{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:M32AOFV4B2ETJB6Y5IJLRNKMBI","short_pith_number":"pith:M32AOFV4","canonical_record":{"source":{"id":"2607.07032","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-07-08T06:05:54Z","cross_cats_sorted":["stat.ML"],"title_canon_sha256":"59a6a0d4f5974b2b902ded16c276f0a44c25658748eb13d604b167a7cfc6c103","abstract_canon_sha256":"8c0bbb9244adfdfbb6be3c2c7c648db97be33e3779f37b3792e940e8e1d5cab3"},"schema_version":"1.0"},"canonical_sha256":"66f40716bc0e893487d8ea12b8b54c0a16a57e10571f130d8381c4a9a6672970","source":{"kind":"arxiv","id":"2607.07032","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.07032","created_at":"2026-07-09T01:20:12Z"},{"alias_kind":"arxiv_version","alias_value":"2607.07032v1","created_at":"2026-07-09T01:20:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.07032","created_at":"2026-07-09T01:20:12Z"},{"alias_kind":"pith_short_12","alias_value":"M32AOFV4B2ET","created_at":"2026-07-09T01:20:12Z"},{"alias_kind":"pith_short_16","alias_value":"M32AOFV4B2ETJB6Y","created_at":"2026-07-09T01:20:12Z"},{"alias_kind":"pith_short_8","alias_value":"M32AOFV4","created_at":"2026-07-09T01:20:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:M32AOFV4B2ETJB6Y5IJLRNKMBI","target":"record","payload":{"canonical_record":{"source":{"id":"2607.07032","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-07-08T06:05:54Z","cross_cats_sorted":["stat.ML"],"title_canon_sha256":"59a6a0d4f5974b2b902ded16c276f0a44c25658748eb13d604b167a7cfc6c103","abstract_canon_sha256":"8c0bbb9244adfdfbb6be3c2c7c648db97be33e3779f37b3792e940e8e1d5cab3"},"schema_version":"1.0"},"canonical_sha256":"66f40716bc0e893487d8ea12b8b54c0a16a57e10571f130d8381c4a9a6672970","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-09T01:20:12.860673Z","signature_b64":"DbzseIdlQeXtGbuyca42bMHNKb0Wb4aoaE/Ls90dV+3u3afpLY6/Hop+exROVzYh/aAD+0cSG1wel30kcG3bBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"66f40716bc0e893487d8ea12b8b54c0a16a57e10571f130d8381c4a9a6672970","last_reissued_at":"2026-07-09T01:20:12.860327Z","signature_status":"signed_v1","first_computed_at":"2026-07-09T01:20:12.860327Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.07032","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":"2026-07-09T01:20:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Sv9U+M+ALlXuusaU2hK5KAw0Jn3OUPAEMUucrZNflVgr/Y6tQiTJfF2YPdphMocQ9I1OIoOV4Welavs+MVogCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T20:30:42.715501Z"},"content_sha256":"4ce21900a15532fa3445bd2b1f2dff9de684573064222d5bc3c3a7160ee66c58","schema_version":"1.0","event_id":"sha256:4ce21900a15532fa3445bd2b1f2dff9de684573064222d5bc3c3a7160ee66c58"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:M32AOFV4B2ETJB6Y5IJLRNKMBI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Gauge-Invariant Learnable Spectral Positional Encodings for Directed Graphs via Hermitian Block Krylov Subspaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.ML"],"primary_cat":"cs.LG","authors_text":"Jiaqing Xie, Yuxin Wang","submitted_at":"2026-07-08T06:05:54Z","abstract_excerpt":"Spectral positional encodings (PEs) for \\emph{directed} graphs face two obstacles: magnetic Laplacians require an $O(n^3)$ Hermitian eigendecomposition per potential, and their complex eigenvectors are defined only up to unitary gauge, which prior work handles with basis-invariant architectures. We propose learnable spectral PEs of the form $h_\\theta(A_q)\\,R$, where $A_q$ is a normalized magnetic operator, $h_\\theta$ a learnable scalar spectral response, and $R$ a block of random probes. Because the PE is a \\emph{matrix function} of the operator, it is gauge-invariant by construction. We compu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.07032","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/2607.07032/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":"2026-07-09T01:20:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FGqj0Dedg9majH9LvRbRT6z3U+Lohdn1Fd4cp6l71s4G1x2QoVL2o66Dc/O0WPu2RUEO0yN+yhbdEzyW+xjJBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T20:30:42.716443Z"},"content_sha256":"4ddc57c18b60080fa921cfcc3a8240f5b39649e917278f78b327ca5ed4c0e769","schema_version":"1.0","event_id":"sha256:4ddc57c18b60080fa921cfcc3a8240f5b39649e917278f78b327ca5ed4c0e769"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/M32AOFV4B2ETJB6Y5IJLRNKMBI/bundle.json","state_url":"https://pith.