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We present a mathematically rigorous formulation of the Fundamental Speed Theory (FST), a vector-tensor theory of gravity featuring a dimensionless vector field νᵐ. The theory introduces characteristic scales M₀ = ħ/(cL₀) and L₀ = 10 kpc to ensure complete dimensional consistency, with explicit inclusion of ħ and c in all physical expressions. Galactic dynamics obeyd²ν̃/dξ² + (2/ξ) dν̃/dξ = β_eff ν̃³where ξ = r/L₀ and β_eff = (λν₀²)/(6c₁) = 2.0×10⁷.We perform a hierarchical validation at three distinct levels of parameter freedom: • Level 3 (Zero Free Parameters): Fixed M = 1.0×10¹⁰ M⊙ and r_d = 3.0 kpc for all 175 galaxies. Even with no galaxy-specific parameters, FST correctly describes 65.7% of galaxies with mean χ²_ν = 0.809. • Level 2 (Estimated Parameters): Mass and scale length estimated from scaling relations (no fitting). Success rate reaches 93.6% with mean χ²_ν = 0.347 for the 160 galaxies with χ²_ν < 3.0. • Level 1 (Fully Fitted): Mass and scale length fitted per galaxy. Success rate reaches 100% with mean χ²_ν = 0.170.This hierarchical validation demonstrates that FST captures the essential physics of galactic rotation without overfitting. The theory achieves a mean reduced chi-squared of ⟨χ²_ν⟩ = 0.170 across all 171 SPARC galaxies, with 91.2% of galaxies having χ²_ν < 0.5 (excellent fit) and only 1.8% (three galaxies) having χ²_ν > 1.0. The characteristic transition scale is ξ_c = √(2/β_eff) = 3.16×10⁻⁴, corresponding to a fundamental scale r_c = ξ_c L₀ ≈ 3.16 pc.Remarkably, we discover that all five field parameters (c₁, c₂, c₃, λ, ν₀) unify into a single fundamental acceleration scale:A₀ = (c₁ + c₃) ν₀² c² / L₀ = 2.42 × 10⁻¹⁰ m/s²This unified parameter reproduces the full 5-parameter theory identically for all 171 galaxies, demonstrating that FST is fundamentally a one-parameter theory.Cluster analysis reveals three distinct dynamical families of galaxies. Solar System constraints are satisfied through the galactic field gradient, with the local FST acceleration at Earth being ~8×10⁻¹⁵ of Newtonian acceleration—more than 100,000 times below current observational limits. Complete mathematical derivation and an open-source implementation ensure full reproducibility. Extension to cosmological scales is planned for future work.