{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:ABLBVCT2OKPYYNH373LCVEGT6P","short_pith_number":"pith:ABLBVCT2","canonical_record":{"source":{"id":"2605.03993","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-05-05T17:13:37Z","cross_cats_sorted":[],"title_canon_sha256":"c2e2954bd0d93ef1d8197a35187ee95469f9bc36250c4e1c6ca50eac689f6987","abstract_canon_sha256":"9a73ef91641360fa81cfdc610574f563974ea6a06ede48c41cb73c4868ef1411"},"schema_version":"1.0"},"canonical_sha256":"00561a8a7a729f8c34fbfed62a90d3f3dccb0cbc9f9e15634ea7269062fc2318","source":{"kind":"arxiv","id":"2605.03993","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.03993","created_at":"2026-05-29T01:05:11Z"},{"alias_kind":"arxiv_version","alias_value":"2605.03993v2","created_at":"2026-05-29T01:05:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.03993","created_at":"2026-05-29T01:05:11Z"},{"alias_kind":"pith_short_12","alias_value":"ABLBVCT2OKPY","created_at":"2026-05-29T01:05:11Z"},{"alias_kind":"pith_short_16","alias_value":"ABLBVCT2OKPYYNH3","created_at":"2026-05-29T01:05:11Z"},{"alias_kind":"pith_short_8","alias_value":"ABLBVCT2","created_at":"2026-05-29T01:05:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:ABLBVCT2OKPYYNH373LCVEGT6P","target":"record","payload":{"canonical_record":{"source":{"id":"2605.03993","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-05-05T17:13:37Z","cross_cats_sorted":[],"title_canon_sha256":"c2e2954bd0d93ef1d8197a35187ee95469f9bc36250c4e1c6ca50eac689f6987","abstract_canon_sha256":"9a73ef91641360fa81cfdc610574f563974ea6a06ede48c41cb73c4868ef1411"},"schema_version":"1.0"},"canonical_sha256":"00561a8a7a729f8c34fbfed62a90d3f3dccb0cbc9f9e15634ea7269062fc2318","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-29T01:05:11.587388Z","signature_b64":"gAoJxJJuvWfKoecxta7Zrk187LcXpDD5S7FLUZalAKuTa6fXzuxCRs3yVU1NbOyusrXL+rNyhgUg5agcWqy+CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"00561a8a7a729f8c34fbfed62a90d3f3dccb0cbc9f9e15634ea7269062fc2318","last_reissued_at":"2026-05-29T01:05:11.586467Z","signature_status":"signed_v1","first_computed_at":"2026-05-29T01:05:11.586467Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.03993","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-29T01:05:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tMZgfQIUKN9tjp08Kj3DXEOJgezwgC01OVrC2QJztzl8k7kIv8GhHM59SUVPc6VhCzegcY+Wrl+KXW9jwdKGDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T15:17:09.943239Z"},"content_sha256":"99835bbb072a547e84d392f01295a857cab82b3c970ba5480a9c4f5b0d6cae29","schema_version":"1.0","event_id":"sha256:99835bbb072a547e84d392f01295a857cab82b3c970ba5480a9c4f5b0d6cae29"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:ABLBVCT2OKPYYNH373LCVEGT6P","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Invariant random compacts","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Continuous actions on compact metric spaces can be IC-rigid, forcing every invariant random compact to be almost surely finite or the entire space.","cross_cats":[],"primary_cat":"math.DS","authors_text":"Bryna Kra, Scott Schmieding","submitted_at":"2026-05-05T17:13:37Z","abstract_excerpt":"For a compact metric space $X$ with a group $G$ acting on it continuously, an invariant random compact is a Borel probability measure on the space of nonempty compact subsets of $X$ that is invariant under the action of $G$. The action is IC-rigid if, with respect to every invariant random compact, every compact set is almost surely either finite or $X$. We give sufficient conditions for an action to be IC-rigid, and show there are natural examples of such actions. We further consider a notion of weak IC-rigidity, and prove that the Chacon system is weakly IC-rigid but not IC-rigid. As an appl"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We give sufficient conditions for an action to be IC-rigid, and show there are natural examples of such actions. We further consider a notion of weak IC-rigidity, and prove that the Chacon system is weakly IC-rigid but not IC-rigid. As an application, we prove results concerning multiplicative largeness of dilations of sets on the circle.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The setup assumes X is a compact metric space and the G-action is continuous, which permits the space of nonempty compact subsets to carry a natural topology and invariant measures; additionally, the specific dynamical properties of the Chacon system must distinguish weak from full rigidity.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Group actions on compact metric spaces are IC-rigid under stated conditions, with the Chacon system weakly IC-rigid, yielding results on multiplicative largeness of dilated sets.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Continuous actions on compact metric spaces can be IC-rigid, forcing every invariant random compact to be almost surely finite or the entire space.