Energy-Gated Attention and Wavelet Positional Encoding: Complementary Inductive Biases for Transformer Attention

Standard transformer attention computes pairwise token similarity but treats all tokens as equally salient and all positions as equally local, regardless of the informational structure of the input. We identify two complementary inductive biases that standard attention lacks: energy salience (which tokens concentrate informational energy, learned end-to-end without explicit frequency decomposition) and scale-selective locality (how far positional influence extends at each frequency, implemented via Morlet wavelet encoding). We address both with two simple components. Energy-Gated Attention (EGA) gates value aggregation by a learned energy estimate of key token embeddings, computed via a single linear projection; it selects what to attend to. Morlet Positional Encoding (MoPE) replaces fixed sinusoidal encodings with learned Gaussian-windowed wavelets that adapt the joint position-frequency localization to the corpus; it specifies where attention operates at each scale. On TinyShakespeare, EGA alone achieves +0.092 validation loss improvement over standard attention (+0.103 over Phase 1-3 baseline); MoPE alone is -0.032 (below baseline as a standalone encoding); but their combination achieves +0.119 -- more than the sum of parts. This superadditivity, observed across two independent training runs, is the central empirical finding: salience and locality are complementary inductive biases, each addressing a gap the other cannot fill alone. Ablations confirm that structured spectral priors (Morlet wavelet gates, scale-initialized heads, fixed sinusoidal PE) consistently underperform their unconstrained learned counterparts, while complementary learned components interact superadditively. All experiments are at small scale (<=6M parameters, character-level benchmarks, single seed); larger-scale multi-seed validation is the most important direction for future work.