{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:NXGTI226MJQEIBK2YZMASK2WLG","short_pith_number":"pith:NXGTI226","canonical_record":{"source":{"id":"2604.01975","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-04-02T12:33:33Z","cross_cats_sorted":["cs.NA","quant-ph"],"title_canon_sha256":"c236750e53c5154e792ffd5cd2db55fb821f03b459be0566efe190cb0829a3b0","abstract_canon_sha256":"094c957a37c3a66b5a4a297a902bea93d640976cb86428477f44006b3b961d20"},"schema_version":"1.0"},"canonical_sha256":"6dcd346b5e626044055ac658092b565998bf49a52f8536e249e080f4e74e039c","source":{"kind":"arxiv","id":"2604.01975","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.01975","created_at":"2026-05-25T02:02:14Z"},{"alias_kind":"arxiv_version","alias_value":"2604.01975v2","created_at":"2026-05-25T02:02:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.01975","created_at":"2026-05-25T02:02:14Z"},{"alias_kind":"pith_short_12","alias_value":"NXGTI226MJQE","created_at":"2026-05-25T02:02:14Z"},{"alias_kind":"pith_short_16","alias_value":"NXGTI226MJQEIBK2","created_at":"2026-05-25T02:02:14Z"},{"alias_kind":"pith_short_8","alias_value":"NXGTI226","created_at":"2026-05-25T02:02:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:NXGTI226MJQEIBK2YZMASK2WLG","target":"record","payload":{"canonical_record":{"source":{"id":"2604.01975","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-04-02T12:33:33Z","cross_cats_sorted":["cs.NA","quant-ph"],"title_canon_sha256":"c236750e53c5154e792ffd5cd2db55fb821f03b459be0566efe190cb0829a3b0","abstract_canon_sha256":"094c957a37c3a66b5a4a297a902bea93d640976cb86428477f44006b3b961d20"},"schema_version":"1.0"},"canonical_sha256":"6dcd346b5e626044055ac658092b565998bf49a52f8536e249e080f4e74e039c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-25T02:02:14.865241Z","signature_b64":"lmyjJM+7IsTBNkvLoO0myVkAjL14B1Vxl9S79kG6Nk6m/5f0V2tHxIeI9GZ06TDsapnQwRtTLcuf5ZXFpU4QDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6dcd346b5e626044055ac658092b565998bf49a52f8536e249e080f4e74e039c","last_reissued_at":"2026-05-25T02:02:14.864700Z","signature_status":"signed_v1","first_computed_at":"2026-05-25T02:02:14.864700Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2604.01975","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":"2026-05-25T02:02:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TmkkYvAc80Q9XK1veNkjeYX1cR2MQ25hUo4tw9L/6txFe1DKYlIP12bLPtTCL+gTtshD4aHvJSuL9hxRxGfIDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T15:55:00.563784Z"},"content_sha256":"3d4db19bd64240df4c2692d2d23e44ffcce45910b42d71ea17a08531e3f3b4e3","schema_version":"1.0","event_id":"sha256:3d4db19bd64240df4c2692d2d23e44ffcce45910b42d71ea17a08531e3f3b4e3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:NXGTI226MJQEIBK2YZMASK2WLG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Efficient construction of Lie group-equivariant and permutation-invariant spaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.NA","quant-ph"],"primary_cat":"math.NA","authors_text":"Camille Hernandez, Elo\\\"ise Barthelemy, Genevi\\`eve Dusson, Liwei Zhang","submitted_at":"2026-04-02T12:33:33Z","abstract_excerpt":"We introduce a practical construction of group-equivariant and permutation-invariant functions of $N$ variables given a finite-dimensional space stable with respect to the group action. The construction applies to any connected linear Lie group and relies on leveraging the Lie algebra to build a matrix $M$ whose kernel is in one-to-one correspondence with the subspace with desired equivariance and invariance properties, removing the need for prior knowledge of Clebsch--Gordan coefficients. A similar construction is proposed for group-equivariant functions alone, without imposing permutation-in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.01975","kind":"arxiv","version":2},"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/2604.01975/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":"2026-05-25T02:02:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IGTX/WQpx/yL+6CcPqXCXjFu1qF/qhnYaVrXY84RlYHMFsZCmvUhfzitzF+GnjG/TSrZWZen1JSCDxI9bLfXCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T15:55:00.564171Z"},"content_sha256":"e7a79e85efac88d668c1feab277730a897c4fd324fef7c147865d1e5bb67877f","schema_version":"1.0","event_id":"sha256:e7a79e85efac88d668c1feab277730a897c4fd324fef7c147865d1e5bb67877f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NXGTI226MJQEIBK2YZMASK2WLG/bundle.json","state_url":"https://pith.science/