{"paper":{"title":"A Unified Framework for Critical Scaling of Inverse Temperature in Self-Attention","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The critical inverse-temperature scale for self-attention concentration is fixed by an upper-tail accumulation scale derived from the gap-counting function of each attention row.","cross_cats":["cs.LG","math.PR"],"primary_cat":"stat.ML","authors_text":"Ryo Karakida, Tomohiro Hayase","submitted_at":"2026-05-12T19:48:36Z","abstract_excerpt":"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 $N_n$ 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 thi"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the gap-counting function N_n of each attention row fully determines the critical scale and that the attention-score distributions admit well-defined successive gaps from the maximum.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The upper-tail accumulation scale derived from the gap-counting function N_n sets the critical inverse temperature for softmax attention concentration, unifying prior conflicting laws as special cases of different N_n.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The critical inverse-temperature scale for self-attention concentration is fixed by an upper-tail accumulation scale derived from the gap-counting function of each attention row.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"af513fa9b8c116bca3bf7ede604e09971e8c2013a855088d9e39b9f009f27f80"},"source":{"id":"2605.12697","kind":"arxiv","version":1},"verdict":{"id":"8cc0e077-44f0-4d3f-aca3-acba8844ef1e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T19:50:11.335422Z","strongest_claim":"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.","one_line_summary":"The upper-tail accumulation scale derived from the gap-counting function N_n sets the critical inverse temperature for softmax attention concentration, unifying prior conflicting laws as special cases of different N_n.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the gap-counting function N_n of each attention row fully determines the critical scale and that the attention-score distributions admit well-defined successive gaps from the maximum.","pith_extraction_headline":"The critical inverse-temperature scale for self-attention concentration is fixed by an upper-tail accumulation scale derived from the gap-counting function of each attention row."},"references":{"count":32,"sample":[{"doi":"","year":2017,"title":"Attention is all you need","work_id":"4c6ff6f5-f994-47e1-b9da-213c378d21b8","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2020,"title":"Infinite attention: NNGP and NTK for deep attention networks","work_id":"5edec609-57d6-4fff-a106-35d1df33eb4b","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Infinite limits of multi-head transformer dynamics","work_id":"455a8000-ba63-4352-a5f6-6c98bb18e2d6","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"Infinite-width limit of a single attention layer: Analysis via tensor programs","work_id":"5998f624-fb37-489c-a484-ada3b2168e32","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"A mathematical perspective on transformers","work_id":"d081e886-f9fa-4d25-9355-b6f6d2e2af62","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":32,"snapshot_sha256":"e0407cc42983bdab86c360498ce54eddd54b63baa391db3ca6e5cd661a1b5a3c","internal_anchors":3},"formal_canon":{"evidence_count":3,"snapshot_sha256":"dab92689155357b0c3c51b06698660927100bb511d22f38635a9f2d1129f175c"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}