{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:YSLIF2YIX6LEAQKYZSYMORRZ4Y","short_pith_number":"pith:YSLIF2YI","canonical_record":{"source":{"id":"2109.14790","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2021-09-30T01:34:32Z","cross_cats_sorted":["cond-mat.dis-nn","math-ph","math.MP"],"title_canon_sha256":"efa92493e6da1a091d9cdf929f0212c592ebdc1f0b71b17e9e81e609bf2aae6c","abstract_canon_sha256":"ee6fbb05c18b571ba3f3c3bf442575f493dc5997580b3bfbe6d323421b7f6b0f"},"schema_version":"1.0"},"canonical_sha256":"c49682eb08bf96404158ccb0c74639e62b5fd5cf0894422e776386a01ecf3ef0","source":{"kind":"arxiv","id":"2109.14790","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2109.14790","created_at":"2026-07-05T11:33:11Z"},{"alias_kind":"arxiv_version","alias_value":"2109.14790v4","created_at":"2026-07-05T11:33:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2109.14790","created_at":"2026-07-05T11:33:11Z"},{"alias_kind":"pith_short_12","alias_value":"YSLIF2YIX6LE","created_at":"2026-07-05T11:33:11Z"},{"alias_kind":"pith_short_16","alias_value":"YSLIF2YIX6LEAQKY","created_at":"2026-07-05T11:33:11Z"},{"alias_kind":"pith_short_8","alias_value":"YSLIF2YI","created_at":"2026-07-05T11:33:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:YSLIF2YIX6LEAQKYZSYMORRZ4Y","target":"record","payload":{"canonical_record":{"source":{"id":"2109.14790","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2021-09-30T01:34:32Z","cross_cats_sorted":["cond-mat.dis-nn","math-ph","math.MP"],"title_canon_sha256":"efa92493e6da1a091d9cdf929f0212c592ebdc1f0b71b17e9e81e609bf2aae6c","abstract_canon_sha256":"ee6fbb05c18b571ba3f3c3bf442575f493dc5997580b3bfbe6d323421b7f6b0f"},"schema_version":"1.0"},"canonical_sha256":"c49682eb08bf96404158ccb0c74639e62b5fd5cf0894422e776386a01ecf3ef0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:33:11.822607Z","signature_b64":"ZGc51qPRz2Bi906yx0UKvxQdW4ftnKBpvGQcHQAbo/wN/XUbZYMUqCN4Nr/bpRAFVMH+JMGQwcIrYIrxIqICAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c49682eb08bf96404158ccb0c74639e62b5fd5cf0894422e776386a01ecf3ef0","last_reissued_at":"2026-07-05T11:33:11.822051Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:33:11.822051Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2109.14790","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:33:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6zeBD29eA5x/obMihXsyZvc5uhFZsSWHfaRacsWPmq1KKPHelNWV/ugWtRXfEyyU5HXityE2M81k9u2Tof3CBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T04:32:46.363898Z"},"content_sha256":"715be20a735c808c6059f8310f0a0f5aefe32f27794e0601ec0b7595db2cb522","schema_version":"1.0","event_id":"sha256:715be20a735c808c6059f8310f0a0f5aefe32f27794e0601ec0b7595db2cb522"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:YSLIF2YIX6LEAQKYZSYMORRZ4Y","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Free energy in multi-species mixed $p$-spin spherical models","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":"Erik Bates, Youngtak Sohn","submitted_at":"2021-09-30T01:34:32Z","abstract_excerpt":"We prove a Parisi formula for the limiting free energy of multi-species spherical spin glasses with mixed $p$-spin interactions. The upper bound involves a Guerra-style interpolation and requires a convexity assumption on the model's covariance function. Meanwhile, the lower bound adapts the cavity method of Chen so that it can be combined with the synchronization technique of Panchenko; this part requires no convexity assumption. In order to guarantee that the resulting Parisi formula has a minimizer, we formalize the pairing of synchronization maps with overlap measures so that the constrain"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.14790","kind":"arxiv","version":4},"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/2109.14790/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":"2026-07-05T11:33:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U4tCFe5kektsW0v0bKeMOIxEicNoKibtMXRR6j05oPkMb9keNCJ7J70qHen1QKJvzuq1HBnmVeSMhxGqHO7yDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T04:32:46.364431Z"},"content_sha256":"e82b7fe25c6da43a2bbed99ebe02639d11c73c57fe528311f01095b464cb1a22","schema_version":"1.0","event_id":"sha256:e82b7fe25c6da43a2bbed99ebe02639d11c73c57fe528311f01095b464cb1a22"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YSLIF2YIX6LEAQKYZSYMORRZ4Y/bundle.json","state_url":