{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:3H344HOWQ46EXPX4ZNNQXC5KFV","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":"2fdba31594504cd7f150d8ffb050af35983693407a4cec517be182ab3e2a1041","cross_cats_sorted":["math.DG","math.FA","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2015-05-16T22:55:42Z","title_canon_sha256":"2eb87b75bd1a2f6299ec65a993002c47010fdda3d1a0cd44ce0372082b05f825"},"schema_version":"1.0","source":{"id":"1505.04335","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.04335","created_at":"2026-05-18T02:07:25Z"},{"alias_kind":"arxiv_version","alias_value":"1505.04335v1","created_at":"2026-05-18T02:07:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.04335","created_at":"2026-05-18T02:07:25Z"},{"alias_kind":"pith_short_12","alias_value":"3H344HOWQ46E","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_16","alias_value":"3H344HOWQ46EXPX4","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_8","alias_value":"3H344HOW","created_at":"2026-05-18T12:29:02Z"}],"graph_snapshots":[{"event_id":"sha256:860927ae092e0a66f9850ea7320581615356478b2290b4ebb907cdb64917775f","target":"graph","created_at":"2026-05-18T02:07:25Z","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"},"paper":{"abstract_excerpt":"We show that the family of probability measures on the $n$-dimensional unit sphere, having density proportional to: \\[ S^n \\ni y \\mapsto \\frac{1}{|y - x|^{n+\\alpha}}, \\] satisfies the Curvature-Dimension condition $CD(n-1-\\frac{n+\\alpha}{4},-\\alpha)$, for all $|x| < 1$, $\\alpha \\geq -n$ and $n\\geq 2$. The case $\\alpha = 1$ corresponds to the hitting distribution of the sphere by Brownian motion started at $x$ (so-called \"harmonic measure\" on the sphere). Applications involving isoperimetric, spectral-gap and concentration estimates, as well as potential extensions, are discussed.","authors_text":"Emanuel Milman","cross_cats":["math.DG","math.FA","math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2015-05-16T22:55:42Z","title":"Harmonic Measures on the Sphere via Curvature-Dimension"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.04335","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":"sha256:01a35d22ec7bdc7308a56749d8ee020d5fb2a45a14f689d2884c5eadcdd3437e","target":"record","created_at":"2026-05-18T02:07:25Z","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":"2fdba31594504cd7f150d8ffb050af35983693407a4cec517be182ab3e2a1041","cross_cats_sorted":["math.DG","math.FA","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2015-05-16T22:55:42Z","title_canon_sha256":"2eb87b75bd1a2f6299ec65a993002c47010fdda3d1a0cd44ce0372082b05f825"},"schema_version":"1.0","source":{"id":"1505.04335","kind":"arxiv","version":1}},"canonical_sha256":"d9f7ce1dd6873c4bbefccb5b0b8baa2d5271abae8c401b4e6fea40046188b7e1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d9f7ce1dd6873c4bbefccb5b0b8baa2d5271abae8c401b4e6fea40046188b7e1","first_computed_at":"2026-05-18T02:07:25.652909Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:07:25.652909Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5mbVugvQwSUwcXjhPMWcbyxWWDv3PSrSrC6JRxyQkvyfuCUVmXMBX7VcwmUQQDlU1eXw7tp1e6xO4E3/z7jDAg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:07:25.653771Z","signed_message":"canonical_sha256_bytes"},"source_id":"1505.04335","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:01a35d22ec7bdc7308a56749d8ee020d5fb2a45a14f689d2884c5eadcdd3437e","sha256:860927ae092e0a66f9850ea7320581615356478b2290b4ebb907cdb64917775f"],"state_sha256":"5d08adedfac3250f1a996bde3e317274a4e15fc94d74c9bc696a4815f663de51"}