The Möbius-Kerr-Bautista Topological Framework: An Optical-Geometric Testbed for Cosmological Tensions

Context. The standard cosmological model (ΛCDM) requires the introduction of dark energy and cold dark matter to reconcile theory with observations. The persistent Hubble Tension at cosmic scales and the missing mass problem in galactic rotation curves are traditionally treated as distinct phenomena requiring independent theoretical patches. Aims. We explore a single geometric testbed, formulated within General Relativity, for connecting cosmological distance anomalies and galactic rotation phenomenology. By substituting the flat FLRW background with an inverted topology enveloped by an exponential conformal factor Ω(r) = e^−γr, we test whether the same conformal topology can reproduce luminosity-distance anomalies and galactic rotation signatures without introducing an explicit dark sector. Methods. In Block I, we review the empirical LOG²-Decay luminosity-distance correction, which statistically alleviates the Hubble Tension. In Block II, we construct the first-principles geometric bridge, evaluating the Raychaudhuri expansion scalar for null congruences to identify the topological optical shear. In Block III, we introduce baryonic mass into this topological background, utilizing Mach’s Principle, Lense-Thirring gravitomagnetic perturbation, and scalar-tensor trace coupling to derive MOND/Tully-Fisher-like empirical scalings. Results. The topological shear acts as a 4D geometric lens, reproducing the empirical LOG²-Decay signal. The global optimization reveals a “transparent” superspinning topology (a > m), removing event horizons and allowing undisturbed CMB photon propagation. Furthermore, the induced local frame-dragging by baryonic halos recovers the local rotational coupling κ ∝ M^α. The empirically observed α ≈ 0.68 originates from the scalar-tensor screening of the conformal background by the trace of the baryonic energy-momentum tensor. In the current joint run, the engine finds H₀ ≃ 73 km s⁻¹ Mpc⁻¹, Ωₘ ≃ 0.36 with a passive CMB prediction R ≃ 1.75 compared with R_Planck = 1.7502 ± 0.0046. The symbolic-to-numerical engine compiles the optical and orbital scalars into reusable NumPy callables and returns a complete forward model for luminosity distances, the compressed CMB shift parameter, and local rotation curves. We therefore present the framework as an auditable first-principles candidate for reinterpreting part of dark-sector phenomenology through conformal optical and gravitomagnetic geometry.