Instance-optimal identity testing asks how many samples are needed to test an unknown discrete distribution against a specific known null. Existing characterizations truncate a constant multiple of the testing radius. This is usually harmless, but it creates a polynomial gap for the harmonic null $q_i\propto1/i$, the canonical obstruction highlighted in a 2024 COLT open problem. We give a constant-factor characterization for the full Zipfian class $q_i\asymp1/(Li)$. Let $m_q(\varepsilon)$ be the last index whose strict suffix has null mass at least $\varepsilon$. Throughout the nondegenerate range, $N^\star(q,\varepsilon) =\Theta \left(L\sqrt{m_q(\varepsilon)} \max\{1,(\varepsilon L)^{-2}\}\right),$ where constants depend only on the fixed Zipf envelope. For the exact harmonic law, $L=H_k$ and $m_q(\varepsilon)=\Theta(k\exp(-\varepsilon H_k))$, so $N^\star(q,\varepsilon) =\Theta \left(H_k\sqrt{k e^{-\varepsilon H_k}} \max\{1,(\varepsilon H_k)^{-2}\}\right).$ Thus, for every fixed $\varepsilon\in(0,1/3]$, the answer is $\Theta((\log k)k^{(1-\varepsilon)/2})$. The upper bound combines a coarsened instance-optimal test with one unweighted tail-collision statistic. Its key step is a water-filling inequality: if almost all of a Zipfian tail is deleted, its squared discrepancy cannot fall below the tail's collision mass. The lower bound introduces a graded-thinning prior. Tail coordinate $i$ is retained with probability proportional to $i^{-1/3}$ and inflated when retained. This profile makes the alternatives $\varepsilon$-far while keeping the mixture second moment bounded up to the matching sample size. Beyond settling the harmonic example robustly, the result isolates graded thinning as the mechanism missed by constant-radius truncations.
Keywords: identity testing, distribution testing, Zipfian distributions, property testing, sample complexity, power laws