{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:TBMERZ44RA2X3PMATK4EJBMK2V","short_pith_number":"pith:TBMERZ44","canonical_record":{"source":{"id":"2605.25666","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2026-05-25T10:15:34Z","cross_cats_sorted":[],"title_canon_sha256":"34b6c7c5fe9d8ebae074a9e6055c6a7699d2d7758a99b0086f38704b12471074","abstract_canon_sha256":"d2eb2f37e8a76d1712e4b36c7b53f6c159941583a1163868886d2ad733a5ac1f"},"schema_version":"1.0"},"canonical_sha256":"985848e79c88357dbd809ab844858ad554fdef87ae5e776688210f9189641abc","source":{"kind":"arxiv","id":"2605.25666","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.25666","created_at":"2026-05-26T02:04:49Z"},{"alias_kind":"arxiv_version","alias_value":"2605.25666v1","created_at":"2026-05-26T02:04:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.25666","created_at":"2026-05-26T02:04:49Z"},{"alias_kind":"pith_short_12","alias_value":"TBMERZ44RA2X","created_at":"2026-05-26T02:04:49Z"},{"alias_kind":"pith_short_16","alias_value":"TBMERZ44RA2X3PMA","created_at":"2026-05-26T02:04:49Z"},{"alias_kind":"pith_short_8","alias_value":"TBMERZ44","created_at":"2026-05-26T02:04:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:TBMERZ44RA2X3PMATK4EJBMK2V","target":"record","payload":{"canonical_record":{"source":{"id":"2605.25666","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2026-05-25T10:15:34Z","cross_cats_sorted":[],"title_canon_sha256":"34b6c7c5fe9d8ebae074a9e6055c6a7699d2d7758a99b0086f38704b12471074","abstract_canon_sha256":"d2eb2f37e8a76d1712e4b36c7b53f6c159941583a1163868886d2ad733a5ac1f"},"schema_version":"1.0"},"canonical_sha256":"985848e79c88357dbd809ab844858ad554fdef87ae5e776688210f9189641abc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-26T02:04:49.047057Z","signature_b64":"OMNRAOkiGMlYymFCf/E0qZP0hfWaYbqIXbhPNEv0ZR9qO2uj8Xa5c34mxyaL5nbQ23lXUycpl8X9Bs3CXajVCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"985848e79c88357dbd809ab844858ad554fdef87ae5e776688210f9189641abc","last_reissued_at":"2026-05-26T02:04:49.046309Z","signature_status":"signed_v1","first_computed_at":"2026-05-26T02:04:49.046309Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.25666","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":"2026-05-26T02:04:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/6rqfkgDsQ9r7fZL5Lt01phNFx5+8Y71C1me4ZIyo85pOGZ227K+CcW9XCIbP7e80CdFnBP7eLp/xUzyqHvaCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T21:36:04.964759Z"},"content_sha256":"360893deaa38ac19f98c3dce6bbab7b1ebef9fb115e7b89b5959a41b0bf3c84c","schema_version":"1.0","event_id":"sha256:360893deaa38ac19f98c3dce6bbab7b1ebef9fb115e7b89b5959a41b0bf3c84c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:TBMERZ44RA2X3PMATK4EJBMK2V","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Fixed Point Rigidity of the Operator $\\Gamma_p\\Pi_p^\\ast$ and the LYZ Conjecture","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Sudan Xing, Youjiang Lin","submitted_at":"2026-05-25T10:15:34Z","abstract_excerpt":"Motivated by the recent approach of Milman, Shabelman, and Yehudayoff \\cite{MilmanShabelmanYehudayoff2025}, we establish, for $p\\geq 1$, a complete characterization of the fixed points of the composition of the $L_p$-centroid operator and the polar $L_p$-projection operator. More precisely, we prove that if a convex body $K \\in \\mathcal{K}_o^n$ satisfies \\[\\Gamma_p \\Pi_p^* K = cK\\] for some constant $c>0$, then $K$ must be an ellipsoid. Conversely, ellipsoids are the only such fixed points of convex bodies up to dilation. This confirms a conjecture of Lutwak, Yang, and Zhang \\cite{LutwakYangZh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.25666","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/2605.25666/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":"2026-05-26T02:04:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"V/FN8RxUYiR7Ku6jKaKVN736gTi6y6itpqwQA32HsEkcexcX8z6umgJWY5JiQsiKmPvxu0AqxYYOptcslu0tAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T21:36:04.965139Z"},"content_sha256":"e5fa294a1e250877c202175efc5923c0bb0e8070287ca28c3c06a1b34b822b74","schema_version":"1.0","event_id":"sha256:e5fa294a1e250877c202175efc5923c0bb0e8070287ca28c3c06a1b34b822b74"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TBMERZ44RA2X3PMATK4EJBMK2V/bundle.json","state_url":"https://pith.science