the exact Stokes identity to machine precision. They also show that raw FMT doublets lie near the symmetric rather than the antisymmetric pole, that high coherence can be reproduced by matched-arc and shuffled controls, and that the tested statistics do not uniquely select φ over √2, √3, or 2. These negative findings are retained explicitly. The resulting framework provides an exact two-component dictionary, a declared gauge convention, a generic recurrence null, a Fibonacci spectral specialization, matched-frame controls, and a falsifiable testing program. The two constructions remain mathematically complementary and are not claimed to be identical or to establish a new physical symmetry.