{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:NDQSEKNJSFHWLEG6YXNMWQIIMR","short_pith_number":"pith:NDQSEKNJ","canonical_record":{"source":{"id":"2607.10510","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2026-07-12T00:02:11Z","cross_cats_sorted":["cond-mat.stat-mech","math.MP","math.PR"],"title_canon_sha256":"0fc4c316342ea624ad935e4a964cace63ee53d565c7ae1ddcb655f3f1109f290","abstract_canon_sha256":"3a2d0b542d37997323fbf262a5af5160d773ad8cb06dc228d7c5eb00a7e45cac"},"schema_version":"1.0"},"canonical_sha256":"68e12229a9914f6590dec5dacb4108646e9543ba1532843166c63f6af3d40440","source":{"kind":"arxiv","id":"2607.10510","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.10510","created_at":"2026-07-14T01:21:18Z"},{"alias_kind":"arxiv_version","alias_value":"2607.10510v1","created_at":"2026-07-14T01:21:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.10510","created_at":"2026-07-14T01:21:18Z"},{"alias_kind":"pith_short_12","alias_value":"NDQSEKNJSFHW","created_at":"2026-07-14T01:21:18Z"},{"alias_kind":"pith_short_16","alias_value":"NDQSEKNJSFHWLEG6","created_at":"2026-07-14T01:21:18Z"},{"alias_kind":"pith_short_8","alias_value":"NDQSEKNJ","created_at":"2026-07-14T01:21:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:NDQSEKNJSFHWLEG6YXNMWQIIMR","target":"record","payload":{"canonical_record":{"source":{"id":"2607.10510","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2026-07-12T00:02:11Z","cross_cats_sorted":["cond-mat.stat-mech","math.MP","math.PR"],"title_canon_sha256":"0fc4c316342ea624ad935e4a964cace63ee53d565c7ae1ddcb655f3f1109f290","abstract_canon_sha256":"3a2d0b542d37997323fbf262a5af5160d773ad8cb06dc228d7c5eb00a7e45cac"},"schema_version":"1.0"},"canonical_sha256":"68e12229a9914f6590dec5dacb4108646e9543ba1532843166c63f6af3d40440","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-14T01:21:18.908760Z","signature_b64":"8OWubXdlleCJaTC6kycOCi4Vt6pzIXZKENwuoxuBQ0ZAhCXDRwd7QMMBPhCQmcsbddel53l/KveOLK9vMC3lBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"68e12229a9914f6590dec5dacb4108646e9543ba1532843166c63f6af3d40440","last_reissued_at":"2026-07-14T01:21:18.907901Z","signature_status":"signed_v1","first_computed_at":"2026-07-14T01:21:18.907901Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.10510","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":"2026-07-14T01:21:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5Tdw4ONaCzPchdeokDZU43htXvW3Ipfua5uXHZ6/h4Dz6kzGAk/is94NxM4/pgN4OCSWwF3bxe6jYFVL3ZvLAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T23:21:49.271418Z"},"content_sha256":"5773e9d15c4be465db714864e94304b44aac3270c4d44d2b7c15ebf5e8b44c1e","schema_version":"1.0","event_id":"sha256:5773e9d15c4be465db714864e94304b44aac3270c4d44d2b7c15ebf5e8b44c1e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:NDQSEKNJSFHWLEG6YXNMWQIIMR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\\mathbb{Z}^2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.stat-mech","math.MP","math.PR"],"primary_cat":"math-ph","authors_text":"Qidong He","submitted_at":"2026-07-12T00:02:11Z","abstract_excerpt":"We prove the existence of a nematic phase in a model of $l\\times w$ hard rectangles on the square lattice with two allowed orientations and a large aspect ratio $k:=l/w$. The proof is based on a two-scale cluster expansion method developed previously by Disertori--Giuliani for hard rods in 2D and Disertori--Giuliani--Jauslin for hard plates in 3D. Our main contributions lie in explicitly tracking the constants and parameters and completing the arguments left implicit in these works. Hence, the proof produces a sufficient set of quantitative conditions from which estimates for the required aspe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.10510","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/2607.10510/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":"2026-07-14T01:21:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SCB38Sn8JjN/taTLMmxKWMPHQiUd7ZJHo90q+nV6cKObsGe9Y8r/DlzXVNg+cSSbGaUalgQAL9O+oEN2FrMQCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T23:21:49.272287Z"},"content_sha256":"787900839a26ce5c179f7d5d57657ecc58e5c47a59a35d1abdc68aec09f9d469","schema_version":"1.0","event_id":"sha256:787900839a26ce5c179f7d5d57657ecc58e5c47a59a35d1abdc68aec09f9d469"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NDQSEKNJSFHWLEG6YXNMWQIIMR/bundle.json","state_url":"https://pith.science/pith/NDQSEKNJSFHWLEG6YXNMWQIIMR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NDQSEKNJSFHWLEG6YXNMWQIIMR/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":"2026-08-13T23:21:49Z","links":{"resolver":"https://pith.science/pith/NDQSEKNJSFHWLEG6YXNMWQIIMR","bundle":"https://pith.science/pith/NDQSEKNJSFHWLEG6YXNMWQIIMR/bundle.json","state":"https://pith.science/pith/NDQSEKNJSFHWLEG6YXNMWQIIMR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NDQSEKNJSFHWLEG6YXNMWQIIMR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:NDQSEKNJSFHWLEG6YXNMWQIIMR","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":"3a2d0b542d37997323fbf262a5af5160d773ad8cb06dc228d7c5eb00a7e45cac","cross_cats_sorted":["cond-mat.stat-mech","math.MP","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2026-07-12T00:02:11Z","title_canon_sha256":"0fc4c316342ea624ad935e4a964cace63ee53d565c7ae1ddcb655f3f1109f290"},"schema_version":"1.0","source":{"id":"2607.10510","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.10510","created_at":"2026-07-14T01:21:18Z"},{"alias_kind":"arxiv_version","alias_value":"2607.10510v1","created_at":"2026-07-14T01:21:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.10510","created_at":"2026-07-14T01:21:18Z"},{"alias_kind":"pith_short_12","alias_value":"NDQSEKNJSFHW","created_at":"2026-07-14T01:21:18Z"},{"alias_kind":"pith_short_16","alias_value":"NDQSEKNJSFHWLEG6","created_at":"2026-07-14T01:21:18Z"},{"alias_kind":"pith_short_8","alias_value":"NDQSEKNJ","created_at":"2026-07-14T01:21:18Z"}],"graph_snapshots":[{"event_id":"sha256:787900839a26ce5c179f7d5d57657ecc58e5c47a59a35d1abdc68aec09f9d469","target":"graph","created_at":"2026-07-14T01:21:18Z","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/2607.10510/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove the existence of a nematic phase in a model of $l\\times w$ hard rectangles on the square lattice with two allowed orientations and a large aspect ratio $k:=l/w$. The proof is based on a two-scale cluster expansion method developed previously by Disertori--Giuliani for hard rods in 2D and Disertori--Giuliani--Jauslin for hard plates in 3D. Our main contributions lie in explicitly tracking the constants and parameters and completing the arguments left implicit in these works. Hence, the proof produces a sufficient set of quantitative conditions from which estimates for the required aspe","authors_text":"Qidong He","cross_cats":["cond-mat.stat-mech","math.MP","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2026-07-12T00:02:11Z","title":"Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on $\\mathbb{Z}^2$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.10510","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":"sha256:5773e9d15c4be465db714864e94304b44aac3270c4d44d2b7c15ebf5e8b44c1e","target":"record","created_at":"2026-07-14T01:21:18Z","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":"3a2d0b542d37997323fbf262a5af5160d773ad8cb06dc228d7c5eb00a7e45cac","cross_cats_sorted":["cond-mat.stat-mech","math.MP","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2026-07-12T00:02:11Z","title_canon_sha256":"0fc4c316342ea624ad935e4a964cace63ee53d565c7ae1ddcb655f3f1109f290"},"schema_version":"1.0","source":{"id":"2607.10510","kind":"arxiv","version":1}},"canonical_sha256":"68e12229a9914f6590dec5dacb4108646e9543ba1532843166c63f6af3d40440","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"68e12229a9914f6590dec5dacb4108646e9543ba1532843166c63f6af3d40440","first_computed_at":"2026-07-14T01:21:18.907901Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-14T01:21:18.907901Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8OWubXdlleCJaTC6kycOCi4Vt6pzIXZKENwuoxuBQ0ZAhCXDRwd7QMMBPhCQmcsbddel53l/KveOLK9vMC3lBw==","signature_status":"signed_v1","signed_at":"2026-07-14T01:21:18.908760Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.10510","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5773e9d15c4be465db714864e94304b44aac3270c4d44d2b7c15ebf5e8b44c1e","sha256:787900839a26ce5c179f7d5d57657ecc58e5c47a59a35d1abdc68aec09f9d469"],"state_sha256":"56e26430186d0a1f8eddaea0b3be2f76df33b3daab0636d7560606b804eba1c3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"f17XahibSBetf83P4J0QfQw7FXG8NEba5T74VxgfQ1jrxJdbckMjHShm2hYI+ywFYR/f8DDZrZn9HzNXfEJjBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T23:21:49.281463Z","bundle_sha256":"222f888df52b088dcc3da7729584fe0a77303439454028a127038a9959a4127c"}}