{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:U6XJYVVTXZXEDJ4B42GGKM2BJ4","short_pith_number":"pith:U6XJYVVT","canonical_record":{"source":{"id":"2508.02675","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2025-07-10T09:47:41Z","cross_cats_sorted":["cs.NA","math-ph","math.MP","physics.class-ph"],"title_canon_sha256":"9a683faa9563a75a5e220e2e954016c529f0e49235a5c6648087be8bbe497b6c","abstract_canon_sha256":"ced3808c9073c83cfff60df4727855f78a7067bb694d8e36ec7b4f326cf7ab17"},"schema_version":"1.0"},"canonical_sha256":"a7ae9c56b3be6e41a781e68c6533414f0d72a6940f4181e6cccee73be36a7888","source":{"kind":"arxiv","id":"2508.02675","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2508.02675","created_at":"2026-07-05T11:48:28Z"},{"alias_kind":"arxiv_version","alias_value":"2508.02675v1","created_at":"2026-07-05T11:48:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.02675","created_at":"2026-07-05T11:48:28Z"},{"alias_kind":"pith_short_12","alias_value":"U6XJYVVTXZXE","created_at":"2026-07-05T11:48:28Z"},{"alias_kind":"pith_short_16","alias_value":"U6XJYVVTXZXEDJ4B","created_at":"2026-07-05T11:48:28Z"},{"alias_kind":"pith_short_8","alias_value":"U6XJYVVT","created_at":"2026-07-05T11:48:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:U6XJYVVTXZXEDJ4B42GGKM2BJ4","target":"record","payload":{"canonical_record":{"source":{"id":"2508.02675","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2025-07-10T09:47:41Z","cross_cats_sorted":["cs.NA","math-ph","math.MP","physics.class-ph"],"title_canon_sha256":"9a683faa9563a75a5e220e2e954016c529f0e49235a5c6648087be8bbe497b6c","abstract_canon_sha256":"ced3808c9073c83cfff60df4727855f78a7067bb694d8e36ec7b4f326cf7ab17"},"schema_version":"1.0"},"canonical_sha256":"a7ae9c56b3be6e41a781e68c6533414f0d72a6940f4181e6cccee73be36a7888","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:48:28.876751Z","signature_b64":"r7Z6rQ78k6sePFc72VIU8jzwYVbuYVjB8kIMlZ1IBIlAD1vdUzzodKDvqY4ZpAYwlMmokVGCEAT2NewSeUaRAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a7ae9c56b3be6e41a781e68c6533414f0d72a6940f4181e6cccee73be36a7888","last_reissued_at":"2026-07-05T11:48:28.876028Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:48:28.876028Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2508.02675","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":"2026-07-05T11:48:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OWxIAlo4q++tzHnuSFin+FYod5cso2qvfKPaJaXRPNTjJS+5iOd6joVsPoNRkoQi4WBZYC7zPw96JhikWKkjCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T02:40:13.165905Z"},"content_sha256":"00ba65ff1c141c5cde2ba1459093e71e4c3387ee9aaed60a9832e67bb13d14b0","schema_version":"1.0","event_id":"sha256:00ba65ff1c141c5cde2ba1459093e71e4c3387ee9aaed60a9832e67bb13d14b0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:U6XJYVVTXZXEDJ4B42GGKM2BJ4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Full Vectorial Maxwell Equations with Continuous Angular Indices","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.NA","math-ph","math.MP","physics.class-ph"],"primary_cat":"math.NA","authors_text":"Mustafa Bakr","submitted_at":"2025-07-10T09:47:41Z","abstract_excerpt":"This article presents a mathematical framework for solving Maxwell's equations in cylindrical and spherical geometries with continuous angular indices. We extend beyond standard discrete harmonic decomposition to a continuous spectral representation using generalized spectral integrals, capturing electromagnetic solutions that exhibit singular behavoiur yet yield finite-energy fields at the geometric center. For continuous angular indices $\\ell, m \\in \\mathbb{R}$, we study existence and uniqueness of solutions in weighted Sobolev spaces $H^s_{\\alpha(\\ell,m)}(\\Omega)$ following the framework es"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.02675","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/2508.02675/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":"2026-07-05T11:48:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JPyQqJILF5wTZxzCfDfhSVtRGh/Yg3upUs6pIrJBgPkLzO9YI2Ft0L3Z/LPvnQ71echQnpo9rLCNg9DmE4JkDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T02:40:13.166558Z"},"content_sha256":"5805672083fc9bcebdc37f357ea52b5617c7c1fe39507e79417032aa850dc609","schema_version":"1.0","event_id":"sha256:5805672083fc9bcebdc37f357ea52b5617c7c1fe39507e79417032aa850dc609"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/U6XJYVVTXZXEDJ4B42GGKM2BJ4/bundle.json","state_url":"https://pith.science/pith/U6XJYVVTXZXEDJ4B42GGKM2BJ4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/U6XJYVVTXZXEDJ4B42GGKM2BJ4/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":"2026-07-07T02:40:13Z","links":{"resolver":"https://pith.science/pith/U6XJYVVTXZXEDJ4B42GGKM2BJ4","bundle":"https://pith.science/pith/U6XJYVVTXZXEDJ4B42GGKM2BJ4/bundle.json","state":"https://pith.science/pith/U6XJYVVTXZXEDJ4B42GGKM2BJ4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/U6XJYVVTXZXEDJ4B42GGKM2BJ4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:U6XJYVVTXZXEDJ4B42GGKM2BJ4","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":"ced3808c9073c83cfff60df4727855f78a7067bb694d8e36ec7b4f326cf7ab17","cross_cats_sorted":["cs.NA","math-ph","math.MP","physics.class-ph"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2025-07-10T09:47:41Z","title_canon_sha256":"9a683faa9563a75a5e220e2e954016c529f0e49235a5c6648087be8bbe497b6c"},"schema_version":"1.0","source":{"id":"2508.02675","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2508.02675","created_at":"2026-07-05T11:48:28Z"},{"alias_kind":"arxiv_version","alias_value":"2508.02675v1","created_at":"2026-07-05T11:48:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.02675","created_at":"2026-07-05T11:48:28Z"},{"alias_kind":"pith_short_12","alias_value":"U6XJYVVTXZXE","created_at":"2026-07-05T11:48:28Z"},{"alias_kind":"pith_short_16","alias_value":"U6XJYVVTXZXEDJ4B","created_at":"2026-07-05T11:48:28Z"},{"alias_kind":"pith_short_8","alias_value":"U6XJYVVT","created_at":"2026-07-05T11:48:28Z"}],"graph_snapshots":[{"event_id":"sha256:5805672083fc9bcebdc37f357ea52b5617c7c1fe39507e79417032aa850dc609","target":"graph","created_at":"2026-07-05T11:48:28Z","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/2508.02675/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This article presents a mathematical framework for solving Maxwell's equations in cylindrical and spherical geometries with continuous angular indices. We extend beyond standard discrete harmonic decomposition to a continuous spectral representation using generalized spectral integrals, capturing electromagnetic solutions that exhibit singular behavoiur yet yield finite-energy fields at the geometric center. For continuous angular indices $\\ell, m \\in \\mathbb{R}$, we study existence and uniqueness of solutions in weighted Sobolev spaces $H^s_{\\alpha(\\ell,m)}(\\Omega)$ following the framework es","authors_text":"Mustafa Bakr","cross_cats":["cs.NA","math-ph","math.MP","physics.class-ph"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2025-07-10T09:47:41Z","title":"Full Vectorial Maxwell Equations with Continuous Angular Indices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.02675","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":"sha256:00ba65ff1c141c5cde2ba1459093e71e4c3387ee9aaed60a9832e67bb13d14b0","target":"record","created_at":"2026-07-05T11:48:28Z","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":"ced3808c9073c83cfff60df4727855f78a7067bb694d8e36ec7b4f326cf7ab17","cross_cats_sorted":["cs.NA","math-ph","math.MP","physics.class-ph"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2025-07-10T09:47:41Z","title_canon_sha256":"9a683faa9563a75a5e220e2e954016c529f0e49235a5c6648087be8bbe497b6c"},"schema_version":"1.0","source":{"id":"2508.02675","kind":"arxiv","version":1}},"canonical_sha256":"a7ae9c56b3be6e41a781e68c6533414f0d72a6940f4181e6cccee73be36a7888","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a7ae9c56b3be6e41a781e68c6533414f0d72a6940f4181e6cccee73be36a7888","first_computed_at":"2026-07-05T11:48:28.876028Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:48:28.876028Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"r7Z6rQ78k6sePFc72VIU8jzwYVbuYVjB8kIMlZ1IBIlAD1vdUzzodKDvqY4ZpAYwlMmokVGCEAT2NewSeUaRAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T11:48:28.876751Z","signed_message":"canonical_sha256_bytes"},"source_id":"2508.02675","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:00ba65ff1c141c5cde2ba1459093e71e4c3387ee9aaed60a9832e67bb13d14b0","sha256:5805672083fc9bcebdc37f357ea52b5617c7c1fe39507e79417032aa850dc609"],"state_sha256":"50760ae7eaa7a25f1c7d6f72afbb02b2656bb5dce1cb58251491616ced831563"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LozuYZws2Y29R1C0b4bT7Iwf7aDq7G87h2JQkk7S4v0S17dvsM0ORGjyWLfpZcLcy2PieFED+4eqykp35WDwCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-07T02:40:13.170001Z","bundle_sha256":"51f66ef02d4db534b95366b71f70225839eedf5c1dc7caa82ac52062ac04ef76"}}