Unified Einstein–Hilbert–Kayad Model: Topological Dynamics of the Laplacian L(96) and Dark-Energy Stabilization

Mohamed Nour kayad

This manuscript presents a theoretical and numerical framework that uniesthe micro-topological structure of a discrete 96-zone network, dened by a complex Laplacian operator L(96), with the macroscopic expansion dynamics of the Universe in an EinsteinHilbert extension. By replacing arbitrary scalar elds with a spectral renormalization ow controlled by the topological rigidity of the network, wederive a stable eective equation of state w(a) that remains compatible with con-temporary observational results from Planck, ACT, and DESI. The introduction of a topological cuto function Θkayad removes phantom-type pathologies and guarantees an asymptotic transition toward a topologically locked de Sitter attractor. Finally, we formalize the loss function Lkayad, coupled to an active digital twin for real-time gauge-parameter optimization, and prove the linear stability of the attractor through a dedicated theorem and a technical lemma establishing a spectral lower bound.