There is a communication problem with constant randomized cost that requires Ω(√n) deterministic queries to an Equality oracle, so constant-cost randomness cannot be efficiently derandomized by equality checks.
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Equality is Far Weaker than Constant-Cost Communication
There is a communication problem with constant randomized cost that requires Ω(√n) deterministic queries to an Equality oracle, so constant-cost randomness cannot be efficiently derandomized by equality checks.