Dimension-aware power-mean fusion

Modify a multi-branch fusion point such as multi-sensor feature fusion, confidence aggregation, or an MoE gate when every branch emits a nonnegative scalar or channel-wise activation. Set a_i=softplus(h_i)+epsilon, use fixed convex weights lambda_i or obtain them from a softmax router, and compute y=F_{p,d}(a_1,...,a_m). For a chosen dimension d and exponent p<1/d, calculate q=p/(1-dp), then compute y=(sum_i lambda_i a_i^q)^(1/q). Use y=exp(sum_i lambda_i log(a_i)) when absolute q is below 1e-4. For large positive q, evaluate in the log domain using log-sum-exp; for negative q, clamp a_i to epsilon. A learnable version can parameterize p=(1/d-epsilon)tanh(s), where s is a scalar or per-layer parameter, and add a weak penalty encouraging s near zero. The paper supplies the exponent map and endpoint behavior; the experiment must determine whether the effective dimension should be one, the spatial dimension, or the feature dimension. First compare arithmetic mean, geometric mean, fixed quadratic mean, minimum, and F_{p,d} in a two-sensor CIFAR-10 classifier and a small segmentation network with one randomly corrupted sensor. Sweep p in {-1,-0.25,0,0.1,0.25} for d=1 and d=2. Record clean accuracy, corrupted-sensor accuracy, gradient variance across branches, and sensitivity to branch dropout. Success is higher robustness or lower gradient imbalance at matched parameter count and FLOPs.