Sharp Hölder Stability of Truncated Power-Sum Fibers at Diagonal Collision Points

Let$$\\Phi_J(x,y)=\\left(\\sum_{i=1}^M x_i^k-\\sum_{i=1}^M y_i^k\\right)_{k=1}^J$$be the equal-weight truncated power-sum relation on ordered real configurations. At a fixed interior diagonal point $(r^0,r^0)$, let $\\mathbf{m}=(m_1,\\ldots,m_K)$ be the multiplicity profile of the distinct coordinates of $r^0$, and define$$q_J(\\mathbf{m})=\\min\\left\\{q\\ge1:\\sum_{a=1}^K\\min(m_a,q)\\ge J\\right\\}.$$We prove that the exact local Hölder exponent for distance to the zero fiber $\\Phi_J^{-1}(0)$ is $1/q_J(\\mathbf{m})$. The upper bound is obtained by a common quantitative Nuij–Wakabayashi opening, an exact coefficient-tail filtration, a $q_J$-restricted Vandermonde inverse, and an exact target ODE. Sharpness follows from Hermite-jet rank minimality and a pure grade-$q_J$ power-sum witness. Constants are pointwise-local in the fixed collision point; no uniformity is asserted as distinct collision centers merge.