A Theory of Polarization
summary
Constructive proof that repeated two-channel transform produces synthetic channels whose capacities polarize to near 0 or 1 and whose fraction of near-1 channels equals the original channel capacity. Key specifics: for a BEC(p) the pair becomes w+ becomes BEC of p^2 and w- becomes BEC of 2p−p^2, mutual information is conserved (conservation of mutual information), and a second-moment growth argument replaces heavy martingale machinery. For larger alphabets polarization requires group/quasigroup operations and a strong ergodicity criterion; the construction yields usable polar codes.
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