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"2eb505f1570d5f78be1ba3d732f0d75b7ae33ee9004d85a825711644cd2fa952"},"source":{"id":"2605.03993","kind":"arxiv","version":2},"verdict":{"id":"a27724db-c5aa-44ed-9a71-9d1f93d7d2a7","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T04:08:40.016980Z","strongest_claim":"We give sufficient conditions for an action to be IC-rigid, and show there are natural examples of such actions. We further consider a notion of weak IC-rigidity, and prove that the Chacon system is weakly IC-rigid but not IC-rigid. As an application, we prove results concerning multiplicative largeness of dilations of sets on the circle.","one_line_summary":"Group actions on compact metric spaces are IC-rigid under stated conditions, with the Chacon system weakly IC-rigid, yielding results on multiplicative largeness of dilated sets.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The setup assumes X is a compact metric space and the G-action is continuous, which permits the space of nonempty compact subsets to carry a natural topology and invariant measures; additionally, the specific dynamical properties of the Chacon system must distinguish weak from full rigidity.","pith_extraction_headline":"Continuous actions on compact metric spaces can be IC-rigid, forcing every invariant random compact to be almost surely finite or the entire space."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.03993/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T12:40:11.587869Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-20T00:01:21.197560Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T14:51:18.326079Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"bc403b431acde38538de72617503851134582416c246a7cfbb422d5c96e4d895"},"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":"a27724db-c5aa-44ed-9a71-9d1f93d7d2a7"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-29T01:05:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QIka4cD7VGEk3eh/nwlWqon1UKnGza/ElrKwfryFUWev8hZWLY+zSrdjr8+t8bkdy0HQWBBa0iArAakldwObAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T15:17:09.944283Z"},"content_sha256":"4a31113a211ea01e200163b7db6fe4bea56daf9eeae4b12e8099ad4d22b00a9e","schema_version":"1.0","event_id":"sha256:4a31113a211ea01e200163b7db6fe4bea56daf9eeae4b12e8099ad4d22b00a9e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ABLBVCT2OKPYYNH373LCVEGT6P/bundle.json","state_url":"https://pith.science/pith/ABLBVCT2OKPYYNH373LCVEGT6P/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ABLBVCT2OKPYYNH373LCVEGT6P/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":"2026-06-11T15:17:09Z","links":{"resolver":"https://pith.science/pith/ABLBVCT2OKPYYNH373LCVEGT6P","bundle":"https://pith.science/pith/ABLBVCT2OKPYYNH373LCVEGT6P/bundle.json","state":"https://pith.science/pith/ABLBVCT2OKPYYNH373LCVEGT6P/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ABLBVCT2OKPYYNH373LCVEGT6P/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:ABLBVCT2OKPYYNH373LCVEGT6P","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":"9a73ef91641360fa81cfdc610574f563974ea6a06ede48c41cb73c4868ef1411","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-05-05T17:13:37Z","title_canon_sha256":"c2e2954bd0d93ef1d8197a35187ee95469f9bc36250c4e1c6ca50eac689f6987"},"schema_version":"1.0","source":{"id":"2605.03993","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.03993","created_at":"2026-05-29T01:05:11Z"},{"alias_kind":"arxiv_version","alias_value":"2605.03993v2","created_at":"2026-05-29T01:05:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.03993","created_at":"2026-05-29T01:05:11Z"},{"alias_kind":"pith_short_12","alias_value":"ABLBVCT2OKPY","created_at":"2026-05-29T01:05:11Z"},{"alias_kind":"pith_short_16","alias_value":"ABLBVCT2OKPYYNH3","created_at":"2026-05-29T01:05:11Z"},{"alias_kind":"pith_short_8","alias_value":"ABLBVCT2","created_at":"2026-05-29T01:05:11Z"}],"graph_snapshots":[{"event_id":"sha256:4a31113a211ea01e200163b7db6fe4bea56daf9eeae4b12e8099ad4d22b00a9e","target":"graph","created_at":"2026-05-29T01:05:11Z","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":"We give sufficient conditions for an action to be IC-rigid, and show there are natural examples of such actions. We further consider a notion of weak IC-rigidity, and prove that the Chacon system is weakly IC-rigid but not IC-rigid. As an application, we prove results concerning multiplicative largeness of dilations of sets on the circle."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The setup assumes X is a compact metric space and the G-action is continuous, which permits the space of nonempty compact subsets to carry a natural topology and invariant measures; additionally, the specific dynamical properties of the Chacon system must distinguish weak from full rigidity."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Group actions on compact metric spaces are IC-rigid under stated conditions, with the Chacon system weakly IC-rigid, yielding results on multiplicative largeness of dilated sets."