Möbius Reduction, Exact Dynamics, and Fractional-Order Extensions of Arbitrary-Delay Rational Difference Equations

We present a unified framework for two nonlinear rational difference equations with arbitrary delay. A residue-class decomposition reduces each delayed equation to independent second-order recurrences. Product and quotient transformations then reduce these recurrences to first-order Möbius maps. This structure yields explicit solutions through powers of 2×2 matrices and supports a systematic analysis of admissibility, singular cases, equilibria, periodic solutions, multiplier spectra, and asymptotic behavior. A spectral lifting relation connects the multipliers of each residue map to those of the full delayed system and identifies the neutral directions introduced by reconstruction. We then define a Caputo fractional difference extension of the reduced Möbius dynamics. At order one, the ordinary reduced map is recovered exactly and the admissible fixed points are unchanged. For 0<ν<1, the extension is studied only at the reduced-map level and is not asserted to reconstruct a solution of the original delayed equations. For fractional orders below one, stability depends on the order. In particular, a negative attracting Möbius multiplier produces an explicit critical-order threshold, and locally stable fractional solutions converge algebraically rather than geometrically. Numerical examples verify the exact reconstruction, spectral lifting, ordinary asymptotic regimes, and fractional-order threshold.