STT — A Conservative Total Action Formulation: Gravity, Gauge Fields and Scalars with a Dynamic Boundary Term

We present the complete mathematical formulation of the Selector-Time Theory (STT), a conservative extension of Einsteinian gravity based on a total action that includes, in addition to the usual volumetric terms, a dynamic boundary term with explicit physical meaning. The action incorporates five contributions: Einstein–Hilbert, Gibbons–Hawking–York, Yang–Mills, scalar field, and a quadratic boundary term that couples the scalar field to the spacetime boundary through a coupling constant κ. Variation of the action yields standard field equations in the bulk and a Robin-type boundary condition, naturally emerging from the conservative structure of the theory. We adopt the choice κ = H₀/c, linking the boundary coupling to the present cosmological scale without introducing additional free parameters. In cosmological applications with FLRW symmetry, the boundary condition selects specific scalar field trajectories, leading to stable predictions for the dark energy equation of state, distinct from ΛCDM. The STT is mathematically consistent, free from pathological instabilities, and explicitly testable by next-generation observations, reinterpreting dark energy as a structural effect of dynamic boundary conditions rather than as a new fundamental field.