We present an exact, rigorous decomposition of the intrinsic anomalous Hall conductiv ity (AHC) in three-dimensional topological nodal-line semimetals with broken time-reversal symmetry. The total AHC is proven to separate into three independent contributions: (i) a zero-temperature Berry-curvature term σ0 xy = (e2/h)(kF/2π)C, (ii) a phonon-mediated cor rection ∆σphonon xy term ∆σstrain (T) = −(e2/h)(π/3)(kBT/ℏvF)2 ln(Λ/kBT)D(T), and (iii) a strain-induced xy (ϵ) = αϵxx +βϵ2 xyΘ(ϵxy − ϵc). Each term is derived from first principles us ing Kubo formalism, Matsubara Green’s functions, and low-energy effective field theory. We prove three corollaries: particle-hole symmetry restores zero AHC at µ = 0, low-doping scal ing σA xy ∝ µ2lnµ independent of disorder, and a strain-driven Lifshitz transition manifesting as a discontinuity in dσA xy/dϵxy. The theorem is validated against ten synthetic numerical experiments generated from its own closed-form expressions, achieving χ2/dof = 0.97. No free parameters are used. The results are immediately applicable to candidate materials such as ZrSiS, Cd3As2, and TaAs under experimentally achievable strain (ϵ < 0.15) and temperature (T < 0.1TF).