Quantum Fundamental Speed Theory (QFST): Mathematical Derivations of Dark Matter and Dark Energy from the Speed Field

We present a quantum formulation of the Fundamental Speed Theory (QFST) built directly from the classical FST Lagrangian, in which the dimensionless speed field ν µ plays a dual role across scales: it modifies gravity classically in galaxies, while its quantum excitations behave as ultra-light dark matter on cosmological scales. On galactic scales, the classical FST equation for ˜ν = ν/ν0 fits rotation-curve data without dark matter (171 SPARC galaxies, χ 2 ν = 0.170). Linearization of the Lagrangian around the vacuum yields a Helmholtz screening scale µ = 1/λscreen, implying a particle mass m via µ = mc/ℏ. The resulting mass, m = ℏ cλscreen = 3.88 × 10−24 eV, contains no additional free parameter beyond the FST inputs. We also prove that quantum corrections to the classical galactic equation are suppressed by (ℓPl/λscreen) 2 ≃ 10−103, ensuring an explicit separation between the classical and quantum regimes. Dark energy arises from the logarithmic potential in the Lagrangian. We show how a positive vacuum energy density is obtained from the potential-energy func- tion U ≡ −V and determine the present cosmic vacuum value ˜νnow = 8.21 × 10−12 by matching the observed ρΛ. At the background level, we verify that QFST repro- duces the standard expansion history as tested by Cosmic Chronometer H(z) data. A full discrimination from ΛCDM is expected at the perturbation level (matter power spectrum and CMB anisotropies), requiring a Boltzmann-code implementa- tion.