{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:RM4RTHCAUCTBRZAVSUYPRKGRNB","short_pith_number":"pith:RM4RTHCA","canonical_record":{"source":{"id":"2601.10950","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-01-16T02:34:19Z","cross_cats_sorted":["cs.NA","math.NA"],"title_canon_sha256":"99f74b296825a85714fd5c9541634499b16965eb22b1df577db0bade639fd73f","abstract_canon_sha256":"f609b1ad902107bd1bdbee435fc995d4f5af4b47aeec874bcd506a44d25d1c8f"},"schema_version":"1.0"},"canonical_sha256":"8b39199c40a0a618e4159530f8a8d1684af64b63ec86c6db50d996d04ff441f9","source":{"kind":"arxiv","id":"2601.10950","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2601.10950","created_at":"2026-05-22T01:03:16Z"},{"alias_kind":"arxiv_version","alias_value":"2601.10950v2","created_at":"2026-05-22T01:03:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2601.10950","created_at":"2026-05-22T01:03:16Z"},{"alias_kind":"pith_short_12","alias_value":"RM4RTHCAUCTB","created_at":"2026-05-22T01:03:16Z"},{"alias_kind":"pith_short_16","alias_value":"RM4RTHCAUCTBRZAV","created_at":"2026-05-22T01:03:16Z"},{"alias_kind":"pith_short_8","alias_value":"RM4RTHCA","created_at":"2026-05-22T01:03:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:RM4RTHCAUCTBRZAVSUYPRKGRNB","target":"record","payload":{"canonical_record":{"source":{"id":"2601.10950","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-01-16T02:34:19Z","cross_cats_sorted":["cs.NA","math.NA"],"title_canon_sha256":"99f74b296825a85714fd5c9541634499b16965eb22b1df577db0bade639fd73f","abstract_canon_sha256":"f609b1ad902107bd1bdbee435fc995d4f5af4b47aeec874bcd506a44d25d1c8f"},"schema_version":"1.0"},"canonical_sha256":"8b39199c40a0a618e4159530f8a8d1684af64b63ec86c6db50d996d04ff441f9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-22T01:03:16.984457Z","signature_b64":"zhPTv5RQ6ekxKkkhmb9lw6CkYJP6Z1jEoED45g7I1OBlVuluN/tM84FJBWm08c3CWI3vpTZM4ozGPO3J4CDKDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8b39199c40a0a618e4159530f8a8d1684af64b63ec86c6db50d996d04ff441f9","last_reissued_at":"2026-05-22T01:03:16.983563Z","signature_status":"signed_v1","first_computed_at":"2026-05-22T01:03:16.983563Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2601.10950","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":"2026-05-22T01:03:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"meY4jx0YYIcgA3YYQ+CoN0cy+XtFkVzGeRVwvODCGaPt3lZa0pZiDap1JnatPs9yK4rbCdIzIq1UqSx5URz1Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T18:12:16.746683Z"},"content_sha256":"edf91daf227a11f1f3e1e7aadbdfb6ed716c8f6b952aae7ac89c1040c6920562","schema_version":"1.0","event_id":"sha256:edf91daf227a11f1f3e1e7aadbdfb6ed716c8f6b952aae7ac89c1040c6920562"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:RM4RTHCAUCTBRZAVSUYPRKGRNB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Specular differentiation in normed vector spaces: Quasi-Mean Value and Quasi-Fermat Theorems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA","math.NA"],"primary_cat":"math.OC","authors_text":"Kiyuob Jung","submitted_at":"2026-01-16T02:34:19Z","abstract_excerpt":"This paper introduces specular differentiation, which generalizes G\\^ateaux and Fr\\'echet differentiation in normed vector spaces. We investigate its fundamental theoretical properties and establish weak forms of the Mean Value Theorem and Fermat's Theorem in the specular sense. Finally, we identify a distinguished element of the Fr\\'echet subdifferential of a convex function through specular differentiation."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2601.10950","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/2601.10950/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":"2026-05-22T01:03:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VDfkcivvMUQwWbkj98I4g81lc2Te/lVSKJ0F2j04sRcKG2Ots/xLVAVNscJhDRtUNaHtuJLDiKyRqysZvBrEBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T18:12:16.747056Z"},"content_sha256":"6b795611b3db4cade9c13ae3aeea6bcfdb27b7f003aa875b156e617787df2807","schema_version":"1.0","event_id":"sha256:6b795611b3db4cade9c13ae3aeea6bcfdb27b7f003aa875b156e617787df2807"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RM4RTHCAUCTBRZAVSUYPRKGRNB/bundle.json","state_url":"https://pith.science