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

Manuscript StructureRegularizing Relativistic Singularities through a C -Smooth Extension of the d'AlembertInvariance: Resolving the Hubble Tension and the Non-Zero Photon Mass1. Introduction2. Baseline Mathematics of the Kinematic Boost 2.1 The Lorentz Function Family: Denition and Kinematic Limit 2.2 The Lorentz Function Family: Intrinsic Geometric Properties3. Regularized Relativistic Kinematics 3.1 Asymptotic Invariance of the d'Alembert Operator 3.2 The Deviation Factor D(β): Higher-Order Regularization 3.3 Preservation of the Causal Light-Cone Structure 3.4 Inheritance of Classical Relativity 3.4.1 Relativistic Velocity Composition: The Illusion of Group Closure 3.4.2 Hierarchy of Relativities: Truncation of Kinematic Information4. ALR Dynamics at Light Barrier 4.1 Kinematic Tunneling: Transition at the Light-Barrier 4.2 The Superluminal Regime: Four Stable States 4.3 Convergence of Trajectories and Finite Temporal Flow 4.4 The Architecture of Finite Energy States 4.4.1 Bounded Mass, Energy and Momentum at the β = 1 Limit 4.4.2 Preservation of the Finite Length Contraction 4.5 ALR-Upgrade of Lemaître-Einstein Law5. Application 1: Resolving the Hubble Tension 5.1 The Kinematic Gap: Signicance of Truncated Energy in the Limit Regime 5.2 Quantifying the Hubble Matching Scale (HMS): The Kinematic Breaking Point 5.3 Corresponding Distances: The Hubble-Lemaître Relation at the HMS 5.4 The Hubble Tension in a Nutshell: No Physical Discrepancies 5.5 The Invariance Boundary: Regime Bifurcation at the HMS6. Application 2: The Non-Zero Photon Mass 6.1 Beyond the 0 · ∞ Ambiguity 6.2 Numerical Estimation of Dynamic Photon Mass Benchmark 6.3 The Footnote Scale of Photon Mass 6.4 Photon in a Nutshell:Comparisson with Electron and Neutrino 6.5 Consequences for the Fine-Structure Constant α 6.6 The Footnote-Scale Photon Mass in the LQG Big Bounce7.