High-Precision Gravitational Calibration and Nonlinear Trace-Squared Deformation of Einstein–Skyrme Solitons

We construct static, spherically symmetric, asymptotically flat, unit-winding backgrounds in a two-coupling Einstein–Skyrme system used as the Phase III gravitational benchmark of the Adaptive Manifold Framework (AMF). The undeformed face reproduces the standard Einstein-Skyrme lower–fold–upper geometry at $\alpha_{G,\text{fold}} = 0.1615141012$, with conservative uncertainty $10^{-9}$; a separate 50-decimal Richardson calculation agrees with the binary64 estimator to $5.2 \times 10^{-13}$, and the rescaled upper family approaches the lowest Bartnik–McKinnon control. This is a calibration of the normalization, topology, boundary conditions, and continuation machinery.

For the established squared-trace ($V_2$, or loosely-bound) Skyrme potential, $\bar{V} = \bar{\lambda}(1-\cos F)^2/4$, we then map the inherited static spherical $B = 1$ fold through $\bar{\lambda} = 0.1$. The regular-origin expansion contains genuine $\alpha_G \bar{\lambda}$ and $\alpha_G^2 \bar{\lambda}^2$ terms, so it cannot be obtained by adding gravity-only and deformation-only series. On the component inherited from the undeformed calibration, the fold moves to $\alpha_{G,\text{fold}}(0.1) = 0.14739780$, with a sampled-curve uncertainty of $2 \times 10^{-9}$. A dedicated differential audit gives:

d(α_G,fold) / d(λ_bar) |(0+) = -0.175858 ± 0.000012 d²(α_G,fold) / d(λ_bar)² |(0+) = 0.94 ± 0.04

The positive curvature and the weakening negative slope establish what we call a nonlinear braking response on the sampled interval: this is an author-defined descriptor of a static fold geometry, not a dynamical braking mechanism. A literature and priority audit current through 13 September 2026 found no direct prior report of this fold curve or its one-sided susceptibility and curvature for the specified system.

The two-parameter closure contains 44 sampled $(\alpha_G, \bar{\lambda})$ nodes and 88 lower/upper-candidate backgrounds. Expanded DOP853 shooting and independently coded conservative-flux collocation pass at all 88 backgrounds; all 44 cell boundaries satisfy topology, constraint, horizon, fold-margin, and branch-separation gates. Each accepted background retains a parent link and complete field snapshot, restart identity is bitwise exact in the frozen environment, and the released genealogy is content-addressed by SHA-256. The result is a reproducible static-background response study, not a perturbative-stability, dynamical-collapse, astrophysical, global-component-count, or completed-unification claim.