{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:B5BDMF3WQMHH3VYXBGJN7HNN6Y","short_pith_number":"pith:B5BDMF3W","canonical_record":{"source":{"id":"2606.05619","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2026-06-04T02:43:30Z","cross_cats_sorted":[],"title_canon_sha256":"2bacbdf0816bf5660d56f12240e8651b68c0da168d66f55aede1b43ce22429d7","abstract_canon_sha256":"f849f9c41272f5dad6bf1b1c5fe0ec45e6874b97f49272bee853daf8da4856b8"},"schema_version":"1.0"},"canonical_sha256":"0f42361776830e7dd7170992df9dadf60b21c939cc033b0262a07488c3557912","source":{"kind":"arxiv","id":"2606.05619","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.05619","created_at":"2026-06-05T01:14:56Z"},{"alias_kind":"arxiv_version","alias_value":"2606.05619v1","created_at":"2026-06-05T01:14:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.05619","created_at":"2026-06-05T01:14:56Z"},{"alias_kind":"pith_short_12","alias_value":"B5BDMF3WQMHH","created_at":"2026-06-05T01:14:56Z"},{"alias_kind":"pith_short_16","alias_value":"B5BDMF3WQMHH3VYX","created_at":"2026-06-05T01:14:56Z"},{"alias_kind":"pith_short_8","alias_value":"B5BDMF3W","created_at":"2026-06-05T01:14:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:B5BDMF3WQMHH3VYXBGJN7HNN6Y","target":"record","payload":{"canonical_record":{"source":{"id":"2606.05619","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2026-06-04T02:43:30Z","cross_cats_sorted":[],"title_canon_sha256":"2bacbdf0816bf5660d56f12240e8651b68c0da168d66f55aede1b43ce22429d7","abstract_canon_sha256":"f849f9c41272f5dad6bf1b1c5fe0ec45e6874b97f49272bee853daf8da4856b8"},"schema_version":"1.0"},"canonical_sha256":"0f42361776830e7dd7170992df9dadf60b21c939cc033b0262a07488c3557912","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-05T01:14:56.631002Z","signature_b64":"BXDrsH00zNJcCMYVliaJLTZNLZztBlfi1CdrcvoIhBjE7hSem/mvpglqYR4MKfeWqmkm9K2fgsKaj8WVyM1wCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0f42361776830e7dd7170992df9dadf60b21c939cc033b0262a07488c3557912","last_reissued_at":"2026-06-05T01:14:56.630526Z","signature_status":"signed_v1","first_computed_at":"2026-06-05T01:14:56.630526Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.05619","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":"2026-06-05T01:14:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w+bNMz4JooRjbon8OxR5DZqDiNUozL4XIgAFX2ekhKB6V1LbVi2eZEP2y3kQYoxYRy4IJmlwArflSxBa4EbRAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T18:21:02.790639Z"},"content_sha256":"5457c9a460bd0f3db160c4d47659ef1f3dba7480ba08678a1e76ab273c80cbb9","schema_version":"1.0","event_id":"sha256:5457c9a460bd0f3db160c4d47659ef1f3dba7480ba08678a1e76ab273c80cbb9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:B5BDMF3WQMHH3VYXBGJN7HNN6Y","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Asymmetry of Entropy Invariants for Generic Mixing $Z^n$-Actions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Mikhail V. Engelgardt, Valery V. Ryzhikov","submitted_at":"2026-06-04T02:43:30Z","abstract_excerpt":"Suppose that for a mixing $Z^n$-action, $n>1$, there exists a Kirillov-Kushnirenko entropy that is zero for this action and completely positive for its inverse. We prove that this property is generic for the mixing $Z^n$-actions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.05619","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/2606.05619/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":"2026-06-05T01:14:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9AkmAIU3K9yYeQv0OziPD/HfQr+XFKJ0gFsSCx+79bnUsIDqJFEtZLisIjdTQEYPjV+V/GTO1xvmuIe/w312Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T18:21:02.791411Z"},"content_sha256":"55dc4df60922e35f706a7637943361d2aabc60401abf4997afea178743823704","schema_version":"1.0","event_id":"sha256:55dc4df60922e35f706a7637943361d2aabc60401abf4997afea178743823704"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/B5BDMF3WQMHH3VYXBGJN7HNN6Y/bundle.json","state_url":"https://pith.science/pith/