Spectral Scaling Laws in Language Models: emphHow Effectively Do Feed-Forward Networks Use Their Latent Space?

As Large Language Models (LLMs) scale, the question is not just how large they become, but how much of their capacity is effectively utilized. Existing scaling laws relate model size to loss, yet overlook how components exploit their latent space. In this work, we focus on Feed-Forward Networks (FFNs) and recast width selection as a spectral utilization optimization problem. Using a lightweight diagnostic suite: Hard Rank (participation ratio), Soft Rank (Shannon Rank), Spectral Concentration, and the composite Spectral Utilization Index (SUI), we quantify how many latent directions are meaningfully activated across LLaMA, GPT-2, and nGPT families. Our key finding is an Asymmetric Spectral Scaling Law: soft rank follows an almost perfect power law with FFN width, while hard rank grows only sublinearly, with high variance. This asymmetry suggests that widening FFNs mostly adds low-energy tail directions, while dominant-mode subspaces saturate early. Moreover, at larger widths, variance further collapses into a narrow subspace, leaving much of the latent space under-utilized. These results recast FFN width selection as a principled trade-off between tail capacity and dominant-mode capacity, offering concrete guidance for inference-efficient LLM design.