{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:BIBPRKXUCBIRFMVVEUCC7HDHKF","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":"32d9bfde26b0adfe691f4f3e8ca302963c7316a949bc972cb122621b757170b0","cross_cats_sorted":["cond-mat.mes-hall","math.FA","math.MP","math.SP","quant-ph"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math-ph","submitted_at":"2026-07-09T10:03:51Z","title_canon_sha256":"fbd153cfe9a8c8ce453a6a0329410b91577c156db0674e90d746ac34ac0fef8c"},"schema_version":"1.0","source":{"id":"2607.08320","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.08320","created_at":"2026-07-10T01:19:47Z"},{"alias_kind":"arxiv_version","alias_value":"2607.08320v1","created_at":"2026-07-10T01:19:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.08320","created_at":"2026-07-10T01:19:47Z"},{"alias_kind":"pith_short_12","alias_value":"BIBPRKXUCBIR","created_at":"2026-07-10T01:19:47Z"},{"alias_kind":"pith_short_16","alias_value":"BIBPRKXUCBIRFMVV","created_at":"2026-07-10T01:19:47Z"},{"alias_kind":"pith_short_8","alias_value":"BIBPRKXU","created_at":"2026-07-10T01:19:47Z"}],"graph_snapshots":[{"event_id":"sha256:2fb7cdfa1bf8068345f02a190b6da493e96db73ac159f9604f77ab6d1071bc71","target":"graph","created_at":"2026-07-10T01:19:47Z","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/2607.08320/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we consider Hamiltonians for aperiodic crystals of the form \\begin{align*}\n  H_\\varepsilon:=T(-i\\nabla_x+{\\mathbf A}(x,\\varepsilon x))+V(x,\\varepsilon x),\\qquad x\\in {\\mathbb R}^d \\end{align*} where $T$ represents either a Dirac operators or a Schr\\\"odinger operator, and $x\\mapsto {\\mathbf A}(x,X)$ and $x\\mapsto V(x,X)$ are $\\mathbb L$-periodic with respect to some lattice $\\mathbb L\\subset{\\mathbb R}^d$.\n  Let \\begin{align*}\n  (k,X)\\ni {\\mathbb R}^d\\times {\\mathbb R}^d\\mapsto h(k,X):=T(-i\\nabla_x+k+{\\mathbf A}(x,X))+V(x,X) \\end{align*} be a family of operators acting on $L^2_{\\","authors_text":"Long Meng","cross_cats":["cond-mat.mes-hall","math.FA","math.MP","math.SP","quant-ph"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math-ph","submitted_at":"2026-07-09T10:03:51Z","title":"Approximate eigenfunctions for some aperiodic crystals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.08320","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:ce1428a4d24016a6a8c9a7359de9b6b69576535aba1e1227985008d3e0879691","target":"record","created_at":"2026-07-10T01:19:47Z","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":"32d9bfde26b0adfe691f4f3e8ca302963c7316a949bc972cb122621b757170b0","cross_cats_sorted":["cond-mat.mes-hall","math.FA","math.MP","math.SP","quant-ph"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math-ph","submitted_at":"2026-07-09T10:03:51Z","title_canon_sha256":"fbd153cfe9a8c8ce453a6a0329410b91577c156db0674e90d746ac34ac0fef8c"},"schema_version":"1.0","source":{"id":"2607.08320","kind":"arxiv","version":1}},"canonical_sha256":"0a02f8aaf4105112b2b525042f9c6751698fbab2d5d7bf0e843711cdf457f6fc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0a02f8aaf4105112b2b525042f9c6751698fbab2d5d7bf0e843711cdf457f6fc","first_computed_at":"2026-07-10T01:19:47.181812Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-10T01:19:47.181812Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NYfdl7caC9JNxwmdb01jDsTuhKY90dGCs+3km/oAxAP1hh/EggEypnadS9wOOj4usi8lSFlPDr1bPOvxcmJICw==","signature_status":"signed_v1","signed_at":"2026-07-10T01:19:47.182168Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.08320","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ce1428a4d24016a6a8c9a7359de9b6b69576535aba1e1227985008d3e0879691","sha256:2fb7cdfa1bf8068345f02a190b6da493e96db73ac159f9604f77ab6d1071bc71"],"state_sha256":"7e81ba1013c2b4fa51d075e76e9ea67a2d0670af27ff9fc44290631a7ebe4647"}