Erdős–Szekeres products: the true order of $\log f(n)$ for $\min\max_{|z|=1}|\prod_i(1-z^{a_i})|$ (Erdős #256)

Let $n\geq 1$ and let $f(n)$ be maximal such that for any integers $1\leq a_1\leq\cdots\leq a_n$ we have $$\max_{\lvert z\rvert=1}\left\lvert \prod_{i=1}^{n}(1-z^{a_i})\right\rvert\geq f(n).$$ Estimate $f(n)$: determine the true order of growth of $\log f(n)$. (Erdős's specific sub-question — is $\log f(n)\gg n^{c}$ for some constant $c>0$? — was answered negatively by Belov and Konyagin, who proved $\log f(n)\ll(\log n)^4$; what remains open is where the truth lies in the window $\log n\ll \log f(n)\ll(\log n)^4$.)

Acceptance. FULLY RESOLVES: a proof determining the order of $\log f(n)$ up to constants — i.e. an explicit function $g$ with matching proofs $\log f(n)\gg g(n)$ and $\log f(n)\ll g(n)$ (machine-checkable preferred, else complete written proofs). No finite computation alone can close this. ADVANCES: (a) an upper bound improving the exponent in the $(\log n)^4$ bound stated in the background, with proof; (b) a lower bound $\log f(n)\gg (\log n)^{1+c}$ for some $c>0$, improving the $\log n$ lower bound stated in the background, with proof; (c) an improvement of the Bourgain–Chang bound for the distinct-exponents variant $f^*(n)$ stated in the background, with proof; (d) a reproducible computational study exhibiting explicit exponent multisets whose products are certifiably flatter (smaller sup-norm on $|z|=1$, verified by interval arithmetic or an equivalent rigorous method) than any published example at comparable $n$, with code and certificates; (e) a transfer result making the Atkinson bridge to Chowla's cosine problem quantitative in a new regime. Deliver the proof file, or the code plus certified sup-norm evaluations and the exponent multisets.