Radiation Memory and Refined Scale-Interaction Laws in a Persistent Two-Bubble Regime of the 2-Equivariant Harmonic-Map Heat Flow

This preprint studies the two-dimensional, 2-equivariant harmonic-map heat flow from R2R^2 to S2S^2 under a conditional persistent pure exactly-two-bubble regime with well-separated scales λ(t)≪μ(t)≪tλ(t)μ(t)√ t and a vanishing energy remainder. Within this regime, the analysis derives an intrinsic leading scale-interaction law for the inner bubble, consisting of a radiation-memory contribution and a deterministic nearest-neighbor drift with coefficient −16σ/π-16σ/π. It also establishes finiteness of the logarithmic second interaction clock, constructs a free critical-radiation representation along an admissible contour, and derives a refined second-order interaction law with canonical logarithmic-neck coefficient 128/π128/π. The results are conditional internal-dynamics statements. This work does not prove the existence of a persistent pure exactly-two-bubble solution, does not classify all 2-equivariant harmonic-map heat-flow dynamics, and does not claim an unconditional resolution of the full problem.