Rank-safe Bernoulli layer initialization

Integrate this at initialization of binary or sparse fully connected layers, convolutional pointwise matrices, and low-bit MLP blocks. Let W have shape m x d, with entries sampled independently as Bernoulli(p) and optionally mapped to signed weights W_{ij}=s(2B_{ij}-1), where B_{ij}~Bernoulli(p); retain a separate scale s=1/\sqrt{mp} so activation variance is comparable across p. First choose a desired structural-defect tolerance delta and an estimated defect count k_target, then set p=max(p_min, 1-delta**(1/(m*k_target))). Do not interpret this rectangular rule as a theorem: it is the paper's square-matrix asymptotic used as a conservative proxy. Sample W, count zero columns, and estimate numerical rank by computing singular values for small layers or by randomized range finding for large layers. If any column is zero or the estimated corank exceeds k_target, resample only the affected columns; alternatively force each column to contain at least one nonzero entry and then rescale. Pseudocode: sample B; while zero_columns(B) or numerical_corank(W,tau)>k_target: resample offending columns; scale W; run a forward variance check. For training, compare this initializer against standard iid Bernoulli, dense Gaussian, and orthogonal initialization in a 2-layer MLP on MNIST or Fashion-MNIST, plus a width- and sparsity-matched binary network. Measure initial numerical rank, fraction of dead hidden units, gradient norm dispersion, validation accuracy, and loss after a fixed number of updates. The expected signal is fewer dead channels and more stable early optimization at the same parameter count; if rank screening adds overhead without improving these metrics, reject the method.