Scalar-Sign Rigidity and Topological Bubble Counting for the 2-Equivariant Harmonic-Map Heat Flow

We study global finite-energy 2-equivariant harmonic-map heat flow from R2\mathbb{R}^2 to S2\mathbb{S}^2. Starting from the continuous asymptotic multi-bubble decomposition established by Jendrej and Lawrie, we develop a single-boundary, single-scaling-mode rigidity argument for the scalar signs of adjacent bubbles. The analysis combines an explicit nearest-neighbor interaction force, exterior and weighted active-core first-order coercivity, and a zero-moment critical bridge designed to preserve the leading sign information through the radiation dynamics. For every adjacent pair we derive a renormalized boundary law whose leading term is proportional to the scalar-sign product. The opposite-sign branch is incompatible with the resulting long-time clock growth. Consequently, all bubbles in the asymptotic decomposition have the same scalar sign. Combined with the endpoint topological identity, this yields the bubble-counting relation N=∣m−ℓ∣N = |m-\ell|. The result provides a rigidity mechanism for the critical 2-equivariant regime and constrains the multi-bubble configurations compatible with global finite-energy harmonic-map heat flow. This record contains the final preprint manuscript (v0.30).