{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:4WEIV733CUVA72G7OYQ2D3C6UD","short_pith_number":"pith:4WEIV733","canonical_record":{"source":{"id":"2606.05213","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GM","submitted_at":"2026-05-27T14:03:36Z","cross_cats_sorted":[],"title_canon_sha256":"93ad054d47101c4ad88b63d8ecfc19a327d2ed55657fd57be38040e5a7d7c22e","abstract_canon_sha256":"84bd8df58ba2dcd8043bb10ccf1484399424512a9b5f3e6cea1d925316b76562"},"schema_version":"1.0"},"canonical_sha256":"e5888aff7b152a0fe8df7621a1ec5ea0eaa3990858674cc124c6a8c3383626a7","source":{"kind":"arxiv","id":"2606.05213","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.05213","created_at":"2026-06-05T00:13:48Z"},{"alias_kind":"arxiv_version","alias_value":"2606.05213v1","created_at":"2026-06-05T00:13:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.05213","created_at":"2026-06-05T00:13:48Z"},{"alias_kind":"pith_short_12","alias_value":"4WEIV733CUVA","created_at":"2026-06-05T00:13:48Z"},{"alias_kind":"pith_short_16","alias_value":"4WEIV733CUVA72G7","created_at":"2026-06-05T00:13:48Z"},{"alias_kind":"pith_short_8","alias_value":"4WEIV733","created_at":"2026-06-05T00:13:48Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:4WEIV733CUVA72G7OYQ2D3C6UD","target":"record","payload":{"canonical_record":{"source":{"id":"2606.05213","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GM","submitted_at":"2026-05-27T14:03:36Z","cross_cats_sorted":[],"title_canon_sha256":"93ad054d47101c4ad88b63d8ecfc19a327d2ed55657fd57be38040e5a7d7c22e","abstract_canon_sha256":"84bd8df58ba2dcd8043bb10ccf1484399424512a9b5f3e6cea1d925316b76562"},"schema_version":"1.0"},"canonical_sha256":"e5888aff7b152a0fe8df7621a1ec5ea0eaa3990858674cc124c6a8c3383626a7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-05T00:13:48.919033Z","signature_b64":"l5WImCYBZA9OQYS2PPyNbwrYcxclGb3OSpQVrF3k/ivnp599ekBRxgZqHKESvJOZJIhbG4cOjUv2Vz/01YmJAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e5888aff7b152a0fe8df7621a1ec5ea0eaa3990858674cc124c6a8c3383626a7","last_reissued_at":"2026-06-05T00:13:48.918579Z","signature_status":"signed_v1","first_computed_at":"2026-06-05T00:13:48.918579Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.05213","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":"2026-06-05T00:13:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nXhIoYcrVZy65Q1V2/QqeTmp+LM4zdYE5Xu8nuDxPOcxALGyQc0iiaXv1+ivy8hEJS/eY9eapGxzzIjc4UWUDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T23:21:15.113240Z"},"content_sha256":"dc8ef8f0b4de98831f043717a62916c481b21d90091e8d266d365f363e7e2a97","schema_version":"1.0","event_id":"sha256:dc8ef8f0b4de98831f043717a62916c481b21d90091e8d266d365f363e7e2a97"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:4WEIV733CUVA72G7OYQ2D3C6UD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Umbral methods, function factorisation and generalisation of the Fourier transform method","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Giuseppe Dattoli, Roberto Ricci, Tommaso Severati","submitted_at":"2026-05-27T14:03:36Z","abstract_excerpt":"We propose a systematic way to construct trigonometric-like functions beyond the classical sine--cosine pair by factorising rational umbral operators. The guiding idea is simple: the usual trigonometric functions may be viewed as cyclic components arising from a finite factorisation, and the same principle can be extended to an $n$-fold decomposition of rational umbral expressions. For each integer $n\\geq 2$, the construction produces $n$ functions which play the role of higher-order trigonometric components: their sum reconstructs the corresponding umbral function, while the individual compon"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.05213","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/2606.05213/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":"2026-06-05T00:13:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wIcvS1TB8mHYPKLHV0/YrC3NS6jLIz6Wq9GMNADlB/asgJnRHB80qfCo+V+7/AGQ47sa29dGhH33rvwQKI+SCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T23:21:15.113638Z"},"content_sha256":"5024ea2772d0730ddf0807569d741abc86ac8bd2ccc77f6bc86628d585d69041","schema_version":"1.0","event_id":"sha256:5024ea2772d0730ddf0807569d741abc86ac8bd2ccc77f6bc86628d585d69041"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4WEIV733CUVA72G7OYQ2D3C6UD/bundle.json","state_url":"https://pith.science