{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:BJ3Q2SKVTOQZZB4YXGQIGH2EIJ","short_pith_number":"pith:BJ3Q2SKV","canonical_record":{"source":{"id":"1507.07993","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-07-29T00:22:03Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"036f24261576d001e0b601d0b74159a95e77cf8e99b17286f06fc6123fece235","abstract_canon_sha256":"70d6f0f298602efc0c7860c04cf5aca2b04aa516be1ddcd3c438c098cb76ff4b"},"schema_version":"1.0"},"canonical_sha256":"0a770d49559ba19c8798b9a0831f44424f39032f886163041f35edc9bb940cf7","source":{"kind":"arxiv","id":"1507.07993","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.07993","created_at":"2026-05-18T01:36:08Z"},{"alias_kind":"arxiv_version","alias_value":"1507.07993v1","created_at":"2026-05-18T01:36:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.07993","created_at":"2026-05-18T01:36:08Z"},{"alias_kind":"pith_short_12","alias_value":"BJ3Q2SKVTOQZ","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_16","alias_value":"BJ3Q2SKVTOQZZB4Y","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_8","alias_value":"BJ3Q2SKV","created_at":"2026-05-18T12:29:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:BJ3Q2SKVTOQZZB4YXGQIGH2EIJ","target":"record","payload":{"canonical_record":{"source":{"id":"1507.07993","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-07-29T00:22:03Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"036f24261576d001e0b601d0b74159a95e77cf8e99b17286f06fc6123fece235","abstract_canon_sha256":"70d6f0f298602efc0c7860c04cf5aca2b04aa516be1ddcd3c438c098cb76ff4b"},"schema_version":"1.0"},"canonical_sha256":"0a770d49559ba19c8798b9a0831f44424f39032f886163041f35edc9bb940cf7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:08.716189Z","signature_b64":"n91zSsfMo9Gop/GW5jIXXG3TpwloNsFKmAZrX2a9herXH5PZC8WKNh9JK4GTv3m/KNezEJqTNw1F4Ab14ZsQAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0a770d49559ba19c8798b9a0831f44424f39032f886163041f35edc9bb940cf7","last_reissued_at":"2026-05-18T01:36:08.715608Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:08.715608Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.07993","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":"2026-05-18T01:36:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TXkg0M7W6AYEEluE5JQ2m2T60HDnrSZBiLjnHxKGL8nbn3zD2E/4Bd1ZlRQRVo7tkJ2eoZuLQIAWiNLmJ3dxAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T09:50:57.986134Z"},"content_sha256":"31004abe00e1147ef095c5e1a6bfc1a0109bf5e97be23c2b32006499376eb1ab","schema_version":"1.0","event_id":"sha256:31004abe00e1147ef095c5e1a6bfc1a0109bf5e97be23c2b32006499376eb1ab"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:BJ3Q2SKVTOQZZB4YXGQIGH2EIJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Thermodynamic expansion to arbitrary moduli","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Alex Kontorovich, Jean Bourgain, Michael Magee","submitted_at":"2015-07-29T00:22:03Z","abstract_excerpt":"We extend the thermodynamic expansion results in [BourgainGamburdSarnak2011, MageeOhWinter2015] from square-free to arbitrary moduli by developing a novel decoupling technique and applying [BourgainVarju2012]."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.07993","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":""},"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":"2026-05-18T01:36:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JVeud8mWcq2GblbpjX0DGFqxk3CXUDY8v64jGIvRWwe6Yw5Rmdb9kT/FtPmrFcu0B/V8f8jauOgjlHpnpaD/AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T09:50:57.986502Z"},"content_sha256":"5adbe3675c17763eb0bbdff32e1147563723e6b3bea20d449217931c156f618b","schema_version":"1.0","event_id":"sha256:5adbe3675c17763eb0bbdff32e1147563723e6b3bea20d449217931c156f618b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BJ3Q2SKVTOQZZB4YXGQIGH2EIJ/bundle.json","state