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In the graded case, we define the notion of diagonal $F$-threshold of $R$ as the limit of the top socle degree of $R/m^{[q]}$ over $q$ when $q \\to \\infty$. Diagonal $F$-threshold exists as a positive number (rational number in the latter case) when: (1) $R$ is either a complete intersection or $R$ is $F$-pure on the punctured spectrum; (2) $R$ is a two dimensional normal domain. 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