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 \\frac{\\lambda(t)}{\\lambda(T)} = M_{\\rm rad}^{\\rm cl}(T,t) - \\frac{16\\sigma}{\\pi}\\Theta(T,t) + o(\\Theta(T,t)), where Θ(T,t)=∫Ttμ(s)−2 ds.\\Theta(T,t)=\\int_T^t \\mu(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/\\pi. 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.\\Theta_2(T,t) = \\int_T^t \\frac{\\lambda(s)^2}{\\mu(s)^4} \\log\\frac{\\mu(s)}{\\lambda(s)} \\,ds. The 128/π128/\\pi term is not claimed to define a new asymptotic growth slope, and the present result does not assert an o(Θ2)o(\\Theta_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.