THE CHRONO-GEOMETRY OF THE β-MATERIAL: GLOBAL EXISTENCE, HYPERBOLICITY AND VISCOUS SATURATION FOR A CUTOFF QUARTIC OPERATOR

We study an incompressible MHD system coupled to a scalar field Φ through a degenerate fourth-order energy term F_β = (β/4)(∂μ Φ ∂^μ Φ)² χ_R with cutoff χ_R of order q=5. The operator Δ̂{β,R}(Φ) := -∇·( β |∇Φ|² ∇Φ χ_R ) is of p-Laplacian type with p=5. The single parameter β has dimension [J·m] = [kg·m³·s⁻²]. We prove: 1. sharp monotonicity (F(a)-F(b))·(a-b) ≥ ¼ |a-b|⁴ with optimal constant ¼, 2. reentrant corner analysis for α=π/15 giving Sobolev exponent λ=15/29=0.5172 and L⁴-integrability margin 0.0172, 3. uniform hyperbolicity N²=1-2ε ≥ ¾ for q=5 and failure for q=4, 4. viscous saturation analysis with 𝒜_crit=231 for ⁴He and 𝒜_crit=18 for GaInSn, 5. H-convergence bounds β_min/4 ≤ aʰᵒᵐ ≤ 27β_max/4 with Voigt-Reuss derivation and corrector |∇χ_δ| ≤ 4/δ, 6. global existence of weak solutions for fixed hyper-viscosity ν₄>0 via Galerkin, Aubin-Lions and Minty-Browder, 7. corrected Hartmann prediction δv/v =1.23×10⁻¹⁴ for β=0.127 J·m and detectable threshold β_det=2.88×10¹⁰ J·m for 0.28% flattening, τ_coh=0.061 s, ℱ=99.9669%, Q_ac=49.92. The limit ν₄→0⁺ remains open.