{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:ZWOG46ZSGSSFQ2DXNBRQMGDUEJ","short_pith_number":"pith:ZWOG46ZS","canonical_record":{"source":{"id":"1602.02681","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-08T18:27:50Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"ec1a8d325ddf145121d70976a26b09cf5f26fdb89a0b880927b957764f4f7212","abstract_canon_sha256":"7bd978e7503a9b9723c41b2e75dc7397797f0d3aa64ea25175799f71fc3bc2b2"},"schema_version":"1.0"},"canonical_sha256":"cd9c6e7b3234a45868776863061874226f4bcfb6b42a397cc809667f4506b8ac","source":{"kind":"arxiv","id":"1602.02681","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.02681","created_at":"2026-05-18T01:21:09Z"},{"alias_kind":"arxiv_version","alias_value":"1602.02681v1","created_at":"2026-05-18T01:21:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.02681","created_at":"2026-05-18T01:21:09Z"},{"alias_kind":"pith_short_12","alias_value":"ZWOG46ZSGSSF","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_16","alias_value":"ZWOG46ZSGSSFQ2DX","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_8","alias_value":"ZWOG46ZS","created_at":"2026-05-18T12:30:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:ZWOG46ZSGSSFQ2DXNBRQMGDUEJ","target":"record","payload":{"canonical_record":{"source":{"id":"1602.02681","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-08T18:27:50Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"ec1a8d325ddf145121d70976a26b09cf5f26fdb89a0b880927b957764f4f7212","abstract_canon_sha256":"7bd978e7503a9b9723c41b2e75dc7397797f0d3aa64ea25175799f71fc3bc2b2"},"schema_version":"1.0"},"canonical_sha256":"cd9c6e7b3234a45868776863061874226f4bcfb6b42a397cc809667f4506b8ac","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:09.173896Z","signature_b64":"5qgyBXdKFjzOdGHA25M+GzBjnxpgQYT/vF1hYSuBwXj8ffcg5gJtIc3ehEsk83S2fF3VExEnysdGYGSWrx1SBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cd9c6e7b3234a45868776863061874226f4bcfb6b42a397cc809667f4506b8ac","last_reissued_at":"2026-05-18T01:21:09.173103Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:09.173103Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1602.02681","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":"2026-05-18T01:21:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AK5k0Wr3l1dqurJlytAR1Pg7sWf6DIzwLchNb8JFLSSWAw8AN0Sep+dtD8PZECUR5hNShU4mJDdfliIbBKmfDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T20:31:52.101192Z"},"content_sha256":"e2fa4c7c80589be14ce949aa84d39222d436cf57eef423cdb1a0244a58071be4","schema_version":"1.0","event_id":"sha256:e2fa4c7c80589be14ce949aa84d39222d436cf57eef423cdb1a0244a58071be4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:ZWOG46ZSGSSFQ2DXNBRQMGDUEJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Macdonald's solid-angle sum for real dilations of rational polygons","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Quang-Nhat Le, Sinai Robins","submitted_at":"2016-02-08T18:27:50Z","abstract_excerpt":"The solid-angle sum $A_{\\mathcal{P}} (t)$ of a rational polytope ${\\mathcal{P}} \\subset \\mathbb{R}^d$, with $t \\in \\mathbb{Z}$ was first investigated by I.G. Macdonald. Using our Fourier-analytic methods, we are able to establish an explicit formula for $A_{\\mathcal{P}} (t)$, for any real dilation $t$ and any rational polygon ${\\mathcal{P}} \\subset \\mathbb{R}^2$. Our formulation sheds additional light on previous results, for lattice-point enumerating functions of triangles, which are usually confined to the case of integer dilations. Our approach differs from that of Hardy and Littlewood in 1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.02681","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":"2026-05-18T01:21:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RLwNQhld0aWt1nlsuDyXdEZNHuwepJ+319C8NjBM9L90oDAqR8fWhvmVu+cv5d65AzhI2iT+yEEBHIW2iuzyCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T20:31:52.101796Z"},"content_sha256":"5685732367427b603f01d2af46ec009fb07ed6b5e1fb8329340b112e18aef0b5","schema_version":"1.0","event_id":"sha256:5685732367427b603f01d2af46ec009fb07ed6b5e1fb8329340b112e18aef0b5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZWOG46ZSGSSFQ2DXNBRQMGDUEJ/bundle.json","state_url":"https://