science/pith/M32AOFV4B2ETJB6Y5IJLRNKMBI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/M32AOFV4B2ETJB6Y5IJLRNKMBI/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":"2026-08-14T20:30:42Z","links":{"resolver":"https://pith.science/pith/M32AOFV4B2ETJB6Y5IJLRNKMBI","bundle":"https://pith.science/pith/M32AOFV4B2ETJB6Y5IJLRNKMBI/bundle.json","state":"https://pith.science/pith/M32AOFV4B2ETJB6Y5IJLRNKMBI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/M32AOFV4B2ETJB6Y5IJLRNKMBI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:M32AOFV4B2ETJB6Y5IJLRNKMBI","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":"8c0bbb9244adfdfbb6be3c2c7c648db97be33e3779f37b3792e940e8e1d5cab3","cross_cats_sorted":["stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-07-08T06:05:54Z","title_canon_sha256":"59a6a0d4f5974b2b902ded16c276f0a44c25658748eb13d604b167a7cfc6c103"},"schema_version":"1.0","source":{"id":"2607.07032","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.07032","created_at":"2026-07-09T01:20:12Z"},{"alias_kind":"arxiv_version","alias_value":"2607.07032v1","created_at":"2026-07-09T01:20:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.07032","created_at":"2026-07-09T01:20:12Z"},{"alias_kind":"pith_short_12","alias_value":"M32AOFV4B2ET","created_at":"2026-07-09T01:20:12Z"},{"alias_kind":"pith_short_16","alias_value":"M32AOFV4B2ETJB6Y","created_at":"2026-07-09T01:20:12Z"},{"alias_kind":"pith_short_8","alias_value":"M32AOFV4","created_at":"2026-07-09T01:20:12Z"}],"graph_snapshots":[{"event_id":"sha256:4ddc57c18b60080fa921cfcc3a8240f5b39649e917278f78b327ca5ed4c0e769","target":"graph","created_at":"2026-07-09T01:20:12Z","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/2607.07032/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Spectral positional encodings (PEs) for \\emph{directed} graphs face two obstacles: magnetic Laplacians require an $O(n^3)$ Hermitian eigendecomposition per potential, and their complex eigenvectors are defined only up to unitary gauge, which prior work handles with basis-invariant architectures. We propose learnable spectral PEs of the form $h_\\theta(A_q)\\,R$, where $A_q$ is a normalized magnetic operator, $h_\\theta$ a learnable scalar spectral response, and $R$ a block of random probes. Because the PE is a \\emph{matrix function} of the operator, it is gauge-invariant by construction. We compu","authors_text":"Jiaqing Xie, Yuxin Wang","cross_cats":["stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-07-08T06:05:54Z","title":"Gauge-Invariant Learnable Spectral Positional Encodings for Directed Graphs via Hermitian Block Krylov Subspaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.07032","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":"sha256:4ce21900a15532fa3445bd2b1f2dff9de684573064222d5bc3c3a7160ee66c58","target":"record","created_at":"2026-07-09T01:20:12Z","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":"8c0bbb9244adfdfbb6be3c2c7c648db97be33e3779f37b3792e940e8e1d5cab3","cross_cats_sorted":["stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-07-08T06:05:54Z","title_canon_sha256":"59a6a0d4f5974b2b902ded16c276f0a44c25658748eb13d604b167a7cfc6c103"},"schema_version":"1.0","source":{"id":"2607.07032","kind":"arxiv","version":1}},"canonical_sha256":"66f40716bc0e893487d8ea12b8b54c0a16a57e10571f130d8381c4a9a6672970","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"66f40716bc0e893487d8ea12b8b54c0a16a57e10571f130d8381c4a9a6672970","first_computed_at":"2026-07-09T01:20:12.860327Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-09T01:20:12.860327Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DbzseIdlQeXtGbuyca42bMHNKb0Wb4aoaE/Ls90dV+3u3afpLY6/Hop+exROVzYh/aAD+0cSG1wel30kcG3bBw==","signature_status":"signed_v1","signed_at":"2026-07-09T01:20:12.860673Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.07032","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4ce21900a15532fa3445bd2b1f2dff9de684573064222d5bc3c3a7160ee66c58","sha256:4ddc57c18b60080fa921cfcc3a8240f5b39649e917278f78b327ca5ed4c0e769"],"state_sha256":"feec4aa7cb37a70f187de23d5df8abed4d05b2fcb14c54ee8943fe91da3c507c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ksjd6c+a2MLrsYy0/QAOdbu7PdDNnp6UvR5FbhzSDCMBYpy0k6ptzfV5GTJfreWCaGqwwhbPKoaWq+Pgj58KDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T20:30:42.721586Z","bundle_sha256":"286b345cf4ac379be31fde708e557f5e38e7a20fcc69d8636205438cf4fe1695"}}