{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:7NUZYYBB7ILYBVGVQX4JFGHX5J","short_pith_number":"pith:7NUZYYBB","canonical_record":{"source":{"id":"2608.08893","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-08-09T20:14:04Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"cea2b52e8a4e44950c4e8f40172218e37c7a11bee67bf21d21948165adc0fbae","abstract_canon_sha256":"a3b8155ef65f0b0eb84d019d80d127cb078c23ea34a24828a557845909a620d0"},"schema_version":"1.0"},"canonical_sha256":"fb699c6021fa1780d4d585f89298f7ea6e47bb6e9ad855bb2e0effe12cee40e9","source":{"kind":"arxiv","id":"2608.08893","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.08893","created_at":"2026-08-11T01:25:38Z"},{"alias_kind":"arxiv_version","alias_value":"2608.08893v1","created_at":"2026-08-11T01:25:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.08893","created_at":"2026-08-11T01:25:38Z"},{"alias_kind":"pith_short_12","alias_value":"7NUZYYBB7ILY","created_at":"2026-08-11T01:25:38Z"},{"alias_kind":"pith_short_16","alias_value":"7NUZYYBB7ILYBVGV","created_at":"2026-08-11T01:25:38Z"},{"alias_kind":"pith_short_8","alias_value":"7NUZYYBB","created_at":"2026-08-11T01:25:38Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:7NUZYYBB7ILYBVGVQX4JFGHX5J","target":"record","payload":{"canonical_record":{"source":{"id":"2608.08893","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-08-09T20:14:04Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"cea2b52e8a4e44950c4e8f40172218e37c7a11bee67bf21d21948165adc0fbae","abstract_canon_sha256":"a3b8155ef65f0b0eb84d019d80d127cb078c23ea34a24828a557845909a620d0"},"schema_version":"1.0"},"canonical_sha256":"fb699c6021fa1780d4d585f89298f7ea6e47bb6e9ad855bb2e0effe12cee40e9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-11T01:25:38.642153Z","signature_b64":"oWbVLEHM9G4Ep4lwPdj2oE3BMIIhXW8YA76RewqL2cSCZFRRTTk5l4xeqXlGvoceGqAye/sjMGLWgpSp9Jr2Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fb699c6021fa1780d4d585f89298f7ea6e47bb6e9ad855bb2e0effe12cee40e9","last_reissued_at":"2026-08-11T01:25:38.639603Z","signature_status":"signed_v1","first_computed_at":"2026-08-11T01:25:38.639603Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2608.08893","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":"2026-08-11T01:25:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SdGjIFmK8dmpHqeFHvNH6CCOM65o6Gqd3ozuU9ZG/GkepvsMTYnx5zT20htwK4m8NcyzXK/deAlfoasH+NH+Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T12:23:36.573002Z"},"content_sha256":"6cc33c7a472fc33a924c4649898c8a96c64fff96f91a6ddb4d8a2ae494895b6e","schema_version":"1.0","event_id":"sha256:6cc33c7a472fc33a924c4649898c8a96c64fff96f91a6ddb4d8a2ae494895b6e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:7NUZYYBB7ILYBVGVQX4JFGHX5J","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sharp $\\ell^p$-Improving Estimates for Fixed-Radius Discrete Spherical Averages","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CA","authors_text":"Fan Yang, Rui Han","submitted_at":"2026-08-09T20:14:04Z","abstract_excerpt":"Let $d\\geq 4$ and let $R>0$. When $d=4$, assume that $R^2\\in\\mathbb{N}\\setminus 4\\mathbb{N}$; when $d\\geq 5$, let $R^2\\in\\mathbb{N}$ be arbitrary. We prove the fixed-radius estimate $$\\|A_R f\\|_{\\ell^{p'}(\\mathbb{Z}^d)}\\leq C_{d,p,\\varepsilon}R^{-d(2/p-1)+\\varepsilon}\\|f\\|_{\\ell^p(\\mathbb{Z}^d)}$$ for $(d+2)/d\\leq p\\leq 2$, where $p'$ is the H\\\"older conjugate exponent of $p$ and $A_R$ is the probability average over the lattice sphere of radius $R$. This extends the fixed-radius estimates of Kesler--Lacey and Hughes to the sharp lower endpoint $p=(d+2)/d$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.08893","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/2608.08893/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":"2026-08-11T01:25:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Y3pKkR6wiIgpP8reQblBnN5qqthee6x+jva2hhPBzIxp2R6JopR6iyN3hQET9VEOiQRvh9RBEqtv6Pq6APYoBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T12:23:36.573593Z"},"content_sha256":"0f7ac6e70f770edbbfa0113b4b07106cedb522b13914456d01cc22ccfe5c8ca8","schema_version":"1.0","event_id":"sha256:0f7ac6e70f770edbbfa0113b4b07106cedb522b13914456d01cc22ccfe5c8ca8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7NUZYYBB7ILYBVGVQX4JFGHX5J/bundle.json","state_url":"https