{"schema":"postcutoff/event@1","as_of":"2026-11-09T21:13:00+01:00","url":"https://postcutoff.com/e/2026-11-05-hopfield-critical-capacity-claude-opus-5-5/","md":"https://postcutoff.com/e/2026-11-05-hopfield-critical-capacity-claude-opus-5-5/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-11-05-hopfield-critical-capacity-claude-opus-5-5","date":"2026-11-05","date_precision":"day","short_title":"Hopfield critical capacity bracketed, by Claude Opus 5.5","deck":null,"takeaway":"A Hexagon deposit contributed by Yuwen Zhou claims to prove that the zero-temperature critical capacity of the Hopfield model lies in [0.138198129, 0.13820106], against a previous best rigorous lower bound of 0.113.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"Hexagon 2611.00105: The critical capacity of the Hopfield model","url":"https://hexagonmath.org/abs/2611.00105","type":"paper","group":"primary","domain":"hexagonmath.org"},{"n":2,"title":"Code: OctopusTide/hopfield-capacity","url":"https://github.com/OctopusTide/hopfield-capacity","type":"code","group":"primary","domain":"github.com"}],"official":2,"filed":"2026-11-06","updated":"2026-11-06","orgs":["Anthropic","OpenAI"],"title":"Hexagon deposit claims a rigorous bracket [0.138198, 0.138201] for the Hopfield model's critical capacity, proofs derived by Claude Opus 5.5","summary":"A Hexagon deposit contributed by Yuwen Zhou claims to prove that the zero-temperature critical capacity of the Hopfield model lies in [0.138198129, 0.13820106], against a previous best rigorous lower bound of 0.113. It states that Claude Opus 5.5 derived the proofs and wrote the text, and GPT-6.1 Sol wrote an independent report on each result.","key_facts":["Hexagon 2611.00105, 5 Nov 2026; authorship 'no-human-author-claimed', human understanding 'some'","Claimed interval [0.138198129, 0.13820106]; best rigorous lower bound before: 0.113; the lower end exceeds the published one- and two-step replica-symmetry-breaking values","The physics prediction αc ≈ 0.138 dates to Amit, Gutfreund and Sompolinsky (1985)","Provenance: 'Claude Opus 5.5 (Anthropic) derived the proofs and wrote the rest, and GPT-6.1 Sol (OpenAI) wrote an independent report on each result. A final audit by fresh AI readers covered the longer version'","Code and data: github.com/OctopusTide/hopfield-capacity"],"key_numbers":[],"tags":["math","probability","spin-glasses","neural-networks","claude-opus-5-5","gpt-6-1-sol","hexagon"],"science":{"field":"mathematics","subfield":"probability (spin glasses)","problem":"Rigorous critical storage capacity of the Hopfield associative-memory model at zero temperature","result":"Claimed bracket αc ∈ [0.138198129, 0.13820106]","open_since":"1985","ai_system":["Claude Opus 5.5","GPT-6.1 Sol"],"human_role":"Mainly AI-generated; human contributor checked parts","verification":{"key":"none","label":"Unverified claim","text":"Hexagon deposit, unrefereed, AI cross-review only"},"status":"pending","shock":""},"body_md":"## What happened\n\n[Hexagon 2611.00105](https://hexagonmath.org/abs/2611.00105) claims a rigorous narrow interval for the capacity value physicists predicted in 1985 (about 0.138). The record says the work was generated mainly by AI systems, with a human checking parts. Hexagon does not judge correctness.\n\n## Why it matters\n\nTurning the replica prediction for the Hopfield model into a theorem has been a long-standing goal in the mathematics of spin glasses. If correct, this is a large jump from the 0.113 lower bound. No expert comment was found on 6 Nov.","disputed":[{"label":"Verification","text":"Hexagon deposit, unrefereed, AI cross-review only","sources":[]}],"related":[{"id":"2024-10-08-nobel-physics-hopfield-hinton","url":"https://postcutoff.com/e/2024-10-08-nobel-physics-hopfield-hinton/","date":"2024-10-08","date_precision":"day","short_title":"Nobel Prize in Physics awarded to John Hopfield and Geoffrey Hinton","deck":null,"takeaway":"Together with the Chemistry prize a day later, it marked unprecedented recognition of AI by science's most prestigious award.","category":"milestone","category_label":"Milestones","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":3,"official":2,"filed":"2026-09-29","updated":"2026-09-29","orgs":["Royal Swedish Academy of Sciences"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-11-06","type":"filed","text":"Created from the Hexagon record"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-11-06","run":null,"sources_read":"The Hexagon record","updated":"2026-11-06","human_review":null,"version":{"date":"2026-11-06"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":189,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":128,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":219,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":158,"in_training_data":false}],"short_url":"https://postcutoff.com/s/hopfield-critical-capacity"}