{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:E7UTZKXITA4PUDWKQIB65JYPBE","short_pith_number":"pith:E7UTZKXI","canonical_record":{"source":{"id":"2608.08752","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-08-09T15:03:26Z","cross_cats_sorted":["cond-mat.dis-nn","math-ph","math.MP"],"title_canon_sha256":"167837fb50a43bcfa34713444bf0a2d06ccc38c14ea9a21e6c4abef319023e45","abstract_canon_sha256":"3dcb482404248a25df76632b89894ae321f6cdd977701c8abcbbd794ffeb01d1"},"schema_version":"1.0"},"canonical_sha256":"27e93caae89838fa0eca8203eea70f0922b0428427668cf7875c31f23e720f2c","source":{"kind":"arxiv","id":"2608.08752","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.08752","created_at":"2026-08-11T01:25:13Z"},{"alias_kind":"arxiv_version","alias_value":"2608.08752v1","created_at":"2026-08-11T01:25:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.08752","created_at":"2026-08-11T01:25:13Z"},{"alias_kind":"pith_short_12","alias_value":"E7UTZKXITA4P","created_at":"2026-08-11T01:25:13Z"},{"alias_kind":"pith_short_16","alias_value":"E7UTZKXITA4PUDWK","created_at":"2026-08-11T01:25:13Z"},{"alias_kind":"pith_short_8","alias_value":"E7UTZKXI","created_at":"2026-08-11T01:25:13Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:E7UTZKXITA4PUDWKQIB65JYPBE","target":"record","payload":{"canonical_record":{"source":{"id":"2608.08752","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-08-09T15:03:26Z","cross_cats_sorted":["cond-mat.dis-nn","math-ph","math.MP"],"title_canon_sha256":"167837fb50a43bcfa34713444bf0a2d06ccc38c14ea9a21e6c4abef319023e45","abstract_canon_sha256":"3dcb482404248a25df76632b89894ae321f6cdd977701c8abcbbd794ffeb01d1"},"schema_version":"1.0"},"canonical_sha256":"27e93caae89838fa0eca8203eea70f0922b0428427668cf7875c31f23e720f2c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-11T01:25:13.783570Z","signature_b64":"qq/1nPeiZ9YszNR3BPzuC3eTMeFpyQqhId8xLhPmTmu7kcjRSK5OheG2OV3CFnnE2EZJozRlsElVq5SVJ9ViDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"27e93caae89838fa0eca8203eea70f0922b0428427668cf7875c31f23e720f2c","last_reissued_at":"2026-08-11T01:25:13.780971Z","signature_status":"signed_v1","first_computed_at":"2026-08-11T01:25:13.780971Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2608.08752","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-08-11T01:25:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"i5M8lMtpL/gI7+eehblVN0rxcIG3aQd03+/rvZrJavnJ1dLgpcdAaDakb283ofsb9B/7Gov6CvnoBkw5d9vkDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T05:52:59.961033Z"},"content_sha256":"6ef85a1cd0a129baf3a80a61004bf9dc2ad49a5ee3439ef82a6713dbce2d6985","schema_version":"1.0","event_id":"sha256:6ef85a1cd0a129baf3a80a61004bf9dc2ad49a5ee3439ef82a6713dbce2d6985"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:E7UTZKXITA4PUDWKQIB65JYPBE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Overlap distribution of the critical Sherrington-Kirkpatrick model","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.dis-nn","math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Brice Huang, Hang Du","submitted_at":"2026-08-09T15:03:26Z","abstract_excerpt":"We study the distribution of the two-replica overlap $R_{1,2}$ in the Ising and spherical Sherrington-Kirkpatrick models at the critical inverse temperature $\\beta = 1$. Our main result shows that in both models, $R_{1,2}$ has scale $N^{-1/3}$, and the quenched distribution of $N^{1/3} R_{1,2}$ converges to an explicit random probability measure defined in terms of the reflected $\\mathrm{Airy}_1$ point process. As a consequence, we characterize the limiting value of $N^{2/3} \\mathbb{E} \\langle R_{1,2}^2 \\rangle$, answering a question of Talagrand \\cite{talagrand2011mean2}.\n  For the spherical "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.08752","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.08752/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-08-11T01:25:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Akbf5WVYmPaSxJ2tKYLZ1RvhsEyIQ/Slw2yAdWDGvgGxyGzB7+5LfZx2sVhUVHAv8LD8/N4Mj0OapSXOtc8OBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T05:52:59.961523Z"},"content_sha256":"385d79a40421ab15db7443f824f16ebeb40d0096387cb8e5bb338617c306e527","schema_version":"1.0","event_id":"sha256:385d79a40421ab15db7443f824f16ebeb40d0096387cb8e5bb338617c306e527"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/E7UTZKXITA4PUDWKQIB65JYPBE/bundle.json","state_url