Formalizing the SV0 Action Principle and Deformed Lorentz-Invariant Quantum Geometry: A Rigorous Field-Theoretic Foundation (Preprint)

While the Signed-Volume (SV0) framework has successfully resolved numerous cosmological and particle physics anomalies, a fundamental critique remains: the phenomenological exchange law dx/dt = γ H x(1-x) requires a rigorous Action Principle, and the discrete "Space Packet" hypothesis must preserve Lorentz Invariance. In this paper, we address both challenges. First, we formulate the complete covariant action for the SV0 model using a non-canonical K-essence Lagrangian. We demonstrate mathematically that the Euler-Lagrange equations derived from this action naturally yield the logistic exchange law as an exact dynamical attractor in the slow-roll regime, elevating it from a postulate to a derived result. Second, we resolve the Lorentz violation problem in discrete spacetime by employing Deformed Special Relativity (DSR) and the κ-Poincaré algebra. We derive a modified dispersion relation, which predicts a tiny, testable time-delay in Gamma-Ray Bursts (GRBs). This work solidifies the SV0 framework as a mathematically rigorous, deformed Lorentz-invariant quantum field theory. References:[1] M. U. Sandaruwan, "Structurally Guaranteed Phantom Crossing in a Signed-Volume Effective Fluid..." (2026). Zenodo: https://doi.org/10.5281/zenodo.24026844[2] M. U. Sandaruwan, "Discrete Spacetime and Space Packets..." (2026). Zenodo: https://doi.org/10.5281/zenodo.24046352[3] M. U. Sandaruwan, "The SV0 Theory of Everything: A Comprehensive Review..." (2026). Zenodo: https://doi.org/10.5281/zenodo.24050022[4] G. Amelino-Camelia, "Testable scenario for relativity with minimum length", Phys. Lett. B 510, 255 (2001).[5] J. Kowalski-Glikman, "Introduction to doubly special relativity", Lect. Notes Phys. 669, 131 (2005).