Constructive MTT Quantum Gravity I: Borel Summability of the SPT-Filtered Transverse–Traceless Sector

Modal Triplet Theory (MTT) predicts a universal proper-time (SPT) damping of ultraviolet modes in the gravitational sector. In this paper we establish the nonperturbative existence of the resulting quantum theory in the transverse–traceless (TT) graviton sector using methods from constructive quantum field theory. Working on bounded-geometry spacetime slabs, we show that SPT filtering yields a Gaussian ultraviolet bound on the TT propagator with a strictly positive infrared shift. The resulting covariance is Hilbert–Schmidt, allowing a convergent Loop Vertex Expansion and uniform control of perturbative coefficients. We prove analyticity in a cardioid domain and apply the Nevanlinna–Sokal criterion to demonstrate Borel summability of the TT sector to all orders. The analysis is fully nonperturbative, regulator-controlled, and does not assume unitarity or scattering. This paper provides the constructive ultraviolet foundation for the subsequent gauge-invariant and infrared analyses of MTT quantum gravity.