{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:IIKRGJRORWIBC4MPB2VVYWHCMM","short_pith_number":"pith:IIKRGJRO","canonical_record":{"source":{"id":"2604.11481","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"astro-ph.CO","submitted_at":"2026-04-13T13:47:35Z","cross_cats_sorted":["math-ph","math.MP","nlin.PS","physics.data-an","stat.AP"],"title_canon_sha256":"96ed80d205fac666dad17f338a12a351ebd2157df52d6d4715ab22a4130e4e27","abstract_canon_sha256":"83bcc42c653344a90c45bb78a0c85879bd05f85ebf9c88cc07fdebfb4faa6659"},"schema_version":"1.0"},"canonical_sha256":"421513262e8d9011718f0eab5c58e26308f26499a17377895da93b3ae04f8bfc","source":{"kind":"arxiv","id":"2604.11481","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.11481","created_at":"2026-05-27T01:05:54Z"},{"alias_kind":"arxiv_version","alias_value":"2604.11481v2","created_at":"2026-05-27T01:05:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.11481","created_at":"2026-05-27T01:05:54Z"},{"alias_kind":"pith_short_12","alias_value":"IIKRGJRORWIB","created_at":"2026-05-27T01:05:54Z"},{"alias_kind":"pith_short_16","alias_value":"IIKRGJRORWIBC4MP","created_at":"2026-05-27T01:05:54Z"},{"alias_kind":"pith_short_8","alias_value":"IIKRGJRO","created_at":"2026-05-27T01:05:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:IIKRGJRORWIBC4MPB2VVYWHCMM","target":"record","payload":{"canonical_record":{"source":{"id":"2604.11481","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"astro-ph.CO","submitted_at":"2026-04-13T13:47:35Z","cross_cats_sorted":["math-ph","math.MP","nlin.PS","physics.data-an","stat.AP"],"title_canon_sha256":"96ed80d205fac666dad17f338a12a351ebd2157df52d6d4715ab22a4130e4e27","abstract_canon_sha256":"83bcc42c653344a90c45bb78a0c85879bd05f85ebf9c88cc07fdebfb4faa6659"},"schema_version":"1.0"},"canonical_sha256":"421513262e8d9011718f0eab5c58e26308f26499a17377895da93b3ae04f8bfc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-27T01:05:54.394453Z","signature_b64":"a+B+aCSfw6HRHwYBoiAsrNaaMqY7P72hIuthw4utAn9RRPMLJ6dAwyuWB93fGEwyvG8uhfasYHeF74jzodiTCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"421513262e8d9011718f0eab5c58e26308f26499a17377895da93b3ae04f8bfc","last_reissued_at":"2026-05-27T01:05:54.393507Z","signature_status":"signed_v1","first_computed_at":"2026-05-27T01:05:54.393507Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2604.11481","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-27T01:05:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XHTcFEkbFMrLhbsJ/x3/oFToFTpdUaP6xkMWIGfl29AVHRkv58uptuzwLyz8aawXlpUfSuCbtvuz1YKYYHXFDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T01:36:57.842351Z"},"content_sha256":"f5dc22219fbf4bd5769aa6b1fff36ac153ffb83b18aa26970695bd7efe597d07","schema_version":"1.0","event_id":"sha256:f5dc22219fbf4bd5769aa6b1fff36ac153ffb83b18aa26970695bd7efe597d07"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:IIKRGJRORWIBC4MPB2VVYWHCMM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Emergence of Complex Web Structures","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A Lagrangian-Eulerian transport map governs density amplification through its Jacobian while anisotropic collapse follows from the eigenvalues of successive deformation tensors.","cross_cats":["math-ph","math.MP","nlin.PS","physics.data-an","stat.AP"],"primary_cat":"astro-ph.CO","authors_text":"Francisco-Shu Kitaura","submitted_at":"2026-04-13T13:47:35Z","abstract_excerpt":"Complex structures often emerge from initially homogeneous or weakly correlated states. We address the apparent tension between this ordering and entropy growth through a unified framework combining semi-microscopic phase-space dynamics, transport geometry, information theory, and coarse-grained effective modeling. The key point is that entropy depends on the level of description: a coarse-grained spatial field may become more ordered as structure forms, even while the full phase-space description becomes more complex through shell crossing, multistreaming, and the activation of velocity degre"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Using a Lagrangian--Eulerian transport map, we show how density amplification is governed by the Jacobian of the deformation and how anisotropic collapse arises from the eigenvalues of a hierarchy of deformation tensors... the nonlocal tidal level becomes relevant already at moderate overdensity.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the proposed Lagrangian-Eulerian transport geometry and its connection to a maximum-entropy Gaussian baseline plus Landau-Ginzburg effective description accurately capture the dynamics and entropy behavior across the relevant scales without additional unstated assumptions about the underlying microscopic interactions.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Coarse-grained spatial ordering can increase during structure formation even as full phase-space entropy grows through nonlocal transport, Jacobian-governed density amplification, and activation of lower free-energy branches in a Landau-Ginzburg description.