A Principal-Symbol Subluminality Constraint on Disformal Couplings in Quasi-Static Dust Backgrounds

We perform an explicit derivation of the principal symbol of the scalar perturbation in a smooth-inhomogeneous quasi-static dust background under the decoupling limit, for scalar-tensor theories with a disformal coupling of the form g̃_μν = A(C) g_μν + B(C) ∂_μC ∂_νC. Working in the eikonal regime and in the local orthonormal frame comoving with the dust, the resulting phase velocity is c_s² = 1/(1−ξ) with ξ ≡ A_0^(3/2) √Z_0 B_0 ρ̃_0 / K_0 evaluated on the background, where Z = A − 2BX and X = −½ g^μν ∂_μC ∂_νC. Within this declared regime, subluminal causal propagation of the scalar mode requires B(C) ≤ 0 in regions with positive dust density. The constraint is sharp, falsifiable, and operationally testable. We discuss the strict validity domain of this result, the operational (non-causal) status of the perturbative regime |ξ| < 1, and the potential tension with disformal parametrizations that adopt B(C) > 0 in regions of phenomenological interest. The result is offered as a diagnostic tool for disformal scalar-tensor models coupled to dust, not as a verdict on any specific construction. We do not claim universality; we claim transparency. The analysis is restricted to: pressureless dust matter, quasi-static backgrounds with ũ^μ ∇_μ C_0 = 0, decoupling limit (δg_μν = 0), perturbative-control regime |ξ| < 1, and strict invertibility Z_0 > 0. Cosmological backgrounds with nonzero Hubble flow, ghost-free conditions, post-Newtonian compatibility, and the fully coupled scalar-tensor analysis lie outside the present scope and are noted as natural extensions.