A fundamental question in streaming complexity is whether every space-efficient turnstile algorithm is implicitly a linear sketch. The work of Li, Nguyen, and Woodruff (STOC~2014) established an equivalence between the two, but their reduction requires a stream length that is at least doubly exponential in the dimension $n$. In the opposite direction, Kallaugher and Price (STOC~2020) showed a separation for streams of linear length, so the equivalence does not hold in general. The most natural and practically relevant regime, that of polynomial-length streams, has therefore remained open. We show that polynomial-length turnstile algorithms admit linear-sketch simulations. More precisely, if a turnstile algorithm uses $S$ bits of space and succeeds on all streams of length $\textrm{poly}(D, n)$, then on final vectors $x$ with $\|x\|_2 \le D$, its output can be recovered from $O(S)$ linear measurements of $x$, using $O(S)$ bits overall. For smooth problems under appropriate input distributions, a mollified version of the reduction yields a bounded-entry sketch with $O(S / \log D)$ measurements and optimal $O(S)$ total space. Our results extend to strict turnstile streams and non-uniform Read-Once Branching Programs (ROBPs). Instead of the transition-graph arguments of prior work, our proof uses Fourier analysis and tools from additive combinatorics to extract discrete linear measurements. We show that an $S$-bit algorithm can only be sensitive to a small set of heavy Fourier frequencies, which lie close to a finite group of order $2^{O(S)}$, and we use generators of this group as the rows of the sketching matrix. Combined with existing lower bounds for real sketches and for communication, this gives new lower bounds for polynomial-length turnstile streams.

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