Spectral pinning of neural modules

Use this mechanism for distributed training, mixture-of-experts routing, or a deep residual network whose blocks exchange representations. Construct a graph with one node per worker, expert, or module. Set edge weight A_ij to the measured communication frequency, routing probability, attention exchange, or an exponentially transformed representation similarity between nodes i and j. A pinned node receives an additional anchor loss p_i ||h_i-h_i_teacher||^2/2, periodic synchronization with a reference model, or a stronger local correction; unpinned nodes receive only graph-consensus regularization proportional to sum over edges of A_ij||h_i-h_j||^2. Given a budget of m pinned nodes, initialize P=0 and greedily evaluate each unpinned candidate by computing the smallest eigenvalue of L+P with Lanczos. Pin the candidate with the largest eigenvalue increase and repeat until m nodes are selected. During training, log the disagreement energy V=one half times the sum of squared deviations of h_i from their graph-weighted mean. The paper supplies the spectral stability mechanism; graph weights and effective coupling c must be estimated from the implementation. First test four or eight small MLP modules trained asynchronously on MNIST or CIFAR-10, comparing spectral-greedy pins, random pins, and highest-degree pins at equal anchor budget. The quantitative prediction is that the measured disagreement decay slope will scale as -2c lambda_min(L+P) times lambda_min(Gamma). Increasing pin strength should produce diminishing returns after the grounded-Laplacian gap saturates, and spectral pinning should yield a faster decay slope than random pinning. Require the measured slope to track the eigenvalue prediction within 25 percent before assessing accuracy or throughput.