Boundary Asymptotics and Quadratic Landing in Degenerate Mixed-Gradient Dissipative Fields

This technical note investigates the asymptotic boundary structure of a degenerate mixed-gradient dissipative partial differential equation associated with the Boundary Information Geometry (BIG) framework. The study focuses on the emergence of compact-support-like localized steady states generated by the interplay of: degenerate field-dependent transport, quartic gradient flux terms, linear dissipation. High-resolution numerical simulations were performed in spherical symmetry using finite-volume discretization and optimized C-accelerated computation. The analysis includes: threshold-radius saturation tests, grid convergence studies, Richardson extrapolation, asymptotic boundary fitting, amplitude-law verification. The numerical results consistently indicate a quadratic boundary landing of the form Φ(r)∼A(Rc−r)2,\Phi(r) \sim A(R_c-r)^2,Φ(r)∼A(Rc−r)2, which is independently derived through a dominant-balance asymptotic analysis. The predicted amplitude scaling A = sqrt(mu / (96 gamma))=\sqrt{\mu/(96\gamma)}A=μ/(96γ) shows strong agreement with numerical measurements for moderate stiffness parameters. This release should be regarded as a technical preprint and numerical research note rather than a finalized mathematical proof. In particular, rigorous existence theory, exact compact-support proofs, multidimensional stability, and the exclusion of ultra-weak exponential tails remain open problems. Nevertheless, the combined asymptotic and numerical evidence suggests that the BIG equation may belong to a distinct class of degenerate dissipative systems exhibiting smooth free-boundary-like structures. The present upload intentionally restricts itself to mathematical and numerical aspects. Broader philosophical interpretations and speculative physical extensions are considered outside the scope of this technical note, though possible connections to dissipative localization phenomena in other systems are briefly noted as future directions. This work is intended as an open technical research record and may evolve in future revisions.