{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:FWIKX6SLRII2ZEJUQQQAUUWNKM","short_pith_number":"pith:FWIKX6SL","canonical_record":{"source":{"id":"1804.10360","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LO","submitted_at":"2018-04-27T06:52:42Z","cross_cats_sorted":["math.CT","quant-ph"],"title_canon_sha256":"a9b2b5c2e64defcbc1524d0016f167f1985732410e60b3c31cb890216bfb45c0","abstract_canon_sha256":"681f84412ea108fdf6356fedc33eb2943ab2cb5e596c931f4fc7edb07a890bd7"},"schema_version":"1.0"},"canonical_sha256":"2d90abfa4b8a11ac913484200a52cd53391627cf492f753ce96b990e22431773","source":{"kind":"arxiv","id":"1804.10360","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.10360","created_at":"2026-05-17T23:55:20Z"},{"alias_kind":"arxiv_version","alias_value":"1804.10360v4","created_at":"2026-05-17T23:55:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.10360","created_at":"2026-05-17T23:55:20Z"},{"alias_kind":"pith_short_12","alias_value":"FWIKX6SLRII2","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_16","alias_value":"FWIKX6SLRII2ZEJU","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_8","alias_value":"FWIKX6SL","created_at":"2026-05-18T12:32:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:FWIKX6SLRII2ZEJUQQQAUUWNKM","target":"record","payload":{"canonical_record":{"source":{"id":"1804.10360","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LO","submitted_at":"2018-04-27T06:52:42Z","cross_cats_sorted":["math.CT","quant-ph"],"title_canon_sha256":"a9b2b5c2e64defcbc1524d0016f167f1985732410e60b3c31cb890216bfb45c0","abstract_canon_sha256":"681f84412ea108fdf6356fedc33eb2943ab2cb5e596c931f4fc7edb07a890bd7"},"schema_version":"1.0"},"canonical_sha256":"2d90abfa4b8a11ac913484200a52cd53391627cf492f753ce96b990e22431773","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:55:20.874494Z","signature_b64":"lJRTcjLmqI8UYidxNDf1OOvfWjBB6Hnf2ZlPlNoSfpd6bn8BrBE5vSHyVgyhGFEe73njbFhydgNChP4vv3hTDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2d90abfa4b8a11ac913484200a52cd53391627cf492f753ce96b990e22431773","last_reissued_at":"2026-05-17T23:55:20.874005Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:55:20.874005Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1804.10360","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:55:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"I/jpbTH1ub09ZyZYpCO9E6vDI/u9rAOYBDoWTSrbdi/1T3JCW0siI2y9VN49PMhnXmr7HX4fziM/Q0kUaE9xAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T03:32:50.780831Z"},"content_sha256":"338f5e34e888511988c5601853071bc4c7dabeda6b5317a41acbf92b70f96477","schema_version":"1.0","event_id":"sha256:338f5e34e888511988c5601853071bc4c7dabeda6b5317a41acbf92b70f96477"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:FWIKX6SLRII2ZEJUQQQAUUWNKM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Category TOF","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CT","quant-ph"],"primary_cat":"cs.LO","authors_text":"Cole Comfort, J.R.B. Cockett","submitted_at":"2018-04-27T06:52:42Z","abstract_excerpt":"We provide a complete set of identities for the symmetric monoidal category, TOF, generated by the Toffoli gate and computational ancillary bits. We do so by demonstrating that the functor which evaluates circuits on total points, is an equivalence into the full subcategory of sets and partial isomorphisms with objects finite powers of the two element set. The structure of the proof builds -- and follows the proof of Cockett et al.-- which provided a full set of identities for the cnot gate with computational ancillary bits. Thus, first it is shown that TOF is a discrete inverse category in wh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.10360","kind":"arxiv","version":4},"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":"2026-05-17T23:55:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ulLMZ229rtnATTFxQM5xoNkG/9q5Kvox7bhhEsLf40o0o+aMnbIIqbZm8jHO79TC3kWSpofGCGGsllkQ6DNqDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T03:32:50.781186Z"},"content_sha256":"43412c326f1d83c61567aeffc2193867c1e28ab5e904ef3882d3cde296bbc8a6","schema_version":"1.0","event_id":"sha256:43412c326f1d83c61567aeffc2193867c1e28ab5e904ef3882d3cde296bbc8a6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FWIKX6SLRII2ZEJUQQQAUUWNKM/bundle.json","state_url":"https://pith.science/pith/FWIKX6SLRII2ZEJUQQQAUUWNKM