{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:GIGCQLZVK5E6M7CDHR2YWEYBTP","short_pith_number":"pith:GIGCQLZV","canonical_record":{"source":{"id":"2502.15513","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2025-02-21T15:10:10Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"88981ce0188a8b77f4de8c351f056c52647b11f03d6434d800ec3f20187f780e","abstract_canon_sha256":"84cb9fd01b0b93e3da924cf7603117f6a6a7eb96dc263982e07bdb5b4f00b110"},"schema_version":"1.0"},"canonical_sha256":"320c282f355749e67c433c758b13019befca0ccd28bba8e29357a478822bd1d3","source":{"kind":"arxiv","id":"2502.15513","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2502.15513","created_at":"2026-07-05T10:18:02Z"},{"alias_kind":"arxiv_version","alias_value":"2502.15513v1","created_at":"2026-07-05T10:18:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2502.15513","created_at":"2026-07-05T10:18:02Z"},{"alias_kind":"pith_short_12","alias_value":"GIGCQLZVK5E6","created_at":"2026-07-05T10:18:02Z"},{"alias_kind":"pith_short_16","alias_value":"GIGCQLZVK5E6M7CD","created_at":"2026-07-05T10:18:02Z"},{"alias_kind":"pith_short_8","alias_value":"GIGCQLZV","created_at":"2026-07-05T10:18:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:GIGCQLZVK5E6M7CDHR2YWEYBTP","target":"record","payload":{"canonical_record":{"source":{"id":"2502.15513","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2025-02-21T15:10:10Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"88981ce0188a8b77f4de8c351f056c52647b11f03d6434d800ec3f20187f780e","abstract_canon_sha256":"84cb9fd01b0b93e3da924cf7603117f6a6a7eb96dc263982e07bdb5b4f00b110"},"schema_version":"1.0"},"canonical_sha256":"320c282f355749e67c433c758b13019befca0ccd28bba8e29357a478822bd1d3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:18:02.903466Z","signature_b64":"pWEbIg1k2mQbYcnBjf+ds/1qVQh9j2k8Z+USK620LyuHacwi1sGlGENe1v30GYb85u62kCIOM1i+Owmpn4V4AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"320c282f355749e67c433c758b13019befca0ccd28bba8e29357a478822bd1d3","last_reissued_at":"2026-07-05T10:18:02.902979Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:18:02.902979Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2502.15513","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":"2026-07-05T10:18:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LWkj3OXbBveFDztJFNisBMZ10lfvEjLdhr/8Kx8qivYO2unfskarDcLTLELGNOaGInV20dpiwyVCfeAsjel4Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T15:17:53.457051Z"},"content_sha256":"f9dd5861aff62eab180ed414a9dd2f34e1ae064a43b8a446a19b200b98cc9b96","schema_version":"1.0","event_id":"sha256:f9dd5861aff62eab180ed414a9dd2f34e1ae064a43b8a446a19b200b98cc9b96"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:GIGCQLZVK5E6M7CDHR2YWEYBTP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Maximal Representation Dimensions of Algebraic Tori of Fixed Dimension Over Arbitrary Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.AG","authors_text":"Bailey Heath","submitted_at":"2025-02-21T15:10:10Z","abstract_excerpt":"We define the representation dimension of an algebraic torus $T$ to be the minimal positive integer $r$ such that there exists a faithful embedding $T \\hookrightarrow \\operatorname{GL}_r$. Given a positive integer $n$, there exists a maximal representation dimension of all $n$-dimensional algebraic tori over all fields. In this paper, we use the theory of group actions on lattices to find lower bounds on this maximum for all $n$. Further, we find the exact maximum value for irreducible tori for all $n \\in \\left\\lbrace 1, 2, \\dots, 10, 11, 13, 17, 19, 23\\right\\rbrace$ and conjecturally infinite"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.15513","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/2502.15513/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":"2026-07-05T10:18:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MbAV7UMoAJrPrVZDrGYLBKEoyB8B2fwyoccdGpekoSdjwvEkVxs5EIPExx91Hlon7+p73wWzQBuMEjFrXJ9eDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T15:17:53.457950Z"},"content_sha256":"1d13bd8324eab11931d1068ff20081ea46f98914dd2b2386a2a19cf769432772","schema_version":"1.0","event_id":"sha256:1d13bd8324eab11931d1068ff20081ea46f98914dd2b2386a2a19cf769432772"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GIGCQLZVK5E6M7CDHR2YWEYBTP/bundle.json","state_url":"https://