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Sign-coherence and tropical sign pattern for rank $3$ real cluster-cyclic exchange matrices
The sign-coherence about $c$-vectors was conjectured by Fomin-Zelevinsky and proved by Gross-Hacking-Keel-Kontsevich for integer skew-symmetrizable case. We prove this conjecture associated with $c$-vectors for rank 3 real cluster-cyclic skew-symmetrizable case. Simultaneously, we establish their se
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