Gray-coded numeric embeddings

Integrate this at the numeric-feature embedding or tokenizer boundary. For every integer-valued field n, first compute its binary codeword using k=floor(log2(n+1)), r=n-(2^k-1), q=(2^k-1)-r, and codeword bits equal to q xor (q>>1), padded to exactly k bits; represent n=0 by an empty sequence plus a dedicated empty-vector marker. Feed digit embeddings through a two-layer Transformer, GRU, or mean-pooling encoder f, and project the resulting vector to the model dimension. During training, retain the original supervised or language-model loss and sample integer pairs (n,n+1), plus harder pairs at distance d. Apply the displayed adjacent-pair penalty only to valid non-overflowing values; optionally use a hinge version max(0, ||f(V(n))-f(V(m))||-c|n-m|). The code itself is computed exactly; no Gray-code statistics need estimating. Measure sequence length, embedding-table size, and the correlation between embedding distance and |n-m|. The first experiment should use a small decoder Transformer on synthetic arithmetic sequences and a tabular regression/classification benchmark containing counters and indices. Compare one-hot integer embeddings, decimal digit embeddings, and Gray-coded embeddings at equal parameter count. Success is lower validation error and better extrapolation to unseen larger integers, with no increase in training FLOPs; a useful signal is smooth prediction of n+1 from n and reduced sample count to learn monotone arithmetic relations.