{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:CCBSDMYHXNHZR75PZTLAXPG3GN","short_pith_number":"pith:CCBSDMYH","canonical_record":{"source":{"id":"2607.21144","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CV","submitted_at":"2026-07-23T10:24:37Z","cross_cats_sorted":[],"title_canon_sha256":"4ef76067dc158d08d3b87699e57a820d5102e47e645d5dfd385685f4e31e920c","abstract_canon_sha256":"db8b0e6a6f160b4f321b10121974e0f8444aa3d244a3fa40644a7100f8332be7"},"schema_version":"1.0"},"canonical_sha256":"108321b307bb4f98ffafccd60bbcdb336531232646e5f87fa0e6b5e5514e4713","source":{"kind":"arxiv","id":"2607.21144","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.21144","created_at":"2026-07-24T01:23:55Z"},{"alias_kind":"arxiv_version","alias_value":"2607.21144v1","created_at":"2026-07-24T01:23:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.21144","created_at":"2026-07-24T01:23:55Z"},{"alias_kind":"pith_short_12","alias_value":"CCBSDMYHXNHZ","created_at":"2026-07-24T01:23:55Z"},{"alias_kind":"pith_short_16","alias_value":"CCBSDMYHXNHZR75P","created_at":"2026-07-24T01:23:55Z"},{"alias_kind":"pith_short_8","alias_value":"CCBSDMYH","created_at":"2026-07-24T01:23:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:CCBSDMYHXNHZR75PZTLAXPG3GN","target":"record","payload":{"canonical_record":{"source":{"id":"2607.21144","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CV","submitted_at":"2026-07-23T10:24:37Z","cross_cats_sorted":[],"title_canon_sha256":"4ef76067dc158d08d3b87699e57a820d5102e47e645d5dfd385685f4e31e920c","abstract_canon_sha256":"db8b0e6a6f160b4f321b10121974e0f8444aa3d244a3fa40644a7100f8332be7"},"schema_version":"1.0"},"canonical_sha256":"108321b307bb4f98ffafccd60bbcdb336531232646e5f87fa0e6b5e5514e4713","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-24T01:23:55.376462Z","signature_b64":"S3pQnLrsXJ4tX2WxslYucQbKRBgdzoHbJ3RI8hcLyRMSidkPUPS+N1/FW6XfAopXZs62BZJ0P4ywraPCrdHbDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"108321b307bb4f98ffafccd60bbcdb336531232646e5f87fa0e6b5e5514e4713","last_reissued_at":"2026-07-24T01:23:55.375549Z","signature_status":"signed_v1","first_computed_at":"2026-07-24T01:23:55.375549Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.21144","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-24T01:23:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LjSgvpmacdtZ03dolrPqUk0qDFV+SzOXCtcQxZvlNr+gBlSxEAkR8olg3DQimBMn7YuwfFYxwqNA7zhmXYr1CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T21:42:42.193554Z"},"content_sha256":"0809e7f9788301054477cf0e69dcfae998f6c236e5b73d78e8b1b718032d5d1e","schema_version":"1.0","event_id":"sha256:0809e7f9788301054477cf0e69dcfae998f6c236e5b73d78e8b1b718032d5d1e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:CCBSDMYHXNHZR75PZTLAXPG3GN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Some Integrability Properties of $m$-Subharmonic Functions","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Genglong Lin","submitted_at":"2026-07-23T10:24:37Z","abstract_excerpt":"Let $1\\leq m<n$ and let $u$ be an $m$-subharmonic function on a domain in $\\mathbb{C}^n$. We study three related forms of local integrability: exponential integrability, polynomial integrability, and the sharp exponent predicted by B{\\l}ocki's conjecture. Explicit radial examples show that the direct Guan-Zhou strong-openness statement and the direct Skoda criterion in terms of the $m$-Lelong number both fail when $(m<n)$. We classify a family of radial power-logarithmic singularities and compute its exact $L^p$ intervals, including endpoint behavior.\n  We answer both problems posed by Benali-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.21144","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/2607.21144/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":"2026-07-24T01:23:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"37eftxJyL2s7gYw1Q4SDFNAqAaC3s/B3Wp8gA6aTqWltUMBOEkdI5H428gzLY4TSbkPIB80bt0vY2c+JOiN2CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T21:42:42.194433Z"},"content_sha256":"f6d9ae152873be8d09b522f61a559092e0e6533d89c7d6d3df16f39537942e6c","schema_version":"1.0","event_id":"sha256:f6d9ae152873be8d09b522f61a559092e0e6533d89c7d6d3df16f39537942e6c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CCBSDMYHXNHZR75PZTLAXPG3GN/bundle.json","state_url":"https://pith.science/pith