Prove that for every \(\varepsilon>0\) there exist explicit constants \(C_\varepsilon,A_\varepsilon>0\) and an explicit integer-valued function \(J(x)\) satisfying \[ J(x)\le C_\varepsilon(\log x)^{A_\varepsilon} \] such that, for all sufficiently large \(x\), \[ \sum_{n\in\mathcal S_x}|r_{J(x)}(n)-1| \le C_\varepsilon x^{1/2+\varepsilon}. \] The proof must derive this bound from the local arithmetic geometry of \(G_x\) and the stated balancing dynamics, without using estimates for the Mertens function \(M(x)\), locations of zeros of \(\zeta\), or a separately optimized global matching.

Since \(K_j\) is symmetric and every edge joins opposite signs, \[ \sum_{n\in\mathcal S_x^+}r_j(n)=\sum_{n\in\mathcal S_x^-}r_j(n). \] Hence \[ |M(x)|\le \sum_{n\in\mathcal S_x}|1-r_j(n)|. \] Therefore the target estimate implies \[ M(x)=O_\varepsilon(x^{1/2+\varepsilon}), \] and hence the Riemann Hypothesis.

As an intermediate structural target, prove or refute that \(G_x\) has a matching saturating the smaller of \(\mathcal S_x^+\) and \(\mathcal S_x^-\) for every \(x\), equivalently that its matching defect is exactly \(|M(x)|\). More generally, determine the least fixed odd exchange radius \(r\) for which an analogous local balancing mechanism can achieve square-root-scale defect.