":"https://pith.science/pith/E7UTZKXITA4PUDWKQIB65JYPBE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/E7UTZKXITA4PUDWKQIB65JYPBE/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-23T05:52:59Z","links":{"resolver":"https://pith.science/pith/E7UTZKXITA4PUDWKQIB65JYPBE","bundle":"https://pith.science/pith/E7UTZKXITA4PUDWKQIB65JYPBE/bundle.json","state":"https://pith.science/pith/E7UTZKXITA4PUDWKQIB65JYPBE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/E7UTZKXITA4PUDWKQIB65JYPBE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:E7UTZKXITA4PUDWKQIB65JYPBE","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"3dcb482404248a25df76632b89894ae321f6cdd977701c8abcbbd794ffeb01d1","cross_cats_sorted":["cond-mat.dis-nn","math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-08-09T15:03:26Z","title_canon_sha256":"167837fb50a43bcfa34713444bf0a2d06ccc38c14ea9a21e6c4abef319023e45"},"schema_version":"1.0","source":{"id":"2608.08752","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.08752","created_at":"2026-08-11T01:25:13Z"},{"alias_kind":"arxiv_version","alias_value":"2608.08752v1","created_at":"2026-08-11T01:25:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.08752","created_at":"2026-08-11T01:25:13Z"},{"alias_kind":"pith_short_12","alias_value":"E7UTZKXITA4P","created_at":"2026-08-11T01:25:13Z"},{"alias_kind":"pith_short_16","alias_value":"E7UTZKXITA4PUDWK","created_at":"2026-08-11T01:25:13Z"},{"alias_kind":"pith_short_8","alias_value":"E7UTZKXI","created_at":"2026-08-11T01:25:13Z"}],"graph_snapshots":[{"event_id":"sha256:385d79a40421ab15db7443f824f16ebeb40d0096387cb8e5bb338617c306e527","target":"graph","created_at":"2026-08-11T01:25:13Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2608.08752/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the distribution of the two-replica overlap $R_{1,2}$ in the Ising and spherical Sherrington-Kirkpatrick models at the critical inverse temperature $\\beta = 1$. Our main result shows that in both models, $R_{1,2}$ has scale $N^{-1/3}$, and the quenched distribution of $N^{1/3} R_{1,2}$ converges to an explicit random probability measure defined in terms of the reflected $\\mathrm{Airy}_1$ point process. As a consequence, we characterize the limiting value of $N^{2/3} \\mathbb{E} \\langle R_{1,2}^2 \\rangle$, answering a question of Talagrand \\cite{talagrand2011mean2}.\n  For the spherical ","authors_text":"Brice Huang, Hang Du","cross_cats":["cond-mat.dis-nn","math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-08-09T15:03:26Z","title":"Overlap distribution of the critical Sherrington-Kirkpatrick model"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.08752","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:6ef85a1cd0a129baf3a80a61004bf9dc2ad49a5ee3439ef82a6713dbce2d6985","target":"record","created_at":"2026-08-11T01:25:13Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"3dcb482404248a25df76632b89894ae321f6cdd977701c8abcbbd794ffeb01d1","cross_cats_sorted":["cond-mat.dis-nn","math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-08-09T15:03:26Z","title_canon_sha256":"167837fb50a43bcfa34713444bf0a2d06ccc38c14ea9a21e6c4abef319023e45"},"schema_version":"1.0","source":{"id":"2608.08752","kind":"arxiv","version":1}},"canonical_sha256":"27e93caae89838fa0eca8203eea70f0922b0428427668cf7875c31f23e720f2c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"27e93caae89838fa0eca8203eea70f0922b0428427668cf7875c31f23e720f2c","first_computed_at":"2026-08-11T01:25:13.780971Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-11T01:25:13.780971Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qq/1nPeiZ9YszNR3BPzuC3eTMeFpyQqhId8xLhPmTmu7kcjRSK5OheG2OV3CFnnE2EZJozRlsElVq5SVJ9ViDg==","signature_status":"signed_v1","signed_at":"2026-08-11T01:25:13.783570Z","signed_message":"canonical_sha256_bytes"},"source_id":"2608.08752","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6ef85a1cd0a129baf3a80a61004bf9dc2ad49a5ee3439ef82a6713dbce2d6985","sha256:385d79a40421ab15db7443f824f16ebeb40d0096387cb8e5bb338617c306e527"],"state_sha256":"979b9fcc3529d91a0d59c57089f901feac0f9ded4d60ffab09252c776b3034b1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TVT22fARQWsQFYvxt6n2E86KUxquYRGwNUMqJ0naSCtxi01ah2I5Kp4dZ3Rch+8pWKXw1XHitMcGi+mnTiJyAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-23T05:52:59.965579Z","bundle_sha256":"31f331d913a9d422f6955d8912e962a9adc5e1f5588d6345b2202df1f96425ac"}}