Emergent Lorentzian Geometry from Scalar Temporal Dynamics: A Non-Degenerate Metric Construction in Time–Scalar Field Theory

We present a revised formulation of Time–Scalar Field Theory (TSFT) in which spacetime geometry emerges from a scalar temporal field Θ(xμ) while avoiding the degeneracy inherent in gradient-only metric constructions. Previous approaches based on direct algebraic closure of the form gμν ∝ ∂μΘ∂νΘ are shown to yield rank-deficient metrics incapable of supporting a Lorentzian spacetime structure. To resolve this, we introduce a non-degenerate metric ansatz of disformal type, gμν = A(Θ) ημν + B(Θ) ∂μΘ∂νΘ, which preserves invertibility while retaining dependence on temporal gradients. The scalar field obeys a covariant wave equation □Θ = S, providing the fundamental dynamical content of the theory. We show that, under weak-field conditions, this structure reproduces the Newtonian potential and standard post-Newtonian parameter γ = 1 for a constrained class of coupling functions. The role of temporal coherence is reinterpreted as a selection principle constraining admissible metric functionals rather than directly generating geometry. This formulation clarifies the metric closure problem in TSFT, replacing algebraic degeneracy with a well-defined class of non-degenerate geometries and establishing a consistent foundation for further development in strong-field regimes and observational tests.