{"paper":{"title":"Quantitative Spectral Rigidity and Finite-Time Spectral Thermodynamics in Reversible Markov Chains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Qiao Wang","submitted_at":"2026-05-16T16:58:23Z","abstract_excerpt":"We study finite-time spectral rigidity in reversible Markov chains via exact spectral relaxation dynamics. While the underlying identities follow classically from self-adjointness on $L^2(\\pi)$, organizing the dynamics around the relaxation operator $\\mathcal{G}=I-P$ reveals finite-time structures invisible to traditional asymptotic estimates.\n  For chains with $\\lambda_2>\\lambda_3$, we establish explicit two-sided bounds on the rigidity time $T_{\\mathrm{rigid}}(\\delta)$, the first moment the slowest mode captures a fraction $1-\\delta$ of the total spectral energy. The bounds differ by at most"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.17082","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/2605.17082/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-19T22:33:23.807177Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T22:21:57.743757Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"e1c3ffb1bf781d9531ab36afc7f9ee0b10f55caddd32a7bc0abee26313762baf"},"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"}