{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:RYP62CPM3LEZSZDUFHJCIG4T63","short_pith_number":"pith:RYP62CPM","canonical_record":{"source":{"id":"2502.05875","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-02-09T12:24:03Z","cross_cats_sorted":[],"title_canon_sha256":"69c56c880753b4f866bf955587a214ac3f8847bb1613c2f6c48cc183539ccfab","abstract_canon_sha256":"37c1d88be63cfd36544d20d9302a73e5eb89b4bc4618193e23c05eb9a6a8c1cd"},"schema_version":"1.0"},"canonical_sha256":"8e1fed09ecdac999647429d2241b93f6c12d051277cfb39a0444825fccd81efc","source":{"kind":"arxiv","id":"2502.05875","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2502.05875","created_at":"2026-07-05T10:11:44Z"},{"alias_kind":"arxiv_version","alias_value":"2502.05875v1","created_at":"2026-07-05T10:11:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2502.05875","created_at":"2026-07-05T10:11:44Z"},{"alias_kind":"pith_short_12","alias_value":"RYP62CPM3LEZ","created_at":"2026-07-05T10:11:44Z"},{"alias_kind":"pith_short_16","alias_value":"RYP62CPM3LEZSZDU","created_at":"2026-07-05T10:11:44Z"},{"alias_kind":"pith_short_8","alias_value":"RYP62CPM","created_at":"2026-07-05T10:11:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:RYP62CPM3LEZSZDUFHJCIG4T63","target":"record","payload":{"canonical_record":{"source":{"id":"2502.05875","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-02-09T12:24:03Z","cross_cats_sorted":[],"title_canon_sha256":"69c56c880753b4f866bf955587a214ac3f8847bb1613c2f6c48cc183539ccfab","abstract_canon_sha256":"37c1d88be63cfd36544d20d9302a73e5eb89b4bc4618193e23c05eb9a6a8c1cd"},"schema_version":"1.0"},"canonical_sha256":"8e1fed09ecdac999647429d2241b93f6c12d051277cfb39a0444825fccd81efc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:11:44.561416Z","signature_b64":"tMfvd8HrrcJKrkFsgtnNfSwt/BiK9m+UXwS8vBXBOhPnJsJxf44IvjCxsY0OANQaOL2N3S5JXXexM1Qk+BBdAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8e1fed09ecdac999647429d2241b93f6c12d051277cfb39a0444825fccd81efc","last_reissued_at":"2026-07-05T10:11:44.560924Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:11:44.560924Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2502.05875","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":"2026-07-05T10:11:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sI/fiAClCDwY/keCYHSVjCvi4FQC4J+OpXSeVPJ47LL4xKS98qTTXgUf6Qdgug9q31kQQmgWlG2xtVug8msRDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T13:20:01.918934Z"},"content_sha256":"d5fccb82dbc7bea51142da3e3a945553cb1c03fada18bf23a2fc426c0af157f8","schema_version":"1.0","event_id":"sha256:d5fccb82dbc7bea51142da3e3a945553cb1c03fada18bf23a2fc426c0af157f8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:RYP62CPM3LEZSZDUFHJCIG4T63","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Extended weak order for the affine symmetric group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Grant T. Barkley","submitted_at":"2025-02-09T12:24:03Z","abstract_excerpt":"The extended weak order on a Coxeter group $W$ is the poset of biclosed sets in its root system. In (Barkley-Speyer 2024), it was shown that when $W=\\widetilde{S}_n$ is the affine symmetric group, then the extended weak order is a quotient of the lattice $L_n$ of translation-invariant total orderings of the integers. In this article, we give a combinatorial introduction to $L_n$ and the extended weak order on $\\widetilde{S}_n$. We show that $L_n$ is an algebraic completely semidistributive lattice. We describe its canonical join representations using a cyclic version of Reading's non-crossing "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.05875","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/2502.05875/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":"2026-07-05T10:11:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rXrtZcVM3R427CQj/80KtjdBV0F0lySVnYD6k/2gp+xOZ4ZUISKqVxCOfZ+3wX/g8smoJ4ZBuXgC5SopVyMTAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T13:20:01.919446Z"},"content_sha256":"98d1e253799b622cf7a8f03bc505b497d5a2393033c804e4b8f0f5e20612dc60","schema_version":"1.0","event_id":"sha256:98d1e253799b622cf7a8f03bc505b497d5a2393033c804e4b8f0f5e20612dc60"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RYP62CPM3LEZSZDUFHJCIG4T63/bundle.json","state_url":"https://pith.science