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It is conjectured by Krauth and M\\'ezard (1989) that the (random) ratio $\\alpha_N$ converges in probability to an explicit constant $\\alpha_\\star \\doteq 0.83$. Kim and Roche (1998) proved the existence of a positive constant $\\gamma$ such that $\\gamma \\le \\alpha_N \\le 1-\\gamma$ with high probability; see also Talagrand (1999"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1809.07742","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-09-20T17:14:51Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"719c012dc8081fbae7f9049a12b93ff7945ab7768bc70c50d2cb2fe4ccbe7aab","abstract_canon_sha256":"799ce440f0d333f341255a318915a23e43fdea7eb6e83440949457bcf7b6f96e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:05:15.277291Z","signature_b64":"pac+hVZFZfhlRco8o7SXiCk0mHrH7yemBRDsbTrxjhc506cELLgEdz0enT6dQWpeqv9GDKXxgsxoZRqfPqqFCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"be6c8c7309de426b87cbfe74a350fb95cc9d6a4763a56a75367a237bfbb3b546","last_reissued_at":"2026-05-18T00:05:15.276666Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:05:15.276666Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Capacity lower bound for the Ising perceptron","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Jian Ding, Nike Sun","submitted_at":"2018-09-20T17:14:51Z","abstract_excerpt":"We consider the Ising perceptron with gaussian disorder, which is equivalent to the discrete cube $\\{-1,+1\\}^N$ intersected by $M$ random half-spaces. 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