{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:MTZY5ASQJCG4A7VAMSTMIHIX4L","short_pith_number":"pith:MTZY5ASQ","canonical_record":{"source":{"id":"2607.24381","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-07-27T12:57:42Z","cross_cats_sorted":["math.LO"],"title_canon_sha256":"9d918ffe559953c17a651909edb407b5c2c57d040f0b5cae5d7b14f01bd1af13","abstract_canon_sha256":"370cb4f9813bf5dfa36be37f2376d8bc05499e86461529ccd64145a69f345b98"},"schema_version":"1.0"},"canonical_sha256":"64f38e8250488dc07ea064a6c41d17e2cfc7ff275a0069b773d5e096d08555cf","source":{"kind":"arxiv","id":"2607.24381","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.24381","created_at":"2026-07-28T02:24:00Z"},{"alias_kind":"arxiv_version","alias_value":"2607.24381v1","created_at":"2026-07-28T02:24:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.24381","created_at":"2026-07-28T02:24:00Z"},{"alias_kind":"pith_short_12","alias_value":"MTZY5ASQJCG4","created_at":"2026-07-28T02:24:00Z"},{"alias_kind":"pith_short_16","alias_value":"MTZY5ASQJCG4A7VA","created_at":"2026-07-28T02:24:00Z"},{"alias_kind":"pith_short_8","alias_value":"MTZY5ASQ","created_at":"2026-07-28T02:24:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:MTZY5ASQJCG4A7VAMSTMIHIX4L","target":"record","payload":{"canonical_record":{"source":{"id":"2607.24381","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-07-27T12:57:42Z","cross_cats_sorted":["math.LO"],"title_canon_sha256":"9d918ffe559953c17a651909edb407b5c2c57d040f0b5cae5d7b14f01bd1af13","abstract_canon_sha256":"370cb4f9813bf5dfa36be37f2376d8bc05499e86461529ccd64145a69f345b98"},"schema_version":"1.0"},"canonical_sha256":"64f38e8250488dc07ea064a6c41d17e2cfc7ff275a0069b773d5e096d08555cf","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-28T02:24:00.188717Z","signature_b64":"B6bslqeQ8oDeHNupMLIZg3a2vUvMvKewUuuVioayihoeSPr4utg80G+/7ozkL4bXYdbwV7wmY/Iz4LM1x1kpAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"64f38e8250488dc07ea064a6c41d17e2cfc7ff275a0069b773d5e096d08555cf","last_reissued_at":"2026-07-28T02:24:00.187901Z","signature_status":"signed_v1","first_computed_at":"2026-07-28T02:24:00.187901Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.24381","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":"2026-07-28T02:24:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Di46ETla9IcacgP4KBbxnlG3UVOgpkfanC9vhnBZoPy19pRCwYmGB2ll/Fk52le0pelwNye6y+g7+wMCsbgjDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T23:04:35.467766Z"},"content_sha256":"32cab91ab7334c8eda1fb069fc7c0bde62a3e14c28a57769b939a9d2558f578e","schema_version":"1.0","event_id":"sha256:32cab91ab7334c8eda1fb069fc7c0bde62a3e14c28a57769b939a9d2558f578e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:MTZY5ASQJCG4A7VAMSTMIHIX4L","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Haar decompression and amenability of Ellis flows","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.LO"],"primary_cat":"math.DS","authors_text":"Daniel Max Hoffmann, Krzysztof Krupi\\'nski","submitted_at":"2026-07-27T12:57:42Z","abstract_excerpt":"Let $(X,G)$ be a tame flow and let $K$ be an Ellis group of its enveloping semigroup $E(X,G)$. Although $K$ is a compact Hausdorff topological group in its $\\tau$-topology, the inclusion of $K$ into $E(X,G)$ need not be Borel. We show that normalized Haar measure on $K$ nevertheless determines, via the Riesz--Markov theorem, a canonical regular Borel probability measure $\\mu_K$ on $E(X,G)$, called its Haar decompression.\n  Our principal structural result states that, for every tame flow, $(E(X,G),G)$ is amenable if and only if $(X,G)$ is hereditarily amenable. For a tame hereditarily amenable "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.24381","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/2607.24381/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":"2026-07-28T02:24:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"k8nbjDE79xc6F9SyQhHg/fZsn/BoOnGN+Bjm5kWCbr5lR1ZGeW5nIPh0zrD/WPp5FxWJPqaC4e4tWPimQtylBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T23:04:35.468269Z"},"content_sha256":"88575cdccea733ce2fb6ddbe4136f763b3e16f3cc2c754cef0a9edca6e4611a4","schema_version":"1.0","event_id":"sha256:88575cdccea733ce2fb6ddbe4136f763b3e16f3cc2c754cef0a9edca6e4611a4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MTZY5ASQJCG4A7VAMSTMIHIX4L/bundle.json","state_url":"https://pith.science