{"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"}