A Unified Framework for Critical Scaling of Inverse Temperature in Self-Attention

Length-dependent logit rescaling is widely used to stabilize long-context self-attention, but existing analyses and methods suggest conflicting inverse-temperature laws for the context length n, ranging from (log n)1/2 to log n and (log n)2. We provide a general theory showing that the desirable scale is determined by the gap-counting function Nn of each attention row. Counting how many competitors lie within each gap from the maximum, we define an upper-tail accumulation scale and prove that it gives the critical inverse-temperature scale for softmax concentration: below this scale, the top competitors remain unseparated, whereas above it, the attention entropy collapses. This framework unifies prior scaling laws as different Nn and yields a direct diagnostic for attention-score families, from idealized theoretical models to more practical transformers.