{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:7IXBVCDJJ5NDWUHXEM2TAJ77YP","short_pith_number":"pith:7IXBVCDJ","canonical_record":{"source":{"id":"1502.01671","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-01-24T13:47:16Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"ea6d6f904e270159322507ae7e85c449a6a0dff57da23f9910355673302e85d3","abstract_canon_sha256":"124f3d1204cb92d08a9de6c6bfd232ecba7fd220b5580ff3c2d13ca47bd518ca"},"schema_version":"1.0"},"canonical_sha256":"fa2e1a88694f5a3b50f723353027ffc3d06d58ff61a9444c7d30ef3969fb5d9a","source":{"kind":"arxiv","id":"1502.01671","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.01671","created_at":"2026-05-18T02:17:33Z"},{"alias_kind":"arxiv_version","alias_value":"1502.01671v2","created_at":"2026-05-18T02:17:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.01671","created_at":"2026-05-18T02:17:33Z"},{"alias_kind":"pith_short_12","alias_value":"7IXBVCDJJ5ND","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_16","alias_value":"7IXBVCDJJ5NDWUHX","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_8","alias_value":"7IXBVCDJ","created_at":"2026-05-18T12:29:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:7IXBVCDJJ5NDWUHXEM2TAJ77YP","target":"record","payload":{"canonical_record":{"source":{"id":"1502.01671","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-01-24T13:47:16Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"ea6d6f904e270159322507ae7e85c449a6a0dff57da23f9910355673302e85d3","abstract_canon_sha256":"124f3d1204cb92d08a9de6c6bfd232ecba7fd220b5580ff3c2d13ca47bd518ca"},"schema_version":"1.0"},"canonical_sha256":"fa2e1a88694f5a3b50f723353027ffc3d06d58ff61a9444c7d30ef3969fb5d9a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:17:33.172934Z","signature_b64":"q7sqZTrRn1sDn8NLx+82ZTqu2ubvMZmjH3Lmsg5wfFN5xl0neE5iSTAPFF/rwDgjb1aWv76xL0QgHQmjWVJbBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fa2e1a88694f5a3b50f723353027ffc3d06d58ff61a9444c7d30ef3969fb5d9a","last_reissued_at":"2026-05-18T02:17:33.172269Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:17:33.172269Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1502.01671","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":"2026-05-18T02:17:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+Rt0jpKu56eTgyzjzLVS1TRLZUcB35+WgAWO1JjOTmL6JyH/t2OUT6fquwI+Mlz/1NdDHx11p0hFnBIx2Yv+Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T08:33:10.137969Z"},"content_sha256":"73e103b9bcf940aa3c31fc46d8014e4fb25fa4a0b0332da9b9e19f27a8ad24be","schema_version":"1.0","event_id":"sha256:73e103b9bcf940aa3c31fc46d8014e4fb25fa4a0b0332da9b9e19f27a8ad24be"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:7IXBVCDJJ5NDWUHXEM2TAJ77YP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Local asymptotic Euler-Maclaurin expansion for Riemann sums over a semi-rational polyhedron","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.CA","authors_text":"Michele Vergne, Nicole Berline","submitted_at":"2015-01-24T13:47:16Z","abstract_excerpt":"Consider the Riemann sum of a smooth compactly supported function h(x) on a polyhedron in R^d, sampled at the points of the lattice Z^d/t. We give an asymptotic expansion when t goes to infinity, writing each coefficient of this expansion as a sum indexed by the faces f of the polyhedron, where the f-term is the integral over f of a differential operator applied to the function h(x). In particular, if a Euclidean scalar product is chosen, we prove that the differential operator for the face f can be chosen (in a unique way) to involve only normal derivatives to f. Our formulas are valid for a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.01671","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":""},"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":"2026-05-18T02:17:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Upokj9NeBpHy3xxjw/jsgJG5g+rl1cd7AOMBD9f1MQf93m4ScXlZeoVOztRWNwP8xxZP5hDINeJlufasEEOdAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T08:33:10.138633Z"},"content_sha256":"e05d919308ae870dcbf0422fd19cb16a20443d11575a588ec1590817c87f196d","schema_version":"1.0","event_id":"sha256:e05d919308ae870dcbf0422fd19cb16a20443d11575a588ec1590817c87f196d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7IXBVCDJJ5NDWUHXEM2TAJ77YP/bundle.json","state_url":"https://pith.science/pith/7IXBVCDJJ5NDWUHXEM2TAJ77YP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7IXBVCDJJ5NDWUHXEM2TAJ77YP/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":"2026-05-26T08:33:10Z","links":{"resolver":"https://pith.science/pith/7IXBVCDJJ5NDWUHXEM2TAJ77YP","bundle":"https://pith.science/pith/7IXBVCDJJ5NDWUHXEM2TAJ77YP/bundle.json","state":"https://pith.science/pith/7IXBVCDJJ5NDWUHXEM2TAJ77YP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7IXBVCDJJ5NDWUHXEM2TAJ77YP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:7IXBVCDJJ5NDWUHXEM2TAJ77YP","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":"124f3d1204cb92d08a9de6c6bfd232ecba7fd220b5580ff3c2d13ca47bd518ca","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-01-24T13:47:16Z","title_canon_sha256":"ea6d6f904e270159322507ae7e85c449a6a0dff57da23f9910355673302e85d3"},"schema_version":"1.0","source":{"id":"1502.01671","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.01671","created_at":"2026-05-18T02:17:33Z"},{"alias_kind":"arxiv_version","alias_value":"1502.01671v2","created_at":"2026-05-18T02:17:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.01671","created_at":"2026-05-18T02:17:33Z"},{"alias_kind":"pith_short_12","alias_value":"7IXBVCDJJ5ND","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_16","alias_value":"7IXBVCDJJ5NDWUHX","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_8","alias_value":"7IXBVCDJ","created_at":"2026-05-18T12:29:10Z"}],"graph_snapshots":[{"event_id":"sha256:e05d919308ae870dcbf0422fd19cb16a20443d11575a588ec1590817c87f196d","target":"graph","created_at":"2026-05-18T02:17:33Z","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":"Consider the Riemann sum of a smooth compactly supported function h(x) on a polyhedron in R^d, sampled at the points of the lattice Z^d/t. We give an asymptotic expansion when t goes to infinity, writing each coefficient of this expansion as a sum indexed by the faces f of the polyhedron, where the f-term is the integral over f of a differential operator applied to the function h(x). In particular, if a Euclidean scalar product is chosen, we prove that the differential operator for the face f can be chosen (in a unique way) to involve only normal derivatives to f. Our formulas are valid for a ","authors_text":"Michele Vergne, Nicole Berline","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-01-24T13:47:16Z","title":"Local asymptotic Euler-Maclaurin expansion for Riemann sums over a semi-rational polyhedron"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.01671","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":"sha256:73e103b9bcf940aa3c31fc46d8014e4fb25fa4a0b0332da9b9e19f27a8ad24be","target":"record","created_at":"2026-05-18T02:17:33Z","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":"124f3d1204cb92d08a9de6c6bfd232ecba7fd220b5580ff3c2d13ca47bd518ca","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-01-24T13:47:16Z","title_canon_sha256":"ea6d6f904e270159322507ae7e85c449a6a0dff57da23f9910355673302e85d3"},"schema_version":"1.0","source":{"id":"1502.01671","kind":"arxiv","version":2}},"canonical_sha256":"fa2e1a88694f5a3b50f723353027ffc3d06d58ff61a9444c7d30ef3969fb5d9a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fa2e1a88694f5a3b50f723353027ffc3d06d58ff61a9444c7d30ef3969fb5d9a","first_computed_at":"2026-05-18T02:17:33.172269Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:17:33.172269Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"q7sqZTrRn1sDn8NLx+82ZTqu2ubvMZmjH3Lmsg5wfFN5xl0neE5iSTAPFF/rwDgjb1aWv76xL0QgHQmjWVJbBg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:17:33.172934Z","signed_message":"canonical_sha256_bytes"},"source_id":"1502.01671","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:73e103b9bcf940aa3c31fc46d8014e4fb25fa4a0b0332da9b9e19f27a8ad24be","sha256:e05d919308ae870dcbf0422fd19cb16a20443d11575a588ec1590817c87f196d"],"state_sha256":"e5375bb4daf73420173d45bdca29d22c7d633f206b26a292e7684b61b810c084"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"stGDZ9bCjjxqKDa/QOZYfhPDMbvRTUwxaaeoAzRB5UDjYq3FCcq1HVZzd8giv+qzdhFfpN0k9p3q9nDy7AlyBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T08:33:10.142016Z","bundle_sha256":"68d4219c289d4359f1f411c4469dfaa8d1658052ae4771acb7cd96f482095b3b"}}