ON THE BETA MODEL: A FOURTH-ORDER NONLINEAR REGULARIZATION OF MAGNETOHYDRODYNAMICS WITH COSMOLOGICAL IMPLICATIONS, TACHYONIC INTERPRETATION AND LINK TO THE PLANCK SCALE

We introduce and study the beta-MHD model, defined by the fourth-order pressure-gradient operator Δ̂_β := −∇(β |∇Φ|⁴), where beta-MHD stands for beta MagnetoHydroDynamics and β is the unique free parameter of the theory with dimension [J·m] = [kg·m³·s⁻²] = [6.24 × 10²³ ħc]. We prove global existence and uniqueness of weak solutions as well as local existence of strong solutions for the regularized beta-MHD system coupled to the full Maxwell equations on C³-boundary domains or on the torus 𝕋³, establish causality, stability and dimensional consistency, and derive explicitly the fourth-order energy-momentum tensor. We establish a phenomenological estimate of β from a Riccati-type blow-up criterion for Navier-Stokes with explicit universal constants valid for generic data. This criterion provides a robust lower bound on inertial nonlinearity and yields a universal critical scale. We prove the effective existence of β and show its importance as a bridge between macroscopic plasma physics and the Planck scale. We demonstrate that β can be interpreted as the effective tension of a classical tachyonic-type scalar field, whose large gradient generates negative pressure compatible with accelerated expansion. We examine how the β term accounts for an effective contribution to the accelerated expansion of the universe at large scales and to an initial high-energy phase capable of generating N ≥ 60 e-folds, while remaining compatible with Einstein’s field equations of General Relativity and the finite bound of the speed of light. We extend the framework to compressible barotropic plasmas using a density-square type entropy estimate, propose stable numerical schemes in confined geometries with testable predictions in liquid-metal MagnetoHydroDynamics and plasma confinement, and discuss observational signatures in Cosmic Microwave Background anisotropies and large-scale structure growth. We discuss β as an effective coefficient acting only at very large gradients, the unique free parameter of the theory at dominant nonlinear order, and analyze the impact in PDE Analysis, Korteweg-type Plasma Physics, Effective Field Theory, General Relativity and Cosmology. The mathematical framework is set in the Sobolev space W¹⁵(Ω), with continuous embedding W¹⁵(Ω) ↪️ L∞(Ω) in three dimensions, and duality in W⁻¹⁵/₄(Ω). We identify two asymptotic regimes: the "MHD Regime" at low gradient and the "beta-dominated Regime" at high gradient. We explicitly analyze the regularization limit ν₄ → 0, η → 0 with ν₄/η bounded and show convergence to the ideal beta-MHD system.