{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:YAWKKH2DOHOGTFLLISIA742E2T","short_pith_number":"pith:YAWKKH2D","canonical_record":{"source":{"id":"2605.04646","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-05-06T08:52:07Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"ebf278da6723c338feea66c033231e746d0ef68a19198c5d436d5c33b7d381c9","abstract_canon_sha256":"2932158df3b3670630a1f4b083861b6d86bda5a1b5e4ecc0ea952dcc0aa2e4e5"},"schema_version":"1.0"},"canonical_sha256":"c02ca51f4371dc69956b44900ff344d4ee2acc3e4219a86bc3f8bdf404c42cb4","source":{"kind":"arxiv","id":"2605.04646","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.04646","created_at":"2026-05-29T02:05:46Z"},{"alias_kind":"arxiv_version","alias_value":"2605.04646v2","created_at":"2026-05-29T02:05:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.04646","created_at":"2026-05-29T02:05:46Z"},{"alias_kind":"pith_short_12","alias_value":"YAWKKH2DOHOG","created_at":"2026-05-29T02:05:46Z"},{"alias_kind":"pith_short_16","alias_value":"YAWKKH2DOHOGTFLL","created_at":"2026-05-29T02:05:46Z"},{"alias_kind":"pith_short_8","alias_value":"YAWKKH2D","created_at":"2026-05-29T02:05:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:YAWKKH2DOHOGTFLLISIA742E2T","target":"record","payload":{"canonical_record":{"source":{"id":"2605.04646","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-05-06T08:52:07Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"ebf278da6723c338feea66c033231e746d0ef68a19198c5d436d5c33b7d381c9","abstract_canon_sha256":"2932158df3b3670630a1f4b083861b6d86bda5a1b5e4ecc0ea952dcc0aa2e4e5"},"schema_version":"1.0"},"canonical_sha256":"c02ca51f4371dc69956b44900ff344d4ee2acc3e4219a86bc3f8bdf404c42cb4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-29T02:05:46.019537Z","signature_b64":"HGK+QCi8ka80aD/ID4VjzYWT2TMlBqboeo3Ph+EgKGhLhONkQdD+nM+cQkFHm4DehPAVSwu4g2PvH8sATm37Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c02ca51f4371dc69956b44900ff344d4ee2acc3e4219a86bc3f8bdf404c42cb4","last_reissued_at":"2026-05-29T02:05:46.018672Z","signature_status":"signed_v1","first_computed_at":"2026-05-29T02:05:46.018672Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.04646","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":"2026-05-29T02:05:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"txSUCtYTUSLmg0YIruVCw2jjSmiUMHOv1Oi3fb8VHiQ6zif6RujCtiKwCZ31rR14WLA15u60iYeO8hkKNWfzBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T12:37:28.123834Z"},"content_sha256":"d898e45029502b58451d9c9af3dcafe513f0360fa58ad251795585b9c4e15c99","schema_version":"1.0","event_id":"sha256:d898e45029502b58451d9c9af3dcafe513f0360fa58ad251795585b9c4e15c99"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:YAWKKH2DOHOGTFLLISIA742E2T","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The geometry of wreath and semi-direct products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Twisting and wreath products extend to coset geometries while preserving flag-transitivity, residual-connectedness and thinness, yielding regular polytopes and hypertopes for almost-simple groups with sporadic socles.","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Claudio Alexandre Piedade, Philippe Tranchida","submitted_at":"2026-05-06T08:52:07Z","abstract_excerpt":"Coset geometries are incidence geometries constructed from a group $G$ and a system of subgroups $(G_i)_{i \\in I}$ of subgroups of $G$. For any algebraic group operation, it is then natural to wonder whether it can be extended to the framework of coset geometries. This has been achieved in the case of the halving (\\cite{halving}) and in the case of free (amalgamated) products, HNN-extensions, and semi-direct products (\\cite{piedade2025group}). In this article, we explore more deeply two operations related to semi-direct products: the twisting and the wreath product. We show that these operatio"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we show that there exists regular polytopes and hypertopes for almost-simple group with socle a sporadic simple group.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the twisting and wreath product operations can be defined on arbitrary coset geometries so that flag-transitivity, residual-connectedness, and thinness are automatically preserved.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Twisting and wreath products extend to coset geometries preserving key properties, enabling regular polytopes and hypertopes for almost-simple groups with sporadic socles.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Twisting and wreath products extend to coset geometries while preserving flag-transitivity, residual-connectedness and thinness, yielding regular polytopes and hypertopes for almost-simple groups with sporadic socles.