Boundary-Regularized Vacuum Dynamics in Scalar-Tensor Gravity: Sound-Speed Stability, Degeneracy Constraints, and Axion Signatures

We construct a scalar-tensor framework in which a boundary-conditioned Casimir-type vacuum Lagrangian, coupled to a dynamical scalar field through a dimensionless modulation \xi, screens ultraviolet vacuum-energy fluctuations to a stable, non-perturbative scale. We derive the conditions under which the effective sound speed satisfies 0 < c_s^2 \le 1 through a null-energy-condition (NEC) violating bounce, state the degeneracy (Ostrogradsky) constraints on the multifield Galileon-type kinetic completion adopted by the program, and analyze the dynamical attractor of the homogeneous field equations exactly. The analysis yields a sharp, falsifiable partition of the framework: the Casimir term acts as a laboratory-scale coupling (a fractional force modulation of order \xi = 10^{-5} between plates), while any cosmological vacuum-energy screening requires the boundary phase to occupy a volume fraction f \equiv d_c / L \sim O(10^{-4}), an explicit, testable condition. We derive a conservative upper bound on the coupled axion-sector amplification that is directly testable by current and planned haloscopes.