Sharp Feedback--Arity Frontiers for Periodic Quasi-Variational Inequalities

We locate sharp structural and approximation-complexity frontiers for periodic moving-box quasi-variational inequalities (QVIs) encoded by rational piecewise-linear response circuits. If $r$ is maximum response arity and $\tau$ is directed feedback vertex number, the normalization $K(x)=h(x)+[0,1]^n$, $F(x)=x+\one$ makes the natural residual exactly $R(x)=x-h(x)$. Eliminating every acyclic shell yields a residual-isometric return map on any feedback core, implying the sharp bound $\dim\operatorname{SOL}(F,K)\leq\tau$. Exact search is polynomial-time when $\tau\leq1$ or $r\leq1$, whereas the first joint cell $\tau=r=2$ is $\PPAD$-complete. For every fixed rational $\varepsilon<(\sqrt2-1)/4$, hardness survives simultaneously with two phases, bipartiteness, no self-loops, and maximum degree three. Without a feedback bound, weak- and feasible-natural-residual search undergo exact radius transitions at $1/2$ and $1$, respectively. Finally, phase refinement obeys a sharp gain-weighted residual law, and a constraint-only generalized Nash realization preserves equilibria, residuals, graph parameters, and monodromy. Thus endogenous feasibility alone creates hard periodic equilibrium even when every frozen variational inequality is uniformly well conditioned.