Singular Tracking of Fractional Brownian Motion under Proportional Costs

We study adapted finite-variation tracking of fractional Brownian motion under quadratic loss and proportional adjustment costs. For $H\\in(0,1)$, \\[V_\\lambda(T;a,b,\\sigma)\\underset{\\lambda\\downarrow0}{\\sim}\\kappa_HT a^{(1-H)/(1+H)}(\\lambda b)^{2H/(1+H)}\\sigma^{2/(1+H)},\\qquad\\lambda\\downarrow0.\\]The positive long-run cell value $\\kappa_H$ is unchanged by full past information. A bounded full-past reset correction yields finite-horizon bounds and continuity in $H$; $\\kappa_{1/2}=\\frac12(\\frac32)^{2/3}$. A coupling of local paths and filtrations gives the corresponding local limit, verified for Gaussian Volterra targets. Every asymptotically optimal control has the same leading tracking--variation cost split. Primal and dual bounds and a constructive positive-factor approximation for $H<1/2$ quantify the cell value, with an explicit relation between factor resolution and horizon length.