Constructive MTT Quantum Gravity I: Finite-Volume SPT Gaussian Control and a Conditional Borel-Summability Contract

This paper separates three constructive questions: existence of the filtered Gaussian measure, Borel summability at a finite spectral cutoff, and removal of that cutoff for a selected gravitational interaction. On a bounded-geometry Euclidean domain, a positive proper-time gap makes the spectral proper-time (SPT) covariance trace class, not merely Hilbert–Schmidt. It therefore defines a centered Borel Gaussian measure on the stated TT Hilbert space and has almost-sure Sobolev regularity of every finite order. We then prove ordinary Borel summability for finite-dimensional nonnegative quartic interactions and give an exact Borel–Laplace representation on the positive coupling ray. Finally, we prove a sufficient weighted-Borel convergence criterion for cutoff removal. These results do not establish Borel summability of the TT-expanded Einstein–Hilbert action: its sectorial stability, a selected interaction and filter, cutoff-uniform constructive bounds, and convergence of the Borel transforms remain separate obligations. The result is a rigorous finite-volume constructive core and a precise input contract for the BRST/BV successor, rather than a nonperturbative construction of MTT gravity.