{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:3KWMAEAA4Q3WHJFDPD42JOSCF4","short_pith_number":"pith:3KWMAEAA","canonical_record":{"source":{"id":"2608.00900","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-08-02T00:00:20Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"d2bbc5efc40121a7c1df9b57fd3a9292be5e2595527315468aae0eda8414b315","abstract_canon_sha256":"7016e54f703b26b21819b2a64c65de5d93f982cbaaea247cce525ec13a857587"},"schema_version":"1.0"},"canonical_sha256":"daacc01000e43763a4a378f9a4ba422f20de9cc80c41b14ffa42701a285f0fd0","source":{"kind":"arxiv","id":"2608.00900","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.00900","created_at":"2026-08-04T01:55:30Z"},{"alias_kind":"arxiv_version","alias_value":"2608.00900v1","created_at":"2026-08-04T01:55:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.00900","created_at":"2026-08-04T01:55:30Z"},{"alias_kind":"pith_short_12","alias_value":"3KWMAEAA4Q3W","created_at":"2026-08-04T01:55:30Z"},{"alias_kind":"pith_short_16","alias_value":"3KWMAEAA4Q3WHJFD","created_at":"2026-08-04T01:55:30Z"},{"alias_kind":"pith_short_8","alias_value":"3KWMAEAA","created_at":"2026-08-04T01:55:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:3KWMAEAA4Q3WHJFDPD42JOSCF4","target":"record","payload":{"canonical_record":{"source":{"id":"2608.00900","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-08-02T00:00:20Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"d2bbc5efc40121a7c1df9b57fd3a9292be5e2595527315468aae0eda8414b315","abstract_canon_sha256":"7016e54f703b26b21819b2a64c65de5d93f982cbaaea247cce525ec13a857587"},"schema_version":"1.0"},"canonical_sha256":"daacc01000e43763a4a378f9a4ba422f20de9cc80c41b14ffa42701a285f0fd0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-04T01:55:30.429306Z","signature_b64":"AuT5vceSbC7OMbqA5CEr6mRZgUSPbarsPXAeAt9sk3Ff932kPJbfYLkuXfpPMEaRa1Q/4zEd7QopFS47cWu1BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"daacc01000e43763a4a378f9a4ba422f20de9cc80c41b14ffa42701a285f0fd0","last_reissued_at":"2026-08-04T01:55:30.427619Z","signature_status":"signed_v1","first_computed_at":"2026-08-04T01:55:30.427619Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2608.00900","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":"2026-08-04T01:55:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UBJdzDjQ1Q14F104p+zGKARBnmR3mnK8Kf3MArZtNo4tAzxAzMgpVeSlgXu/zyMLcps1gVrZz0h1WrLkWwsZBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T16:36:58.264373Z"},"content_sha256":"b8e533a0c3f85e2941ff03a6d2975b977b7b6ddae62cb383c87ba7048792f862","schema_version":"1.0","event_id":"sha256:b8e533a0c3f85e2941ff03a6d2975b977b7b6ddae62cb383c87ba7048792f862"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:3KWMAEAA4Q3WHJFDPD42JOSCF4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An explicit construction of two completely independent spanning trees in the four-dimensional dual-cube","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Jitendra Prajapati","submitted_at":"2026-08-02T00:00:20Z","abstract_excerpt":"Lalou, Mbarek, Skender and Togni (arXiv:2607.25917) proved that the $n$-dimensional dual-cube $F_n$ admits two completely independent spanning trees for every $n\\ge 5$, observed that none exist for $n\\le 3$, and identified $F_4$ as the first unresolved case, reporting more than 700 hours of inconclusive computation. We settle this case affirmatively by an explicit construction, completing the classification: $F_n$ admits two completely independent spanning trees if and only if $n\\ge 4$. The internal-vertex sets of the two trees are the level sets of a single ten-term cubic polynomial over $\\ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.00900","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/2608.00900/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":"2026-08-04T01:55:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"o2AChEsVvpCkE6gx3ZRg8lHJKt65udrT9WJY2Z/9bvX7glnlIM8ORjhsUs5g6sEKZSgnF73xu4K70Er4PdA4CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T16:36:58.264941Z"},"content_sha256":"3c02d5c028b7d92b4edc0852899e0c05c9f31fd21ae48f6426f1028fbcdc2be0","schema_version":"1.0","event_id":"sha256:3c02d5c028b7d92b4edc0852899e0c05c9f31fd21ae48f6426f1028fbcdc2be0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3KWMAEAA4Q3WHJFDPD42JOSCF4/bundle.json","state_url":"https://pith.science/pith/3KWMAEAA4Q3WHJFDPD