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We think the log rank conjecture is true over $\{0,1\}$ real matrices and over any fixed alphabet matrix.

What is the fastest function $f(r)$ of rank $r$ such that the log rank conjecture over $\{0,1\}$ real matrices holds iff there is an $f(r)$-log rank conjecture that holds over $\{0,1,\dots,f(r)-1,f(r)\}$ real matrices?

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