Locally reversible hierarchical block

Modify a hierarchical Transformer, U-Net, or multiscale CNN at every downsampling stage. Split the activation $h^{(r)}$ into local blocks $x_j$, apply a strided local encoder $E_r$ to produce coarse tokens $h^{(r+1)}_j$, and attach a decoder $D_r$ that reconstructs either the original block or a projected version $P_rx_j$. Use the objective $\mathcal{L}=\mathcal{L}_{task}+\lambda\sum_r\|D_r(E_r(x_j))-P_rx_j\|_2^2$, where $P_r$ is a fixed low-pass convolution, channel projection, or identity map. Keep $D_r$ strictly local: it may read only the corresponding coarse token and positional information, not neighboring blocks. To test the paper's outside-insensitivity property, perturb, drop, or randomly replace all blocks outside $j$ before decoding and verify that reconstruction of block $j$ is unchanged. For memory reduction, checkpoint only coarse activations and recompute local encoders during backward; an alternative is an additive coupling block whose inverse is exact. The exact identity and locality come from the paper, while reconstruction error measures the approximation in the learned network. First run on CIFAR-100 with a 12-layer hierarchical ViT, comparing standard pooling, pooling plus a local autoencoder, and the proposed projected reversible pooling at matched parameter count. Record peak activation memory, examples per second, validation accuracy, reconstruction error under outside perturbations, and gradient-norm variance. Success is at least 25% lower peak memory with no validation-accuracy loss and stable gradients at depth.