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"0129e7c28d2830024b600cb51f8c7b88c9b43ea7cb409fce81cfbf47991acb53"},"source":{"id":"2605.04646","kind":"arxiv","version":2},"verdict":{"id":"21f122cf-58d2-45d4-9a09-8e9813b0d488","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T16:36:14.679471Z","strongest_claim":"we show that there exists regular polytopes and hypertopes for almost-simple group with socle a sporadic simple group.","one_line_summary":"Twisting and wreath products extend to coset geometries preserving key properties, enabling regular polytopes and hypertopes for almost-simple groups with sporadic socles.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the twisting and wreath product operations can be defined on arbitrary coset geometries so that flag-transitivity, residual-connectedness, and thinness are automatically preserved.","pith_extraction_headline":"Twisting and wreath products extend to coset geometries while preserving flag-transitivity, residual-connectedness and thinness, yielding regular polytopes and hypertopes for almost-simple groups with sporadic socles."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.04646/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T11:36:19.371994Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T22:31:19.904865Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T14:15:23.310077Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"759841956606009dcedcba5b33e9470ce530b9d545da5f0c5fa3ea53f541289c"},"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":"21f122cf-58d2-45d4-9a09-8e9813b0d488"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-29T02:05:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+IkjwL6+JLm5CUW2JDkahBa7IN7wq7iRPAL2OZrGT1xoyNFZJyYEP0lcjTtGdIuUNZ/Mm0l4NFdFJS/Lo5amCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T12:37:28.124953Z"},"content_sha256":"5c1945208a95786bb786e5ce6826a70b9285f7215ea56765510c372b2a66fee2","schema_version":"1.0","event_id":"sha256:5c1945208a95786bb786e5ce6826a70b9285f7215ea56765510c372b2a66fee2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YAWKKH2DOHOGTFLLISIA742E2T/bundle.json","state_url":"https://pith.science/pith/YAWKKH2DOHOGTFLLISIA742E2T/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YAWKKH2DOHOGTFLLISIA742E2T/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":"2026-06-10T12:37:28Z","links":{"resolver":"https://pith.science/pith/YAWKKH2DOHOGTFLLISIA742E2T","bundle":"https://pith.science/pith/YAWKKH2DOHOGTFLLISIA742E2T/bundle.json","state":"https://pith.science/pith/YAWKKH2DOHOGTFLLISIA742E2T/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YAWKKH2DOHOGTFLLISIA742E2T/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:YAWKKH2DOHOGTFLLISIA742E2T","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":"2932158df3b3670630a1f4b083861b6d86bda5a1b5e4ecc0ea952dcc0aa2e4e5","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-05-06T08:52:07Z","title_canon_sha256":"ebf278da6723c338feea66c033231e746d0ef68a19198c5d436d5c33b7d381c9"},"schema_version":"1.0","source":{"id":"2605.04646","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.04646","created_at":"2026-05-29T02:05:46Z"},{"alias_kind":"arxiv_version","alias_value":"2605.04646v2","created_at":"2026-05-29T02:05:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.04646","created_at":"2026-05-29T02:05:46Z"},{"alias_kind":"pith_short_12","alias_value":"YAWKKH2DOHOG","created_at":"2026-05-29T02:05:46Z"},{"alias_kind":"pith_short_16","alias_value":"YAWKKH2DOHOGTFLL","created_at":"2026-05-29T02:05:46Z"},{"alias_kind":"pith_short_8","alias_value":"YAWKKH2D","created_at":"2026-05-29T02:05:46Z"}],"graph_snapshots":[{"event_id":"sha256:5c1945208a95786bb786e5ce6826a70b9285f7215ea56765510c372b2a66fee2","target":"graph","created_at":"2026-05-29T02:05:46Z","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 show that there exists regular polytopes and hypertopes for almost-simple group with socle a sporadic simple group."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"That the twisting and wreath product operations can be defined on arbitrary coset geometries so that flag-transitivity, residual-connectedness, and thinness are automatically preserved."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Twisting and wreath products extend to coset geometries preserving key properties, enabling regular polytopes and hypertopes for almost-simple groups with sporadic socles."