Spinning Particle Dynamics and Observational Redshift around an Asymptotically Flat Symmergent Black Hole

We investigate timelike particle dynamics, collision energetics, and photon frequency shifts in the perturbative variable-scalar curvature branch of asymptotically flat symmergent gravity. The low-energy vacuum action contains an $R^{2}$ correction whose coefficient is set by the boson--fermion imbalance of the underlying quantum field theory. At linear order, the exterior geometry is conformal to Schwarzschild spacetime through a radial mode satisfying a linear equation. We retain an independent boundary-condition-dependent amplitude and restrict the analysis to the perturbative domain. The two signs of the symmergent parameter $\gamma$ yield distinct profiles: $\gamma>0$ gives a Yukawa-suppressed deformation, whereas $\gamma<0$ produces an oscillatory inverse-radius deformation. We derive radial equations, effective potentials, circular-orbit and marginal-stability conditions for neutral, electrically charged, and spinning massive particles. Charged particles are treated in the test-field approximation, while spinning particles obey the Mathisson--Papapetrou--Dixon equations with the Tulczyjew condition. We also compute the center-of-mass energy of neutral-particle collisions and the frequency shifts of photons emitted tangentially by circular geodesic sources and detected by a static observer at infinity. The redshift and blueshift factors satisfy $(1+z_{+})(1+z_{-})=1/A(r_e)$, directly linking their product to the lapse function at emission. The $\gamma>0$ branch yields smooth, short-range deviations from Schwarzschild dynamics, whereas the $\gamma<0$ branch can generate oscillatory radial bands admitting circular-orbit solutions whose stability must be tested independently. These observables provide complementary probes of the variable-curvature sector, although their quantitative interpretation also depends on the deformation amplitude and, for the oscillatory branch, its phase.