The Skyrme Model as a Controlled Obstruction Test Topological Invariance and Inequivalent Auxiliary-Metric Response of Static Energy and Translational Inertia Paper I of the VFT Admissibility Programme

We use the Skyrme SU(2) field U : Σ → SU(2) as a controlled obstruction test for the admissibility programme of the Verarmungsfeldtheorie (VFT). On a Riemannian auxiliary three-manifold (Σ,hij ,dµh) we compute three observables: the topological charge BU, the static energy EhU, and the translational moduli inertia G(h) XX U. We probe these under conformal auxiliary-metric variation hij →Φ(r)2δij , for two profile stages: Stage A (analytic hedgehog f(r) = 2 arctan(R2/r2), R = 1) and Stage B (numerical flat-space minimizer obtained from the Skyrme BVP). Four findings are reported. (i) The topological charge is preserved to numerical precision across every Φ tested under both stages: |BA−1|≤3×10−9 and |BB−1|≤5×10−11. (ii) At the flat-space calibration point Φ ≡1, the corrected sectoral relation G(h) 2 XX = 3 E2 + 4 3 E4, combined with Derrick balance E2 ≃E4 at the minimizer, yields GXX,flat/Eflat = 1.0000446 in Stage B: the static energy and the translational inertia are numerically calibrated to coincide at the 5 ×10−5 level. (iii)Undera30-point(a,σ) gridofradialconformalperturbationsΦ(r) = 1+aexp(−(r/σ)2), the mean absolute obstruction contrast ⟨|∆G/G−∆E/E|⟩is 0.2792 for Stage A and 0.3037 for Stage B; the obstruction strengthens by 8.8% on average upon replacement of the analytic profile by the numerical minimizer, with full sign stability sign(C) = sign(a) on 30/30 points in both stages. (iv) The scaling exponents E2 ∼Φ, E4 ∼Φ−1 , G2 ∼Φ3 , G4 ∼Φ are verified exactly by global rescaling tests. We classify the result as Outcome A: structured scaffolding dependence within the tested conformally-flat radial auxiliary class. Skyrme realises the topological content of Axiom II, but its static energy and translational inertia remain auxiliary-geometry functionals and do not provide the auxiliary-independent inertia that Axiom IV requires.