Singular-limit conjecture for adaptive weight dynamics

Let the assumptions of Theorem~ hold. For $\epsilon>0$, let $(\bm{X}^{(\epsilon)}(t),\bm{w}^{(\epsilon)}(t))$ solve, and let $(\bm{X}^{(0)}(t))$ solve the limiting system with the same initial data. Define $\bm{w}^{(0)}_{ij}(t)\vcentcolon=\beta(\bm{X}^{(0)}_i(t),\bm{X}^{(0)}_j(t))$. Singular-limit conjecture for adaptive weight dynamics. There exists $C(t)>0$, independent of $\epsilon$ and dependent on $t$ and the bounds in Assumptions~ and, such that

$$\frac{1}{N^2}\|\bm{w}^{(\epsilon)}(t)-\bm{w}^{(0)}(t)\|_{\ell^1([N]\times[N])}+\frac{1}{N}\|\bm{X}^{(\epsilon)}(t)-\bm{X}^{(0)}(t)\|_{\ell^2([N])}\leq\epsilon C(t).$$

This formalizes the expectation that rapidly relaxing weights become instantaneously determined by the states, reducing the coupled system to the limiting state equation.