Sharp Scalar Reconstruction Rates and Fusion Separation under Lipschitz Constraints

In multimodal prediction, observations may be compressed separately before fusion or jointly after fusion. We ask whether the Euclidean dimension of a nonlinear representation alone imposes a nontrivial information constraint. Without a fixed Lipschitz or network-complexity budget, scalar neural coding extends to arbitrary finite-dimensional observation laws and square-integrable responses. The joint risk infimum equals the full-observation Bayes risk, while separate encoding reduces the dimension budget to modality selection. Under a Lipschitz product budget, however, the approximation problem becomes quantitative. For a uniform input on $[0,1{]}^d$ with $d\ge 2$, the optimal mean squared reconstruction error through a scalar bottleneck is of order ${\Lambda }^{-2/(d+1)}$ when the product of the encoder and decoder Lipschitz constants is at most $\Lambda$. The lower bound holds for all Lipschitz maps, and finite ReLU networks achieve the same rate. For two independent uniform three-dimensional modalities and an additive response, separate encoding with total latent dimension two has risk of order ${\Lambda }^{-1/2}$, whereas joint encoding under the same budgets attains $O({\Lambda }^{-2/3})$. Linear Gaussian frontiers, exact continuous factorization dimensions, and matching finite-bit Gaussian risk exponents provide complementary characterizations of representation cost. Seven controlled experiments examine the constructions and Gaussian benchmarks, showing a decreasing attained joint-to-separate risk ratio under matched Lipschitz budgets and finite-bit decay rates close to the rank-based predictions.