{"paper":{"title":"Solving linear-rate ODE hierarchies (like master equations) using closures and operator splitting","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["cs.NA","math.DS","math.PR","q-bio.QM","stat.CO"],"primary_cat":"math.NA","authors_text":"Joshua C Chang","submitted_at":"2026-05-16T22:46:15Z","abstract_excerpt":"Countably infinite systems of linear ODEs arise as forward equations for many continuous-time Markov processes. The standard recipe -- truncate to a finite cap N and exponentiate -- pays cubic cost in N and a time-growing boundary-feedback bias. We identify a structural condition on the rates, L_{n+r,n} = alpha_r n + beta_r (\"linear-rate\"), under which the generating function G(z,t) = sum_n x_n(t) z^n satisfies a first-order linear PDE in z, and the method of characteristics yields a composition-multiplier representation G(z,t) = K_t(z) G(Phi_t(z), 0). The Taylor coefficients of Phi_t and K_t "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.17186","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/2605.17186/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-19T22:33:23.744104Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T22:01:57.961924Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"5d021cff417f4ae7c4839e9d8fd72c5eadda92be224c2224cb9d1ea89565b8cf"},"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"}