_url":"https://pith.science/pith/BJ3Q2SKVTOQZZB4YXGQIGH2EIJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BJ3Q2SKVTOQZZB4YXGQIGH2EIJ/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":"2026-05-28T09:50:57Z","links":{"resolver":"https://pith.science/pith/BJ3Q2SKVTOQZZB4YXGQIGH2EIJ","bundle":"https://pith.science/pith/BJ3Q2SKVTOQZZB4YXGQIGH2EIJ/bundle.json","state":"https://pith.science/pith/BJ3Q2SKVTOQZZB4YXGQIGH2EIJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BJ3Q2SKVTOQZZB4YXGQIGH2EIJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:BJ3Q2SKVTOQZZB4YXGQIGH2EIJ","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":"70d6f0f298602efc0c7860c04cf5aca2b04aa516be1ddcd3c438c098cb76ff4b","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-07-29T00:22:03Z","title_canon_sha256":"036f24261576d001e0b601d0b74159a95e77cf8e99b17286f06fc6123fece235"},"schema_version":"1.0","source":{"id":"1507.07993","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.07993","created_at":"2026-05-18T01:36:08Z"},{"alias_kind":"arxiv_version","alias_value":"1507.07993v1","created_at":"2026-05-18T01:36:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.07993","created_at":"2026-05-18T01:36:08Z"},{"alias_kind":"pith_short_12","alias_value":"BJ3Q2SKVTOQZ","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_16","alias_value":"BJ3Q2SKVTOQZZB4Y","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_8","alias_value":"BJ3Q2SKV","created_at":"2026-05-18T12:29:14Z"}],"graph_snapshots":[{"event_id":"sha256:5adbe3675c17763eb0bbdff32e1147563723e6b3bea20d449217931c156f618b","target":"graph","created_at":"2026-05-18T01:36:08Z","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 extend the thermodynamic expansion results in [BourgainGamburdSarnak2011, MageeOhWinter2015] from square-free to arbitrary moduli by developing a novel decoupling technique and applying [BourgainVarju2012].","authors_text":"Alex Kontorovich, Jean Bourgain, Michael Magee","cross_cats":["math.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-07-29T00:22:03Z","title":"Thermodynamic expansion to arbitrary moduli"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.07993","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":"sha256:31004abe00e1147ef095c5e1a6bfc1a0109bf5e97be23c2b32006499376eb1ab","target":"record","created_at":"2026-05-18T01:36:08Z","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":"70d6f0f298602efc0c7860c04cf5aca2b04aa516be1ddcd3c438c098cb76ff4b","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-07-29T00:22:03Z","title_canon_sha256":"036f24261576d001e0b601d0b74159a95e77cf8e99b17286f06fc6123fece235"},"schema_version":"1.0","source":{"id":"1507.07993","kind":"arxiv","version":1}},"canonical_sha256":"0a770d49559ba19c8798b9a0831f44424f39032f886163041f35edc9bb940cf7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0a770d49559ba19c8798b9a0831f44424f39032f886163041f35edc9bb940cf7","first_computed_at":"2026-05-18T01:36:08.715608Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:36:08.715608Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"n91zSsfMo9Gop/GW5jIXXG3TpwloNsFKmAZrX2a9herXH5PZC8WKNh9JK4GTv3m/KNezEJqTNw1F4Ab14ZsQAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:36:08.716189Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.07993","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:31004abe00e1147ef095c5e1a6bfc1a0109bf5e97be23c2b32006499376eb1ab","sha256:5adbe3675c17763eb0bbdff32e1147563723e6b3bea20d449217931c156f618b"],"state_sha256":"997a9ba21c11050e0d70e806ce6ac77fee340e6433cf3618aeff22f330937bf5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oORLDHEsmcSXMrxzhwda8jZ33vE0402S94HEUewU/RmQeAwR0DgXt4ERpa7FRiYR1qadp12tJ7dUH4MFAiaQDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T09:50:57.988638Z","bundle_sha256":"3f7c32ef17815f3327357e263c9947736ac15e96ce1f82c480516af21992b76b"}}