Homomorphic Tempo Conditioning

(1) Integrate this at the input or state-conditioning point of a small temporal Transformer or SSM. Each example should carry a positive relative tempo or time-dilation ratio \(\lambda\) between a reference sequence and its target sampling rate. Compute \(g(\lambda)=W\log\lambda\), then add this vector to every token representation, or add it to the query/key projections so temporal interactions are explicitly rate-conditioned. For a discrete experiment, factor rational rates and use \(g(r)=\sum_{p\in P}e_p(r)c_p\) for a finite prime set \(P\). (2) Pseudocode: input tokens x[1:T] and ratio lambda; compute z = W * log(lambda); compute h = encoder(x) + z[None, :]; run the normal Transformer or SSM; predict the target sequence. Sample rates during training from a set of rational values and optionally from a log-uniform continuous distribution. (3) The paper supplies the homomorphic form and exact composition law; estimate \(W\) or the prime vectors \(c_p\) by backpropagation. Empirically test composition by checking whether conditioning for two successive resamplings equals conditioning for their product. (4) First experiment: train a 4-layer causal Transformer or diagonal SSM on synthetic chirps and speech-like sequences with rates \(\{1/2,2/3,1,3/2,2\}\), withholding \(3/4\) and \(4/3\). Compare no conditioning, ordinary categorical rate embeddings, and logarithmic conditioning at equal parameter count. Measure reconstruction or next-token loss on seen and withheld rates, interpolation smoothness, and error after composing two resamplings. Success is lower unseen-rate loss without additional inference FLOPs, ideally at least 10% lower held-out-rate error than categorical embeddings.