Algebraic Consistency of Relativistic Field Theories: Extended CST Framework on FLRW, Exact Solutions, and Gravitational-Wave Phenomenology

CST (Causality-Stability Tester) is a background-dependent consistency audit for relativisticfield theories formulated at the level of an action. It compresses the minimum necessary contentof a perturbative viability claim into a fixed, auditable gate sequence applied to the physicaldegrees of freedom: a well-posed variational principle; consistent constraint reduction and DOFcounting; positivity of the reduced kinetic form (ghost freedom); hyperbolicity of the reducedequations via the principal symbol (causal well-posedness); gradient stability; and controlledinfrared mass behavior on the chosen background timescale. This paper standardizes the CSTgates and conventions and presents a Minkowski reference implementation where all diagnosticsreduce to explicit algebraic conditions on the reduced quadratic action and on the characteristicpolynomial. The CST output is a report that localizes failure modes to gate numbers, supportssymbolic automation, and makes stability and causality claims reproducible.