{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:BDFPYRMJWTEBN4TH44PDFKJXY3","short_pith_number":"pith:BDFPYRMJ","canonical_record":{"source":{"id":"2104.02729","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2021-04-06T18:00:14Z","cross_cats_sorted":[],"title_canon_sha256":"9c94f1fff9818077e1915e0da7cd611193e2d3d00c3f5357d33cf3c664605275","abstract_canon_sha256":"ad0037a24ed1f07bea623ef903128cd0938c0716e74037eb891c1f767ae1c351"},"schema_version":"1.0"},"canonical_sha256":"08cafc4589b4c816f267e71e32a937c6cbfe942d985e6800b1192a55b1e5159b","source":{"kind":"arxiv","id":"2104.02729","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2104.02729","created_at":"2026-07-05T08:24:52Z"},{"alias_kind":"arxiv_version","alias_value":"2104.02729v4","created_at":"2026-07-05T08:24:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2104.02729","created_at":"2026-07-05T08:24:52Z"},{"alias_kind":"pith_short_12","alias_value":"BDFPYRMJWTEB","created_at":"2026-07-05T08:24:52Z"},{"alias_kind":"pith_short_16","alias_value":"BDFPYRMJWTEBN4TH","created_at":"2026-07-05T08:24:52Z"},{"alias_kind":"pith_short_8","alias_value":"BDFPYRMJ","created_at":"2026-07-05T08:24:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:BDFPYRMJWTEBN4TH44PDFKJXY3","target":"record","payload":{"canonical_record":{"source":{"id":"2104.02729","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2021-04-06T18:00:14Z","cross_cats_sorted":[],"title_canon_sha256":"9c94f1fff9818077e1915e0da7cd611193e2d3d00c3f5357d33cf3c664605275","abstract_canon_sha256":"ad0037a24ed1f07bea623ef903128cd0938c0716e74037eb891c1f767ae1c351"},"schema_version":"1.0"},"canonical_sha256":"08cafc4589b4c816f267e71e32a937c6cbfe942d985e6800b1192a55b1e5159b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:24:52.098404Z","signature_b64":"nOeIfD/1wjyNgSai525wzsywhC9/EPdYsY/ipLXAYS11wm5gKIbHKpsPIOc1lg/SMDd3poIqi/dHOpE8jua6Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"08cafc4589b4c816f267e71e32a937c6cbfe942d985e6800b1192a55b1e5159b","last_reissued_at":"2026-07-05T08:24:52.097975Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:24:52.097975Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2104.02729","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T08:24:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"md9QSRY0FFlis99iguBMr0GlGf8Hoq6a+wTPq/sDr5OlFAChCGQ+efBLUz22Ya36n4LPuHW1E9qJ3ET2xkAKDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T22:39:10.199028Z"},"content_sha256":"d207761293ba1c0097c1d66846059538381f9828be2846fd59842f59514ec9c4","schema_version":"1.0","event_id":"sha256:d207761293ba1c0097c1d66846059538381f9828be2846fd59842f59514ec9c4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:BDFPYRMJWTEBN4TH44PDFKJXY3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Configuration spaces of clusters as $E_d$-algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Florian Kranhold","submitted_at":"2021-04-06T18:00:14Z","abstract_excerpt":"It is a classical result that configuration spaces of labelled particles in $\\mathbb{R}^d$ are free $E_d$-algebras and that their $d$-fold bar construction is equivalent to the $d$-fold suspension of the labelling space. In this paper, we study a variation of these spaces, namely configuration spaces of labelled clusters of particles. These configuration spaces are again $E_d$-algebras, and we give geometric models for their iterated bar construction in two different ways: one establishes a description of these configuration spaces of clusters as cellular $E_1$-algebras, and the other one uses"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2104.02729","kind":"arxiv","version":4},"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/2104.02729/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":"2026-07-05T08:24:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"f39QUxA+Qu0u7+X27/Sv4SoXzrQCvUXxt820t86hwZPWK2lQmxpWxh75JOCmK0wnSMD4oTrha9CvDcVAgLh4BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T22:39:10.199521Z"},"content_sha256":"4c135e14e584c6594f8174c55a79fbecd3a854d47683f7058979f47a3c17a503","schema_version":"1.0","event_id":"sha256:4c135e14e584c6594f8174c55a79fbecd3a854d47683f7058979f47a3c17a503"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BDFPYRMJWTEBN4TH44PDFKJXY3/bundle.json","state_url":"https://pith.science/pith/BDFPYRMJWTEBN4TH44PDFKJXY3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BDFPYRMJWTEBN4TH44PDFKJXY3/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":"2026-08-23T22:39:10Z","links":{"resolver":"https://pith.science/pith/BDFPYRMJWTEBN4TH44PDFKJXY3","bundle":"https://pith.science/pith/BDFPYRMJWTEBN4TH44PDFKJXY3/bundle.json","state":"https://pith.science/pith/BDFPYRMJWTEBN4TH44PDFKJXY3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BDFPYRMJWTEBN4TH44PDFKJXY3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:BDFPYRMJWTEBN4TH44PDFKJXY3","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":"ad0037a24ed1f07bea623ef903128cd0938c0716e74037eb891c1f767ae1c351","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2021-04-06T18:00:14Z","title_canon_sha256":"9c94f1fff9818077e1915e0da7cd611193e2d3d00c3f5357d33cf3c664605275"},"schema_version":"1.0","source":{"id":"2104.02729","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2104.02729","created_at":"2026-07-05T08:24:52Z"},{"alias_kind":"arxiv_version","alias_value":"2104.02729v4","created_at":"2026-07-05T08:24:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2104.02729","created_at":"2026-07-05T08:24:52Z"},{"alias_kind":"pith_short_12","alias_value":"BDFPYRMJWTEB","created_at":"2026-07-05T08:24:52Z"},{"alias_kind":"pith_short_16","alias_value":"BDFPYRMJWTEBN4TH","created_at":"2026-07-05T08:24:52Z"},{"alias_kind":"pith_short_8","alias_value":"BDFPYRMJ","created_at":"2026-07-05T08:24:52Z"}],"graph_snapshots":[{"event_id":"sha256:4c135e14e584c6594f8174c55a79fbecd3a854d47683f7058979f47a3c17a503","target":"graph","created_at":"2026-07-05T08:24:52Z","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/2104.02729/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"It is a classical result that configuration spaces of labelled particles in $\\mathbb{R}^d$ are free $E_d$-algebras and that their $d$-fold bar construction is equivalent to the $d$-fold suspension of the labelling space. In this paper, we study a variation of these spaces, namely configuration spaces of labelled clusters of particles. These configuration spaces are again $E_d$-algebras, and we give geometric models for their iterated bar construction in two different ways: one establishes a description of these configuration spaces of clusters as cellular $E_1$-algebras, and the other one uses","authors_text":"Florian Kranhold","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2021-04-06T18:00:14Z","title":"Configuration spaces of clusters as $E_d$-algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2104.02729","kind":"arxiv","version":4},"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:d207761293ba1c0097c1d66846059538381f9828be2846fd59842f59514ec9c4","target":"record","created_at":"2026-07-05T08:24:52Z","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":"ad0037a24ed1f07bea623ef903128cd0938c0716e74037eb891c1f767ae1c351","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2021-04-06T18:00:14Z","title_canon_sha256":"9c94f1fff9818077e1915e0da7cd611193e2d3d00c3f5357d33cf3c664605275"},"schema_version":"1.0","source":{"id":"2104.02729","kind":"arxiv","version":4}},"canonical_sha256":"08cafc4589b4c816f267e71e32a937c6cbfe942d985e6800b1192a55b1e5159b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"08cafc4589b4c816f267e71e32a937c6cbfe942d985e6800b1192a55b1e5159b","first_computed_at":"2026-07-05T08:24:52.097975Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:24:52.097975Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nOeIfD/1wjyNgSai525wzsywhC9/EPdYsY/ipLXAYS11wm5gKIbHKpsPIOc1lg/SMDd3poIqi/dHOpE8jua6Dg==","signature_status":"signed_v1","signed_at":"2026-07-05T08:24:52.098404Z","signed_message":"canonical_sha256_bytes"},"source_id":"2104.02729","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d207761293ba1c0097c1d66846059538381f9828be2846fd59842f59514ec9c4","sha256:4c135e14e584c6594f8174c55a79fbecd3a854d47683f7058979f47a3c17a503"],"state_sha256":"4be7ce080fb2d7ffed49004c1e2e4055d88473fc0afa79301066f212954667e4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hEGz6BgcR383B4zA1A1XmgjwTlLqnMrFZHSvO85GP2Cb/m2lk7O/300Hvsoqj14OAGR9TXdqefyKu4STn1PwCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-23T22:39:10.204647Z","bundle_sha256":"b582c8767a63afc0e9916cf88a4ad0f4445e0ee81a7952b6fa6d19b3cc7b2076"}}