{"paper":{"title":"On the Spectral Synthesis of Lipschitz Persistence Diagram Vectorizations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.FA","authors_text":"Charles Fanning, Mehmet Emin Aktas","submitted_at":"2026-07-18T20:22:28Z","abstract_excerpt":"A persistence diagram represents the birth and death of homology classes along a filtration as a multiset of intervals, and we consider persistence diagram vectorizations to be maps $D(X,A) \\to E$ sending persistence diagrams over a metric pair $(X,A)$ to values in a Banach space $E$. We prove an isometric isomorphism between Lipschitz extensions of vectorizations, modulo constants, on the Grothendieck completion $K(X,A)$ of $D(X,A)$ and the bounded $1$-cocycles for the translation action of $K(X,A)$ on $\\ell^\\infty(K(X,A),E)$. We then prove that if every bounded linear functional on $E$ maps "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.16957","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/2607.16957/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"}