Gravitating Skyrme–Dilaton Solitons

Abstract We organize the Einstein–Skyrme–Dilaton system presented in Addendum IV into two distinct boundary-value problems: globally regular solitons and static black holes carrying Skyrme–Dilaton hair. The Einstein equations and radial matter equations are restated with the parent normalization, and the black-hole horizon constraints follow algebraically from regularity. We also isolate the effect of a prescribed cosmological constant. For Λ = 0 we solve the coupled system numerically, first for horizonless solitons at weak gravitational coupling (Section 8) and then, in Stage 3 (Section 11), for hairy black holes. At G = 0.001, 0.002 and 0.004 the black-hole solutions form a one-parameter family that turns back at a maximal horizon radius, satisfies the first law dM = T dS to within 3×10⁻⁴ (tested along the first branch at G = 0.002), and has lower entropy than a Schwarzschild black hole of equal mass. All results are restricted to static spherical symmetry and Λ = 0; stability, uniqueness, Λ > 0 and an emergent metric remain open. An a = 0 control (Addendum A) shows that at G = 0.002 the dilaton coupling changes the hairy branch only slightly: ADM masses by below 0.25% and the maximal horizon radius by 0.4%.