Causal-Boundary Disformal Gravity: A Matter-Coupled Effective Field Theory

We propose a scalar-tensor effective field theory in which the local causal structure dynamically responds to matter density through a bounded disformal coupling. The theory introduces a scalar field Φ disformally coupled to the matter sector, motivated by the disformal deformation of the physical matter volume element. The resulting effective scalar Lagrangian adopts a Dirac-Born-Infeld (DBI) kinetic structure, whose reality condition provides a fundamental kinematic bound on the scalar kinetic invariant. In the strong-density regime, the scalar characteristic speed asymptotically vanishes (cs2→0), forcing the scalar sector into an ultra-local Carrollian limit. We propose dynamical characteristic degeneration as a potential collapse endpoint. This dynamic acoustic freezing of the scalar sector may significantly alter the coupled Einstein-scalar evolution, potentially bypassing the geometric assumptions required by standard singularity theorems. A homogeneous cosmological application suggests the expansion of the universe is a consequence of scalar relaxation.