{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:6O2HXGFETSGQRUAOIOVLMTQTQV","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"07deee118cbfd06b9977779f492340e9b04c4971ff3e041a0b9ca99a0f0f60fc","cross_cats_sorted":["cs.DM"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2026-06-26T14:43:15Z","title_canon_sha256":"c812c402d70e00475c7809a9e2faa791330b632735a6958b50fafc0625b478de"},"schema_version":"1.0","source":{"id":"2606.28147","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.28147","created_at":"2026-06-29T01:15:07Z"},{"alias_kind":"arxiv_version","alias_value":"2606.28147v1","created_at":"2026-06-29T01:15:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.28147","created_at":"2026-06-29T01:15:07Z"},{"alias_kind":"pith_short_12","alias_value":"6O2HXGFETSGQ","created_at":"2026-06-29T01:15:07Z"},{"alias_kind":"pith_short_16","alias_value":"6O2HXGFETSGQRUAO","created_at":"2026-06-29T01:15:07Z"},{"alias_kind":"pith_short_8","alias_value":"6O2HXGFE","created_at":"2026-06-29T01:15:07Z"}],"graph_snapshots":[{"event_id":"sha256:8b92c455f06d13db407a94def303e73ee4daf7075ecf4fb9554615be205ec217","target":"graph","created_at":"2026-06-29T01:15:07Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2606.28147/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove that for any matrix $A \\in \\mathbb{R}^{m \\times n}$ and any $\\varepsilon \\in (0, 1/2]$ there is a diagonal matrix $D \\in \\mathbb{R}_{\\geq 0}^{m \\times m}$ with at most $O(\\frac{n}{\\varepsilon^2} \\log(\\frac{1}{\\varepsilon}))$ nonzero entries so that \\[(1-\\varepsilon) \\|Ax\\|_1 \\leq \\|DAx\\|_1 \\leq (1+\\varepsilon)\\|Ax\\|_1 \\quad \\forall x \\in \\mathbb{R}^n.\\]In particular, for any zonotope $Z \\subseteq \\mathbb{R}^{n}$ there exists a zonotope $Z' \\subseteq \\mathbb{R}^{n}$ generated by at most $O(\\frac{n}{\\varepsilon^2} \\log(\\frac{1}{\\varepsilon}))$ segments so that $(1-\\varepsilon) Z \\subset","authors_text":"Thomas Rothvoss, Victor Reis","cross_cats":["cs.DM"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2026-06-26T14:43:15Z","title":"Linear-size $\\ell_1$ sparsifiers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.28147","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:a93a59822a6751380eb8f8a54efedbe3680d085faad690aaae8d93345cec9f93","target":"record","created_at":"2026-06-29T01:15:07Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"07deee118cbfd06b9977779f492340e9b04c4971ff3e041a0b9ca99a0f0f60fc","cross_cats_sorted":["cs.DM"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2026-06-26T14:43:15Z","title_canon_sha256":"c812c402d70e00475c7809a9e2faa791330b632735a6958b50fafc0625b478de"},"schema_version":"1.0","source":{"id":"2606.28147","kind":"arxiv","version":1}},"canonical_sha256":"f3b47b98a49c8d08d00e43aab64e1385625842c94bb070ea0cc6a49ee0e17453","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f3b47b98a49c8d08d00e43aab64e1385625842c94bb070ea0cc6a49ee0e17453","first_computed_at":"2026-06-29T01:15:07.393195Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-29T01:15:07.393195Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QhJO4LA6L5mavw2DK01n4pQeIvIyIyzVXv4RUoid1JYo3AalOUbOADQdnSsRkjPrnV+ya/Y8kNpOMgO6qYaHAw==","signature_status":"signed_v1","signed_at":"2026-06-29T01:15:07.393629Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.28147","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a93a59822a6751380eb8f8a54efedbe3680d085faad690aaae8d93345cec9f93","sha256:8b92c455f06d13db407a94def303e73ee4daf7075ecf4fb9554615be205ec217"],"state_sha256":"15e6ae12df26a7ed46e262b19945d7bba780b0dd7e8f35f41fc4df547f53f776"}