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In this sequel, we determine the rest of the phase diagram. We prove that there is a unique second critical temperature $\\beta_2^p>\\beta_1^p$ such that the Parisi measure is 1-RSB for $\\beta_1^p<\\beta\\leq\\beta_2^p$, while for $\\beta>\\beta_2^p$, $\\operatorname{supp}\\mu_\\beta=\\{0\\}\\cup[q_\\beta,q'_\\beta],$ for s"},"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":"2608.06523","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-08-06T19:08:04Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"7cb7804c9530efd945b5c1ab8aba6416e10cd46414400a74bd61cdc8c2586953","abstract_canon_sha256":"b9036d02991b7b554cded9245e2e34904eae3db8dc1e493da903de80afe08f32"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-10T01:10:55.926433Z","signature_b64":"Tpx0gXg+D7IWVVs+17nP5e3q4zFlDlZFczvHWvRQU9JzI6R2uDlqWj61HJ90SYqS9tYRPKnuQjfS+PTCr4UoDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fcbe98434d812121b4ec8fec6380a0d7731fe51c9f0c17bf8fd5ab838e4cb667","last_reissued_at":"2026-08-10T01:10:55.923898Z","signature_status":"signed_v1","first_computed_at":"2026-08-10T01:10:55.923898Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Gardner Transition in the Ising Pure $p$-Spin Glass II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Yuxin Zhou","submitted_at":"2026-08-06T19:08:04Z","abstract_excerpt":"In previous work, we identified, for every $p\\geq3$, a unique first critical temperature $\\beta_1^p$ and proved that the Parisi measure is replica symmetric (RS) for $0<\\beta\\leq\\beta_1^p$ and one-step replica symmetry breaking (1-RSB) on a nonempty interval immediately above $\\beta_1^p$. In this sequel, we determine the rest of the phase diagram. We prove that there is a unique second critical temperature $\\beta_2^p>\\beta_1^p$ such that the Parisi measure is 1-RSB for $\\beta_1^p<\\beta\\leq\\beta_2^p$, while for $\\beta>\\beta_2^p$, $\\operatorname{supp}\\mu_\\beta=\\{0\\}\\cup[q_\\beta,q'_\\beta],$ for s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.06523","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/2608.06523/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2608.06523","created_at":"2026-08-10T01:10:55.924835+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.06523v1","created_at":"2026-08-10T01:10:55.924835+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.06523","created_at":"2026-08-10T01:10:55.924835+00:00"},{"alias_kind":"pith_short_12","alias_value":"7S7JQQ2NQEQS","created_at":"2026-08-10T01:10:55.924835+00:00"},{"alias_kind":"pith_short_16","alias_value":"7S7JQQ2NQEQSDNHM","created_at":"2026-08-10T01:10:55.924835+00:00"},{"alias_kind":"pith_short_8","alias_value":"7S7JQQ2N","created_at":"2026-08-10T01:10:55.924835+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7S7JQQ2NQEQSDNHMR7WGHAFA25","json":"https://pith.science/pith/7S7JQQ2NQEQSDNHMR7WGHAFA25.json","graph_json":"https://pith.science/api/pith-number/7S7JQQ2NQEQSDNHMR7WGHAFA25/graph.json","events_json":"https://pith.science/api/pith-number/7S7JQQ2NQEQSDNHMR7WGHAFA25/events.json","paper":"https://pith.science/paper/7S7JQQ2N"},"agent_actions":{"view_html":"https://pith.science/pith/7S7JQQ2NQEQSDNHMR7WGHAFA25","download_json":"https://pith.science/pith/7S7JQQ2NQEQSDNHMR7WGHAFA25.json","view_paper":"https://pith.science/paper/7S7JQQ2N","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.06523&json=true","fetch_graph":"https://pith.science/api/pith-number/7S7JQQ2NQEQSDNHMR7WGHAFA25/graph.json","fetch_events":"https://pith.science/api/pith-number/7S7JQQ2NQEQSDNHMR7WGHAFA25/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7S7JQQ2NQEQSDNHMR7WGHAFA25/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7S7JQQ2NQEQSDNHMR7WGHAFA25/action/storage_attestation","attest_author":"https://pith.science/pith/7S7JQQ2NQEQSDNHMR7WGHAFA25/action/author_attestation","sign_citation":"https://pith.science/pith/7S7JQQ2NQEQSDNHMR7WGHAFA25/action/citation_signature","submit_replication":"https://pith.science/pith/7S7JQQ2NQEQSDNHMR7WGHAFA25/action/replication_record"}},"created_at":"2026-08-10T01:10:55.924835+00:00","updated_at":"2026-08-10T01:10:55.924835+00:00"}