Topological Model of Spatial Connectivity: Geometric Anisotropy from Gik and Cosmological Consequences of Variable ceff

Since the early twentieth century, the quest to describe all fundamental interactions as manifestations of a single geometric structure of spacetime has remained central to theoretical physics. Einstein’s unified–field program attempted to generalize gravity by including electromagnetism on a geometric basis, and later developments—from Kaluza–Klein models to modern string theory and loop quantum gravity—have continued this line of thought. Yet the reconciliation of general relativity (GR) and quantum mechanics (QM) is still incomplete: GR captures macroscopic curvature but neglects quantum discreteness, while QM relies on a fixed background. Here we propose a minimal geometric framework in which space is characterized by local topological connectivity defined by a universal constant ℓ∗, the fundamental connectivity length. This constant represents the smallest invariant separation between neighboring points of space and serves as the microscopic carrier of geometric information. Local anisotropies of the spatial metric Gik stretch or compress this invariant unit, giving rise to an effective Planck length \ (L₏ (x, n) =*G₈₊ (ₗ) \, ₍^₈ n^{k} \), and to the corresponding effective speed of light \ (c₄₅₅ (x) = (GL₏ (x) ^{2}) ^1/3 \). In the isotropic limit Gik→δik one recovers Lp = ℓ∗ = ℓp and ceff = c0, ensuring full compatibility with local Lorentz invariance. Variations of Gik (t) then induce the apparent evolution of Lp (t) and ceff (t), producing a natural early–universe regime where Lp → 0 and ceff → ∞. This mechanism removes the cosmological horizon problem without requiring an inflationary potential and regularizes curvature singularities by introducing a finite topological cutoff ℓ∗. The construction unifies the geometric content of electromagnetism and gravitation through the symmetric and antisymmetric parts of the local tensor Gik, reproducing the Einstein–Maxwell equations in the quasi–isotropic limit. It further allows a topological interpretation of elementary particles as stable connectivity defects (torus for the electron, trefoil for the proton, balanced knot for the neutron). Observable consequences include direction–dependent gravitational lensing, polarization–dependent propagation of gravitational waves, and potential anisotropic signatures in the cosmic microwave background and the stochastic gravitational–wave background. The proposed framework preserves mathematical minimalism—a single additional constant ℓ∗—while yielding falsifiable predictions that can be tested with current and forthcoming observational data.