Regularizing Relativistic Singularities through a C ∞ -Smooth Extension of the d'Alembert Invariance: Resolving the Hubble Tension and the Non-Zero Photon Mass

Manuscript Structure 1. Introduction 2. Baseline Mathematics of the Kinematic Boost2.1 The Lorentz Function Family: Denition and Kinematic Limit2.2 The Lorentz Function Family: Intrinsic Geometric Properties 3. Regularized Relativistic Kinematics3.1 Asymptotic Invariance of the d'Alembert Operator3.2 The Deviation Factor D(β): Higher-Order Regularization3.3 Preservation of the Causal Light-Cone Structure3.4 Inheritance of Classical Relativity3.4.1 Relativistic Velocity Composition: The Illusion of Group Closure3.4.2 Hierarchy of Relativities: Truncation of Kinematic Information 4. ALR Dynamics at Light Barrier4.1 Kinematic Tunneling: Transition at the Light-Barrier4.2 The Superluminal Regime: Four Stable States4.3 Convergence of Trajectories and Finite Temporal Flow4.4 The Architecture of Finite Energy States4.4.1 Bounded Mass, Energy and Momentum at the β = 1 Limit4.4.2 Preservation of the Finite Length Contraction4.5 ALR-Upgrade of Lemaître-Einstein Law 5. Application 1: Resolving the Hubble Tension5.1 The Kinematic Gap: Signicance of Truncated Energy in the Limit Regime5.2 Quantifying the Hubble Matching Scale (HMS): The Kinematic Breaking Point5.3 Corresponding Distances: The Hubble-Lemaître Relation at the HMS5.4 The Hubble Tension in a Nutshell: No Physical Discrepancies5.5 The Invariance Boundary: Regime Bifurcation at the HMS 6. Application 2: The Non-Zero Photon Mass6.1 Beyond the 0 · ∞ Ambiguity6.2 Numerical Estimation of Dynamic Photon Mass Benchmark6.3 The Footnote Scale of Photon Mass6.4 Photon in a Nutshell:Comparisson with Electron and Neutrino6.5 Consequences for the Fine-Structure Constant α6.6 The Footnote-Scale Photon Mass in the LQG Big Bounce 7. Conclusion and Outlook