We prove a sharp extrapolation bound for $k$-sparse Fourier sums with arbitrary real frequencies and no separation assumption. For every $g(t)=\sum_{j=1}^k v_je^{iλ_jt}$ with $λ_j\in\mathbb{R}$ and every $x\geq 1$, \begin{align*} |g(x)|\le k\exp(O(k\operatorname{arcosh} x))\|g\|_{L^2[-1,1]}. \end{align*} Near the endpoint at $x=1+δ$, this gives an exponential factor of $\exp(O(k\sqrtδ))$, improving on the $\exp(O(k^2\log k\cdotδ))$ estimate of Chen and Price (ICALP 2019), and the exponent is optimal up to a constant factor. Our theorem has algorithmic applications to recovering gapless Fourier-sparse signals: it improves the resolution of cluster-location algorithms and the sample complexity of Fourier interpolation algorithms. In addition, it gives sharp bounds on the error transfer of sparse Fourier models just outside the training interval.

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