Polynomial-expander feature mixer

Integrate this at the input side of a small MLP, factorization machine, or retrieval key generator for examples containing n+2 categorical fields. First assign every category a scalar code e_j in {0,1,...,K-1}, normalize it to u_j = 2e_j/(K-1)-1, and group n+2 fields per mixer. Compute the polynomial by Horner evaluation to avoid separately materializing powers: set q = u_2-u_1, initialize z = 0, and for i = n,n-1,...,1 update z = q*z + u_1^(n-i)*u_{i+2}; alternatively precompute the n powers of u_1 for a vectorized batch. For a useful vector representation, create r channels using independent random permutations of the field order and independent signed affine codes, yielding z_1,...,z_r, then concatenate them with ordinary embeddings before the MLP. Keep the polynomial path fixed initially; optionally learn a small affine scale and bias after it, not the polynomial coefficients, so the discrete expansion property remains interpretable. The formula and cardinality lower bound come from the paper; the normalized coding, multiple channels, and Horner implementation are engineering adaptations. Estimate empirical expansion by enumerating all tuples for small K, counting unique quantized outputs, and measuring collision rates on held-out tuples. The first experiment should use n=3, K in {8,16,32}, and a two-layer MLP on synthetic parity, interaction, and Criteo-style categorical subsets. Compare against concatenated embeddings, a learned linear projection to one scalar, and random hashing at equal stored feature dimension. Success means lower unseen-tuple collision rate and equal-or-better downstream accuracy with less embedding memory; explicitly monitor activation magnitude, gradient norm, and sensitivity to code permutations because the theorem does not guarantee numerical separation after floating-point rounding.