/pith/MTZY5ASQJCG4A7VAMSTMIHIX4L/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MTZY5ASQJCG4A7VAMSTMIHIX4L/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":"2026-08-07T23:04:35Z","links":{"resolver":"https://pith.science/pith/MTZY5ASQJCG4A7VAMSTMIHIX4L","bundle":"https://pith.science/pith/MTZY5ASQJCG4A7VAMSTMIHIX4L/bundle.json","state":"https://pith.science/pith/MTZY5ASQJCG4A7VAMSTMIHIX4L/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MTZY5ASQJCG4A7VAMSTMIHIX4L/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:MTZY5ASQJCG4A7VAMSTMIHIX4L","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":"370cb4f9813bf5dfa36be37f2376d8bc05499e86461529ccd64145a69f345b98","cross_cats_sorted":["math.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-07-27T12:57:42Z","title_canon_sha256":"9d918ffe559953c17a651909edb407b5c2c57d040f0b5cae5d7b14f01bd1af13"},"schema_version":"1.0","source":{"id":"2607.24381","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.24381","created_at":"2026-07-28T02:24:00Z"},{"alias_kind":"arxiv_version","alias_value":"2607.24381v1","created_at":"2026-07-28T02:24:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.24381","created_at":"2026-07-28T02:24:00Z"},{"alias_kind":"pith_short_12","alias_value":"MTZY5ASQJCG4","created_at":"2026-07-28T02:24:00Z"},{"alias_kind":"pith_short_16","alias_value":"MTZY5ASQJCG4A7VA","created_at":"2026-07-28T02:24:00Z"},{"alias_kind":"pith_short_8","alias_value":"MTZY5ASQ","created_at":"2026-07-28T02:24:00Z"}],"graph_snapshots":[{"event_id":"sha256:88575cdccea733ce2fb6ddbe4136f763b3e16f3cc2c754cef0a9edca6e4611a4","target":"graph","created_at":"2026-07-28T02:24:00Z","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/2607.24381/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $(X,G)$ be a tame flow and let $K$ be an Ellis group of its enveloping semigroup $E(X,G)$. Although $K$ is a compact Hausdorff topological group in its $\\tau$-topology, the inclusion of $K$ into $E(X,G)$ need not be Borel. We show that normalized Haar measure on $K$ nevertheless determines, via the Riesz--Markov theorem, a canonical regular Borel probability measure $\\mu_K$ on $E(X,G)$, called its Haar decompression.\n  Our principal structural result states that, for every tame flow, $(E(X,G),G)$ is amenable if and only if $(X,G)$ is hereditarily amenable. For a tame hereditarily amenable ","authors_text":"Daniel Max Hoffmann, Krzysztof Krupi\\'nski","cross_cats":["math.LO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-07-27T12:57:42Z","title":"Haar decompression and amenability of Ellis flows"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.24381","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":"sha256:32cab91ab7334c8eda1fb069fc7c0bde62a3e14c28a57769b939a9d2558f578e","target":"record","created_at":"2026-07-28T02:24:00Z","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":"370cb4f9813bf5dfa36be37f2376d8bc05499e86461529ccd64145a69f345b98","cross_cats_sorted":["math.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-07-27T12:57:42Z","title_canon_sha256":"9d918ffe559953c17a651909edb407b5c2c57d040f0b5cae5d7b14f01bd1af13"},"schema_version":"1.0","source":{"id":"2607.24381","kind":"arxiv","version":1}},"canonical_sha256":"64f38e8250488dc07ea064a6c41d17e2cfc7ff275a0069b773d5e096d08555cf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"64f38e8250488dc07ea064a6c41d17e2cfc7ff275a0069b773d5e096d08555cf","first_computed_at":"2026-07-28T02:24:00.187901Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-28T02:24:00.187901Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"B6bslqeQ8oDeHNupMLIZg3a2vUvMvKewUuuVioayihoeSPr4utg80G+/7ozkL4bXYdbwV7wmY/Iz4LM1x1kpAA==","signature_status":"signed_v1","signed_at":"2026-07-28T02:24:00.188717Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.24381","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:32cab91ab7334c8eda1fb069fc7c0bde62a3e14c28a57769b939a9d2558f578e","sha256:88575cdccea733ce2fb6ddbe4136f763b3e16f3cc2c754cef0a9edca6e4611a4"],"state_sha256":"1735696c688fa1b0dd4202819cde51da7fea9a01605a93ab2aceb66916685b93"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"h2/tVKCk3yLmdKXkHqfGXXqVBjf5CzVsmPYBCtPBsCU2FzAb+aCPqP3M21Z6CSdNy39PNMJu2651Kg3zBI4mCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-07T23:04:35.472254Z","bundle_sha256":"ae542648f7ab87e6ae151c55498062cb7ee2c128a260702e6dc35097417eaadf"}}