Global Closure of the Scalar-Time Potential Through Temporal Efficiency Geometry

Previous formulations of Time–Scalar Field Theory (TSFT) established that temporal advance partitions between internally retained evolution and external propagation according to the conserved temporal-efficiency relation η_int^2 + η_prop^2 = 1, with null propagation corresponding to the maximum-efficiency coherence limit η_int = 0, η_prop = 1. These earlier results successfully recovered relativistic time dilation, photon propagation, temporal inertia, and coherence-driven bound structure, but left the scalar-time potential V (Θ) formally unspecified beyond local asymptotic constraints. In the present work, we demonstrate that the global structure of the scalar-time potential is not arbitrary, but is constrained by temporal-efficiency geometry itself. Beginning from the conserved partition law and the requirement that null propagation represent the maximal coherence state, we derive the admissible curvature structure of V (Θ) from internal temporal-allocation consistency. The conserved temporal partition naturally generates an efficiency ratio measuring internally retained temporal allocation relative to propagative temporal al-location. Combining this ratio with the scalar-time coupling factor α(Θ) = e^−Θ, we derive a scalar-only globally closed temporal-efficiency potential. Imposing nullsector coherence conditions V (0) = 0, V ′(0) = 0, V ′′(0) = 0, naturally selects the null-normalized scalar-time potential V (Θ) = V∗ (e^2Θ − 1 − 2Θ − 2Θ^2), up to an overall coherence-curvature scale V∗. The resulting potential generates asymptotic inverse-radial scalar-time structure, subshell-sensitive coherence deformations, and thresholded shell-crossing behavior in the atomic sector. Both the dominant shell-generating interaction and the subleading coherence-binding deformation arise from the same globally closed scalar-time curvature hierarchy: V^(3)(0) = 8V_∗, V^(4)(0) = 16V_∗. Atomic organization therefore emerges not from empirical shell-ordering rules, but from globally constrained temporal-efficiency geometry within the scalar-time field itself. More broadly, the present work proposes that mass, propagation, binding, and spectral organization are different realizations of a single conserved temporal efficiency structure. The scalar-time potential is therefore interpreted not as an arbitrary phenomenological input, but as the variational curvature geometry generated by admissible temporal allocation itself.