B5BDMF3WQMHH3VYXBGJN7HNN6Y/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/B5BDMF3WQMHH3VYXBGJN7HNN6Y/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":"2026-06-09T18:21:02Z","links":{"resolver":"https://pith.science/pith/B5BDMF3WQMHH3VYXBGJN7HNN6Y","bundle":"https://pith.science/pith/B5BDMF3WQMHH3VYXBGJN7HNN6Y/bundle.json","state":"https://pith.science/pith/B5BDMF3WQMHH3VYXBGJN7HNN6Y/state.json","well_known_bundle":"https://pith.science/.well-known/pith/B5BDMF3WQMHH3VYXBGJN7HNN6Y/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:B5BDMF3WQMHH3VYXBGJN7HNN6Y","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":"f849f9c41272f5dad6bf1b1c5fe0ec45e6874b97f49272bee853daf8da4856b8","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2026-06-04T02:43:30Z","title_canon_sha256":"2bacbdf0816bf5660d56f12240e8651b68c0da168d66f55aede1b43ce22429d7"},"schema_version":"1.0","source":{"id":"2606.05619","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.05619","created_at":"2026-06-05T01:14:56Z"},{"alias_kind":"arxiv_version","alias_value":"2606.05619v1","created_at":"2026-06-05T01:14:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.05619","created_at":"2026-06-05T01:14:56Z"},{"alias_kind":"pith_short_12","alias_value":"B5BDMF3WQMHH","created_at":"2026-06-05T01:14:56Z"},{"alias_kind":"pith_short_16","alias_value":"B5BDMF3WQMHH3VYX","created_at":"2026-06-05T01:14:56Z"},{"alias_kind":"pith_short_8","alias_value":"B5BDMF3W","created_at":"2026-06-05T01:14:56Z"}],"graph_snapshots":[{"event_id":"sha256:55dc4df60922e35f706a7637943361d2aabc60401abf4997afea178743823704","target":"graph","created_at":"2026-06-05T01:14:56Z","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/2606.05619/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Suppose that for a mixing $Z^n$-action, $n>1$, there exists a Kirillov-Kushnirenko entropy that is zero for this action and completely positive for its inverse. We prove that this property is generic for the mixing $Z^n$-actions.","authors_text":"Mikhail V. Engelgardt, Valery V. Ryzhikov","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2026-06-04T02:43:30Z","title":"Asymmetry of Entropy Invariants for Generic Mixing $Z^n$-Actions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.05619","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":"sha256:5457c9a460bd0f3db160c4d47659ef1f3dba7480ba08678a1e76ab273c80cbb9","target":"record","created_at":"2026-06-05T01:14:56Z","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":"f849f9c41272f5dad6bf1b1c5fe0ec45e6874b97f49272bee853daf8da4856b8","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2026-06-04T02:43:30Z","title_canon_sha256":"2bacbdf0816bf5660d56f12240e8651b68c0da168d66f55aede1b43ce22429d7"},"schema_version":"1.0","source":{"id":"2606.05619","kind":"arxiv","version":1}},"canonical_sha256":"0f42361776830e7dd7170992df9dadf60b21c939cc033b0262a07488c3557912","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0f42361776830e7dd7170992df9dadf60b21c939cc033b0262a07488c3557912","first_computed_at":"2026-06-05T01:14:56.630526Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-05T01:14:56.630526Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BXDrsH00zNJcCMYVliaJLTZNLZztBlfi1CdrcvoIhBjE7hSem/mvpglqYR4MKfeWqmkm9K2fgsKaj8WVyM1wCQ==","signature_status":"signed_v1","signed_at":"2026-06-05T01:14:56.631002Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.05619","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5457c9a460bd0f3db160c4d47659ef1f3dba7480ba08678a1e76ab273c80cbb9","sha256:55dc4df60922e35f706a7637943361d2aabc60401abf4997afea178743823704"],"state_sha256":"4f04669c473fd578b1ced7ebc110469cec1ac55bc6ec8cef53bab26d14930dc0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vPd4IYTXC2ag8wnv2rqh5T1GTbc/KyhDsE6Hoqq5xmIXDkbVZ1SqCe+m3vpcXeslemQlKKC9axEjAcDwEK7ADg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T18:21:02.795678Z","bundle_sha256":"596a21825bb2b59504bbe061ebd5504f912f79ed023c612df23d5786eec521a9"}}