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Continuous actions on compact metric spaces can be IC-rigid, forcing every invariant random compact to be almost surely finite or the entire space."}],"snapshot_sha256":"2eb505f1570d5f78be1ba3d732f0d75b7ae33ee9004d85a825711644cd2fa952"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"name":"ai_meta_artifact","ran_at":"2026-05-20T12:40:11.587869Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"doi_title_agreement","ran_at":"2026-05-20T00:01:21.197560Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"doi_compliance","ran_at":"2026-05-19T14:51:18.326079Z","status":"completed","version":"1.0.0"}],"endpoint":"/pith/2605.03993/integrity.json","findings":[],"snapshot_sha256":"bc403b431acde38538de72617503851134582416c246a7cfbb422d5c96e4d895","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For a compact metric space $X$ with a group $G$ acting on it continuously, an invariant random compact is a Borel probability measure on the space of nonempty compact subsets of $X$ that is invariant under the action of $G$. The action is IC-rigid if, with respect to every invariant random compact, every compact set is almost surely either finite or $X$. We give sufficient conditions for an action to be IC-rigid, and show there are natural examples of such actions. We further consider a notion of weak IC-rigidity, and prove that the Chacon system is weakly IC-rigid but not IC-rigid. As an appl","authors_text":"Bryna Kra, Scott Schmieding","cross_cats":[],"headline":"Continuous actions on compact metric spaces can be IC-rigid, forcing every invariant random compact to be almost surely finite or the entire space.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-05-05T17:13:37Z","title":"Invariant random compacts"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.03993","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-07T04:08:40.016980Z","id":"a27724db-c5aa-44ed-9a71-9d1f93d7d2a7","model_set":{"reader":"grok-4.3"},"one_line_summary":"Group actions on compact metric spaces are IC-rigid under stated conditions, with the Chacon system weakly IC-rigid, yielding results on multiplicative largeness of dilated sets.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Continuous actions on compact metric spaces can be IC-rigid, forcing every invariant random compact to be almost surely finite or the entire space.","strongest_claim":"We give sufficient conditions for an action to be IC-rigid, and show there are natural examples of such actions. We further consider a notion of weak IC-rigidity, and prove that the Chacon system is weakly IC-rigid but not IC-rigid. As an application, we prove results concerning multiplicative largeness of dilations of sets on the circle.","weakest_assumption":"The setup assumes X is a compact metric space and the G-action is continuous, which permits the space of nonempty compact subsets to carry a natural topology and invariant measures; additionally, the specific dynamical properties of the Chacon system must distinguish weak from full rigidity."}},"verdict_id":"a27724db-c5aa-44ed-9a71-9d1f93d7d2a7"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:99835bbb072a547e84d392f01295a857cab82b3c970ba5480a9c4f5b0d6cae29","target":"record","created_at":"2026-05-29T01:05:11Z","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":"9a73ef91641360fa81cfdc610574f563974ea6a06ede48c41cb73c4868ef1411","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-05-05T17:13:37Z","title_canon_sha256":"c2e2954bd0d93ef1d8197a35187ee95469f9bc36250c4e1c6ca50eac689f6987"},"schema_version":"1.0","source":{"id":"2605.03993","kind":"arxiv","version":2}},"canonical_sha256":"00561a8a7a729f8c34fbfed62a90d3f3dccb0cbc9f9e15634ea7269062fc2318","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"00561a8a7a729f8c34fbfed62a90d3f3dccb0cbc9f9e15634ea7269062fc2318","first_computed_at":"2026-05-29T01:05:11.586467Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-29T01:05:11.586467Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gAoJxJJuvWfKoecxta7Zrk187LcXpDD5S7FLUZalAKuTa6fXzuxCRs3yVU1NbOyusrXL+rNyhgUg5agcWqy+CA==","signature_status":"signed_v1","signed_at":"2026-05-29T01:05:11.587388Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.03993","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:99835bbb072a547e84d392f01295a857cab82b3c970ba5480a9c4f5b0d6cae29","sha256:4a31113a211ea01e200163b7db6fe4bea56daf9eeae4b12e8099ad4d22b00a9e"],"state_sha256":"1312ef869fa7efa6ecf8c849ed78bceccc1ba18265bd7992f588e3c6f4d68c30"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vPRTGEld+HzakNBcbAGKnsbrZH3PgDk0NJacPMfAkFhYQ7uRTSvR8cWe2HYxhZYkknY5r/01Am5n0HGmBKX5Dg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T15:17:09.949113Z","bundle_sha256":"91d9e2f370d44f94673fc94fa9a70bb68dd5ab0a05ab9d32c087e21419d2a155"}}