{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:FJNAVABMM7LSIOXBQUPEHHQMTN","short_pith_number":"pith:FJNAVABM","canonical_record":{"source":{"id":"2606.03389","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RA","submitted_at":"2026-06-02T09:33:07Z","cross_cats_sorted":[],"title_canon_sha256":"ba7c75d5064ce5652f04d370085337b6fdbdff5e1bf02bd002493d5eb72b983f","abstract_canon_sha256":"f9d85fb99b207d67ce399fefb9d128d911eb639b40f9249ed0cc8144642562f4"},"schema_version":"1.0"},"canonical_sha256":"2a5a0a802c67d7243ae1851e439e0c9b577a4c31cad88bbbd468c7f23f8e030f","source":{"kind":"arxiv","id":"2606.03389","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.03389","created_at":"2026-06-03T01:05:56Z"},{"alias_kind":"arxiv_version","alias_value":"2606.03389v1","created_at":"2026-06-03T01:05:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.03389","created_at":"2026-06-03T01:05:56Z"},{"alias_kind":"pith_short_12","alias_value":"FJNAVABMM7LS","created_at":"2026-06-03T01:05:56Z"},{"alias_kind":"pith_short_16","alias_value":"FJNAVABMM7LSIOXB","created_at":"2026-06-03T01:05:56Z"},{"alias_kind":"pith_short_8","alias_value":"FJNAVABM","created_at":"2026-06-03T01:05:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:FJNAVABMM7LSIOXBQUPEHHQMTN","target":"record","payload":{"canonical_record":{"source":{"id":"2606.03389","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RA","submitted_at":"2026-06-02T09:33:07Z","cross_cats_sorted":[],"title_canon_sha256":"ba7c75d5064ce5652f04d370085337b6fdbdff5e1bf02bd002493d5eb72b983f","abstract_canon_sha256":"f9d85fb99b207d67ce399fefb9d128d911eb639b40f9249ed0cc8144642562f4"},"schema_version":"1.0"},"canonical_sha256":"2a5a0a802c67d7243ae1851e439e0c9b577a4c31cad88bbbd468c7f23f8e030f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-03T01:05:56.624505Z","signature_b64":"o2QSC9sZYW0M/a/DBSnbsJr0qhdqLtJIC4AoNrEJIWzBb3YCAqXyjuFBeJl1VTKV7xU5Np+uOys5k67Zf/kVAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2a5a0a802c67d7243ae1851e439e0c9b577a4c31cad88bbbd468c7f23f8e030f","last_reissued_at":"2026-06-03T01:05:56.623967Z","signature_status":"signed_v1","first_computed_at":"2026-06-03T01:05:56.623967Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.03389","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":"2026-06-03T01:05:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GIKs6Ysq5t/vzzbrPBBaQaO2oxGXggjKR+4ux+6bBt90FO8UCMae8jOnTwvavB+rppjbkRA3WgZnvjJqxs5RDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T10:31:03.670343Z"},"content_sha256":"86272736aa0c62e8a419565552be1dc85d80e60c8d2d5a93e652c5d885fe88a9","schema_version":"1.0","event_id":"sha256:86272736aa0c62e8a419565552be1dc85d80e60c8d2d5a93e652c5d885fe88a9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:FJNAVABMM7LSIOXBQUPEHHQMTN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Spectral duality for some modal and residuated groupoid expansions of De Morgan algebras","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Joseph McDonald","submitted_at":"2026-06-02T09:33:07Z","abstract_excerpt":"Stone demonstrated that the category $\\mathbf{DLATT_{0,1}}$ of bounded distributive lattices is dually equivalent to the category $\\mathbf{Spec}$ of spectral spaces and Priestley showed that $\\mathbf{DLatt_{0,1}}$ is dually equivalent to the category $\\mathbf{Priest}$ of Priestley spaces so that $\\mathbf{Spec}$ is equivalent $\\mathbf{Priest}$. Cornish strengthened this by showing that $\\mathbf{Spec}$ and $\\mathbf{Priest}$ are in fact isomorphic. In this study, we investigate the duality theory of various lattice expansions of certain bounded distributive lattice-ordered algebras, known as De M"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.03389","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/2606.03389/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":"2026-06-03T01:05:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YWPbIonU1wT8u12VYASOTyLZ9leWonJrF3OHjRyySpeUefdoqR44L7sQiwbE6ncVTU0/BaFHRZ/DIC/Pt4c4AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T10:31:03.670812Z"},"content_sha256":"dcae88cce72fd1b5281b4440ab629bcda5a5b82b7d622cfe4e5051436a9c1cf2","schema_version":"1.0","event_id":"sha256:dcae88cce72fd1b5281b4440ab629bcda5a5b82b7d622cfe4e5051436a9c1cf2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FJNAVABMM7LSIOXBQUPEHHQMTN/bundle.json","state_url":"https://pith.science/pith