QUANTUM FIELD THEORY FROM SUPERALGEBRAIC GEOMETRY: GRAVITON PROPAGATOR, COSMOLOGICAL CONSTANT, AND UV FINITENESS

We derive the quantum field-theoretic consequences of the superalgebraic curvature decomposition established in our previous work. Using the effective action obtained from the superpermanent of the bosonic curvature on the noncommutativetorus T 2 θ , we compute the exact graviton propagator and prove its UV finiteness andunitarity. The propagator takes the form D(k) ∼ 1/(k2 exp Θ^2Λ^2 NCk^2), which has no ghost poles and renders all loop integrals finite.We compute the one-loop vacuum energy and show that the supertrace over the graded algebra A = A0 ⊕ A1 cancels the leading quartic divergences, leaving a residual cosmological constant determined by the Atiyah–Singer index of the Dirac operator onT 2 θ . We establish the consistency of this result with the KO-dimension condition d ≡ 0 (mod 8) from our earlier work on permanent gravity, which forces the bare cosmological constant to vanish identically. The observed small value arises from the slow-rollevolution of the modular field after Starobinsky inflation.We derive the modified Newtonian potential from the non-local propagator and show that it is finite at r → 0, eliminating the classical singularity. We compute the one-loop beta function for the Newton constant and demonstrate the existence of a non-trivial UV fixed point (asymptotic safety).We identify the Higgs field as the A1-component of the superconnection, with its mass fixed by the topology of the torus, resolving the hierarchy problem. Finally, we analyze the modified Schwarzschild solution and show that black hole evaporation leavesa stable Planck-scale remnant topologically equivalent to T 2 θ , preserving information and resolving the information paradox.