{"paper":{"title":"First Law of Proto-Area Entropy from Modular Spectral Geometry","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["gr-qc","quant-ph"],"primary_cat":"hep-th","authors_text":"Ling-Zheng Xia, Lixin Xu","submitted_at":"2026-07-31T13:55:16Z","abstract_excerpt":"We derive a first law of proto-area entropy in the CCKLP--Witten framework for approximate holographic entanglement wedge reconstruction. The central spectral function admits a modular Hamiltonian representation with kernel $L(x)=x\\coth(x/2)$. For near-maximally-mixed bulk states, and within the unstructured Gaussian-unitary-ensemble (GUE) model of the encoding perturbation, the ensemble-averaged proto-area entropy varies linearly with bulk entropy to leading order, with a response coefficient containing a universal factor~$1/3$, traced to $L''(0)/2=1/6$ and independent of the bulk spectrum's "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.29432","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/2607.29432/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"}