{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:3HHX47UKIDRA2GS5LTIOPEVXC4","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":"ef67a7fef3e5fde2ca114be7bf62fd3db05aa50308748920b958902508940747","cross_cats_sorted":["math-ph","math.MP","nlin.AO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-04-29T22:01:02Z","title_canon_sha256":"273db5ce5942eddacf7879b2aaef5c7db767e3f95c6fa8062d0c4ff6a0785eeb"},"schema_version":"1.0","source":{"id":"1605.00034","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.00034","created_at":"2026-05-18T00:39:02Z"},{"alias_kind":"arxiv_version","alias_value":"1605.00034v1","created_at":"2026-05-18T00:39:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.00034","created_at":"2026-05-18T00:39:02Z"},{"alias_kind":"pith_short_12","alias_value":"3HHX47UKIDRA","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"3HHX47UKIDRA2GS5","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"3HHX47UK","created_at":"2026-05-18T12:29:55Z"}],"graph_snapshots":[{"event_id":"sha256:20e3d18f0f4562730908e945452d852fdd2a923dcc44fc432a617b5c300c1a6f","target":"graph","created_at":"2026-05-18T00:39:02Z","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 show that the emerging field of discrete differential geometry can be usefully brought to bear on crystallization problems. In particular, we give a simplified proof of the Heitmann-Radin crystallization theorem (R. C. Heitmann, C. Radin, J. Stat. Phys. 22, 281-287, 1980), which concerns a system of $N$ identical atoms in two dimensions interacting via the idealized pair potential $V(r)=+\\infty$ if $r<1$, $-1$ if $r=1$, $0$ if $r>1$. This is done by endowing the bond graph of a general particle configuration with a suitable notion of {\\it discrete curvature}, and appealing to a {\\it discret","authors_text":"Gero Friesecke, Lucia De Luca","cross_cats":["math-ph","math.MP","nlin.AO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-04-29T22:01:02Z","title":"Crystallization in two dimensions and a discrete Gauss-Bonnet theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.00034","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:e9586dd1826eeac491119dbdf489af479488aa443530abe846271dc2dee1b6e0","target":"record","created_at":"2026-05-18T00:39:02Z","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":"ef67a7fef3e5fde2ca114be7bf62fd3db05aa50308748920b958902508940747","cross_cats_sorted":["math-ph","math.MP","nlin.AO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-04-29T22:01:02Z","title_canon_sha256":"273db5ce5942eddacf7879b2aaef5c7db767e3f95c6fa8062d0c4ff6a0785eeb"},"schema_version":"1.0","source":{"id":"1605.00034","kind":"arxiv","version":1}},"canonical_sha256":"d9cf7e7e8a40e20d1a5d5cd0e792b7173482216f963a65fd51b4eab52cc5ebf1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d9cf7e7e8a40e20d1a5d5cd0e792b7173482216f963a65fd51b4eab52cc5ebf1","first_computed_at":"2026-05-18T00:39:02.902046Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:39:02.902046Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4eBmF5Tf/89/gkamkgRjGBFYZkJNnfnmKuXcWJ8rtCgi72HLt5blhQA4bIOXvqrrrEq3cO4km3gs9DLaUIYXCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:39:02.902728Z","signed_message":"canonical_sha256_bytes"},"source_id":"1605.00034","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e9586dd1826eeac491119dbdf489af479488aa443530abe846271dc2dee1b6e0","sha256:20e3d18f0f4562730908e945452d852fdd2a923dcc44fc432a617b5c300c1a6f"],"state_sha256":"fff48733b9aec295aa24ec6ed3ea0ff33a385cd631b4bd74f37d7f069ae8d579"}