A Local Tree Structure is NOT Sufficient for the Local Optimality of Message-Passing Decoding in Low Density Parity Check Codes

We address the problem,`Is a local tree structure sufficient for the local optimality of message passing algorithm in low density parity check codes?'.It is shown that the answer is negative. Using this observation, we pinpoint a flaw in the proof of Theorem 1 in the paper `The Capacity of Low-Density Parity-Check Codes Under Message-Passing Decoding' by Thomas J. Richardson and R\"udiger L.Urbanke\cite{RUCapacity}. We further provide a new proof of that theorem based on a different argument.