We study peeling decoding on the binary erasure channel when redundant parity checks may be added without restriction in number, but each check is subject to a row-weight constraint. For random simple binary regular low-density parity-check (LDPC) codes, we consider the parity-check matrix formed by all nonzero dual codewords up to a prescribed weight. We show that, throughout an explicit logarithmic row-weight range, this complete family of redundant checks has the same asymptotic normalized-residual threshold as peeling with the original LDPC matrix. In the same range, with probability tending to one over the random code, it still has a linear-size stopping set that is uniquely decodable. Consequently, every parity-check representation whose peeling decoder succeeds on every uniquely decodable erasure pattern must contain checks above this logarithmic scale, and the supports of those heavier checks must collectively contain a linear number of coordinates. Thus row weight remains an asymptotic limitation even when the number and selection of redundant checks are unrestricted.

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