PERTURBATIVE CONTROL OF DIRAC EIGENVALUES IN NON-UNITAL SPECTRAL TRIPLES: ENGINEERING FINITE-RESOLUTION TOPOLOGICAL INVARIANTS WITHIN PLANCK-SCALE C*-ALGEBRAIC LATTICES VIA CYCLIC COHOMOLOGY DESCENTS

Mathematical Applications of Science Fiction Abstract. We present a rigorous constructive framework for the perturbative control of the spectrumof Dirac operators associated with non-unital spectral triples (A, H, D). By embedding the non-compactmanifold data into a sequence of finite-dimensional C*-algebraic lattices (Planck-scale approximations),we derive explicit bounds on the eigenvalue shifts under inner fluctuations of the metric. We intro-duce the Selene-Dirac Perturbation Series, a convergent expansion in the Banach space of boundedoperators, to engineer specific spectral gaps. Furthermore, we establish a descent mechanism in cycliccohomology, proving that the Chern-Connes character of the perturbed lattice system converges to thetopological invariants of the continuum limit in the weak-∗ topology. This provides a mechanism for"finite-resolution" topological protection, with direct applications to error correction in quantum gravitymodels and topological quantum computing.