Longitudinal Scalar Perturbations in a SynchronoGeometry-Inspired Effective Four-Dimensional Model: Radion Dynamics, the Fixed-Flux Electric Sector, and Kinetic Degeneracy

We study a restricted scalar perturbation sector of a SynchronoGeometry-inspired effective four-dimensional model containing a canonical radion-like field, a non-canonical scalar, and an electric gauge sector. The background geometry is an axisymmetric Bianchi I spacetime, and the homogeneous electric field is aligned with its distinguished spatial direction. The four-dimensional action is treated as an adopted effective model; an explicit derivation from a five-dimensional Kaluza-Klein ansatz is not claimed. For the scalar Lagrangian L(Φ,X) = −cRX + DΦX2 − W(Φ), with X = gμν∇μΘ∇νΘ, we derive the background-dependent derivatives entering the longitudinal perturbation equations. We formulate the perturbations in ADM variables and state the linear momentum and Hamiltonian constraints in the spatially flat gauge. The electric sector is eliminated at fixed electric flux by a Routhian transformation. This choice of ensemble is essential: direct substitution of the electric solution into the original Lagrangian does not give the fixed-flux effective functional. For nonzero wave number parallel to the electric direction, the reduced two-field principal matrices are K = diag(1, 2cR(1−12q)) and G∥ = diag(1, 2cR(1−4q)). For cR > 0, positivity of both matrices requires 0 < q < 1/12. The longitudinal high-frequency characteristic speed of the non-canonical scalar is cπ,∥2 = (1−4q)/(1−12q). This result is confined to the longitudinal channel and does not determine the characteristic structure for general wave-vector orientations. We also give the reduced mixing and potential matrices, formulate the corresponding action equations, and compare them with the independently perturbed radion and scalar-current equations. The comparison is an algebraic consistency check within the stated truncation, not an independent numerical or observational validation. Finally, we derive the cubic and quartic self-interactions of the non-canonical scalar in a locally fixed metric and examine their temporal canonical scaling near the kinetic-degeneracy surface. These matter-only coefficients do not determine a physical strong-coupling cutoff. The analysis therefore establishes a restricted analytical description of the longitudinal scalar sector. Explicit five-dimensional reduction, the full three-dimensional gradient matrix, complete nonlinear ADM constraint elimination, and a physical perturbative-unitarity analysis remain open.