{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:7EHRGU3EA3TENOAX2BCKFP3RIR","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":"80f369f55965a7141f43a9b6a891135af82eeb233b691fc8c51dca0278306ca1","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2026-05-15T12:21:59Z","title_canon_sha256":"b324d359802272decb4005d00675ca97431d67ea38dd7752b42b586bf4c37221"},"schema_version":"1.0","source":{"id":"2605.15891","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.15891","created_at":"2026-05-20T00:01:23Z"},{"alias_kind":"arxiv_version","alias_value":"2605.15891v1","created_at":"2026-05-20T00:01:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.15891","created_at":"2026-05-20T00:01:23Z"},{"alias_kind":"pith_short_12","alias_value":"7EHRGU3EA3TE","created_at":"2026-05-20T00:01:23Z"},{"alias_kind":"pith_short_16","alias_value":"7EHRGU3EA3TENOAX","created_at":"2026-05-20T00:01:23Z"},{"alias_kind":"pith_short_8","alias_value":"7EHRGU3E","created_at":"2026-05-20T00:01:23Z"}],"graph_snapshots":[{"event_id":"sha256:1868c66eb63411d826c2df9b2f1a3db517b6fbd42893d2ba97df5dc75d1b2865","target":"graph","created_at":"2026-05-20T00:01:23Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"For 0<q≤n, we give a complete existence characterization in the framework of G-invariant convex bodies, recovering the origin-symmetric setting when G={±I}. The necessary and sufficient conditions concern the concentration of the measure on G-invariant subspaces, both in the range 0<q<n and at the critical endpoint q=n."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The given measure must obey specific concentration restrictions on the G-invariant subspaces; if this concentration condition fails, no G-invariant solution exists, as this forms the necessary and sufficient criterion stated for both the subcritical and critical cases."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"The paper establishes necessary and sufficient conditions for the existence of G-invariant convex bodies solving the dual Minkowski problem, with the conditions depending on measure concentration on G-invariant subspaces, including the logarithmic case at q = n."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"The dual Minkowski problem has a complete existence characterization for G-invariant convex bodies when measures concentrate properly on invariant subspaces."}],"snapshot_sha256":"658b026a43e6b32055c09d8a64ab818c4c0b1ebf082d49ef0abfcf9271b2e8c8"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"3363e61c38af867d481e7ab638f9a92b12e58d8093f3625d48a4dd10c9b5e069"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"name":"doi_compliance","ran_at":"2026-05-19T17:36:25.372690Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"ai_meta_artifact","ran_at":"2026-05-19T17:33:47.518818Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"doi_title_agreement","ran_at":"2026-05-19T17:31:18.441954Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"claim_evidence","ran_at":"2026-05-19T17:01:55.783361Z","status":"completed","version":"1.0.0"}],"endpoint":"/pith/2605.15891/integrity.json","findings":[],"snapshot_sha256":"2ba631560c0b873825f261966b51f4fa73522d596f7156ab9a9a2db1c747e1d7","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we study the dual Minkowski problem under group symmetry. For $0<q\\le n$, we give a complete existence characterization in the framework of $G$-invariant convex bodies, recovering the origin-symmetric setting when $G=\\{\\pm I\\}$. The necessary and sufficient conditions concern the concentration of the measure on $G$-invariant subspaces, both in the range $0<q<n$ and at the critical endpoint $q=n$, where the problem becomes the logarithmic Minkowski problem.","authors_text":"Junjie Shan","cross_cats":[],"headline":"The dual Minkowski problem has a complete existence characterization for G-invariant convex bodies when measures concentrate properly on invariant subspaces.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2026-05-15T12:21:59Z","title":"The Dual Minkowski Problem under Group Actions"},"references":{"count":46,"internal_anchors":0,"resolved_work":46,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"B¨ or¨ oczky, E","work_id":"12909b98-f0eb-42c0-b397-fba5844c8784","year":2013},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"B¨ or¨ oczky, F","work_id":"6408a013-364f-4306-8d95-6e6ee6a291cf","year":2019},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"B¨ or¨ oczky, P","work_id":"61cddad1-986d-4241-ab97-70f939d19466","year":2016},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"B¨ or¨ oczky, M","work_id":"e0ef2f07-2eb8-4c2b-8916-00aa6b7ba221","year":2016},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"B¨ or¨ oczky, M","work_id":"f1577371-a099-4392-ba54-781cc8f93d8b","year":2018}],"snapshot_sha256":"fa80c2cfd3704ade4efc7ebfde9ebbd2a4964827764691bb5a6f7d7371231c6d"},"source":{"id":"2605.15891","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-19T17:24:34.779056Z","id":"8c50d871-cf3f-4ade-89f4-294fe3b2b9d1","model_set":{"reader":"grok-4.3"},"one_line_summary":"The paper establishes necessary and sufficient conditions for the existence of G-invariant convex bodies solving the dual Minkowski problem, with the conditions depending on measure concentration on G-invariant subspaces, including the logarithmic case at q = n.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"The dual Minkowski problem has a complete existence characterization for G-invariant convex bodies when measures concentrate properly on invariant subspaces.","strongest_claim":"For 0<q≤n, we give a complete existence characterization in the framework of G-invariant convex bodies, recovering the origin-symmetric setting when G={±I}. The necessary and sufficient conditions concern the concentration of the measure on G-invariant subspaces, both in the range 0<q<n and at the critical endpoint q=n.","weakest_assumption":"The given measure must obey specific concentration restrictions on the G-invariant subspaces; if this concentration condition fails, no G-invariant solution exists, as this forms the necessary and sufficient criterion stated for both the subcritical and critical cases."}},"verdict_id":"8c50d871-cf3f-4ade-89f4-294fe3b2b9d1"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:d5cebb87f19e78bdbe84ae22c7dc908b144c1c8c8f407ba92adc14e286e08c1f","target":"record","created_at":"2026-05-20T00:01:23Z","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":"80f369f55965a7141f43a9b6a891135af82eeb233b691fc8c51dca0278306ca1","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2026-05-15T12:21:59Z","title_canon_sha256":"b324d359802272decb4005d00675ca97431d67ea38dd7752b42b586bf4c37221"},"schema_version":"1.0","source":{"id":"2605.15891","kind":"arxiv","version":1}},"canonical_sha256":"f90f13536406e646b817d044a2bf714455c94fc1e31a769291852e313cad7cd6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f90f13536406e646b817d044a2bf714455c94fc1e31a769291852e313cad7cd6","first_computed_at":"2026-05-20T00:01:23.986223Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:01:23.986223Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NQiiGcrT1vG1RHsZtjmAbQPPZU7jFn4v19QVTgJWtVNngfKVxBt9t0aNxSUi0B8/hY7Mdf/AL3kKVdz0jx2uDg==","signature_status":"signed_v1","signed_at":"2026-05-20T00:01:23.987118Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.15891","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d5cebb87f19e78bdbe84ae22c7dc908b144c1c8c8f407ba92adc14e286e08c1f","sha256:1868c66eb63411d826c2df9b2f1a3db517b6fbd42893d2ba97df5dc75d1b2865"],"state_sha256":"f173ad1a6ac493f509b7ecbe7b48bb2cc61d45bce9f076cfa6f4b97915346513"}