{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:432E7WOU4V7QXW3N2ARYNRDCCA","short_pith_number":"pith:432E7WOU","canonical_record":{"source":{"id":"2507.02638","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-07-03T14:02:52Z","cross_cats_sorted":[],"title_canon_sha256":"7c909887633eaaba4753a41a259b4497511bf5a8a19637762655819c9dab8b44","abstract_canon_sha256":"b02ab3d35319407978024d2903b9dbfe22d403462bb5ae417742c72728bfcf56"},"schema_version":"1.0"},"canonical_sha256":"e6f44fd9d4e57f0bdb6dd02386c462102c08f3a8e9d9e0c5259594b884d3476f","source":{"kind":"arxiv","id":"2507.02638","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.02638","created_at":"2026-07-05T11:31:32Z"},{"alias_kind":"arxiv_version","alias_value":"2507.02638v1","created_at":"2026-07-05T11:31:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.02638","created_at":"2026-07-05T11:31:32Z"},{"alias_kind":"pith_short_12","alias_value":"432E7WOU4V7Q","created_at":"2026-07-05T11:31:32Z"},{"alias_kind":"pith_short_16","alias_value":"432E7WOU4V7QXW3N","created_at":"2026-07-05T11:31:32Z"},{"alias_kind":"pith_short_8","alias_value":"432E7WOU","created_at":"2026-07-05T11:31:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:432E7WOU4V7QXW3N2ARYNRDCCA","target":"record","payload":{"canonical_record":{"source":{"id":"2507.02638","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-07-03T14:02:52Z","cross_cats_sorted":[],"title_canon_sha256":"7c909887633eaaba4753a41a259b4497511bf5a8a19637762655819c9dab8b44","abstract_canon_sha256":"b02ab3d35319407978024d2903b9dbfe22d403462bb5ae417742c72728bfcf56"},"schema_version":"1.0"},"canonical_sha256":"e6f44fd9d4e57f0bdb6dd02386c462102c08f3a8e9d9e0c5259594b884d3476f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:31:32.185826Z","signature_b64":"RMQicetsQTbnKhCMsEMf+yZCZL7iCJfqeB+ot4umLC+KxLCt0/p1QaFjW6nvA3XRPcMEaUDRnovPUDW7MNu5Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e6f44fd9d4e57f0bdb6dd02386c462102c08f3a8e9d9e0c5259594b884d3476f","last_reissued_at":"2026-07-05T11:31:32.185367Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:31:32.185367Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2507.02638","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:31:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+/E8oGXjTsvDiFYYVa/DIqgwWB6yoB/T83NEOoHP+NyFy8yZdjSDs/2SjzCN+6gHAnLTLsszjnU4vwKaMIx6BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T16:41:02.740231Z"},"content_sha256":"3fe7568f4a095ba36a9f2a76e39dd7a8125644a8a9e051a4726ce20a9a24ea18","schema_version":"1.0","event_id":"sha256:3fe7568f4a095ba36a9f2a76e39dd7a8125644a8a9e051a4726ce20a9a24ea18"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:432E7WOU4V7QXW3N2ARYNRDCCA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Homogenisation and spectral convergence of high-contrast convolution type operators","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Andrey Piatnitski, Igor Velcic, Mikhail Cherdantsev","submitted_at":"2025-07-03T14:02:52Z","abstract_excerpt":"The paper deals with homogenisation problems for high-contrast symmetric convolution-type operators with integrable kernels in media with a periodic microstructure. We adapt the two-scale convergence method to nonlocal convolution-type operators and obtain the homogenisation result both for problems stated in the whole space and in bounded domains with the homogeneous Dirichlet boundary condition.\n  Our main focus is on spectral analysis. We describe the spectrum of the limit two-scale operator and characterize the limit behaviour of the spectrum of the original problem as the microstructure p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.02638","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.02638/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T11:31:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Lh8mZUhSliGPmb7Z6ac5kQebZsjdZ0xAK38WWzOu47zNpTMaLUJLvv5oqdZgixenLaZeLptINoYLTiy+8JNJDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T16:41:02.740801Z"},"content_sha256":"850c6e785d62fa55a85d9e9d3e94e883507cbe6f1aaf926c1fd23ecc03de14bf","schema_version":"1.0","event_id":"sha256:850c6e785d62fa55a85d9e9d3e94e883507cbe6f1aaf926c1fd23ecc03de14bf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/432E7WOU4V7QXW3N2ARYNRDCCA/bundle.json","state_url":"https://pith.science/pith/432E7WOU4V7QXW3N2ARYNRDCCA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/432E7WOU4V7QXW3N2ARYNRDCCA/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":"2026-08-09T16:41:02Z","links":{"resolver":"https://pith.science/pith/432E7WOU4V7QXW3N2ARYNRDCCA","bundle":"https://pith.science/pith/432E7WOU4V7QXW3N2ARYNRDCCA/bundle.json","state":"https://pith.science/pith/432E7WOU4V7QXW3N2ARYNRDCCA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/432E7WOU4V7QXW3N2ARYNRDCCA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:432E7WOU4V7QXW3N2ARYNRDCCA","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":"b02ab3d35319407978024d2903b9dbfe22d403462bb5ae417742c72728bfcf56","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-07-03T14:02:52Z","title_canon_sha256":"7c909887633eaaba4753a41a259b4497511bf5a8a19637762655819c9dab8b44"},"schema_version":"1.0","source":{"id":"2507.02638","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.02638","created_at":"2026-07-05T11:31:32Z"},{"alias_kind":"arxiv_version","alias_value":"2507.02638v1","created_at":"2026-07-05T11:31:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.02638","created_at":"2026-07-05T11:31:32Z"},{"alias_kind":"pith_short_12","alias_value":"432E7WOU4V7Q","created_at":"2026-07-05T11:31:32Z"},{"alias_kind":"pith_short_16","alias_value":"432E7WOU4V7QXW3N","created_at":"2026-07-05T11:31:32Z"},{"alias_kind":"pith_short_8","alias_value":"432E7WOU","created_at":"2026-07-05T11:31:32Z"}],"graph_snapshots":[{"event_id":"sha256:850c6e785d62fa55a85d9e9d3e94e883507cbe6f1aaf926c1fd23ecc03de14bf","target":"graph","created_at":"2026-07-05T11:31:32Z","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/2507.02638/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The paper deals with homogenisation problems for high-contrast symmetric convolution-type operators with integrable kernels in media with a periodic microstructure. We adapt the two-scale convergence method to nonlocal convolution-type operators and obtain the homogenisation result both for problems stated in the whole space and in bounded domains with the homogeneous Dirichlet boundary condition.\n  Our main focus is on spectral analysis. We describe the spectrum of the limit two-scale operator and characterize the limit behaviour of the spectrum of the original problem as the microstructure p","authors_text":"Andrey Piatnitski, Igor Velcic, Mikhail Cherdantsev","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-07-03T14:02:52Z","title":"Homogenisation and spectral convergence of high-contrast convolution type operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.02638","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":"sha256:3fe7568f4a095ba36a9f2a76e39dd7a8125644a8a9e051a4726ce20a9a24ea18","target":"record","created_at":"2026-07-05T11:31:32Z","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":"b02ab3d35319407978024d2903b9dbfe22d403462bb5ae417742c72728bfcf56","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-07-03T14:02:52Z","title_canon_sha256":"7c909887633eaaba4753a41a259b4497511bf5a8a19637762655819c9dab8b44"},"schema_version":"1.0","source":{"id":"2507.02638","kind":"arxiv","version":1}},"canonical_sha256":"e6f44fd9d4e57f0bdb6dd02386c462102c08f3a8e9d9e0c5259594b884d3476f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e6f44fd9d4e57f0bdb6dd02386c462102c08f3a8e9d9e0c5259594b884d3476f","first_computed_at":"2026-07-05T11:31:32.185367Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:31:32.185367Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RMQicetsQTbnKhCMsEMf+yZCZL7iCJfqeB+ot4umLC+KxLCt0/p1QaFjW6nvA3XRPcMEaUDRnovPUDW7MNu5Cw==","signature_status":"signed_v1","signed_at":"2026-07-05T11:31:32.185826Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.02638","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3fe7568f4a095ba36a9f2a76e39dd7a8125644a8a9e051a4726ce20a9a24ea18","sha256:850c6e785d62fa55a85d9e9d3e94e883507cbe6f1aaf926c1fd23ecc03de14bf"],"state_sha256":"305f5d7d373152ecf909a7826f94143a43a71643638745d6952be9d5ce79e12d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qCp27rZuIXHfcEBJjDKQByjzmSzG2TL4Th3GjBPmGk245WEDxqN68UmRUfQ3uamU3i5hjBFAVP6QEYg3VwKLBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T16:41:02.744544Z","bundle_sha256":"e42427d3fd2ae1f15caf6bcf619ed7221c8a1d67cfc676660cbdf6495dc7e551"}}