Quantitative bounds imply exponent $1/2$

For each natural number $n$, let $I_n=\{k\in\mathbb N:n<k<2n\}$ and $J_n=\{k\in\mathbb N:2n<k<4n\}$, and let $f_{\mathbb N}(n)$ be the greatest natural number $t\le |J_n|+|I_n|$ such that, for every finite set $B\subseteq J_n$, there exists a finite set $D\subseteq I_n$ for which no sum $d+d'$ of two distinct members $d,d'\in D$ belongs to $B$, and $t\le |B|+|D|$; let $f(n)\in\mathbb Z$ be the integer obtained from $f_{\mathbb N}(n)$. The declaration asserts the following implication: if there exist strictly positive real constants $c$ and $C$, and a natural number $n_0\ge 1$, such that for every natural number $n\ge n_0$, both $c\sqrt{n\log n}\le f(n)$ and $f(n)\le n^{\,1/2+C(\log(\log n)/\log n)^{1/3}}$ hold after viewing $f(n)$ as a real number, then for every real $\varepsilon>0$ there exists a natural number $N_\varepsilon\ge 1$ such that for every natural number $n\ge N_\varepsilon$, $n^{\,1/2-\varepsilon}\le f(n)\le n^{\,1/2+\varepsilon}$, again with $f(n)$ viewed as real. The two thresholds are permitted to equal $1$, so $n=1$ is not intrinsically excluded; all logarithms, divisions, and real powers in the displayed correction use their totalized real-valued conventions, including $\log 0=0$ and division by zero equal to $0$, while $n=0$ is excluded from both families of eventual inequalities by the requirement that the relevant threshold be at least $1$.