An Unconditional Persistent Two-Bubble Solution for the 2-Equivariant Harmonic-Map Heat Flow

We construct a family of smooth finite-energy solutions to the 2-equivariant harmonic-map heat flow in the topological sector (0,2), exhibiting a persistent two-bubble regime without assuming such a regime a priori. For sufficiently small ε > 0, the initial data are given by an explicitly corrected two-bubble profile with canonical scales λ(0) = ε and μ(0) = 1. The resulting solution is global and admits the asymptotic decomposition u(t) = Q_lambda(t) + Q_mu(t) + g(t), with ||g(t)||_E → 0. The outer scale converges to a finite nonzero limit μ_infinity,ε, while the inner scale satisfieslim t→∞ [(1/t) log(λ(t)/λ(0))] = −16 / (π μ_infinity,ε²). The proof combines quantitative Bogomolny trapping, a gauge-restored first interaction corrector, adaptive two-chart coercivity, a finite-horizon inner current, a renormalized outer Hasimoto normal form, bootstrap closure, finite-time singularity exclusion, and continuous bubble decomposition. The manuscript has undergone extensive internal line-by-line verification; independent external peer review has not yet been completed.