BIG-B4.1: Universal Quadratic Boundary Layers and Boundary-Layer Collapse in Degenerate Mixed-Gradient Fields

This report presents a reproducible numerical investigation of boundary-layer structures in a degenerate mixed-gradient dissipative field model developed within the Boundary Information Geometry (BIG) framework.

The study examines stationary solutions of

∂tφ = ∇·(φ²∇φ) − μφ − γ∇·(|∇φ|²∇φ) + S(x),

with fixed quartic-gradient stiffness γ = 1 and varying dissipation parameter μ.

A shell-based local asymptotic fitting procedure is used to analyze free-boundary behavior. The local boundary profile is fitted by

φ(s) ~ A s^ν,

allowing extraction of the boundary exponent ν together with layer-quality diagnostics.

Three numerical stages were performed:

• Reproducibility audit
• Extended μ scan
• Collapse-region investigation

The principal observation is the persistence of a quadratic boundary exponent

ν ≈ 2

throughout the interval

0.1 ≤ μ ≤ 1.1.

While the exponent remains remarkably stable, the coherent quadratic-layer width decreases as dissipation increases. Near μ ≈ 1, coherent quadratic layers disappear according to amplitude-consistency criteria, although local quadratic traces remain detectable.

The results suggest a distinction between boundary geometry and boundary-layer state. The quadratic exponent appears to represent a robust local geometric feature, whereas layer width and amplitude consistency behave as state-dependent quantities.

This work provides a numerical foundation for future two-parameter investigations in (μ, γ)-space and subsequent studies within the Boundary Information Geometry (BIG) program.