Radiation Memory, Nearest-Neighbor Drift, and a Refined Logarithmic Interaction Law in a Persistent Two-Bubble Regime of the 2-Equivariant Harmonic-Map Heat Flow

We study the fine asymptotic dynamics of a persistent pure exactly-two-bubble regime for the 2-equivariant harmonic-map heat flow. Under this conditional two-bubble hypothesis, we separate the motion of the inner scale into a deterministic nearest-neighbor interaction and a critical radiation-memory contribution. The leading interaction law is log⁡λ(t)λ(T)=Mradcl(T,t)−16σπΘ(T,t)+o(Θ(T,t)),log λ(t)/λ(T) = Mradᶜˡ(T,t) - 16σ/πΘ(T,t) + o(Θ(T,t)), where Θ(T,t)=∫Ttμ(s)−2 ds.Θ(T,t)=∫_T^t μ(s)⁻²\,ds. The radiation memory is formulated through a renormalized storage–flux balance along admissible moving contours and is further related to a free order-one heat evolution of a profile-subtracted Hasimoto variable. We also identify a universal local logarithmic-neck resonance coefficient 128/π128/π. After matching across the neck region, this produces a finite accumulated second-order correction associated with Θ2(T,t)=∫Ttλ(s)2μ(s)4log⁡μ(s)λ(s) ds.Θ_2(T,t) = ∫_T^t λ(s)^2/μ(s)^4 logμ(s)/λ(s) \,ds. The 128/π128/π term is not claimed to define a new asymptotic growth slope, and the present result does not assert an o(Θ2)o(Θ_2) relative second-order asymptotic expansion. The paper is conditional on the existence of the persistent pure exactly-two-bubble regime; the construction of such a regime is the subject of Paper I. This is Paper III of a three-paper series on the borderline 2-equivariant harmonic-map heat flow.