{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:AXD3QBNQENU4GDXAV65XQZ4AT6","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":"4a4858c0ae9fa86417898c65dcfb461e1115eea5234e3a6d7f59da846762fbd2","cross_cats_sorted":["math-ph","math.MP","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-02-06T17:00:48Z","title_canon_sha256":"a7aac62eab10b1efdc12e944483d712b86a29a8847609f7fe4b9ce55193fce33"},"schema_version":"1.0","source":{"id":"1802.02072","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1802.02072","created_at":"2026-06-09T02:06:57Z"},{"alias_kind":"arxiv_version","alias_value":"1802.02072v1","created_at":"2026-06-09T02:06:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.02072","created_at":"2026-06-09T02:06:57Z"},{"alias_kind":"pith_short_12","alias_value":"AXD3QBNQENU4","created_at":"2026-06-09T02:06:57Z"},{"alias_kind":"pith_short_16","alias_value":"AXD3QBNQENU4GDXA","created_at":"2026-06-09T02:06:57Z"},{"alias_kind":"pith_short_8","alias_value":"AXD3QBNQ","created_at":"2026-06-09T02:06:57Z"}],"graph_snapshots":[{"event_id":"sha256:9006bd7506f81b35a68fca48ef26af0a70e2551d49dd8b65622c7d5501079084","target":"graph","created_at":"2026-06-09T02:06:57Z","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/1802.02072/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the local optimality of periodic point sets in $\\mathbb{R}^n$ for energy minimization in the Gaussian core model, that is, for radial pair potential functions $f_c(r)=e^{-c r}$ with $c>0$. By considering suitable parameter spaces for $m$-periodic sets, we can locally rigorously analyze the energy of point sets, within the family of periodic sets having the same point density. We derive a characterization of periodic point sets being $f_c$-critical for all $c$ in terms of weighted spherical $2$-designs contained in the set. Especially for $2$-periodic sets like the family $\\mathsf{D}^+","authors_text":"Achill Sch\\\"urmann, Renaud Coulangeon","cross_cats":["math-ph","math.MP","math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-02-06T17:00:48Z","title":"Local Energy Optimality of Periodic Sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.02072","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":"sha256:3355005cdfa99db0b77a7333b5e930d4b721a8418abb9f795516aeb63e763f72","target":"record","created_at":"2026-06-09T02:06:57Z","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":"4a4858c0ae9fa86417898c65dcfb461e1115eea5234e3a6d7f59da846762fbd2","cross_cats_sorted":["math-ph","math.MP","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-02-06T17:00:48Z","title_canon_sha256":"a7aac62eab10b1efdc12e944483d712b86a29a8847609f7fe4b9ce55193fce33"},"schema_version":"1.0","source":{"id":"1802.02072","kind":"arxiv","version":1}},"canonical_sha256":"05c7b805b02369c30ee0afbb7867809f82490f231fb295ebb7dfcce61db48e7b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"05c7b805b02369c30ee0afbb7867809f82490f231fb295ebb7dfcce61db48e7b","first_computed_at":"2026-06-09T02:06:57.285947Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-09T02:06:57.285947Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZNjaLbl52WEOexuWoeyYKpSad+zhEGZRNi/8IYtfHeXK0g6QxcZQzirfaQRHl7rJ9R2ULe2AKq4Z82KW5y+0CA==","signature_status":"signed_v1","signed_at":"2026-06-09T02:06:57.288585Z","signed_message":"canonical_sha256_bytes"},"source_id":"1802.02072","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3355005cdfa99db0b77a7333b5e930d4b721a8418abb9f795516aeb63e763f72","sha256:9006bd7506f81b35a68fca48ef26af0a70e2551d49dd8b65622c7d5501079084"],"state_sha256":"13d5f1928730228149ffae159738ac1a756642030d6af4f8b255782b4b6d5150"}