Unpredictability from Parity: Understanding Linear Correctors for Markovian Sources

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Linear correctors reduce the bias of random bits via parity (XOR), but their guarantees are known mostly for independent bits, an assumption violated by many physical entropy sources. We give a general, exact formula for the parity bias of dependent bits from a (hidden, possibly non-homogeneous) Markov chain, as a product of twisted transfer kernels. Under next-bit unpredictability, i.e. conditional bias at most $\beta$, the decay is governed by $\beta$ itself: sharp for a one-bit context,