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

We construct smooth finite-energy global solutions of the 2-equivariant harmonic-map heat flow in the radial endpoint sector (0,2) that converge in the reduced energy topology to a concentric equal-sign two-bubble configuration with separated scales. For sufficiently small ε > 0, the canonical scales satisfy λ(0) = ε and μ(0) = 1. The outer scale converges to a finite nonzero limit μ∞,ε, with μ∞,ε = 1 + o(1) as ε → 0, while for each fixed ε the inner scale obeys lim_{t→∞} (1/t) log(λ(t)/λ(0)) = -16 / (π μ∞,ε²). The proof is fully forward: persistent two-bubble purity is not assumed. The construction is based on quantitative Bogomolny trapping, a gauge-restored elliptic interaction corrector, canonical modulation, adaptive two-chart spacetime coercivity, a finite-horizon inner current, and a renormalized outer Hasimoto normal form. Global existence is established before invoking the equivariant bubble-decomposition theorem. The latter is used only in the endgame to determine the final bubble count and signs and to identify the asymptotic canonical scales. This upload is the final full archival version of the manuscript. It contains the complete proofs, technical estimates, normalization checks, quantitative error bookkeeping, proof-dependency maps, and appendices used in the construction. This version is intended to serve as the canonical mathematical master for any later journal submission or shortened presentation.