/pith/TBMERZ44RA2X3PMATK4EJBMK2V/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TBMERZ44RA2X3PMATK4EJBMK2V/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":"2026-06-24T21:36:04Z","links":{"resolver":"https://pith.science/pith/TBMERZ44RA2X3PMATK4EJBMK2V","bundle":"https://pith.science/pith/TBMERZ44RA2X3PMATK4EJBMK2V/bundle.json","state":"https://pith.science/pith/TBMERZ44RA2X3PMATK4EJBMK2V/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TBMERZ44RA2X3PMATK4EJBMK2V/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:TBMERZ44RA2X3PMATK4EJBMK2V","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":"d2eb2f37e8a76d1712e4b36c7b53f6c159941583a1163868886d2ad733a5ac1f","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2026-05-25T10:15:34Z","title_canon_sha256":"34b6c7c5fe9d8ebae074a9e6055c6a7699d2d7758a99b0086f38704b12471074"},"schema_version":"1.0","source":{"id":"2605.25666","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.25666","created_at":"2026-05-26T02:04:49Z"},{"alias_kind":"arxiv_version","alias_value":"2605.25666v1","created_at":"2026-05-26T02:04:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.25666","created_at":"2026-05-26T02:04:49Z"},{"alias_kind":"pith_short_12","alias_value":"TBMERZ44RA2X","created_at":"2026-05-26T02:04:49Z"},{"alias_kind":"pith_short_16","alias_value":"TBMERZ44RA2X3PMA","created_at":"2026-05-26T02:04:49Z"},{"alias_kind":"pith_short_8","alias_value":"TBMERZ44","created_at":"2026-05-26T02:04:49Z"}],"graph_snapshots":[{"event_id":"sha256:e5fa294a1e250877c202175efc5923c0bb0e8070287ca28c3c06a1b34b822b74","target":"graph","created_at":"2026-05-26T02:04:49Z","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/2605.25666/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Motivated by the recent approach of Milman, Shabelman, and Yehudayoff \\cite{MilmanShabelmanYehudayoff2025}, we establish, for $p\\geq 1$, a complete characterization of the fixed points of the composition of the $L_p$-centroid operator and the polar $L_p$-projection operator. More precisely, we prove that if a convex body $K \\in \\mathcal{K}_o^n$ satisfies \\[\\Gamma_p \\Pi_p^* K = cK\\] for some constant $c>0$, then $K$ must be an ellipsoid. Conversely, ellipsoids are the only such fixed points of convex bodies up to dilation. This confirms a conjecture of Lutwak, Yang, and Zhang \\cite{LutwakYangZh","authors_text":"Sudan Xing, Youjiang Lin","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2026-05-25T10:15:34Z","title":"Fixed Point Rigidity of the Operator $\\Gamma_p\\Pi_p^\\ast$ and the LYZ Conjecture"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.25666","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":"sha256:360893deaa38ac19f98c3dce6bbab7b1ebef9fb115e7b89b5959a41b0bf3c84c","target":"record","created_at":"2026-05-26T02:04:49Z","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":"d2eb2f37e8a76d1712e4b36c7b53f6c159941583a1163868886d2ad733a5ac1f","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2026-05-25T10:15:34Z","title_canon_sha256":"34b6c7c5fe9d8ebae074a9e6055c6a7699d2d7758a99b0086f38704b12471074"},"schema_version":"1.0","source":{"id":"2605.25666","kind":"arxiv","version":1}},"canonical_sha256":"985848e79c88357dbd809ab844858ad554fdef87ae5e776688210f9189641abc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"985848e79c88357dbd809ab844858ad554fdef87ae5e776688210f9189641abc","first_computed_at":"2026-05-26T02:04:49.046309Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-26T02:04:49.046309Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OMNRAOkiGMlYymFCf/E0qZP0hfWaYbqIXbhPNEv0ZR9qO2uj8Xa5c34mxyaL5nbQ23lXUycpl8X9Bs3CXajVCA==","signature_status":"signed_v1","signed_at":"2026-05-26T02:04:49.047057Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.25666","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:360893deaa38ac19f98c3dce6bbab7b1ebef9fb115e7b89b5959a41b0bf3c84c","sha256:e5fa294a1e250877c202175efc5923c0bb0e8070287ca28c3c06a1b34b822b74"],"state_sha256":"b69a20b749cf18563b7ecf4f83e21c6d4fe2e9d077493418dd0acafa660e62a6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5ZdEuiZdX6MZxtWr9lYrJbd4giC/k2EW3cqW3+tUG9lXW3WzT80CjmRXDx8gQl/wIem4TA5pcalBKzm7axxvBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T21:36:04.967178Z","bundle_sha256":"d303a508abdfd2ae03a94311320b08799a1288903d0ccacd7c3b3b84110659a9"}}