/pith/RM4RTHCAUCTBRZAVSUYPRKGRNB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RM4RTHCAUCTBRZAVSUYPRKGRNB/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":"2026-06-10T18:12:16Z","links":{"resolver":"https://pith.science/pith/RM4RTHCAUCTBRZAVSUYPRKGRNB","bundle":"https://pith.science/pith/RM4RTHCAUCTBRZAVSUYPRKGRNB/bundle.json","state":"https://pith.science/pith/RM4RTHCAUCTBRZAVSUYPRKGRNB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RM4RTHCAUCTBRZAVSUYPRKGRNB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:RM4RTHCAUCTBRZAVSUYPRKGRNB","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":"f609b1ad902107bd1bdbee435fc995d4f5af4b47aeec874bcd506a44d25d1c8f","cross_cats_sorted":["cs.NA","math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-01-16T02:34:19Z","title_canon_sha256":"99f74b296825a85714fd5c9541634499b16965eb22b1df577db0bade639fd73f"},"schema_version":"1.0","source":{"id":"2601.10950","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2601.10950","created_at":"2026-05-22T01:03:16Z"},{"alias_kind":"arxiv_version","alias_value":"2601.10950v2","created_at":"2026-05-22T01:03:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2601.10950","created_at":"2026-05-22T01:03:16Z"},{"alias_kind":"pith_short_12","alias_value":"RM4RTHCAUCTB","created_at":"2026-05-22T01:03:16Z"},{"alias_kind":"pith_short_16","alias_value":"RM4RTHCAUCTBRZAV","created_at":"2026-05-22T01:03:16Z"},{"alias_kind":"pith_short_8","alias_value":"RM4RTHCA","created_at":"2026-05-22T01:03:16Z"}],"graph_snapshots":[{"event_id":"sha256:6b795611b3db4cade9c13ae3aeea6bcfdb27b7f003aa875b156e617787df2807","target":"graph","created_at":"2026-05-22T01:03:16Z","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/2601.10950/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This paper introduces specular differentiation, which generalizes G\\^ateaux and Fr\\'echet differentiation in normed vector spaces. We investigate its fundamental theoretical properties and establish weak forms of the Mean Value Theorem and Fermat's Theorem in the specular sense. Finally, we identify a distinguished element of the Fr\\'echet subdifferential of a convex function through specular differentiation.","authors_text":"Kiyuob Jung","cross_cats":["cs.NA","math.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-01-16T02:34:19Z","title":"Specular differentiation in normed vector spaces: Quasi-Mean Value and Quasi-Fermat Theorems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2601.10950","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:edf91daf227a11f1f3e1e7aadbdfb6ed716c8f6b952aae7ac89c1040c6920562","target":"record","created_at":"2026-05-22T01:03:16Z","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":"f609b1ad902107bd1bdbee435fc995d4f5af4b47aeec874bcd506a44d25d1c8f","cross_cats_sorted":["cs.NA","math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-01-16T02:34:19Z","title_canon_sha256":"99f74b296825a85714fd5c9541634499b16965eb22b1df577db0bade639fd73f"},"schema_version":"1.0","source":{"id":"2601.10950","kind":"arxiv","version":2}},"canonical_sha256":"8b39199c40a0a618e4159530f8a8d1684af64b63ec86c6db50d996d04ff441f9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8b39199c40a0a618e4159530f8a8d1684af64b63ec86c6db50d996d04ff441f9","first_computed_at":"2026-05-22T01:03:16.983563Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-22T01:03:16.983563Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zhPTv5RQ6ekxKkkhmb9lw6CkYJP6Z1jEoED45g7I1OBlVuluN/tM84FJBWm08c3CWI3vpTZM4ozGPO3J4CDKDg==","signature_status":"signed_v1","signed_at":"2026-05-22T01:03:16.984457Z","signed_message":"canonical_sha256_bytes"},"source_id":"2601.10950","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:edf91daf227a11f1f3e1e7aadbdfb6ed716c8f6b952aae7ac89c1040c6920562","sha256:6b795611b3db4cade9c13ae3aeea6bcfdb27b7f003aa875b156e617787df2807"],"state_sha256":"8779b6373971935256022cc78573037087e60de455b1e286230089a6907fe174"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QuVuWnrbcV03SDFiMOth7Y2DFTI8RHGc9XYG1xrPU3W9AVhGpKhofmk9J1t6H53mQs/2Z6WZgv5y4FvJmUSNDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T18:12:16.749014Z","bundle_sha256":"ca525db5d8491c442100ac99f3f7fb8b60ebcffaf48d1510266d3c7a929198f1"}}