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Can we recover the macroscopic observables of $\\phi$ up to $o(1)$ precision? We prove that this statistical reconstruction problem undergoes the following Kosterlitz-Thouless type phase transition:\n  -) If $T<T_{rec}^-$ , one can fully recover $\\phi$ from the knowledge of $\\phi \\pmod{\\frac {2\\pi} T}$. 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Assume $\\phi$ is a discrete Gaussian free field (GFF) on $\\Lambda \\subset \\frac 1 n \\mathbb{Z}^2$ and that we are given $e^{iT \\phi}$, or equivalently $\\phi \\pmod{\\frac {2\\pi} T}$. Can we recover the macroscopic observables of $\\phi$ up to $o(1)$ precision? We prove that this statistical reconstruction problem undergoes the following Kosterlitz-Thouless type phase transition:\n  -) If $T<T_{rec}^-$ , one can fully recover $\\phi$ from the knowledge of $\\phi \\pmod{\\frac {2\\pi} T}$. 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