he manuscript introduces the Asymptotic Lorentz Relativity (ALR) framework. The core of this work is to present a new, C-infinity smooth regularized relativistic kinematics that remains valid even at the light-speed limit (v -> c). While this framework approximates Einsteinian relativity up to order O(beta^4) in the intermediate velocity regime, it substitutes the standard singular behavior with a regularized description as beta approaches 1, where classical theory ceases to be valid. Key scientific highlights of this manuscript include: First-Principles Resolution of the Hubble Tension: A geometric reconciliation of the 5-sigma Hubble tension between Planck and SH0ES data by identifying the Hubble Matching Scale (HMS) at beta_HMS≈0.527 (z_HMS≈ 0.726). d'Alembert Form Invariance: Demonstrates full asymptotic invariance of the differential wave operator across all velocity regimes, eliminating unphysical coordinate and boost divergences without modifying the standard Einsteinian domain at intermediate velocities (beta 0. Applications to Foundational Field Theories & Quantum Cosmology: Provides the mathematical basis for an analytically continuous transition at boundary limits, offering a non-singular background compatible with Big Bounce models (Loop Quantum Gravity) and finite vacuum impedance structures (aligning with research on electromagnetic field bounds by Prof. Gerd Leuchs, FAU Erlangen).

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. First-Principle Application: 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. Conclusion and Outlook

Update Note (July 2026): Version 2.0: Comprehensive structural update. This version streamlines the core manuscript to focus exclusively on the foundational mathematical physics, C-infinity smooth d'Alembert form-invariance, and the first-principles resolution of the Hubble Tension. Secondary phenomenological applications (such as photon mass heuristics) have been removed to provide the most concise and rigorous formulation of the ALR framework, and are planned for a dedicated future publication.Note: For the full extended version including Section 6 on the photon mass derivation and particle mechanics comparison, please refer to Version 5 of this repository : https://doi.org/10.5281/zenodo.20348597