Tight Characterization of Instance-Optimal Identity Testing

Let $\Delta(X)$ be the distributions on a discrete domain $X$. For known $q\in\Delta(X)$ and $\varepsilon\in(0,1]$, the tester receives i.i.d. samples from $p\in\Delta(X)$ and must distinguish $p=q$ from $d_{\rm TV}(p,q)>\varepsilon$ with constant error probability.

1. Is there a succinct functional $\Phi\colon\Delta(X)\times(0,1]\to\mathbb N$ that characterizes the sample complexity up to constant factors? Ideally $\Phi$ has a closed form or is efficiently computable from an explicit description of $q$. If none exists, prove its impossibility.

2. Give a direct proof relating the truncated $2/3$-quasinorm in $\Phi_{\rm VV}$ to the $K$-functional in $\Phi_{\rm BCG}$.

3. Fix the small gap, described in the source, in the lower-bound proof of Valiant and Valiant [Valiant and Valiant, 2014].