/pith/4WEIV733CUVA72G7OYQ2D3C6UD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4WEIV733CUVA72G7OYQ2D3C6UD/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":"2026-06-22T23:21:15Z","links":{"resolver":"https://pith.science/pith/4WEIV733CUVA72G7OYQ2D3C6UD","bundle":"https://pith.science/pith/4WEIV733CUVA72G7OYQ2D3C6UD/bundle.json","state":"https://pith.science/pith/4WEIV733CUVA72G7OYQ2D3C6UD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4WEIV733CUVA72G7OYQ2D3C6UD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:4WEIV733CUVA72G7OYQ2D3C6UD","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":"84bd8df58ba2dcd8043bb10ccf1484399424512a9b5f3e6cea1d925316b76562","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GM","submitted_at":"2026-05-27T14:03:36Z","title_canon_sha256":"93ad054d47101c4ad88b63d8ecfc19a327d2ed55657fd57be38040e5a7d7c22e"},"schema_version":"1.0","source":{"id":"2606.05213","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.05213","created_at":"2026-06-05T00:13:48Z"},{"alias_kind":"arxiv_version","alias_value":"2606.05213v1","created_at":"2026-06-05T00:13:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.05213","created_at":"2026-06-05T00:13:48Z"},{"alias_kind":"pith_short_12","alias_value":"4WEIV733CUVA","created_at":"2026-06-05T00:13:48Z"},{"alias_kind":"pith_short_16","alias_value":"4WEIV733CUVA72G7","created_at":"2026-06-05T00:13:48Z"},{"alias_kind":"pith_short_8","alias_value":"4WEIV733","created_at":"2026-06-05T00:13:48Z"}],"graph_snapshots":[{"event_id":"sha256:5024ea2772d0730ddf0807569d741abc86ac8bd2ccc77f6bc86628d585d69041","target":"graph","created_at":"2026-06-05T00:13:48Z","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/2606.05213/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We propose a systematic way to construct trigonometric-like functions beyond the classical sine--cosine pair by factorising rational umbral operators. The guiding idea is simple: the usual trigonometric functions may be viewed as cyclic components arising from a finite factorisation, and the same principle can be extended to an $n$-fold decomposition of rational umbral expressions. For each integer $n\\geq 2$, the construction produces $n$ functions which play the role of higher-order trigonometric components: their sum reconstructs the corresponding umbral function, while the individual compon","authors_text":"Giuseppe Dattoli, Roberto Ricci, Tommaso Severati","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GM","submitted_at":"2026-05-27T14:03:36Z","title":"Umbral methods, function factorisation and generalisation of the Fourier transform method"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.05213","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:dc8ef8f0b4de98831f043717a62916c481b21d90091e8d266d365f363e7e2a97","target":"record","created_at":"2026-06-05T00:13:48Z","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":"84bd8df58ba2dcd8043bb10ccf1484399424512a9b5f3e6cea1d925316b76562","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GM","submitted_at":"2026-05-27T14:03:36Z","title_canon_sha256":"93ad054d47101c4ad88b63d8ecfc19a327d2ed55657fd57be38040e5a7d7c22e"},"schema_version":"1.0","source":{"id":"2606.05213","kind":"arxiv","version":1}},"canonical_sha256":"e5888aff7b152a0fe8df7621a1ec5ea0eaa3990858674cc124c6a8c3383626a7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e5888aff7b152a0fe8df7621a1ec5ea0eaa3990858674cc124c6a8c3383626a7","first_computed_at":"2026-06-05T00:13:48.918579Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-05T00:13:48.918579Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"l5WImCYBZA9OQYS2PPyNbwrYcxclGb3OSpQVrF3k/ivnp599ekBRxgZqHKESvJOZJIhbG4cOjUv2Vz/01YmJAQ==","signature_status":"signed_v1","signed_at":"2026-06-05T00:13:48.919033Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.05213","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dc8ef8f0b4de98831f043717a62916c481b21d90091e8d266d365f363e7e2a97","sha256:5024ea2772d0730ddf0807569d741abc86ac8bd2ccc77f6bc86628d585d69041"],"state_sha256":"18eadea64f384c15721ef07c1b26056e607c91343d12fb72f3c76179ca5c2d70"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1/F4Ft2Sn3MWJ1cGrdztG8v0DkDBghSeU296R3RMo1MFrabWFyALaseuv1fjbG+WpRLsTEsJTPWs1MtFin97BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T23:21:15.115919Z","bundle_sha256":"bf62c20c7d73cbc38ab6ce4d07d60bf71150c39e66392cb1b4f878ce70e16781"}}