pith/NXGTI226MJQEIBK2YZMASK2WLG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NXGTI226MJQEIBK2YZMASK2WLG/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":"2026-06-03T15:55:00Z","links":{"resolver":"https://pith.science/pith/NXGTI226MJQEIBK2YZMASK2WLG","bundle":"https://pith.science/pith/NXGTI226MJQEIBK2YZMASK2WLG/bundle.json","state":"https://pith.science/pith/NXGTI226MJQEIBK2YZMASK2WLG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NXGTI226MJQEIBK2YZMASK2WLG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:NXGTI226MJQEIBK2YZMASK2WLG","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":"094c957a37c3a66b5a4a297a902bea93d640976cb86428477f44006b3b961d20","cross_cats_sorted":["cs.NA","quant-ph"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-04-02T12:33:33Z","title_canon_sha256":"c236750e53c5154e792ffd5cd2db55fb821f03b459be0566efe190cb0829a3b0"},"schema_version":"1.0","source":{"id":"2604.01975","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.01975","created_at":"2026-05-25T02:02:14Z"},{"alias_kind":"arxiv_version","alias_value":"2604.01975v2","created_at":"2026-05-25T02:02:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.01975","created_at":"2026-05-25T02:02:14Z"},{"alias_kind":"pith_short_12","alias_value":"NXGTI226MJQE","created_at":"2026-05-25T02:02:14Z"},{"alias_kind":"pith_short_16","alias_value":"NXGTI226MJQEIBK2","created_at":"2026-05-25T02:02:14Z"},{"alias_kind":"pith_short_8","alias_value":"NXGTI226","created_at":"2026-05-25T02:02:14Z"}],"graph_snapshots":[{"event_id":"sha256:e7a79e85efac88d668c1feab277730a897c4fd324fef7c147865d1e5bb67877f","target":"graph","created_at":"2026-05-25T02:02:14Z","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/2604.01975/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We introduce a practical construction of group-equivariant and permutation-invariant functions of $N$ variables given a finite-dimensional space stable with respect to the group action. The construction applies to any connected linear Lie group and relies on leveraging the Lie algebra to build a matrix $M$ whose kernel is in one-to-one correspondence with the subspace with desired equivariance and invariance properties, removing the need for prior knowledge of Clebsch--Gordan coefficients. A similar construction is proposed for group-equivariant functions alone, without imposing permutation-in","authors_text":"Camille Hernandez, Elo\\\"ise Barthelemy, Genevi\\`eve Dusson, Liwei Zhang","cross_cats":["cs.NA","quant-ph"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-04-02T12:33:33Z","title":"Efficient construction of Lie group-equivariant and permutation-invariant spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.01975","kind":"arxiv","version":2},"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":"sha256:3d4db19bd64240df4c2692d2d23e44ffcce45910b42d71ea17a08531e3f3b4e3","target":"record","created_at":"2026-05-25T02:02:14Z","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":"094c957a37c3a66b5a4a297a902bea93d640976cb86428477f44006b3b961d20","cross_cats_sorted":["cs.NA","quant-ph"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2026-04-02T12:33:33Z","title_canon_sha256":"c236750e53c5154e792ffd5cd2db55fb821f03b459be0566efe190cb0829a3b0"},"schema_version":"1.0","source":{"id":"2604.01975","kind":"arxiv","version":2}},"canonical_sha256":"6dcd346b5e626044055ac658092b565998bf49a52f8536e249e080f4e74e039c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6dcd346b5e626044055ac658092b565998bf49a52f8536e249e080f4e74e039c","first_computed_at":"2026-05-25T02:02:14.864700Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-25T02:02:14.864700Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lmyjJM+7IsTBNkvLoO0myVkAjL14B1Vxl9S79kG6Nk6m/5f0V2tHxIeI9GZ06TDsapnQwRtTLcuf5ZXFpU4QDQ==","signature_status":"signed_v1","signed_at":"2026-05-25T02:02:14.865241Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.01975","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3d4db19bd64240df4c2692d2d23e44ffcce45910b42d71ea17a08531e3f3b4e3","sha256:e7a79e85efac88d668c1feab277730a897c4fd324fef7c147865d1e5bb67877f"],"state_sha256":"0ad9ae02859ca97ff223c0098bf6ae93d2e68dbf04b7c0e83290fd714c5ae4a7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OdiCYzk3pAMZyvzj3+t53E0JpU06QET0/qyLMzi4WtJolgdSoLpwZqdTDtzsXMpezjcHEJc592Y0S1n0SvzfBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T15:55:00.566458Z","bundle_sha256":"8f7adea6aee0c72f8ceb8c067b8810487275f489abc135828b720a42269289cd"}}