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FWIKX6SLRII2ZEJUQQQAUUWNKM/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":"2026-06-23T03:32:50Z","links":{"resolver":"https://pith.science/pith/FWIKX6SLRII2ZEJUQQQAUUWNKM","bundle":"https://pith.science/pith/FWIKX6SLRII2ZEJUQQQAUUWNKM/bundle.json","state":"https://pith.science/pith/FWIKX6SLRII2ZEJUQQQAUUWNKM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FWIKX6SLRII2ZEJUQQQAUUWNKM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:FWIKX6SLRII2ZEJUQQQAUUWNKM","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":"681f84412ea108fdf6356fedc33eb2943ab2cb5e596c931f4fc7edb07a890bd7","cross_cats_sorted":["math.CT","quant-ph"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LO","submitted_at":"2018-04-27T06:52:42Z","title_canon_sha256":"a9b2b5c2e64defcbc1524d0016f167f1985732410e60b3c31cb890216bfb45c0"},"schema_version":"1.0","source":{"id":"1804.10360","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.10360","created_at":"2026-05-17T23:55:20Z"},{"alias_kind":"arxiv_version","alias_value":"1804.10360v4","created_at":"2026-05-17T23:55:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.10360","created_at":"2026-05-17T23:55:20Z"},{"alias_kind":"pith_short_12","alias_value":"FWIKX6SLRII2","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_16","alias_value":"FWIKX6SLRII2ZEJU","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_8","alias_value":"FWIKX6SL","created_at":"2026-05-18T12:32:25Z"}],"graph_snapshots":[{"event_id":"sha256:43412c326f1d83c61567aeffc2193867c1e28ab5e904ef3882d3cde296bbc8a6","target":"graph","created_at":"2026-05-17T23:55:20Z","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":"We provide a complete set of identities for the symmetric monoidal category, TOF, generated by the Toffoli gate and computational ancillary bits. We do so by demonstrating that the functor which evaluates circuits on total points, is an equivalence into the full subcategory of sets and partial isomorphisms with objects finite powers of the two element set. The structure of the proof builds -- and follows the proof of Cockett et al.-- which provided a full set of identities for the cnot gate with computational ancillary bits. Thus, first it is shown that TOF is a discrete inverse category in wh","authors_text":"Cole Comfort, J.R.B. Cockett","cross_cats":["math.CT","quant-ph"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LO","submitted_at":"2018-04-27T06:52:42Z","title":"The Category TOF"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.10360","kind":"arxiv","version":4},"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":"sha256:338f5e34e888511988c5601853071bc4c7dabeda6b5317a41acbf92b70f96477","target":"record","created_at":"2026-05-17T23:55:20Z","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":"681f84412ea108fdf6356fedc33eb2943ab2cb5e596c931f4fc7edb07a890bd7","cross_cats_sorted":["math.CT","quant-ph"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LO","submitted_at":"2018-04-27T06:52:42Z","title_canon_sha256":"a9b2b5c2e64defcbc1524d0016f167f1985732410e60b3c31cb890216bfb45c0"},"schema_version":"1.0","source":{"id":"1804.10360","kind":"arxiv","version":4}},"canonical_sha256":"2d90abfa4b8a11ac913484200a52cd53391627cf492f753ce96b990e22431773","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2d90abfa4b8a11ac913484200a52cd53391627cf492f753ce96b990e22431773","first_computed_at":"2026-05-17T23:55:20.874005Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:55:20.874005Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lJRTcjLmqI8UYidxNDf1OOvfWjBB6Hnf2ZlPlNoSfpd6bn8BrBE5vSHyVgyhGFEe73njbFhydgNChP4vv3hTDg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:55:20.874494Z","signed_message":"canonical_sha256_bytes"},"source_id":"1804.10360","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:338f5e34e888511988c5601853071bc4c7dabeda6b5317a41acbf92b70f96477","sha256:43412c326f1d83c61567aeffc2193867c1e28ab5e904ef3882d3cde296bbc8a6"],"state_sha256":"fe82f72ce3b30c2f03fdb9a6ebf6214b99303bc0940f7d5012b38badd184eb8a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"boAu/g/9GMvjzo+vUpon4MysbKGCLFsZDckQ/DlbGRtsuE2I3refB/6zgNsvzHY/8k4PeKht3sgH9Z0jyG1iAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T03:32:50.783214Z","bundle_sha256":"5ffd234c754fd25cf4227c77010d6ea9c5ab89bb26b42051762ed2ff187f0e50"}}