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Twisting and wreath products extend to coset geometries while preserving flag-transitivity, residual-connectedness and thinness, yielding regular polytopes and hypertopes for almost-simple groups with sporadic socles."}],"snapshot_sha256":"0129e7c28d2830024b600cb51f8c7b88c9b43ea7cb409fce81cfbf47991acb53"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"name":"ai_meta_artifact","ran_at":"2026-05-20T11:36:19.371994Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"doi_title_agreement","ran_at":"2026-05-19T22:31:19.904865Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"doi_compliance","ran_at":"2026-05-19T14:15:23.310077Z","status":"completed","version":"1.0.0"}],"endpoint":"/pith/2605.04646/integrity.json","findings":[],"snapshot_sha256":"759841956606009dcedcba5b33e9470ce530b9d545da5f0c5fa3ea53f541289c","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Coset geometries are incidence geometries constructed from a group $G$ and a system of subgroups $(G_i)_{i \\in I}$ of subgroups of $G$. For any algebraic group operation, it is then natural to wonder whether it can be extended to the framework of coset geometries. This has been achieved in the case of the halving (\\cite{halving}) and in the case of free (amalgamated) products, HNN-extensions, and semi-direct products (\\cite{piedade2025group}). In this article, we explore more deeply two operations related to semi-direct products: the twisting and the wreath product. We show that these operatio","authors_text":"Claudio Alexandre Piedade, Philippe Tranchida","cross_cats":["math.CO"],"headline":"Twisting and wreath products extend to coset geometries while preserving flag-transitivity, residual-connectedness and thinness, yielding regular polytopes and hypertopes for almost-simple groups with sporadic socles.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-05-06T08:52:07Z","title":"The geometry of wreath and semi-direct products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.04646","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-08T16:36:14.679471Z","id":"21f122cf-58d2-45d4-9a09-8e9813b0d488","model_set":{"reader":"grok-4.3"},"one_line_summary":"Twisting and wreath products extend to coset geometries preserving key properties, enabling regular polytopes and hypertopes for almost-simple groups with sporadic socles.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Twisting and wreath products extend to coset geometries while preserving flag-transitivity, residual-connectedness and thinness, yielding regular polytopes and hypertopes for almost-simple groups with sporadic socles.","strongest_claim":"we show that there exists regular polytopes and hypertopes for almost-simple group with socle a sporadic simple group.","weakest_assumption":"That the twisting and wreath product operations can be defined on arbitrary coset geometries so that flag-transitivity, residual-connectedness, and thinness are automatically preserved."}},"verdict_id":"21f122cf-58d2-45d4-9a09-8e9813b0d488"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:d898e45029502b58451d9c9af3dcafe513f0360fa58ad251795585b9c4e15c99","target":"record","created_at":"2026-05-29T02:05:46Z","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":"2932158df3b3670630a1f4b083861b6d86bda5a1b5e4ecc0ea952dcc0aa2e4e5","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-05-06T08:52:07Z","title_canon_sha256":"ebf278da6723c338feea66c033231e746d0ef68a19198c5d436d5c33b7d381c9"},"schema_version":"1.0","source":{"id":"2605.04646","kind":"arxiv","version":2}},"canonical_sha256":"c02ca51f4371dc69956b44900ff344d4ee2acc3e4219a86bc3f8bdf404c42cb4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c02ca51f4371dc69956b44900ff344d4ee2acc3e4219a86bc3f8bdf404c42cb4","first_computed_at":"2026-05-29T02:05:46.018672Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-29T02:05:46.018672Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HGK+QCi8ka80aD/ID4VjzYWT2TMlBqboeo3Ph+EgKGhLhONkQdD+nM+cQkFHm4DehPAVSwu4g2PvH8sATm37Ag==","signature_status":"signed_v1","signed_at":"2026-05-29T02:05:46.019537Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.04646","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d898e45029502b58451d9c9af3dcafe513f0360fa58ad251795585b9c4e15c99","sha256:5c1945208a95786bb786e5ce6826a70b9285f7215ea56765510c372b2a66fee2"],"state_sha256":"ed8955e828cbb7e40395a050d137e648606ed8ab0c6152b48a638328ba382c37"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uV+f/CXgCh0La0ikvugWqpa2S3QyQp+xSEueoSA4dS6xrzXy6ugG+MtZLavLDvkK5XvwN4x3gVxe5+reBDAfBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T12:37:28.129982Z","bundle_sha256":"52fe6c157b2323c430128866cf2b6a5bf083dfb0870cd48ca7d7765fa4d86aa8"}}