"https://pith.science/pith/YSLIF2YIX6LEAQKYZSYMORRZ4Y/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YSLIF2YIX6LEAQKYZSYMORRZ4Y/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":"2026-08-10T04:32:46Z","links":{"resolver":"https://pith.science/pith/YSLIF2YIX6LEAQKYZSYMORRZ4Y","bundle":"https://pith.science/pith/YSLIF2YIX6LEAQKYZSYMORRZ4Y/bundle.json","state":"https://pith.science/pith/YSLIF2YIX6LEAQKYZSYMORRZ4Y/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YSLIF2YIX6LEAQKYZSYMORRZ4Y/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:YSLIF2YIX6LEAQKYZSYMORRZ4Y","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":"ee6fbb05c18b571ba3f3c3bf442575f493dc5997580b3bfbe6d323421b7f6b0f","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":"2021-09-30T01:34:32Z","title_canon_sha256":"efa92493e6da1a091d9cdf929f0212c592ebdc1f0b71b17e9e81e609bf2aae6c"},"schema_version":"1.0","source":{"id":"2109.14790","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2109.14790","created_at":"2026-07-05T11:33:11Z"},{"alias_kind":"arxiv_version","alias_value":"2109.14790v4","created_at":"2026-07-05T11:33:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2109.14790","created_at":"2026-07-05T11:33:11Z"},{"alias_kind":"pith_short_12","alias_value":"YSLIF2YIX6LE","created_at":"2026-07-05T11:33:11Z"},{"alias_kind":"pith_short_16","alias_value":"YSLIF2YIX6LEAQKY","created_at":"2026-07-05T11:33:11Z"},{"alias_kind":"pith_short_8","alias_value":"YSLIF2YI","created_at":"2026-07-05T11:33:11Z"}],"graph_snapshots":[{"event_id":"sha256:e82b7fe25c6da43a2bbed99ebe02639d11c73c57fe528311f01095b464cb1a22","target":"graph","created_at":"2026-07-05T11:33:11Z","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/2109.14790/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove a Parisi formula for the limiting free energy of multi-species spherical spin glasses with mixed $p$-spin interactions. The upper bound involves a Guerra-style interpolation and requires a convexity assumption on the model's covariance function. Meanwhile, the lower bound adapts the cavity method of Chen so that it can be combined with the synchronization technique of Panchenko; this part requires no convexity assumption. In order to guarantee that the resulting Parisi formula has a minimizer, we formalize the pairing of synchronization maps with overlap measures so that the constrain","authors_text":"Erik Bates, Youngtak Sohn","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":"2021-09-30T01:34:32Z","title":"Free energy in multi-species mixed $p$-spin spherical models"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.14790","kind":"arxiv","version":4},"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":"sha256:715be20a735c808c6059f8310f0a0f5aefe32f27794e0601ec0b7595db2cb522","target":"record","created_at":"2026-07-05T11:33:11Z","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":"ee6fbb05c18b571ba3f3c3bf442575f493dc5997580b3bfbe6d323421b7f6b0f","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":"2021-09-30T01:34:32Z","title_canon_sha256":"efa92493e6da1a091d9cdf929f0212c592ebdc1f0b71b17e9e81e609bf2aae6c"},"schema_version":"1.0","source":{"id":"2109.14790","kind":"arxiv","version":4}},"canonical_sha256":"c49682eb08bf96404158ccb0c74639e62b5fd5cf0894422e776386a01ecf3ef0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c49682eb08bf96404158ccb0c74639e62b5fd5cf0894422e776386a01ecf3ef0","first_computed_at":"2026-07-05T11:33:11.822051Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:33:11.822051Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZGc51qPRz2Bi906yx0UKvxQdW4ftnKBpvGQcHQAbo/wN/XUbZYMUqCN4Nr/bpRAFVMH+JMGQwcIrYIrxIqICAw==","signature_status":"signed_v1","signed_at":"2026-07-05T11:33:11.822607Z","signed_message":"canonical_sha256_bytes"},"source_id":"2109.14790","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:715be20a735c808c6059f8310f0a0f5aefe32f27794e0601ec0b7595db2cb522","sha256:e82b7fe25c6da43a2bbed99ebe02639d11c73c57fe528311f01095b464cb1a22"],"state_sha256":"b21e17834221c17027fdb9af0ef723618f2cf12f6ea0df2168faffb299c9d966"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Uer9k7P9c2KGTg+fHxvayGI00t4/PayuibEs99Up2Ga+bif5Qey6mkmnGr8kI7iMfGK1WIq9sSJiCv4gk+bRAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-10T04:32:46.368631Z","bundle_sha256":"85c7a4c07028285215c4c268a32abc7824b8573afa7ffb3b0a6c1d0445d02c0d"}}