{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:JWT3ATPE3P442DGWCYPHLDP2FV","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":"011911fab16ccbdaaa75a0687e9ef4a621558c4e5d19f197ece9383a775a775c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-06-09T02:51:36Z","title_canon_sha256":"cd04ac70eb4b59abb7dbddf1ef4f352992ce94c22a36833063672b41b4191203"},"schema_version":"1.0","source":{"id":"2506.07377","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.07377","created_at":"2026-07-05T11:18:26Z"},{"alias_kind":"arxiv_version","alias_value":"2506.07377v1","created_at":"2026-07-05T11:18:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.07377","created_at":"2026-07-05T11:18:26Z"},{"alias_kind":"pith_short_12","alias_value":"JWT3ATPE3P44","created_at":"2026-07-05T11:18:26Z"},{"alias_kind":"pith_short_16","alias_value":"JWT3ATPE3P442DGW","created_at":"2026-07-05T11:18:26Z"},{"alias_kind":"pith_short_8","alias_value":"JWT3ATPE","created_at":"2026-07-05T11:18:26Z"}],"graph_snapshots":[{"event_id":"sha256:fed08d16505596ffa20aaf880457ea80c15f239b69d904257296f0f136127e59","target":"graph","created_at":"2026-07-05T11:18:26Z","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/2506.07377/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we establish the theory of $p$-modulus of a family of infinite paths on an infinite-rooted tree and then explore its interpretation and properties. One key result is the formulation of $p$-modulus on the infinite tree as a limit of $p$-modulus on truncated trees, with a formula given in terms of a series. Analogous to the existing theory for finite graphs, the $1$-modulus of a family of descending paths in an infinite tree is related to the minimum cut problem, the $2$-modulus is related to effective resistance, and the $\\infty$-modulus is related to the length of shortest paths","authors_text":"Prem Raj Prasain","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-06-09T02:51:36Z","title":"$p$-Modulus on radially symmetric trees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.07377","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":"sha256:52c256bc24dcebafbb8960ec5e2d3d5e716dff83fbbe9d55830afda17eb8784b","target":"record","created_at":"2026-07-05T11:18:26Z","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":"011911fab16ccbdaaa75a0687e9ef4a621558c4e5d19f197ece9383a775a775c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-06-09T02:51:36Z","title_canon_sha256":"cd04ac70eb4b59abb7dbddf1ef4f352992ce94c22a36833063672b41b4191203"},"schema_version":"1.0","source":{"id":"2506.07377","kind":"arxiv","version":1}},"canonical_sha256":"4da7b04de4dbf9cd0cd6161e758dfa2d4d0bb48be4171b46b0effc54495169da","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4da7b04de4dbf9cd0cd6161e758dfa2d4d0bb48be4171b46b0effc54495169da","first_computed_at":"2026-07-05T11:18:26.127415Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:18:26.127415Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"M5T2bCkt38p9JntjEeocDfod0kqW5pXmeFvHKZOcKz9u+GrNkt8servAh9UJ+Cjl7Ha7FQrAY3MxGqKf5NfTAg==","signature_status":"signed_v1","signed_at":"2026-07-05T11:18:26.128041Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.07377","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:52c256bc24dcebafbb8960ec5e2d3d5e716dff83fbbe9d55830afda17eb8784b","sha256:fed08d16505596ffa20aaf880457ea80c15f239b69d904257296f0f136127e59"],"state_sha256":"15bf5bdd426c05f53966a52ceeee2ad1ad7711eed270f1b5d4f39112c33ac567"}