Inverse-square count-aware Gaussian release

Integrate this at the release boundary of a private federated learner, or at the all-reduce that publishes per-expert token counts in a private MoE router. For a federated experiment, let each round aggregate a clipped group sum G=\sum_{j=1}^{n}g_j, with per-example clipping ||g_j||_2\leq L. Use sensitivity \Delta=L for add/remove adjacency on the sum, or \Delta=L/n for releasing an average with a fixed public n. First choose a target asymptotic zCDP constant C and a numerical shift s, then compute sigma=Delta*(n+s)/sqrt(2*C). Pseudocode: (1) clip each gradient g_j to norm L; (2) form G=sum(g_j); (3) obtain n from a public group-size field or from a separately privatized lagged count; (4) set sigma as above; (5) release G+Normal(0,sigma^2 I); (6) update the server model with the released aggregate. For MoE, apply the same operation to the vector of expert counts before capacity-factor or auxiliary-load-balancing decisions. The mathematical quantities are the inverse-square budget law, the Gaussian zCDP identity, and the relative-error calculation. Estimate empirically the achieved relative error E||noise||/||G||, privacy accountant composition across rounds, and sensitivity after clipping. Begin with CIFAR-10 or FEMNIST using a 2-layer CNN and 20-client FedAvg, comparing standard constant Gaussian noise against count-aware noise at matched average relative error and matched total zCDP. Success is a lower validation-loss variance for heterogeneous client sizes, or equal accuracy at a smaller privacy budget; also test whether expert-load estimates have fewer false overload decisions at equal privacy.