A Spectral Closure Operator in Time–Scalar Field Theory and the Riemann Hypothesis

The Riemann Hypothesis (RH) asserts that every nontrivial zero of the Riemann zeta function ζ(s) lies on the critical line Re(s) = 1/2. A longstanding program initiated by Hilbert and P´olya suggests that RH would follow if the imaginary parts of the nontrivial zeros correspond to the spectrum of a self-adjoint operator. In this work we construct a spectral operator within the framework of Time–Scalar Field Theory (TSFT), where physical and mathematical structures arise as coherent eigenmodes of a scalar-time field. We show that under natural TSFT closure conditions the spectral determinant of the resulting operator corresponds to the completed Riemann function ξ(s). If the operator is shown to be selfadjoint on a suitable domain, its eigenvalues are real, implying that the admissible zeros of ξ(s) must lie on the critical line. The construction is motivated by three independent structures within the TSFT program: (i) spectral projection operators governing coherent temporal modes, (ii) prime-power weighting arising from irreducible harmonic structure, and (iii) logarithmic localization consistent with the explicit formula for prime distributions. These ingredients lead naturally to a logarithmic dilation operator supplemented by a scalar-time closure potential. The resulting framework provides a candidate Hilbert–P´olya realization emerging from TSFT spectral geometry. Supporting analysis