Scalar-Sign Rigidity and Bubble-Number Selection for the 2-Equivariant Harmonic-Map Heat Flow

We study global finite-energy solutions of the 2-equivariant harmonic-map heat flow through their continuous asymptotic multi-bubble decomposition. Starting from the Jendrej–Lawrie decomposition, we prove a scalar-sign rigidity theorem: every adjacent bubble has the same scalar sign. As a consequence, all bubbles share a common scalar sign and the asymptotic bubble number is uniquely determined by the endpoint sector, N=∣m−ℓ∣.N = |m-\\ell|. The proof uses a single-boundary induction based on the exact nearest-neighbor interaction, localized first-order coercivity, an explicit zero-moment bridge, and a renormalized current argument. This result is distinct from the specific persistent two-bubble construction of Paper I: it applies to a general global asymptotic multi-bubble decomposition and establishes a structural rigidity and bubble-number selection law for the borderline 2-equivariant harmonic-map heat flow. This is Paper II of a three-paper series on the 2-equivariant harmonic-map heat flow.