pith.science/pith/GIGCQLZVK5E6M7CDHR2YWEYBTP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GIGCQLZVK5E6M7CDHR2YWEYBTP/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":"2026-08-09T15:17:53Z","links":{"resolver":"https://pith.science/pith/GIGCQLZVK5E6M7CDHR2YWEYBTP","bundle":"https://pith.science/pith/GIGCQLZVK5E6M7CDHR2YWEYBTP/bundle.json","state":"https://pith.science/pith/GIGCQLZVK5E6M7CDHR2YWEYBTP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GIGCQLZVK5E6M7CDHR2YWEYBTP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:GIGCQLZVK5E6M7CDHR2YWEYBTP","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":"84cb9fd01b0b93e3da924cf7603117f6a6a7eb96dc263982e07bdb5b4f00b110","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2025-02-21T15:10:10Z","title_canon_sha256":"88981ce0188a8b77f4de8c351f056c52647b11f03d6434d800ec3f20187f780e"},"schema_version":"1.0","source":{"id":"2502.15513","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2502.15513","created_at":"2026-07-05T10:18:02Z"},{"alias_kind":"arxiv_version","alias_value":"2502.15513v1","created_at":"2026-07-05T10:18:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2502.15513","created_at":"2026-07-05T10:18:02Z"},{"alias_kind":"pith_short_12","alias_value":"GIGCQLZVK5E6","created_at":"2026-07-05T10:18:02Z"},{"alias_kind":"pith_short_16","alias_value":"GIGCQLZVK5E6M7CD","created_at":"2026-07-05T10:18:02Z"},{"alias_kind":"pith_short_8","alias_value":"GIGCQLZV","created_at":"2026-07-05T10:18:02Z"}],"graph_snapshots":[{"event_id":"sha256:1d13bd8324eab11931d1068ff20081ea46f98914dd2b2386a2a19cf769432772","target":"graph","created_at":"2026-07-05T10:18:02Z","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/2502.15513/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We define the representation dimension of an algebraic torus $T$ to be the minimal positive integer $r$ such that there exists a faithful embedding $T \\hookrightarrow \\operatorname{GL}_r$. Given a positive integer $n$, there exists a maximal representation dimension of all $n$-dimensional algebraic tori over all fields. In this paper, we use the theory of group actions on lattices to find lower bounds on this maximum for all $n$. Further, we find the exact maximum value for irreducible tori for all $n \\in \\left\\lbrace 1, 2, \\dots, 10, 11, 13, 17, 19, 23\\right\\rbrace$ and conjecturally infinite","authors_text":"Bailey Heath","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2025-02-21T15:10:10Z","title":"Maximal Representation Dimensions of Algebraic Tori of Fixed Dimension Over Arbitrary Fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.15513","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:f9dd5861aff62eab180ed414a9dd2f34e1ae064a43b8a446a19b200b98cc9b96","target":"record","created_at":"2026-07-05T10:18:02Z","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":"84cb9fd01b0b93e3da924cf7603117f6a6a7eb96dc263982e07bdb5b4f00b110","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2025-02-21T15:10:10Z","title_canon_sha256":"88981ce0188a8b77f4de8c351f056c52647b11f03d6434d800ec3f20187f780e"},"schema_version":"1.0","source":{"id":"2502.15513","kind":"arxiv","version":1}},"canonical_sha256":"320c282f355749e67c433c758b13019befca0ccd28bba8e29357a478822bd1d3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"320c282f355749e67c433c758b13019befca0ccd28bba8e29357a478822bd1d3","first_computed_at":"2026-07-05T10:18:02.902979Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:18:02.902979Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pWEbIg1k2mQbYcnBjf+ds/1qVQh9j2k8Z+USK620LyuHacwi1sGlGENe1v30GYb85u62kCIOM1i+Owmpn4V4AQ==","signature_status":"signed_v1","signed_at":"2026-07-05T10:18:02.903466Z","signed_message":"canonical_sha256_bytes"},"source_id":"2502.15513","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f9dd5861aff62eab180ed414a9dd2f34e1ae064a43b8a446a19b200b98cc9b96","sha256:1d13bd8324eab11931d1068ff20081ea46f98914dd2b2386a2a19cf769432772"],"state_sha256":"aaf252214043b12911bd9418ae6906ef6dbdfe8482eadd21c0a445a481529948"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TszsN5n9j6UmB5TMyvkpQdl9tj8+VK+HfZEDgng6m85ojB5X8277MXq9T9gultpWs6R99jIE0Cb/Lp4WsGMYCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T15:17:53.524824Z","bundle_sha256":"a2c1fe1b3e13efcf61c78d4f29b574c5b41aa866263be8ec334916bde272ea70"}}