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A Lagrangian-Eulerian transport map governs density amplification through its Jacobian while anisotropic collapse follows from the eigenvalues of successive deformation tensors.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"57a3b6e8ea692c57e2c6ced53b012512744d99567201dd3d6d8822bef930c2af"},"source":{"id":"2604.11481","kind":"arxiv","version":2},"verdict":{"id":"9de3f6a6-6fd5-4acd-bc1a-48f61d896820","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T15:36:20.103448Z","strongest_claim":"Using a Lagrangian--Eulerian transport map, we show how density amplification is governed by the Jacobian of the deformation and how anisotropic collapse arises from the eigenvalues of a hierarchy of deformation tensors... the nonlocal tidal level becomes relevant already at moderate overdensity.","one_line_summary":"Coarse-grained spatial ordering can increase during structure formation even as full phase-space entropy grows through nonlocal transport, Jacobian-governed density amplification, and activation of lower free-energy branches in a Landau-Ginzburg description.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the proposed Lagrangian-Eulerian transport geometry and its connection to a maximum-entropy Gaussian baseline plus Landau-Ginzburg effective description accurately capture the dynamics and entropy behavior across the relevant scales without additional unstated assumptions about the underlying microscopic interactions.","pith_extraction_headline":"A Lagrangian-Eulerian transport map governs density amplification through its Jacobian while anisotropic collapse follows from the eigenvalues of successive deformation tensors."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.11481/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":"9de3f6a6-6fd5-4acd-bc1a-48f61d896820"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-27T01:05:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5TQ/mVACKvy8OIYVtGt30MlCqB+D9/2B8F0aTq0pMdSHNG+9GK1uM60xznetAdlMz4ZPnq17hKOejSv6EP5HCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T01:36:57.842836Z"},"content_sha256":"24e2b5077dbfd956e5d1da5ed7f29de9f89bf057598a8823b2e7f59f96e10f96","schema_version":"1.0","event_id":"sha256:24e2b5077dbfd956e5d1da5ed7f29de9f89bf057598a8823b2e7f59f96e10f96"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IIKRGJRORWIBC4MPB2VVYWHCMM/bundle.json","state_url":"https://pith.science/pith/IIKRGJRORWIBC4MPB2VVYWHCMM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IIKRGJRORWIBC4MPB2VVYWHCMM/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":"2026-06-12T01:36:57Z","links":{"resolver":"https://pith.science/pith/IIKRGJRORWIBC4MPB2VVYWHCMM","bundle":"https://pith.science/pith/IIKRGJRORWIBC4MPB2VVYWHCMM/bundle.json","state":"https://pith.science/pith/IIKRGJRORWIBC4MPB2VVYWHCMM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IIKRGJRORWIBC4MPB2VVYWHCMM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:IIKRGJRORWIBC4MPB2VVYWHCMM","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":"83bcc42c653344a90c45bb78a0c85879bd05f85ebf9c88cc07fdebfb4faa6659","cross_cats_sorted":["math-ph","math.MP","nlin.PS","physics.data-an","stat.AP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"astro-ph.CO","submitted_at":"2026-04-13T13:47:35Z","title_canon_sha256":"96ed80d205fac666dad17f338a12a351ebd2157df52d6d4715ab22a4130e4e27"},"schema_version":"1.0","source":{"id":"2604.11481","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.11481","created_at":"2026-05-27T01:05:54Z"},{"alias_kind":"arxiv_version","alias_value":"2604.11481v2","created_at":"2026-05-27T01:05:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.11481","created_at":"2026-05-27T01:05:54Z"},{"alias_kind":"pith_short_12","alias_value":"IIKRGJRORWIB","created_at":"2026-05-27T01:05:54Z"},{"alias_kind":"pith_short_16","alias_value":"IIKRGJRORWIBC4MP","created_at":"2026-05-27T01:05:54Z"},{"alias_kind":"pith_short_8","alias_value":"IIKRGJRO","created_at":"2026-05-27T01:05:54Z"}],"graph_snapshots":[{"event_id":"sha256:24e2b5077dbfd956e5d1da5ed7f29de9f89bf057598a8823b2e7f59f96e10f96","target":"graph","created_at":"2026-05-27T01:05:54Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"Using a Lagrangian--Eulerian transport map, we show how density amplification is governed by the Jacobian of the deformation and how anisotropic collapse arises from the eigenvalues of a hierarchy of deformation tensors... the nonlocal tidal level becomes relevant already at moderate overdensity."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"That the proposed Lagrangian-Eulerian transport geometry and its connection to a maximum-entropy Gaussian baseline plus Landau-Ginzburg effective description accurately capture the dynamics and entropy behavior across the relevant scales without additional unstated assumptions about the underlying microscopic interactions."