GALTTI: A Lie-Algebroid Framework for Gravity and U(1) Gauge Fields with Spectral Control of the One-Loop Determinant and Algebroid Spectral Quantization

We develop a four-dimensional geometric framework, called GALTTI, for gravity and an abelian gauge eld within a single Lie-algebroid setting. The construction does not introduce extra dimensions. Instead, it encodes the coupling between the Lorentz sector and the abelian sector through a global geometric background built into the algebroid extension itself. At the classical level, the theory is formulated in rst-order form and reduces to EinsteinMaxwell theory with a topological ux shift. At the semiclassical level, the gauge- xed uctuation operator belongs to the class of Laplace-type operators on the physical sector. This makes it possible to control the one-loop contribution by means of zeta-regularised determinants, heatkernel methods, and explicit bounded-geometry estimates. A global heat-trace bound links the short-time asymptotic expansion to the long-time spectral decay and yields a concrete volume-and-curvature estimate with an exponentially suppressed tail. The paper is written in a modular but self-contained way. It presents the kinematical construction, the full rst-order variational derivation of the eld equations, the gauge- xed semiclassical operator, a non-vacuous model background for the spectral hypotheses, and the formal quantum extension based on admissible algebroid splittings. The main claim is not a non-perturbative completion of quantum gravity, but a mathematically explicit and auditable semiclassical framework in which the one-loop sector is under analytic control.