{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:FFCBZX6MRVKTSVRH7MZFZXRGV7","short_pith_number":"pith:FFCBZX6M","canonical_record":{"source":{"id":"2108.12034","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-08-26T21:00:52Z","cross_cats_sorted":[],"title_canon_sha256":"a3ef2e08c0a3610a51df96aac8bee5456a2a28725f4ca2f8d9c082b10020f0d2","abstract_canon_sha256":"8a1ffc8bf513074848ec2ffe22f0ce04c4f0e0815ebc03c6b843bbd33b4ead0f"},"schema_version":"1.0"},"canonical_sha256":"29441cdfcc8d55395627fb325cde26afd755ee2b6fa7c89ee45e88c670ffa504","source":{"kind":"arxiv","id":"2108.12034","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2108.12034","created_at":"2026-07-05T05:06:52Z"},{"alias_kind":"arxiv_version","alias_value":"2108.12034v2","created_at":"2026-07-05T05:06:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2108.12034","created_at":"2026-07-05T05:06:52Z"},{"alias_kind":"pith_short_12","alias_value":"FFCBZX6MRVKT","created_at":"2026-07-05T05:06:52Z"},{"alias_kind":"pith_short_16","alias_value":"FFCBZX6MRVKTSVRH","created_at":"2026-07-05T05:06:52Z"},{"alias_kind":"pith_short_8","alias_value":"FFCBZX6M","created_at":"2026-07-05T05:06:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:FFCBZX6MRVKTSVRH7MZFZXRGV7","target":"record","payload":{"canonical_record":{"source":{"id":"2108.12034","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-08-26T21:00:52Z","cross_cats_sorted":[],"title_canon_sha256":"a3ef2e08c0a3610a51df96aac8bee5456a2a28725f4ca2f8d9c082b10020f0d2","abstract_canon_sha256":"8a1ffc8bf513074848ec2ffe22f0ce04c4f0e0815ebc03c6b843bbd33b4ead0f"},"schema_version":"1.0"},"canonical_sha256":"29441cdfcc8d55395627fb325cde26afd755ee2b6fa7c89ee45e88c670ffa504","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:06:52.047274Z","signature_b64":"j2Fqu7amLIXA0nnmsfDRWiTKr2RdrfcD3ZxsqsltcqyfWI4fIssHVaplqUmdwIwrxvg+JuPtJYPVA1Yq9UOWBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"29441cdfcc8d55395627fb325cde26afd755ee2b6fa7c89ee45e88c670ffa504","last_reissued_at":"2026-07-05T05:06:52.046915Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:06:52.046915Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2108.12034","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":"2026-07-05T05:06:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FtpVYae0aBDXfwMsozXWEyDmifOsCQWkvkkkYF8AfaK4tHLH7S5xGnIIYuHsFL1mJcFJu/15ef6V0y8l4QuLCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T18:06:34.367607Z"},"content_sha256":"6ff31ba23694fafafac5229fc8ff573086c3c1763fca55f38d2249b16c13f55f","schema_version":"1.0","event_id":"sha256:6ff31ba23694fafafac5229fc8ff573086c3c1763fca55f38d2249b16c13f55f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:FFCBZX6MRVKTSVRH7MZFZXRGV7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Optimal Point Sets Determining Few Distinct Angles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Charles Wolf, Ethan Pesikoff, Eyvindur A. Palsson, Henry L. Fleischmann, Steven J. Miller","submitted_at":"2021-08-26T21:00:52Z","abstract_excerpt":"We characterize the largest point sets in the plane which define at most 1, 2, and 3 angles. For $P(k)$ the largest size of a point set admitting at most $k$ angles, we prove $P(2)=5$ and $P(3)=5$. We also provide the general bounds of $k+2 \\leq P(k) \\leq 6k$, although the upper bound may be improved pending progress toward the Weak Dirac Conjecture. Notably, it is surprising that $P(k)=\\Theta(k)$ since, in the distance setting, the best known upper bound on the analogous quantity is quadratic and no lower bound is well-understood."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.12034","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/2108.12034/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":"2026-07-05T05:06:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xy2EPBHm35JzZgnw383IrOl6XoR4OJvJ4PRRZ5xVADELBYSRgktT8tik/Z+KUVw8YTwfH2LJ+HJYNEWLWJFHBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T18:06:34.367987Z"},"content_sha256":"e68cc731b074ffc6ff2e715d37334b6fb758f0c726bbb6826111d5604cc0821d","schema_version":"1.0","event_id":"sha256:e68cc731b074ffc6ff2e715d37334b6fb758f0c726bbb6826111d5604cc0821d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FFCBZX6MRVKTSVRH7MZFZXRGV7/bundle.json","state_url":"https://pith.science