Arbitrary-Accuracy Neural Approximation with Optimal Neuron Count and Near-Optimal Bit Complexity

We study the minimum number of hidden neurons required for arbitrary-accuracy approximation of multivariate Hölder-continuous functions on $[0,1{]}^d$ and the associated encoding complexity. For $d\ge 2$, we construct a fixed, explicitly defined activation function for which a closed-form network with two hidden layers of widths $d$ and $1$ achieves arbitrary accuracy in the uniform norm. We prove that $d+1$ is the exact minimum total number of hidden neurons among standard feedforward networks with locally integrable activations and affine outputs. We further give a simpler construction using a single elementary activation that combines the floor and exponential functions. This construction requires three hidden layers of widths $d$, $1$, and $2$, only two neurons above the minimum. If a skip connection is allowed, widths $d$, $1$, and $1$ suffice. These constructions use explicit grid addressing and integer encoding of quantized function values. For a bounded $\alpha$-Hölder class, the target-dependent encoding requires $O({\epsilon }^{-d/\alpha }\log (1/\epsilon ))$ bits, matching the metric-entropy lower bound up to a logarithmic factor.