HyperBlossom formulates most-likely-error decoding as a minimum-weight parity factor (MWPF) problem and generalizes the blossom algorithm to hypergraphs using a primal-dual linear-programming framework. The paper establishes optimality under stated conditions and certifiable proximity bounds. Its open-source implementation, Hyperion, targets general qLDPC codes and uses clustering to achieve almost-linear average runtime scaling in the evaluated settings.