The Equivalence Principle Determines the Quadratic Form in Finslerian Extensions of Gravity

We prove that the equivalence principle, in its Finslerian formulation, forces the effective metric on which matter propagates to be independent of the four-velocity. When applied to Finsler functions of the form F² = H f(β²/H), where H ≡ −g_μν ẋ^μ ẋ^ν and β ≡ A_μ ẋ^μ, this condition imposes f″ = 0, and hence f(x) = 1 + cx. After a suitable renormalisation of the vector field A_μ, one obtains the unique quadratic form F² = α² + β². This result converts the quadratic Finsler function from an arbitrary choice into a necessary consequence of a fundamental physical principle. The mathematical ingredient underlying the proof is the vanishing of the Cartan tensor, which is the standard condition for a Finsler metric to reduce to a Riemannian one. What is new here is not the mathematical fact itself, but its physical interpretation: the equivalence principle provides a fundamental reason why the quadratic form is singled out. The result is established entirely within the ansatz F² = H f(β²/H), and no additional dynamical assumptions are invoked.