A model-agnostic scaling rule behind the critical inverse temperature in softmax self-attention

Length-dependent logit rescaling is widely used to stabilize long-context self-attention, but existing analyses and methods suggest different 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 critical scale is determined by the gap-counting function $N_n$ of each attention row, without specifying the details of the model structure. 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 provides a model-agnostic, higher-level explanation of how scaling laws vary with $N_n$, and yields a direct diagnostic for attention-score families, from idealized theoretical models to practical transformers.