Critical 2d Intensity Bottleneck

Integrate this at the latent bottleneck of a complex-valued autoencoder, VAE, or complex feature extractor. Let the preceding network output z in C^d, represented by two real tensors zr and zi with shape [batch, d]. Store a learnable complex measurement matrix F with shape [m, d], initialize it with independent complex Gaussian entries, and row-normalize it. Compute u = einsum('bd,md->bm', z, conjugate(F)) and b = real(u times conjugate(u)); optionally apply log1p(b) followed by per-channel normalization before feeding b to the decoder. Use m = 2d for the proposed bottleneck. If the decoder must reconstruct a complex vector, compare against a canonical-phase target: rotate each target so its largest-magnitude coordinate is real and positive, because the measurements identify only the equivalence class [z]. The core pseudocode is: u = z @ conjugate(F).T; b = real(u * conjugate(u)); output = decoder(log1p(b)); loss = task_loss(output, target) plus beta times reconstruction_loss(output, target). The theorem determines the width threshold, while F is learned by ordinary backpropagation. First test a small complex autoencoder on synthetic Gaussian complex vectors and CIFAR-10 features with d equal to 8, 16, and 32, comparing m = 2d - 1, 2d, and 2d + 4 against a standard 2d-real-coordinate bottleneck at comparable storage. Record reconstruction MSE, downstream accuracy, activation memory, and optimization stability. The expected falsifiable result is a sharp reconstruction gap for 2d - 1 and recovery of the overcomplete baseline at m = 2d. Also compare frozen random F with learned F to separate identifiability from optimization effects.