{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:PEWQXBIBKJLDZRAQJROJFQSCOC","short_pith_number":"pith:PEWQXBIB","canonical_record":{"source":{"id":"1510.03234","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-10-12T11:24:17Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"86ae6e0d3c4c123a7b34a63a0e7ca141e165f76dc487f185b4a7805dadc30e48","abstract_canon_sha256":"0acd0d48a8c02707766a8d694cc116f80395123244c945658faea3c39d42826f"},"schema_version":"1.0"},"canonical_sha256":"792d0b850152563cc4104c5c92c2427095c10405368198a4afc1331b7a8282bb","source":{"kind":"arxiv","id":"1510.03234","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.03234","created_at":"2026-05-18T01:30:32Z"},{"alias_kind":"arxiv_version","alias_value":"1510.03234v1","created_at":"2026-05-18T01:30:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.03234","created_at":"2026-05-18T01:30:32Z"},{"alias_kind":"pith_short_12","alias_value":"PEWQXBIBKJLD","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"PEWQXBIBKJLDZRAQ","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"PEWQXBIB","created_at":"2026-05-18T12:29:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:PEWQXBIBKJLDZRAQJROJFQSCOC","target":"record","payload":{"canonical_record":{"source":{"id":"1510.03234","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-10-12T11:24:17Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"86ae6e0d3c4c123a7b34a63a0e7ca141e165f76dc487f185b4a7805dadc30e48","abstract_canon_sha256":"0acd0d48a8c02707766a8d694cc116f80395123244c945658faea3c39d42826f"},"schema_version":"1.0"},"canonical_sha256":"792d0b850152563cc4104c5c92c2427095c10405368198a4afc1331b7a8282bb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:30:32.417781Z","signature_b64":"Jz+5XqOFvLoiMKR8eY1xlj/x7nw6hisg8dP0eaj5YIjJ6AhEI/9bMFD7jgr93dQ+ccaIfmSLoHKTATDg5Rz9Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"792d0b850152563cc4104c5c92c2427095c10405368198a4afc1331b7a8282bb","last_reissued_at":"2026-05-18T01:30:32.417096Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:30:32.417096Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1510.03234","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":"2026-05-18T01:30:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0aNZkAyYyGl9S2pex8kfjCKbbdiVn3/aE7N6w+J9RN01CiuFKWO707v8fzgW4B0nfOzEXVePYXBmrZ7GohL8Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T20:00:17.940976Z"},"content_sha256":"6090b7f46a723dd036bff7432bd1602acf052b177ee7fbb225b728a4ce68d344","schema_version":"1.0","event_id":"sha256:6090b7f46a723dd036bff7432bd1602acf052b177ee7fbb225b728a4ce68d344"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:PEWQXBIBKJLDZRAQJROJFQSCOC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Conceptual differential calculus part ii: Cubic higher order calculus","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.DG","authors_text":"Wolfgang Bertram","submitted_at":"2015-10-12T11:24:17Z","abstract_excerpt":"Following the programme set out in Part I of this work, we develop a conceptual higher order differential calculus. The '' local linear algebra '' defined in Part I is generalized by '' higher order local linear algebra ''. The underlying combinatorial object of such higher algebra is the natural n-dimensional hyper-cube, and so we qualify this calculus as '' cubic ''. More precisely, we define two versions of conceptual cubic calculus: '' full '' and '' symmetric cubic ''. The theory thus initiated sheds new light on several foundational issues."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.03234","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":""},"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":"2026-05-18T01:30:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"N6YNhG1WMmjkkFzkoWnF66JLKXOYsKllEd0ybzldcJ5BPCukAicK2suSoipCrhX+K4qSznFkztiHBpLzcqF/CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T20:00:17.941335Z"},"content_sha256":"3dd8dbdade1ccbe40136663e530440fd483cca242eada8a84a13fae0650b6c8d","schema_version":"1.0","event_id":"sha256:3dd8dbdade1ccbe40136663e530440fd483cca242eada8a84a13fae0650b6c8d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PEWQXBIBKJLDZRAQJROJFQSCOC/bundle.json","state