Local Prime-Exchange Balancing and Möbius Cancellation

For $x\ge 1$, let

$$\mathcal S_x=\{n\le x:n\text{ is squarefree}\}.$$

For $n\in\mathcal S_x$, write $P(n)$ for its set of prime divisors, so

$$\mu(n)=(-1)^{|P(n)|}.$$

Let

$$\mathcal S_x^+=\{n\in\mathcal S_x:\mu(n)=+1\},\qquad \mathcal S_x^-=\{n\in\mathcal S_x:\mu(n)=-1\}.$$

Define a bipartite graph $G_x$ on $\mathcal S_x$ by joining $m$ and $n$ exactly when

$$|P(m)\triangle P(n)|\in\{1,3\}.$$

Thus every edge joins opposite Möbius signs. Set $K_0(m,n)=1$ on edges and $0$ otherwise. Given a nonnegative symmetric edge kernel $K_j$, define

$$r_j(n)=\sum_{m\sim n}K_j(n,m),$$

and, whenever all relevant row masses are positive, define

$$K_{j+1}(m,n)=\frac{K_j(m,n)}{\sqrt{r_j(m)r_j(n)}}$$

on edges.

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.