pith.science/pith/ZWOG46ZSGSSFQ2DXNBRQMGDUEJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZWOG46ZSGSSFQ2DXNBRQMGDUEJ/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":"2026-06-06T20:31:52Z","links":{"resolver":"https://pith.science/pith/ZWOG46ZSGSSFQ2DXNBRQMGDUEJ","bundle":"https://pith.science/pith/ZWOG46ZSGSSFQ2DXNBRQMGDUEJ/bundle.json","state":"https://pith.science/pith/ZWOG46ZSGSSFQ2DXNBRQMGDUEJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZWOG46ZSGSSFQ2DXNBRQMGDUEJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:ZWOG46ZSGSSFQ2DXNBRQMGDUEJ","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":"7bd978e7503a9b9723c41b2e75dc7397797f0d3aa64ea25175799f71fc3bc2b2","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-08T18:27:50Z","title_canon_sha256":"ec1a8d325ddf145121d70976a26b09cf5f26fdb89a0b880927b957764f4f7212"},"schema_version":"1.0","source":{"id":"1602.02681","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.02681","created_at":"2026-05-18T01:21:09Z"},{"alias_kind":"arxiv_version","alias_value":"1602.02681v1","created_at":"2026-05-18T01:21:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.02681","created_at":"2026-05-18T01:21:09Z"},{"alias_kind":"pith_short_12","alias_value":"ZWOG46ZSGSSF","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_16","alias_value":"ZWOG46ZSGSSFQ2DX","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_8","alias_value":"ZWOG46ZS","created_at":"2026-05-18T12:30:55Z"}],"graph_snapshots":[{"event_id":"sha256:5685732367427b603f01d2af46ec009fb07ed6b5e1fb8329340b112e18aef0b5","target":"graph","created_at":"2026-05-18T01:21:09Z","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":"The solid-angle sum $A_{\\mathcal{P}} (t)$ of a rational polytope ${\\mathcal{P}} \\subset \\mathbb{R}^d$, with $t \\in \\mathbb{Z}$ was first investigated by I.G. Macdonald. Using our Fourier-analytic methods, we are able to establish an explicit formula for $A_{\\mathcal{P}} (t)$, for any real dilation $t$ and any rational polygon ${\\mathcal{P}} \\subset \\mathbb{R}^2$. Our formulation sheds additional light on previous results, for lattice-point enumerating functions of triangles, which are usually confined to the case of integer dilations. Our approach differs from that of Hardy and Littlewood in 1","authors_text":"Quang-Nhat Le, Sinai Robins","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-08T18:27:50Z","title":"Macdonald's solid-angle sum for real dilations of rational polygons"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.02681","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:e2fa4c7c80589be14ce949aa84d39222d436cf57eef423cdb1a0244a58071be4","target":"record","created_at":"2026-05-18T01:21:09Z","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":"7bd978e7503a9b9723c41b2e75dc7397797f0d3aa64ea25175799f71fc3bc2b2","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-08T18:27:50Z","title_canon_sha256":"ec1a8d325ddf145121d70976a26b09cf5f26fdb89a0b880927b957764f4f7212"},"schema_version":"1.0","source":{"id":"1602.02681","kind":"arxiv","version":1}},"canonical_sha256":"cd9c6e7b3234a45868776863061874226f4bcfb6b42a397cc809667f4506b8ac","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cd9c6e7b3234a45868776863061874226f4bcfb6b42a397cc809667f4506b8ac","first_computed_at":"2026-05-18T01:21:09.173103Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:21:09.173103Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5qgyBXdKFjzOdGHA25M+GzBjnxpgQYT/vF1hYSuBwXj8ffcg5gJtIc3ehEsk83S2fF3VExEnysdGYGSWrx1SBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:21:09.173896Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.02681","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e2fa4c7c80589be14ce949aa84d39222d436cf57eef423cdb1a0244a58071be4","sha256:5685732367427b603f01d2af46ec009fb07ed6b5e1fb8329340b112e18aef0b5"],"state_sha256":"4b493017e14b5b61a607c492676988cf8203d329cce1efe8ef7fdb2507e0510d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"v8iaYn50AuJzNXcp2vM1F4DZKJHl58SnBjf0YVZY8ned+qaZxctaSaNc/u3mJNMYTwq95YGUKDVPR+8r0or9BQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T20:31:52.105307Z","bundle_sha256":"3199135dc59155aa7945c47171110d17aec883e68cdb74cd53b7ad38bc74b03e"}}