/pith/FFCBZX6MRVKTSVRH7MZFZXRGV7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FFCBZX6MRVKTSVRH7MZFZXRGV7/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":"2026-07-06T18:06:34Z","links":{"resolver":"https://pith.science/pith/FFCBZX6MRVKTSVRH7MZFZXRGV7","bundle":"https://pith.science/pith/FFCBZX6MRVKTSVRH7MZFZXRGV7/bundle.json","state":"https://pith.science/pith/FFCBZX6MRVKTSVRH7MZFZXRGV7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FFCBZX6MRVKTSVRH7MZFZXRGV7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:FFCBZX6MRVKTSVRH7MZFZXRGV7","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":"8a1ffc8bf513074848ec2ffe22f0ce04c4f0e0815ebc03c6b843bbd33b4ead0f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-08-26T21:00:52Z","title_canon_sha256":"a3ef2e08c0a3610a51df96aac8bee5456a2a28725f4ca2f8d9c082b10020f0d2"},"schema_version":"1.0","source":{"id":"2108.12034","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2108.12034","created_at":"2026-07-05T05:06:52Z"},{"alias_kind":"arxiv_version","alias_value":"2108.12034v2","created_at":"2026-07-05T05:06:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2108.12034","created_at":"2026-07-05T05:06:52Z"},{"alias_kind":"pith_short_12","alias_value":"FFCBZX6MRVKT","created_at":"2026-07-05T05:06:52Z"},{"alias_kind":"pith_short_16","alias_value":"FFCBZX6MRVKTSVRH","created_at":"2026-07-05T05:06:52Z"},{"alias_kind":"pith_short_8","alias_value":"FFCBZX6M","created_at":"2026-07-05T05:06:52Z"}],"graph_snapshots":[{"event_id":"sha256:e68cc731b074ffc6ff2e715d37334b6fb758f0c726bbb6826111d5604cc0821d","target":"graph","created_at":"2026-07-05T05:06:52Z","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/2108.12034/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We characterize the largest point sets in the plane which define at most 1, 2, and 3 angles. For $P(k)$ the largest size of a point set admitting at most $k$ angles, we prove $P(2)=5$ and $P(3)=5$. We also provide the general bounds of $k+2 \\leq P(k) \\leq 6k$, although the upper bound may be improved pending progress toward the Weak Dirac Conjecture. Notably, it is surprising that $P(k)=\\Theta(k)$ since, in the distance setting, the best known upper bound on the analogous quantity is quadratic and no lower bound is well-understood.","authors_text":"Charles Wolf, Ethan Pesikoff, Eyvindur A. Palsson, Henry L. Fleischmann, Steven J. Miller","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-08-26T21:00:52Z","title":"Optimal Point Sets Determining Few Distinct Angles"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.12034","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":"sha256:6ff31ba23694fafafac5229fc8ff573086c3c1763fca55f38d2249b16c13f55f","target":"record","created_at":"2026-07-05T05:06:52Z","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":"8a1ffc8bf513074848ec2ffe22f0ce04c4f0e0815ebc03c6b843bbd33b4ead0f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-08-26T21:00:52Z","title_canon_sha256":"a3ef2e08c0a3610a51df96aac8bee5456a2a28725f4ca2f8d9c082b10020f0d2"},"schema_version":"1.0","source":{"id":"2108.12034","kind":"arxiv","version":2}},"canonical_sha256":"29441cdfcc8d55395627fb325cde26afd755ee2b6fa7c89ee45e88c670ffa504","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"29441cdfcc8d55395627fb325cde26afd755ee2b6fa7c89ee45e88c670ffa504","first_computed_at":"2026-07-05T05:06:52.046915Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:06:52.046915Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"j2Fqu7amLIXA0nnmsfDRWiTKr2RdrfcD3ZxsqsltcqyfWI4fIssHVaplqUmdwIwrxvg+JuPtJYPVA1Yq9UOWBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T05:06:52.047274Z","signed_message":"canonical_sha256_bytes"},"source_id":"2108.12034","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6ff31ba23694fafafac5229fc8ff573086c3c1763fca55f38d2249b16c13f55f","sha256:e68cc731b074ffc6ff2e715d37334b6fb758f0c726bbb6826111d5604cc0821d"],"state_sha256":"f5982b752ef8e305dc17b0903b5547aebdf520f0b902f578099f1f974af98f1d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fLYIrjHkFUa8Fep9QFWE0dwGVDZMNouOskEf4ljfZmpV60Aj6HXT6XoLOiZYrZCBECS3KmJaheQoNW1nP8sQDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-06T18:06:34.369911Z","bundle_sha256":"24282928c684c74cdbc8a34351a59a696d98d6f47e061e99181a1146d90be25c"}}