Reconstruct one column-parallel linear layer from rank-ordered weight shards. This is the linear projection only, with no nonlinear activation. For rank $r$:

Here, $X$ is the input tensor with shape $(B,D_{\mathrm{in}})$, $W_r$ is its shard with shape $(D_{\mathrm{in}},D_r)$, and $b_r$ is the optional bias of length $D_r$, broadcast across the batch. $B$ is the batch size, $D_{\mathrm{in}}$ is the input width, $D_r$ is rank $r$'s output width, and $Y_r$ is its local result. An absent bias contributes zero. Across $R$ ranks, concatenate output features in rank order:

Return a Python dictionary with local_activations, a rank-ordered list of PyTorch tensors shaped (B, D_r), and full_activation, a PyTorch tensor shaped (B, sum of shard widths). Both preserve the input dtype and device.