{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:PWMEWOTG5MYYF64HKKS5INTIDF","short_pith_number":"pith:PWMEWOTG","canonical_record":{"source":{"id":"2509.01194","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2025-09-01T07:24:40Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"3b417dcbe76c97218db5bab5a7108c10e15cc42593e5d6ba238887ce2395de38","abstract_canon_sha256":"1d8b75ff805ed27f701b50e48ce18c541cd9b1f80d18b8cad0cb2af8b7d28b5d"},"schema_version":"1.0"},"canonical_sha256":"7d984b3a66eb3182fb8752a5d436681953ee5efe65743e874fbcaf486880ce52","source":{"kind":"arxiv","id":"2509.01194","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.01194","created_at":"2026-06-01T02:03:26Z"},{"alias_kind":"arxiv_version","alias_value":"2509.01194v2","created_at":"2026-06-01T02:03:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.01194","created_at":"2026-06-01T02:03:26Z"},{"alias_kind":"pith_short_12","alias_value":"PWMEWOTG5MYY","created_at":"2026-06-01T02:03:26Z"},{"alias_kind":"pith_short_16","alias_value":"PWMEWOTG5MYYF64H","created_at":"2026-06-01T02:03:26Z"},{"alias_kind":"pith_short_8","alias_value":"PWMEWOTG","created_at":"2026-06-01T02:03:26Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:PWMEWOTG5MYYF64HKKS5INTIDF","target":"record","payload":{"canonical_record":{"source":{"id":"2509.01194","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2025-09-01T07:24:40Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"3b417dcbe76c97218db5bab5a7108c10e15cc42593e5d6ba238887ce2395de38","abstract_canon_sha256":"1d8b75ff805ed27f701b50e48ce18c541cd9b1f80d18b8cad0cb2af8b7d28b5d"},"schema_version":"1.0"},"canonical_sha256":"7d984b3a66eb3182fb8752a5d436681953ee5efe65743e874fbcaf486880ce52","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-01T02:03:26.067863Z","signature_b64":"f0ovVoRogpEyQH+QmeKvl3Xa090wf20eznhf1elg2pQnYNfvJaw3FpQzVXNxAQUBe35XnoQjy1dEJ6sP8YY0BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7d984b3a66eb3182fb8752a5d436681953ee5efe65743e874fbcaf486880ce52","last_reissued_at":"2026-06-01T02:03:26.066688Z","signature_status":"signed_v1","first_computed_at":"2026-06-01T02:03:26.066688Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2509.01194","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-01T02:03:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hr5KiBtfUY2Z5y1bofq8fsoQ+PCZb6tybPNNNUZAiZ36T4Bdl1v7BGrmNkEhaSLAWS6KxTwj4UYI90+MJBIeDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T09:57:54.889561Z"},"content_sha256":"1b3bc4285913e4b05857c080d3850f9537be8ecdf8b2f90cb90c2e100728867a","schema_version":"1.0","event_id":"sha256:1b3bc4285913e4b05857c080d3850f9537be8ecdf8b2f90cb90c2e100728867a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:PWMEWOTG5MYYF64HKKS5INTIDF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On infinity thick quasiconvexity and applications","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.MG","authors_text":"Miguel Garc\\'ia-Bravo, Toni Ikonen, Zheng Zhu","submitted_at":"2025-09-01T07:24:40Z","abstract_excerpt":"We investigate geometric properties of a metric measure space where every function in the Newton--Sobolev space $N^{1,\\infty}(Z)$ has a Lipschitz representative. We prove that when the metric space is locally complete and the reference measure is infinitesimally doubling, the above property is equivalent to the space being very $\\infty$-thick quasiconvex up to a scale. That is, up to some scale, every pair of points can be joined by a family of quasiconvex curves that is not negligible for the $\\infty$-modulus.\n  As a first application, we prove a local-to-global improvement for the weak $(1,\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.01194","kind":"arxiv","version":2},"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/2509.01194/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-01T02:03:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jq8XyNqCv96kWgF5QPq1EhdHt9M/M1gf2PHHYlnMK0clafF9+8+vd/0P9bMztz/v/R051ciXisCXBZIBgYidDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T09:57:54.889953Z"},"content_sha256":"33f76ac289e6c5ee2207df59a9bbedb7306fdf8bb926fa919ba6462e2ed8f2dd","schema_version":"1.0","event_id":"sha256:33f76ac289e6c5ee2207df59a9bbedb7306fdf8bb926fa919ba6462e2ed8f2dd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PWMEWOTG5MYYF64HKKS5INTIDF/bundle.json","state_url":"https://pith.science/pith/PWMEWOTG5MYYF64