{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:GLMJPAT6ISROQ2AKS2AVNRT2S6","short_pith_number":"pith:GLMJPAT6","canonical_record":{"source":{"id":"2606.24562","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-23T13:29:58Z","cross_cats_sorted":["math.PR","q-bio.PE"],"title_canon_sha256":"084d3ed627837380150ac228ba4c4fa7fce52351f97eb87155d23b9e54e6cd18","abstract_canon_sha256":"c9ea2fd62660fcb19e1096a0eb3fbac080d225a922e5cabe197c16f89fd5b6ee"},"schema_version":"1.0"},"canonical_sha256":"32d897827e44a2e8680a968156c67a979ece68110f7d315227c72e2d0298d89a","source":{"kind":"arxiv","id":"2606.24562","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.24562","created_at":"2026-06-24T01:15:33Z"},{"alias_kind":"arxiv_version","alias_value":"2606.24562v1","created_at":"2026-06-24T01:15:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.24562","created_at":"2026-06-24T01:15:33Z"},{"alias_kind":"pith_short_12","alias_value":"GLMJPAT6ISRO","created_at":"2026-06-24T01:15:33Z"},{"alias_kind":"pith_short_16","alias_value":"GLMJPAT6ISROQ2AK","created_at":"2026-06-24T01:15:33Z"},{"alias_kind":"pith_short_8","alias_value":"GLMJPAT6","created_at":"2026-06-24T01:15:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:GLMJPAT6ISROQ2AKS2AVNRT2S6","target":"record","payload":{"canonical_record":{"source":{"id":"2606.24562","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-23T13:29:58Z","cross_cats_sorted":["math.PR","q-bio.PE"],"title_canon_sha256":"084d3ed627837380150ac228ba4c4fa7fce52351f97eb87155d23b9e54e6cd18","abstract_canon_sha256":"c9ea2fd62660fcb19e1096a0eb3fbac080d225a922e5cabe197c16f89fd5b6ee"},"schema_version":"1.0"},"canonical_sha256":"32d897827e44a2e8680a968156c67a979ece68110f7d315227c72e2d0298d89a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-24T01:15:33.960433Z","signature_b64":"vyK4HmcBBRtVx08srAS7GiX+/sTvGE0GvhcV4Ib3X45MZaFiX218fWl3P+OLJpOcXGF2adM+WKZqDZ5b8bELDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"32d897827e44a2e8680a968156c67a979ece68110f7d315227c72e2d0298d89a","last_reissued_at":"2026-06-24T01:15:33.960075Z","signature_status":"signed_v1","first_computed_at":"2026-06-24T01:15:33.960075Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.24562","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":"2026-06-24T01:15:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xNvZGE8piixlAm6kcTe9iLz0Z5yQRKizK9/zv7xEk6QPqDVgpg8LnFDkVz/owc1BIWQPtQ1BWVi0+PC0G2fvCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T11:41:08.517214Z"},"content_sha256":"65dba18a7e42e122c04b82159a3d66b73183f7449e26a4bea69d16fc172d21ac","schema_version":"1.0","event_id":"sha256:65dba18a7e42e122c04b82159a3d66b73183f7449e26a4bea69d16fc172d21ac"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:GLMJPAT6ISROQ2AKS2AVNRT2S6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A parameterized family of balance indices for phylogenetic networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR","q-bio.PE"],"primary_cat":"math.CO","authors_text":"Fran\\c{c}ois Bienvenu, Hadrien Maffioli, Jean-Jil Duchamps","submitted_at":"2026-06-23T13:29:58Z","abstract_excerpt":"We introduce a new family of balance indices for phylogenetic networks: the $H_\\alpha$ indices, where $\\alpha$ is a positive real number. This family includes the $B_2$ index as a special case ($\\alpha = 1$) and provides a natural extension of the Sackin index to phylogenetic networks. We show that the $H_\\alpha$ indices share many structural properties with the $B_2$ index, most notably a \"grafting property\" that makes it possible to express the $H_\\alpha$ index of a network in terms of the $H_\\alpha$ indices of its biconnected components. These properties allow us to identify networks that m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.24562","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/2606.24562/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":"2026-06-24T01:15:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xZSP9af+vhAY0kZsW7tDli1aiUzmYLih3ib2ok8GTpqisXJP2oEghuhL3B3zBTTQUQFrC7xHoE/Au/MSPF9vDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T11:41:08.517610Z"},"content_sha256":"20e903a3699cf26e0bf6afeb43d9063f541bbb402111158a1fcb6debfe4b61b3","schema_version":"1.0","event_id":"sha256:20e903a3699cf26e0bf6afeb43d9063f541bbb402111158a1fcb6debfe4b61b3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GLMJPAT6ISROQ2AKS2AVNRT2S6/bundle.json","state_url":"https://pith.science/pith/GLMJPAT6ISROQ2AKS2AVNRT2S6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GLMJPAT6ISROQ2AKS2AVNRT2S6/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":"2026-06-29T11:41:08Z","links":{"resolver":"https://pith.science/pith/GLMJPAT6ISROQ2AKS2AVNRT2S6","bundle":"https://pith.science/pith/GLMJPAT6ISROQ2AKS2AVNRT2S6/bundle.json","state":"https://pith.science/pith/GLMJPAT6ISROQ2AKS2AVNRT2S6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GLMJPAT6ISROQ2AKS2AVNRT2S6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:GLMJPAT6ISROQ2AKS2AVNRT2S6","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":"c9ea2fd62660fcb19e1096a0eb3fbac080d225a922e5cabe197c16f89fd5b6ee","cross_cats_sorted":["math.PR","q-bio.PE"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-23T13:29:58Z","title_canon_sha256":"084d3ed627837380150ac228ba4c4fa7fce52351f97eb87155d23b9e54e6cd18"},"schema_version":"1.0","source":{"id":"2606.24562","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.24562","created_at":"2026-06-24T01:15:33Z"},{"alias_kind":"arxiv_version","alias_value":"2606.24562v1","created_at":"2026-06-24T01:15:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.24562","created_at":"2026-06-24T01:15:33Z"},{"alias_kind":"pith_short_12","alias_value":"GLMJPAT6ISRO","created_at":"2026-06-24T01:15:33Z"},{"alias_kind":"pith_short_16","alias_value":"GLMJPAT6ISROQ2AK","created_at":"2026-06-24T01:15:33Z"},{"alias_kind":"pith_short_8","alias_value":"GLMJPAT6","created_at":"2026-06-24T01:15:33Z"}],"graph_snapshots":[{"event_id":"sha256:20e903a3699cf26e0bf6afeb43d9063f541bbb402111158a1fcb6debfe4b61b3","target":"graph","created_at":"2026-06-24T01:15:33Z","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/2606.24562/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We introduce a new family of balance indices for phylogenetic networks: the $H_\\alpha$ indices, where $\\alpha$ is a positive real number. This family includes the $B_2$ index as a special case ($\\alpha = 1$) and provides a natural extension of the Sackin index to phylogenetic networks. We show that the $H_\\alpha$ indices share many structural properties with the $B_2$ index, most notably a \"grafting property\" that makes it possible to express the $H_\\alpha$ index of a network in terms of the $H_\\alpha$ indices of its biconnected components. These properties allow us to identify networks that m","authors_text":"Fran\\c{c}ois Bienvenu, Hadrien Maffioli, Jean-Jil Duchamps","cross_cats":["math.PR","q-bio.PE"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-23T13:29:58Z","title":"A parameterized family of balance indices for phylogenetic networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.24562","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":"sha256:65dba18a7e42e122c04b82159a3d66b73183f7449e26a4bea69d16fc172d21ac","target":"record","created_at":"2026-06-24T01:15:33Z","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":"c9ea2fd62660fcb19e1096a0eb3fbac080d225a922e5cabe197c16f89fd5b6ee","cross_cats_sorted":["math.PR","q-bio.PE"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-23T13:29:58Z","title_canon_sha256":"084d3ed627837380150ac228ba4c4fa7fce52351f97eb87155d23b9e54e6cd18"},"schema_version":"1.0","source":{"id":"2606.24562","kind":"arxiv","version":1}},"canonical_sha256":"32d897827e44a2e8680a968156c67a979ece68110f7d315227c72e2d0298d89a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"32d897827e44a2e8680a968156c67a979ece68110f7d315227c72e2d0298d89a","first_computed_at":"2026-06-24T01:15:33.960075Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-24T01:15:33.960075Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vyK4HmcBBRtVx08srAS7GiX+/sTvGE0GvhcV4Ib3X45MZaFiX218fWl3P+OLJpOcXGF2adM+WKZqDZ5b8bELDQ==","signature_status":"signed_v1","signed_at":"2026-06-24T01:15:33.960433Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.24562","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:65dba18a7e42e122c04b82159a3d66b73183f7449e26a4bea69d16fc172d21ac","sha256:20e903a3699cf26e0bf6afeb43d9063f541bbb402111158a1fcb6debfe4b61b3"],"state_sha256":"6b7e46e8a534bfd29d47746fa666cc67719b534699ccb79793e4a44e87927eb1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"608flvcjU/ekNs7Zd3dGEN36fmqqjEBX81x30ZkZ1eNKFO2R9bipYjzzn2Txx3atMlXllS6Ja1dpiqetrUeyAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T11:41:08.519654Z","bundle_sha256":"2cfed05c571d25a10f6e1ff0c42fcbaaf6c2f302d3e738cf7253a1e9597ce29b"}}