{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:VF6PNVSARNEUJSAKO63W4IC2EE","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":"ac97a32bc438e8a6d701f8072fbed46ebcd3cfc68c930a4696920edada53c736","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-08-10T07:32:55Z","title_canon_sha256":"576ba4284020f177e8c03f80b42fbdb784e3f4468ca6b9036660412a1945f196"},"schema_version":"1.0","source":{"id":"2608.10035","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.10035","created_at":"2026-08-12T00:22:38Z"},{"alias_kind":"arxiv_version","alias_value":"2608.10035v1","created_at":"2026-08-12T00:22:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.10035","created_at":"2026-08-12T00:22:38Z"},{"alias_kind":"pith_short_12","alias_value":"VF6PNVSARNEU","created_at":"2026-08-12T00:22:38Z"},{"alias_kind":"pith_short_16","alias_value":"VF6PNVSARNEUJSAK","created_at":"2026-08-12T00:22:38Z"},{"alias_kind":"pith_short_8","alias_value":"VF6PNVSA","created_at":"2026-08-12T00:22:38Z"}],"graph_snapshots":[{"event_id":"sha256:dfcb620d82c33838a2290c35c92077ea071a561c28f0cec4b118dd4d930168c4","target":"graph","created_at":"2026-08-12T00:22:38Z","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/2608.10035/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $G$ be a finite group and $H$ be a proper normal subgroup of $G$. In this note, we introduce two new functions $o(G,H)$ and $\\psi''(G,H)$ involving the sums of element orders of $G$ and $H$. We prove that they satisfy certain inequalities, with equality if and only if $(G,H)$ is an equal order pair of a particular type.","authors_text":"Marius T\\u{a}rn\\u{a}uceanu","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-08-10T07:32:55Z","title":"Two new functions related to the sum of element orders of a finite group"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.10035","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":"sha256:93f274cab6ec143bb797a43dda9c1928e42a58c9f8a5ec88a44a21dae1bc3c0a","target":"record","created_at":"2026-08-12T00:22:38Z","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":"ac97a32bc438e8a6d701f8072fbed46ebcd3cfc68c930a4696920edada53c736","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-08-10T07:32:55Z","title_canon_sha256":"576ba4284020f177e8c03f80b42fbdb784e3f4468ca6b9036660412a1945f196"},"schema_version":"1.0","source":{"id":"2608.10035","kind":"arxiv","version":1}},"canonical_sha256":"a97cf6d6408b4944c80a77b76e205a2101ccb0ec7b1aa60102f1aabe26274f9a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a97cf6d6408b4944c80a77b76e205a2101ccb0ec7b1aa60102f1aabe26274f9a","first_computed_at":"2026-08-12T00:22:38.420069Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-12T00:22:38.420069Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"L4l2tDrI4TmAX9napp7h/kmSu0A6gHwm68prWRNQmi9XeuMqhh8YcWhIqfi8MvsaYaOiMBtv/ZZpqIwvqEsHAw==","signature_status":"signed_v1","signed_at":"2026-08-12T00:22:38.423012Z","signed_message":"canonical_sha256_bytes"},"source_id":"2608.10035","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:93f274cab6ec143bb797a43dda9c1928e42a58c9f8a5ec88a44a21dae1bc3c0a","sha256:dfcb620d82c33838a2290c35c92077ea071a561c28f0cec4b118dd4d930168c4"],"state_sha256":"539431db8917d61dfe7dba896a841300964cfee60d9bb7c20425113bf85ee0af"}