/CCBSDMYHXNHZR75PZTLAXPG3GN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CCBSDMYHXNHZR75PZTLAXPG3GN/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":"2026-08-13T21:42:42Z","links":{"resolver":"https://pith.science/pith/CCBSDMYHXNHZR75PZTLAXPG3GN","bundle":"https://pith.science/pith/CCBSDMYHXNHZR75PZTLAXPG3GN/bundle.json","state":"https://pith.science/pith/CCBSDMYHXNHZR75PZTLAXPG3GN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CCBSDMYHXNHZR75PZTLAXPG3GN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:CCBSDMYHXNHZR75PZTLAXPG3GN","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":"db8b0e6a6f160b4f321b10121974e0f8444aa3d244a3fa40644a7100f8332be7","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CV","submitted_at":"2026-07-23T10:24:37Z","title_canon_sha256":"4ef76067dc158d08d3b87699e57a820d5102e47e645d5dfd385685f4e31e920c"},"schema_version":"1.0","source":{"id":"2607.21144","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.21144","created_at":"2026-07-24T01:23:55Z"},{"alias_kind":"arxiv_version","alias_value":"2607.21144v1","created_at":"2026-07-24T01:23:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.21144","created_at":"2026-07-24T01:23:55Z"},{"alias_kind":"pith_short_12","alias_value":"CCBSDMYHXNHZ","created_at":"2026-07-24T01:23:55Z"},{"alias_kind":"pith_short_16","alias_value":"CCBSDMYHXNHZR75P","created_at":"2026-07-24T01:23:55Z"},{"alias_kind":"pith_short_8","alias_value":"CCBSDMYH","created_at":"2026-07-24T01:23:55Z"}],"graph_snapshots":[{"event_id":"sha256:f6d9ae152873be8d09b522f61a559092e0e6533d89c7d6d3df16f39537942e6c","target":"graph","created_at":"2026-07-24T01:23:55Z","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/2607.21144/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $1\\leq m<n$ and let $u$ be an $m$-subharmonic function on a domain in $\\mathbb{C}^n$. We study three related forms of local integrability: exponential integrability, polynomial integrability, and the sharp exponent predicted by B{\\l}ocki's conjecture. Explicit radial examples show that the direct Guan-Zhou strong-openness statement and the direct Skoda criterion in terms of the $m$-Lelong number both fail when $(m<n)$. We classify a family of radial power-logarithmic singularities and compute its exact $L^p$ intervals, including endpoint behavior.\n  We answer both problems posed by Benali-","authors_text":"Genglong Lin","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CV","submitted_at":"2026-07-23T10:24:37Z","title":"Some Integrability Properties of $m$-Subharmonic Functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.21144","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":"sha256:0809e7f9788301054477cf0e69dcfae998f6c236e5b73d78e8b1b718032d5d1e","target":"record","created_at":"2026-07-24T01:23:55Z","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":"db8b0e6a6f160b4f321b10121974e0f8444aa3d244a3fa40644a7100f8332be7","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CV","submitted_at":"2026-07-23T10:24:37Z","title_canon_sha256":"4ef76067dc158d08d3b87699e57a820d5102e47e645d5dfd385685f4e31e920c"},"schema_version":"1.0","source":{"id":"2607.21144","kind":"arxiv","version":1}},"canonical_sha256":"108321b307bb4f98ffafccd60bbcdb336531232646e5f87fa0e6b5e5514e4713","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"108321b307bb4f98ffafccd60bbcdb336531232646e5f87fa0e6b5e5514e4713","first_computed_at":"2026-07-24T01:23:55.375549Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-24T01:23:55.375549Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"S3pQnLrsXJ4tX2WxslYucQbKRBgdzoHbJ3RI8hcLyRMSidkPUPS+N1/FW6XfAopXZs62BZJ0P4ywraPCrdHbDg==","signature_status":"signed_v1","signed_at":"2026-07-24T01:23:55.376462Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.21144","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0809e7f9788301054477cf0e69dcfae998f6c236e5b73d78e8b1b718032d5d1e","sha256:f6d9ae152873be8d09b522f61a559092e0e6533d89c7d6d3df16f39537942e6c"],"state_sha256":"1a0b27d648bf298694cdd496b46666279d71ef8a98507e891767ba3f3a13af7b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EGN8RhbqQNFtD7Sok7gfXXarGgXAs2tnC049bTXRqKDuBgL/Abunyy9zK4iNZVRlar5YThPFnYLtS8Tfl+IZAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T21:42:42.484452Z","bundle_sha256":"b82a0ad059627f54c93d9e6630c4d96743bf07c4d9595bf25cc858afb6716b99"}}