/pith/RYP62CPM3LEZSZDUFHJCIG4T63/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RYP62CPM3LEZSZDUFHJCIG4T63/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":"2026-08-09T13:20:01Z","links":{"resolver":"https://pith.science/pith/RYP62CPM3LEZSZDUFHJCIG4T63","bundle":"https://pith.science/pith/RYP62CPM3LEZSZDUFHJCIG4T63/bundle.json","state":"https://pith.science/pith/RYP62CPM3LEZSZDUFHJCIG4T63/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RYP62CPM3LEZSZDUFHJCIG4T63/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:RYP62CPM3LEZSZDUFHJCIG4T63","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":"37c1d88be63cfd36544d20d9302a73e5eb89b4bc4618193e23c05eb9a6a8c1cd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-02-09T12:24:03Z","title_canon_sha256":"69c56c880753b4f866bf955587a214ac3f8847bb1613c2f6c48cc183539ccfab"},"schema_version":"1.0","source":{"id":"2502.05875","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2502.05875","created_at":"2026-07-05T10:11:44Z"},{"alias_kind":"arxiv_version","alias_value":"2502.05875v1","created_at":"2026-07-05T10:11:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2502.05875","created_at":"2026-07-05T10:11:44Z"},{"alias_kind":"pith_short_12","alias_value":"RYP62CPM3LEZ","created_at":"2026-07-05T10:11:44Z"},{"alias_kind":"pith_short_16","alias_value":"RYP62CPM3LEZSZDU","created_at":"2026-07-05T10:11:44Z"},{"alias_kind":"pith_short_8","alias_value":"RYP62CPM","created_at":"2026-07-05T10:11:44Z"}],"graph_snapshots":[{"event_id":"sha256:98d1e253799b622cf7a8f03bc505b497d5a2393033c804e4b8f0f5e20612dc60","target":"graph","created_at":"2026-07-05T10:11:44Z","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/2502.05875/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The extended weak order on a Coxeter group $W$ is the poset of biclosed sets in its root system. In (Barkley-Speyer 2024), it was shown that when $W=\\widetilde{S}_n$ is the affine symmetric group, then the extended weak order is a quotient of the lattice $L_n$ of translation-invariant total orderings of the integers. In this article, we give a combinatorial introduction to $L_n$ and the extended weak order on $\\widetilde{S}_n$. We show that $L_n$ is an algebraic completely semidistributive lattice. We describe its canonical join representations using a cyclic version of Reading's non-crossing ","authors_text":"Grant T. Barkley","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-02-09T12:24:03Z","title":"Extended weak order for the affine symmetric group"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.05875","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:d5fccb82dbc7bea51142da3e3a945553cb1c03fada18bf23a2fc426c0af157f8","target":"record","created_at":"2026-07-05T10:11:44Z","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":"37c1d88be63cfd36544d20d9302a73e5eb89b4bc4618193e23c05eb9a6a8c1cd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-02-09T12:24:03Z","title_canon_sha256":"69c56c880753b4f866bf955587a214ac3f8847bb1613c2f6c48cc183539ccfab"},"schema_version":"1.0","source":{"id":"2502.05875","kind":"arxiv","version":1}},"canonical_sha256":"8e1fed09ecdac999647429d2241b93f6c12d051277cfb39a0444825fccd81efc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8e1fed09ecdac999647429d2241b93f6c12d051277cfb39a0444825fccd81efc","first_computed_at":"2026-07-05T10:11:44.560924Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:11:44.560924Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tMfvd8HrrcJKrkFsgtnNfSwt/BiK9m+UXwS8vBXBOhPnJsJxf44IvjCxsY0OANQaOL2N3S5JXXexM1Qk+BBdAw==","signature_status":"signed_v1","signed_at":"2026-07-05T10:11:44.561416Z","signed_message":"canonical_sha256_bytes"},"source_id":"2502.05875","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d5fccb82dbc7bea51142da3e3a945553cb1c03fada18bf23a2fc426c0af157f8","sha256:98d1e253799b622cf7a8f03bc505b497d5a2393033c804e4b8f0f5e20612dc60"],"state_sha256":"9778d9290782c2e1e5d52da82a6d231181d7438e874bc2e4fefc676eae303a70"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rAtPjwmQangPqaumZ4SHle75lg3z/7V7gVdzNvCBJ1B1H8ZlEm1tscy17a/nAWtmkmnGFcFZDb5zIDayhHFrBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T13:20:01.924619Z","bundle_sha256":"c8fe1a1d0607d3c780c38a5e886b9da3592988a4425b6db2bef9647330066454"}}