://pith.science/pith/7NUZYYBB7ILYBVGVQX4JFGHX5J/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7NUZYYBB7ILYBVGVQX4JFGHX5J/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":"2026-08-20T12:23:36Z","links":{"resolver":"https://pith.science/pith/7NUZYYBB7ILYBVGVQX4JFGHX5J","bundle":"https://pith.science/pith/7NUZYYBB7ILYBVGVQX4JFGHX5J/bundle.json","state":"https://pith.science/pith/7NUZYYBB7ILYBVGVQX4JFGHX5J/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7NUZYYBB7ILYBVGVQX4JFGHX5J/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:7NUZYYBB7ILYBVGVQX4JFGHX5J","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":"a3b8155ef65f0b0eb84d019d80d127cb078c23ea34a24828a557845909a620d0","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-08-09T20:14:04Z","title_canon_sha256":"cea2b52e8a4e44950c4e8f40172218e37c7a11bee67bf21d21948165adc0fbae"},"schema_version":"1.0","source":{"id":"2608.08893","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.08893","created_at":"2026-08-11T01:25:38Z"},{"alias_kind":"arxiv_version","alias_value":"2608.08893v1","created_at":"2026-08-11T01:25:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.08893","created_at":"2026-08-11T01:25:38Z"},{"alias_kind":"pith_short_12","alias_value":"7NUZYYBB7ILY","created_at":"2026-08-11T01:25:38Z"},{"alias_kind":"pith_short_16","alias_value":"7NUZYYBB7ILYBVGV","created_at":"2026-08-11T01:25:38Z"},{"alias_kind":"pith_short_8","alias_value":"7NUZYYBB","created_at":"2026-08-11T01:25:38Z"}],"graph_snapshots":[{"event_id":"sha256:0f7ac6e70f770edbbfa0113b4b07106cedb522b13914456d01cc22ccfe5c8ca8","target":"graph","created_at":"2026-08-11T01:25:38Z","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/2608.08893/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $d\\geq 4$ and let $R>0$. When $d=4$, assume that $R^2\\in\\mathbb{N}\\setminus 4\\mathbb{N}$; when $d\\geq 5$, let $R^2\\in\\mathbb{N}$ be arbitrary. We prove the fixed-radius estimate $$\\|A_R f\\|_{\\ell^{p'}(\\mathbb{Z}^d)}\\leq C_{d,p,\\varepsilon}R^{-d(2/p-1)+\\varepsilon}\\|f\\|_{\\ell^p(\\mathbb{Z}^d)}$$ for $(d+2)/d\\leq p\\leq 2$, where $p'$ is the H\\\"older conjugate exponent of $p$ and $A_R$ is the probability average over the lattice sphere of radius $R$. This extends the fixed-radius estimates of Kesler--Lacey and Hughes to the sharp lower endpoint $p=(d+2)/d$.","authors_text":"Fan Yang, Rui Han","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-08-09T20:14:04Z","title":"Sharp $\\ell^p$-Improving Estimates for Fixed-Radius Discrete Spherical Averages"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.08893","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":"sha256:6cc33c7a472fc33a924c4649898c8a96c64fff96f91a6ddb4d8a2ae494895b6e","target":"record","created_at":"2026-08-11T01:25:38Z","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":"a3b8155ef65f0b0eb84d019d80d127cb078c23ea34a24828a557845909a620d0","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-08-09T20:14:04Z","title_canon_sha256":"cea2b52e8a4e44950c4e8f40172218e37c7a11bee67bf21d21948165adc0fbae"},"schema_version":"1.0","source":{"id":"2608.08893","kind":"arxiv","version":1}},"canonical_sha256":"fb699c6021fa1780d4d585f89298f7ea6e47bb6e9ad855bb2e0effe12cee40e9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fb699c6021fa1780d4d585f89298f7ea6e47bb6e9ad855bb2e0effe12cee40e9","first_computed_at":"2026-08-11T01:25:38.639603Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-11T01:25:38.639603Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"oWbVLEHM9G4Ep4lwPdj2oE3BMIIhXW8YA76RewqL2cSCZFRRTTk5l4xeqXlGvoceGqAye/sjMGLWgpSp9Jr2Ag==","signature_status":"signed_v1","signed_at":"2026-08-11T01:25:38.642153Z","signed_message":"canonical_sha256_bytes"},"source_id":"2608.08893","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6cc33c7a472fc33a924c4649898c8a96c64fff96f91a6ddb4d8a2ae494895b6e","sha256:0f7ac6e70f770edbbfa0113b4b07106cedb522b13914456d01cc22ccfe5c8ca8"],"state_sha256":"82b32c1dcbdefe1c350cbb070eba762406c2fde1719e1e521c14f493fe96b1b2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"A8Jl+F2eTlE8QSSK9M1mCNi0u8ay+jDnLArrIIvIWfi/+QpZhArWl9J3wWfU/gAXIKqJ7P7e0taZCKoNk4mVAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-20T12:23:36.577514Z","bundle_sha256":"f69decb6ea3d83c0743e5a5e96feba8b9a6c2d39251ec8fe72dce2b98dd50f9b"}}