Topological Coherence Theory: A Scale-Invariant Framework for Structural Integrity and Galactic Dynamics

Abstract Standard cosmological models ($\Lambda$CDM) rely on non-baryonic Dark Matter to account for gravitational discrepancies across galactic scales. This paper introduces Topological Coherence Theory (TCT), a relational framework that redefines structural existence as a dynamic state of phase-locked coherence ($\Xi$) with the vacuum substrate. By establishing a universal scaling law between stochastic tolerance ($w^2$) and algebraic connectivity ($\lambda_2$) through the Basel Resonance ($k = \pi^2/12$), we derive a master velocity equation anchored to the Fine Structure Constant ($\alpha_{fs}$) and the Spherical Flux Dilution of the vacuum manifold ($(4\pi)^2$). Key Findings Galactic Dynamics: TCT recovers the Baryonic Tully-Fisher Relationship ($M \propto v^4$) with an emergent slope of 4.05, resolving galactic rotation curves without Dark Matter. Scale Invariance: The theory demonstrates that vacuum impedance is scale-invariant, accurately predicting the Chandrasekhar Limit ($1.36 M_\odot$), the TOV Limit for neutron stars ($2.19 M_\odot$), and the Proton-to-Electron mass ratio ($1833.5$) from first principles. Inertial Exemption: We propose a modification of inertial mass governed by topological connectivity, preserving Galilean Invariance for isolated bodies while recovering network effects in galactic disks. Keywords: Modified Gravity, Dark Matter Alternatives, Galactic Rotation Curves, Baryonic Tully-Fisher Relation, Information Theory in Physics