42JOSCF4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3KWMAEAA4Q3WHJFDPD42JOSCF4/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":"2026-08-11T16:36:58Z","links":{"resolver":"https://pith.science/pith/3KWMAEAA4Q3WHJFDPD42JOSCF4","bundle":"https://pith.science/pith/3KWMAEAA4Q3WHJFDPD42JOSCF4/bundle.json","state":"https://pith.science/pith/3KWMAEAA4Q3WHJFDPD42JOSCF4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3KWMAEAA4Q3WHJFDPD42JOSCF4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:3KWMAEAA4Q3WHJFDPD42JOSCF4","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":"7016e54f703b26b21819b2a64c65de5d93f982cbaaea247cce525ec13a857587","cross_cats_sorted":["cs.DM"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-08-02T00:00:20Z","title_canon_sha256":"d2bbc5efc40121a7c1df9b57fd3a9292be5e2595527315468aae0eda8414b315"},"schema_version":"1.0","source":{"id":"2608.00900","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.00900","created_at":"2026-08-04T01:55:30Z"},{"alias_kind":"arxiv_version","alias_value":"2608.00900v1","created_at":"2026-08-04T01:55:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.00900","created_at":"2026-08-04T01:55:30Z"},{"alias_kind":"pith_short_12","alias_value":"3KWMAEAA4Q3W","created_at":"2026-08-04T01:55:30Z"},{"alias_kind":"pith_short_16","alias_value":"3KWMAEAA4Q3WHJFD","created_at":"2026-08-04T01:55:30Z"},{"alias_kind":"pith_short_8","alias_value":"3KWMAEAA","created_at":"2026-08-04T01:55:30Z"}],"graph_snapshots":[{"event_id":"sha256:3c02d5c028b7d92b4edc0852899e0c05c9f31fd21ae48f6426f1028fbcdc2be0","target":"graph","created_at":"2026-08-04T01:55:30Z","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/2608.00900/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Lalou, Mbarek, Skender and Togni (arXiv:2607.25917) proved that the $n$-dimensional dual-cube $F_n$ admits two completely independent spanning trees for every $n\\ge 5$, observed that none exist for $n\\le 3$, and identified $F_4$ as the first unresolved case, reporting more than 700 hours of inconclusive computation. We settle this case affirmatively by an explicit construction, completing the classification: $F_n$ admits two completely independent spanning trees if and only if $n\\ge 4$. The internal-vertex sets of the two trees are the level sets of a single ten-term cubic polynomial over $\\ma","authors_text":"Jitendra Prajapati","cross_cats":["cs.DM"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-08-02T00:00:20Z","title":"An explicit construction of two completely independent spanning trees in the four-dimensional dual-cube"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.00900","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:b8e533a0c3f85e2941ff03a6d2975b977b7b6ddae62cb383c87ba7048792f862","target":"record","created_at":"2026-08-04T01:55:30Z","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":"7016e54f703b26b21819b2a64c65de5d93f982cbaaea247cce525ec13a857587","cross_cats_sorted":["cs.DM"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-08-02T00:00:20Z","title_canon_sha256":"d2bbc5efc40121a7c1df9b57fd3a9292be5e2595527315468aae0eda8414b315"},"schema_version":"1.0","source":{"id":"2608.00900","kind":"arxiv","version":1}},"canonical_sha256":"daacc01000e43763a4a378f9a4ba422f20de9cc80c41b14ffa42701a285f0fd0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"daacc01000e43763a4a378f9a4ba422f20de9cc80c41b14ffa42701a285f0fd0","first_computed_at":"2026-08-04T01:55:30.427619Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-04T01:55:30.427619Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AuT5vceSbC7OMbqA5CEr6mRZgUSPbarsPXAeAt9sk3Ff932kPJbfYLkuXfpPMEaRa1Q/4zEd7QopFS47cWu1BQ==","signature_status":"signed_v1","signed_at":"2026-08-04T01:55:30.429306Z","signed_message":"canonical_sha256_bytes"},"source_id":"2608.00900","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b8e533a0c3f85e2941ff03a6d2975b977b7b6ddae62cb383c87ba7048792f862","sha256:3c02d5c028b7d92b4edc0852899e0c05c9f31fd21ae48f6426f1028fbcdc2be0"],"state_sha256":"64e7e0456b226594f9d69b1b83143c141ccb3f8ea06f8e16a9510c2db3f4fda6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GXRROxDrJntMeXF/ojs1YOw6juOpxoGS4vrZM7Oy6b7ZGDgkCGGaQTh6gPxHBDQbZyVxk3x4+GwqixWq4TwTBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-11T16:36:58.270187Z","bundle_sha256":"2439cc36d9237a5bb760eab53769ee3710e20b2b600406250551ea0da5dafd17"}}