_url":"https://pith.science/pith/PEWQXBIBKJLDZRAQJROJFQSCOC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PEWQXBIBKJLDZRAQJROJFQSCOC/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":"2026-06-01T20:00:17Z","links":{"resolver":"https://pith.science/pith/PEWQXBIBKJLDZRAQJROJFQSCOC","bundle":"https://pith.science/pith/PEWQXBIBKJLDZRAQJROJFQSCOC/bundle.json","state":"https://pith.science/pith/PEWQXBIBKJLDZRAQJROJFQSCOC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PEWQXBIBKJLDZRAQJROJFQSCOC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:PEWQXBIBKJLDZRAQJROJFQSCOC","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":"0acd0d48a8c02707766a8d694cc116f80395123244c945658faea3c39d42826f","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-10-12T11:24:17Z","title_canon_sha256":"86ae6e0d3c4c123a7b34a63a0e7ca141e165f76dc487f185b4a7805dadc30e48"},"schema_version":"1.0","source":{"id":"1510.03234","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.03234","created_at":"2026-05-18T01:30:32Z"},{"alias_kind":"arxiv_version","alias_value":"1510.03234v1","created_at":"2026-05-18T01:30:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.03234","created_at":"2026-05-18T01:30:32Z"},{"alias_kind":"pith_short_12","alias_value":"PEWQXBIBKJLD","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"PEWQXBIBKJLDZRAQ","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"PEWQXBIB","created_at":"2026-05-18T12:29:37Z"}],"graph_snapshots":[{"event_id":"sha256:3dd8dbdade1ccbe40136663e530440fd483cca242eada8a84a13fae0650b6c8d","target":"graph","created_at":"2026-05-18T01:30:32Z","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"},"paper":{"abstract_excerpt":"Following the programme set out in Part I of this work, we develop a conceptual higher order differential calculus. The '' local linear algebra '' defined in Part I is generalized by '' higher order local linear algebra ''. The underlying combinatorial object of such higher algebra is the natural n-dimensional hyper-cube, and so we qualify this calculus as '' cubic ''. More precisely, we define two versions of conceptual cubic calculus: '' full '' and '' symmetric cubic ''. The theory thus initiated sheds new light on several foundational issues.","authors_text":"Wolfgang Bertram","cross_cats":["math.CT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-10-12T11:24:17Z","title":"Conceptual differential calculus part ii: Cubic higher order calculus"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.03234","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":"sha256:6090b7f46a723dd036bff7432bd1602acf052b177ee7fbb225b728a4ce68d344","target":"record","created_at":"2026-05-18T01:30:32Z","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":"0acd0d48a8c02707766a8d694cc116f80395123244c945658faea3c39d42826f","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-10-12T11:24:17Z","title_canon_sha256":"86ae6e0d3c4c123a7b34a63a0e7ca141e165f76dc487f185b4a7805dadc30e48"},"schema_version":"1.0","source":{"id":"1510.03234","kind":"arxiv","version":1}},"canonical_sha256":"792d0b850152563cc4104c5c92c2427095c10405368198a4afc1331b7a8282bb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"792d0b850152563cc4104c5c92c2427095c10405368198a4afc1331b7a8282bb","first_computed_at":"2026-05-18T01:30:32.417096Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:30:32.417096Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Jz+5XqOFvLoiMKR8eY1xlj/x7nw6hisg8dP0eaj5YIjJ6AhEI/9bMFD7jgr93dQ+ccaIfmSLoHKTATDg5Rz9Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:30:32.417781Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.03234","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6090b7f46a723dd036bff7432bd1602acf052b177ee7fbb225b728a4ce68d344","sha256:3dd8dbdade1ccbe40136663e530440fd483cca242eada8a84a13fae0650b6c8d"],"state_sha256":"3f9c3e0a211e802df1f26eed6061f087bd72319f4c3f4addc104accf1d7d45df"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AFU1rfJVIRAFL229nBS/J6LsUYOZQaqtsEyn6s0qzM8hseCERw+fxwj4IOgCVvkAFLtA0gVXjrBDnn9tqRNCAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T20:00:17.943204Z","bundle_sha256":"e3940fbb304b30f54459c00bef21a6172243007f726538da7675f54963c79b47"}}