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Then, for every $\\epsilon>0$ and every $\\delta\\in(0,1)$, $$\n  h^K_\\mu(6\\epsilon,\\delta)\\leq \\underline h^{BK}_\\mu(\\epsilon). $$ Combining this estimate with the usual Katok-type variational principle for metric mean dimension gives the corresponding variational principle with lower Brin-Katok local entropy."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.31407","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-05-29T15:11:01Z","cross_cats_sorted":[],"title_canon_sha256":"b90db31926610ffca78becbf53d5f3e5bde54092b3131a173c3c7bc7bc80fce9","abstract_canon_sha256":"597154292141c288c2f8c10ae236c760af77637bf5578026e4e39c9bfc4869d2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-01T02:04:03.812179Z","signature_b64":"wPdc7Bon064sEBLcfrazffNZjXWBGi5MuN/SsVsmbwm5K9BIQbkXEztIDriNHg/Lidp1oEoQ+HxqCQnISiGsDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fbc2e4db24b353b95d7035b8347008a57f70193afa879676ae3727639b62e2b9","last_reissued_at":"2026-06-01T02:04:03.811217Z","signature_status":"signed_v1","first_computed_at":"2026-06-01T02:04:03.811217Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A variational principle for metric mean dimension via lower Brin-Katok local entropy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Ruxi Shi","submitted_at":"2026-05-29T15:11:01Z","abstract_excerpt":"We prove a finite-scale comparison between lower Brin-Katok local entropy and Katok covering entropy. Let $(\\mathcal{X},d,T)$ be a compact metric topological dynamical system and let $\\mu$ be ergodic. Then, for every $\\epsilon>0$ and every $\\delta\\in(0,1)$, $$\n  h^K_\\mu(6\\epsilon,\\delta)\\leq \\underline h^{BK}_\\mu(\\epsilon). $$ Combining this estimate with the usual Katok-type variational principle for metric mean dimension gives the corresponding variational principle with lower Brin-Katok local entropy."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.31407","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/2605.31407/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2605.31407","created_at":"2026-06-01T02:04:03.811385+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.31407v1","created_at":"2026-06-01T02:04:03.811385+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.31407","created_at":"2026-06-01T02:04:03.811385+00:00"},{"alias_kind":"pith_short_12","alias_value":"7PBOJWZEWNJ3","created_at":"2026-06-01T02:04:03.811385+00:00"},{"alias_kind":"pith_short_16","alias_value":"7PBOJWZEWNJ3SXLQ","created_at":"2026-06-01T02:04:03.811385+00:00"},{"alias_kind":"pith_short_8","alias_value":"7PBOJWZE","created_at":"2026-06-01T02:04:03.811385+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7PBOJWZEWNJ3SXLQGW4DI4AIUV","json":"https://pith.science/pith/7PBOJWZEWNJ3SXLQGW4DI4AIUV.json","graph_json":"https://pith.science/api/pith-number/7PBOJWZEWNJ3SXLQGW4DI4AIUV/graph.json","events_json":"https://pith.science/api/pith-number/7PBOJWZEWNJ3SXLQGW4DI4AIUV/events.json","paper":"https://pith.science/paper/7PBOJWZE"},"agent_actions":{"view_html":"https://pith.science/pith/7PBOJWZEWNJ3SXLQGW4DI4AIUV","download_json":"https://pith.science/pith/7PBOJWZEWNJ3SXLQGW4DI4AIUV.json","view_paper":"https://pith.science/paper/7PBOJWZE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.31407&json=true","fetch_graph":"https://pith.science/api/pith-number/7PBOJWZEWNJ3SXLQGW4DI4AIUV/graph.json","fetch_events":"https://pith.science/api/pith-number/7PBOJWZEWNJ3SXLQGW4DI4AIUV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7PBOJWZEWNJ3SXLQGW4DI4AIUV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7PBOJWZEWNJ3SXLQGW4DI4AIUV/action/storage_attestation","attest_author":"https://pith.science/pith/7PBOJWZEWNJ3SXLQGW4DI4AIUV/action/author_attestation","sign_citation":"https://pith.science/pith/7PBOJWZEWNJ3SXLQGW4DI4AIUV/action/citation_signature","submit_replication":"https://pith.science/pith/7PBOJWZEWNJ3SXLQGW4DI4AIUV/action/replication_record"}},"created_at":"2026-06-01T02:04:03.811385+00:00","updated_at":"2026-06-01T02:04:03.811385+00:00"}