{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"5c71567f4b4fc944311e1b6e085d44820b67c9f84cefd855ca920f3cfb6d1140","created":"2026-10-03T07:18:11Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"5cff02f79db7f655e95d30aa6c702c1511e3b86a9877c1eed500e3c70ca424dd","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"value","assisted_by":[],"external_id":"stat.learning-inference.dense-memory-capacity","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"A Hopfield network stores patterns as low points of an energy, and replacing pairwise couplings by couplings among p units at once lets it store vastly more patterns. The exact number of patterns it can store and still recall from a corrupted cue has not been computed with proof.","posed_since":"1987","precise":"Spins $\\sigma_i = \\pm 1, i = 1..N$, energy $E = -N \\sum_{\\mu=1..P} (m_\\mu)^p$ with overlaps $m_\\mu = (1/N) \\sum_i \\xi_i^\\mu \\sigma_i$ and P independent uniform random patterns $\\xi^\\mu$, at load $P = \\alpha N^{(p-1)}$ with $p \\ge 3$ fixed. Determine $\\alpha_c(p)$ such that for $\\alpha < \\alpha_c(p)$ every pattern has a nearby energy minimum with overlap m close to 1 and a basin of attraction of macroscopic size under zero-temperature single-spin-flip dynamics, and for $\\alpha > \\alpha_c(p)$ it does not. The answer is $\\alpha_c(p)$ with a rigorous proof, or a proof that the replica-method prediction is exact.","problem_ref":null,"references":"","settled_by":"A rigorous computation of $\\alpha_{c}(p)$ for the retrieval transition at load $P$ proportional to $N^{p-1}$, with basins of macroscopic size.","status_note":"The scaling $P \\sim N^{p-1}$ and a replica prediction of $\\alpha_{c}(p)$ date to 1987 (Gardner, J. Phys. A 20, 3453, DOI 10.1088/0305-4470/20/11/046), the model was revived as dense associative memory by Krotov and Hopfield (2016), fixed-point capacity of order $N^{p-1}/\\ln N$ is proved, and the exponential-interaction version was solved by replica methods (Lucibello and Mezard, PRL 2024); no rigorous derivation of $\\alpha_{c}(p)$ at P proportional to $N^{p-1}$ was located (2026).","title":"Exact storage capacity of dense associative memories with p-body interactions","topic_ref":"e915a7e12556cfb0adf583a894af7480127cf4267dc4e2458650f4872a864ed1"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"f17940bf88d222f113619b6ae6153e975aa788db5e20d9dfbbe2c8797eb26706","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"87d40fa57ea8709d4c208fe5a30be7d80780fddc544bd8fa8291d88da7817a9c","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"jn9SDwhv-Utq2ygTsW3Ae0a16QiXhRBz2DZysL5iRhk1GWB9G8KtN_i7zzir0mOgivNTTjyEK2MwdRMuBDHhDA"},"schema":"pubphys.envelope/1"},"record_hash":"f17940bf88d222f113619b6ae6153e975aa788db5e20d9dfbbe2c8797eb26706","leaf_index":2091}