/FJNAVABMM7LSIOXBQUPEHHQMTN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FJNAVABMM7LSIOXBQUPEHHQMTN/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":"2026-07-06T10:31:03Z","links":{"resolver":"https://pith.science/pith/FJNAVABMM7LSIOXBQUPEHHQMTN","bundle":"https://pith.science/pith/FJNAVABMM7LSIOXBQUPEHHQMTN/bundle.json","state":"https://pith.science/pith/FJNAVABMM7LSIOXBQUPEHHQMTN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FJNAVABMM7LSIOXBQUPEHHQMTN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:FJNAVABMM7LSIOXBQUPEHHQMTN","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":"f9d85fb99b207d67ce399fefb9d128d911eb639b40f9249ed0cc8144642562f4","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RA","submitted_at":"2026-06-02T09:33:07Z","title_canon_sha256":"ba7c75d5064ce5652f04d370085337b6fdbdff5e1bf02bd002493d5eb72b983f"},"schema_version":"1.0","source":{"id":"2606.03389","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.03389","created_at":"2026-06-03T01:05:56Z"},{"alias_kind":"arxiv_version","alias_value":"2606.03389v1","created_at":"2026-06-03T01:05:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.03389","created_at":"2026-06-03T01:05:56Z"},{"alias_kind":"pith_short_12","alias_value":"FJNAVABMM7LS","created_at":"2026-06-03T01:05:56Z"},{"alias_kind":"pith_short_16","alias_value":"FJNAVABMM7LSIOXB","created_at":"2026-06-03T01:05:56Z"},{"alias_kind":"pith_short_8","alias_value":"FJNAVABM","created_at":"2026-06-03T01:05:56Z"}],"graph_snapshots":[{"event_id":"sha256:dcae88cce72fd1b5281b4440ab629bcda5a5b82b7d622cfe4e5051436a9c1cf2","target":"graph","created_at":"2026-06-03T01:05:56Z","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/2606.03389/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Stone demonstrated that the category $\\mathbf{DLATT_{0,1}}$ of bounded distributive lattices is dually equivalent to the category $\\mathbf{Spec}$ of spectral spaces and Priestley showed that $\\mathbf{DLatt_{0,1}}$ is dually equivalent to the category $\\mathbf{Priest}$ of Priestley spaces so that $\\mathbf{Spec}$ is equivalent $\\mathbf{Priest}$. Cornish strengthened this by showing that $\\mathbf{Spec}$ and $\\mathbf{Priest}$ are in fact isomorphic. In this study, we investigate the duality theory of various lattice expansions of certain bounded distributive lattice-ordered algebras, known as De M","authors_text":"Joseph McDonald","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RA","submitted_at":"2026-06-02T09:33:07Z","title":"Spectral duality for some modal and residuated groupoid expansions of De Morgan algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.03389","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":"sha256:86272736aa0c62e8a419565552be1dc85d80e60c8d2d5a93e652c5d885fe88a9","target":"record","created_at":"2026-06-03T01:05:56Z","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":"f9d85fb99b207d67ce399fefb9d128d911eb639b40f9249ed0cc8144642562f4","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RA","submitted_at":"2026-06-02T09:33:07Z","title_canon_sha256":"ba7c75d5064ce5652f04d370085337b6fdbdff5e1bf02bd002493d5eb72b983f"},"schema_version":"1.0","source":{"id":"2606.03389","kind":"arxiv","version":1}},"canonical_sha256":"2a5a0a802c67d7243ae1851e439e0c9b577a4c31cad88bbbd468c7f23f8e030f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2a5a0a802c67d7243ae1851e439e0c9b577a4c31cad88bbbd468c7f23f8e030f","first_computed_at":"2026-06-03T01:05:56.623967Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-03T01:05:56.623967Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"o2QSC9sZYW0M/a/DBSnbsJr0qhdqLtJIC4AoNrEJIWzBb3YCAqXyjuFBeJl1VTKV7xU5Np+uOys5k67Zf/kVAQ==","signature_status":"signed_v1","signed_at":"2026-06-03T01:05:56.624505Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.03389","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:86272736aa0c62e8a419565552be1dc85d80e60c8d2d5a93e652c5d885fe88a9","sha256:dcae88cce72fd1b5281b4440ab629bcda5a5b82b7d622cfe4e5051436a9c1cf2"],"state_sha256":"2c5f0813b27fcc6489a0543f68ca8107463bcaffb64b27497b9825a9968bcc3c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"i5l4NjNqv/coR5hb/uYJFRrZ9xmlF/gcBrYpfRD31nh8sDAbuZS2vn3l5g1UZ4e/2aJ9iiSOwlTNTkm0Y0f5Aw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-06T10:31:03.672724Z","bundle_sha256":"130d28936d3abdb6db0802c665712a64c79ca80cb2a67557f9a283b6a4383b05"}}