Spatially Coupled MacKay-Neal Codes Achieve Capacity on BMS Channels at Fixed Degrees

We establish two degree-dependent results for spatially coupled MacKay-Neal codes on binary-input memoryless symmetric channels. The ensembles use uniform random smoothing and shorten both variable types outside the active chain. For $r=g=3$, every integer $\mathrm{\ell }\ge 4$, and each channel of capacity greater than $R=r\mathrm{/}\mathrm{\ell }$, there is a sequence of code realizations whose actual transmitted rate tends to $R$ and whose average sum-product bit error tends to zero. For $r=g=2$ and each $\mathrm{\ell }\in \{3,4,5\}$, an interval of binary symmetric channels has capacity greater than $R$ but retains positive transmitted-bit error under terminated density evolution as chain length grows relative to coupling width. Both results follow from the signs of the same density-valued potential at uncoupled fixed points. The degree-three proof combines analytic bounds for $\mathrm{\ell }\ge 33$ with exact interval certificates for $4\le \mathrm{\ell }\le 32$, followed by threshold saturation and a projection argument for the actual rate. The degree-two proof analytically constructs a fixed point with negative potential and controls both boundary contributions. We relate the potential and the degree-two bifurcation condition to earlier statistical-mechanical predictions. The finite certificates and verification software are available in a versioned supplement.