HKKS5INTIDF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PWMEWOTG5MYYF64HKKS5INTIDF/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-05T09:57:54Z","links":{"resolver":"https://pith.science/pith/PWMEWOTG5MYYF64HKKS5INTIDF","bundle":"https://pith.science/pith/PWMEWOTG5MYYF64HKKS5INTIDF/bundle.json","state":"https://pith.science/pith/PWMEWOTG5MYYF64HKKS5INTIDF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PWMEWOTG5MYYF64HKKS5INTIDF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:PWMEWOTG5MYYF64HKKS5INTIDF","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":"1d8b75ff805ed27f701b50e48ce18c541cd9b1f80d18b8cad0cb2af8b7d28b5d","cross_cats_sorted":["math.FA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2025-09-01T07:24:40Z","title_canon_sha256":"3b417dcbe76c97218db5bab5a7108c10e15cc42593e5d6ba238887ce2395de38"},"schema_version":"1.0","source":{"id":"2509.01194","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.01194","created_at":"2026-06-01T02:03:26Z"},{"alias_kind":"arxiv_version","alias_value":"2509.01194v2","created_at":"2026-06-01T02:03:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.01194","created_at":"2026-06-01T02:03:26Z"},{"alias_kind":"pith_short_12","alias_value":"PWMEWOTG5MYY","created_at":"2026-06-01T02:03:26Z"},{"alias_kind":"pith_short_16","alias_value":"PWMEWOTG5MYYF64H","created_at":"2026-06-01T02:03:26Z"},{"alias_kind":"pith_short_8","alias_value":"PWMEWOTG","created_at":"2026-06-01T02:03:26Z"}],"graph_snapshots":[{"event_id":"sha256:33f76ac289e6c5ee2207df59a9bbedb7306fdf8bb926fa919ba6462e2ed8f2dd","target":"graph","created_at":"2026-06-01T02:03:26Z","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/2509.01194/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We investigate geometric properties of a metric measure space where every function in the Newton--Sobolev space $N^{1,\\infty}(Z)$ has a Lipschitz representative. We prove that when the metric space is locally complete and the reference measure is infinitesimally doubling, the above property is equivalent to the space being very $\\infty$-thick quasiconvex up to a scale. That is, up to some scale, every pair of points can be joined by a family of quasiconvex curves that is not negligible for the $\\infty$-modulus.\n  As a first application, we prove a local-to-global improvement for the weak $(1,\\","authors_text":"Miguel Garc\\'ia-Bravo, Toni Ikonen, Zheng Zhu","cross_cats":["math.FA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2025-09-01T07:24:40Z","title":"On infinity thick quasiconvexity and applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.01194","kind":"arxiv","version":2},"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":"sha256:1b3bc4285913e4b05857c080d3850f9537be8ecdf8b2f90cb90c2e100728867a","target":"record","created_at":"2026-06-01T02:03:26Z","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":"1d8b75ff805ed27f701b50e48ce18c541cd9b1f80d18b8cad0cb2af8b7d28b5d","cross_cats_sorted":["math.FA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2025-09-01T07:24:40Z","title_canon_sha256":"3b417dcbe76c97218db5bab5a7108c10e15cc42593e5d6ba238887ce2395de38"},"schema_version":"1.0","source":{"id":"2509.01194","kind":"arxiv","version":2}},"canonical_sha256":"7d984b3a66eb3182fb8752a5d436681953ee5efe65743e874fbcaf486880ce52","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7d984b3a66eb3182fb8752a5d436681953ee5efe65743e874fbcaf486880ce52","first_computed_at":"2026-06-01T02:03:26.066688Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-01T02:03:26.066688Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"f0ovVoRogpEyQH+QmeKvl3Xa090wf20eznhf1elg2pQnYNfvJaw3FpQzVXNxAQUBe35XnoQjy1dEJ6sP8YY0BA==","signature_status":"signed_v1","signed_at":"2026-06-01T02:03:26.067863Z","signed_message":"canonical_sha256_bytes"},"source_id":"2509.01194","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1b3bc4285913e4b05857c080d3850f9537be8ecdf8b2f90cb90c2e100728867a","sha256:33f76ac289e6c5ee2207df59a9bbedb7306fdf8bb926fa919ba6462e2ed8f2dd"],"state_sha256":"5091f81dd4164d059851455cc99f10366a401b54621301f4f7bc79ea6b0bd7b6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Dyidswp+VNc+uVk9l5IGM8e32IxE1zhp8TK7/jNxEdKzhacLda2TmO4FEFGEz65U0PWeTcSbFj7/2Yrw88XRAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T09:57:54.892260Z","bundle_sha256":"8c762e405ecf3923e550488b257a1e7beecff074afb3562f71db26166d24be35"}}