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Coarse-grained spatial ordering can increase during structure formation even as full phase-space entropy grows through nonlocal transport, Jacobian-governed density amplification, and activation of lower free-energy branches in a Landau-Ginzburg description."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"A Lagrangian-Eulerian transport map governs density amplification through its Jacobian while anisotropic collapse follows from the eigenvalues of successive deformation tensors."}],"snapshot_sha256":"57a3b6e8ea692c57e2c6ced53b012512744d99567201dd3d6d8822bef930c2af"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2604.11481/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Complex structures often emerge from initially homogeneous or weakly correlated states. We address the apparent tension between this ordering and entropy growth through a unified framework combining semi-microscopic phase-space dynamics, transport geometry, information theory, and coarse-grained effective modeling. The key point is that entropy depends on the level of description: a coarse-grained spatial field may become more ordered as structure forms, even while the full phase-space description becomes more complex through shell crossing, multistreaming, and the activation of velocity degre","authors_text":"Francisco-Shu Kitaura","cross_cats":["math-ph","math.MP","nlin.PS","physics.data-an","stat.AP"],"headline":"A Lagrangian-Eulerian transport map governs density amplification through its Jacobian while anisotropic collapse follows from the eigenvalues of successive deformation tensors.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"astro-ph.CO","submitted_at":"2026-04-13T13:47:35Z","title":"Emergence of Complex Web Structures"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.11481","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-10T15:36:20.103448Z","id":"9de3f6a6-6fd5-4acd-bc1a-48f61d896820","model_set":{"reader":"grok-4.3"},"one_line_summary":"Coarse-grained spatial ordering can increase during structure formation even as full phase-space entropy grows through nonlocal transport, Jacobian-governed density amplification, and activation of lower free-energy branches in a Landau-Ginzburg description.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"A Lagrangian-Eulerian transport map governs density amplification through its Jacobian while anisotropic collapse follows from the eigenvalues of successive deformation tensors.","strongest_claim":"Using a Lagrangian--Eulerian transport map, we show how density amplification is governed by the Jacobian of the deformation and how anisotropic collapse arises from the eigenvalues of a hierarchy of deformation tensors... the nonlocal tidal level becomes relevant already at moderate overdensity.","weakest_assumption":"That the proposed Lagrangian-Eulerian transport geometry and its connection to a maximum-entropy Gaussian baseline plus Landau-Ginzburg effective description accurately capture the dynamics and entropy behavior across the relevant scales without additional unstated assumptions about the underlying microscopic interactions."}},"verdict_id":"9de3f6a6-6fd5-4acd-bc1a-48f61d896820"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:f5dc22219fbf4bd5769aa6b1fff36ac153ffb83b18aa26970695bd7efe597d07","target":"record","created_at":"2026-05-27T01:05:54Z","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":"83bcc42c653344a90c45bb78a0c85879bd05f85ebf9c88cc07fdebfb4faa6659","cross_cats_sorted":["math-ph","math.MP","nlin.PS","physics.data-an","stat.AP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"astro-ph.CO","submitted_at":"2026-04-13T13:47:35Z","title_canon_sha256":"96ed80d205fac666dad17f338a12a351ebd2157df52d6d4715ab22a4130e4e27"},"schema_version":"1.0","source":{"id":"2604.11481","kind":"arxiv","version":2}},"canonical_sha256":"421513262e8d9011718f0eab5c58e26308f26499a17377895da93b3ae04f8bfc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"421513262e8d9011718f0eab5c58e26308f26499a17377895da93b3ae04f8bfc","first_computed_at":"2026-05-27T01:05:54.393507Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-27T01:05:54.393507Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"a+B+aCSfw6HRHwYBoiAsrNaaMqY7P72hIuthw4utAn9RRPMLJ6dAwyuWB93fGEwyvG8uhfasYHeF74jzodiTCQ==","signature_status":"signed_v1","signed_at":"2026-05-27T01:05:54.394453Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.11481","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f5dc22219fbf4bd5769aa6b1fff36ac153ffb83b18aa26970695bd7efe597d07","sha256:24e2b5077dbfd956e5d1da5ed7f29de9f89bf057598a8823b2e7f59f96e10f96"],"state_sha256":"aed2b0aebd23176a4852c7f6d52f2170ceaeccfb368ee73ab9ae6dee6d6e91bb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aUobPnzdR+yIGhilpSxpUXncE/3fKRSbRo+WdARJ48l5GNhKREh9T8J26XqrHqQBvTV8ePeyNSluWiiJ9sP8Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T01:36:57.845156Z","bundle_sha256":"ca390de8247be2eb326559e346ca8acf9f023982e2033b546c8f78ff10096943"}}