{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:VUYAWQGNK46DONPDEVRLRO7B2N","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":"daf6da93405bec84851eca046faeeb55c3d92b91a4efd58a22601d785c570e8e","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-01-18T17:45:02Z","title_canon_sha256":"4fc8f5cc45ea05060f500c7fdda25fbdcb6b0293e08a0421cde0757b0736ef6c"},"schema_version":"1.0","source":{"id":"1901.06353","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1901.06353","created_at":"2026-05-17T23:52:31Z"},{"alias_kind":"arxiv_version","alias_value":"1901.06353v2","created_at":"2026-05-17T23:52:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.06353","created_at":"2026-05-17T23:52:31Z"},{"alias_kind":"pith_short_12","alias_value":"VUYAWQGNK46D","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"VUYAWQGNK46DONPD","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"VUYAWQGN","created_at":"2026-05-18T12:33:30Z"}],"graph_snapshots":[{"event_id":"sha256:69c4063fd775a8387c91f8832a507b20ab539e2cdf85f3017d5c3ec93f4f263c","target":"graph","created_at":"2026-05-17T23:52:31Z","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":"A biperiodic planar network is a pair $(G,c)$ where $G$ is a graph embedded on the torus and $c$ is a function from the edges of $G$ to non-zero complex numbers. Associated to the discrete Laplacian on a biperiodic planar network is its spectrum: a triple $(C,S,\\nu)$, where $C$ is a curve and $S$ is a divisor on it. We give a complete classification of networks (modulo a natural equivalence) in terms of their spectral data. The space of networks has a large group of cluster automorphisms arising from the $Y-\\Delta$ transformations. We show that the spectrum provides action-angle coordinates fo","authors_text":"Terrence George","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-01-18T17:45:02Z","title":"Spectra of biperiodic planar networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.06353","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":"sha256:0cdf632b7e99129d40032cd1751fa77866a7f0aa7f976524903fe98c1c137f6b","target":"record","created_at":"2026-05-17T23:52:31Z","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":"daf6da93405bec84851eca046faeeb55c3d92b91a4efd58a22601d785c570e8e","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-01-18T17:45:02Z","title_canon_sha256":"4fc8f5cc45ea05060f500c7fdda25fbdcb6b0293e08a0421cde0757b0736ef6c"},"schema_version":"1.0","source":{"id":"1901.06353","kind":"arxiv","version":2}},"canonical_sha256":"ad300b40cd573c3735e32562b8bbe1d340906321ed35cd85b42b1b2e53312947","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ad300b40cd573c3735e32562b8bbe1d340906321ed35cd85b42b1b2e53312947","first_computed_at":"2026-05-17T23:52:31.874378Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:52:31.874378Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ELIIkRvWP3yiyY3DQGtCcpvMmgdPDdMIagD4kUH9sOEWS2jZns5aCZskqIBccvjwcrGZV87JbwHTegML0AgPAw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:52:31.874883Z","signed_message":"canonical_sha256_bytes"},"source_id":"1901.06353","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0cdf632b7e99129d40032cd1751fa77866a7f0aa7f976524903fe98c1c137f6b","sha256:69c4063fd775a8387c91f8832a507b20ab539e2cdf85f3017d5c3ec93f4f263c"],"state_sha256":"75dd6416916e8425e52d7dac70c93d2d6cd6ade3d4b9b50dd8bfd45f0dc90579"}