{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:5JJMCT4S4NAZJ4EHAJQH6PVMVO","short_pith_number":"pith:5JJMCT4S","canonical_record":{"source":{"id":"2403.18415","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2024-03-27T10:06:33Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"3c8d779fd42370a32dadc72440c3460ac7ef8ffe390b366222385230139cb5d1","abstract_canon_sha256":"a0da32a816443d26c41a78b81908a519ee9e4a15e0d8091d31736884a827a6db"},"schema_version":"1.0"},"canonical_sha256":"ea52c14f92e34194f08702607f3eacabb774ff89490706553257002fe167eb0a","source":{"kind":"arxiv","id":"2403.18415","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2403.18415","created_at":"2026-07-05T08:15:46Z"},{"alias_kind":"arxiv_version","alias_value":"2403.18415v3","created_at":"2026-07-05T08:15:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2403.18415","created_at":"2026-07-05T08:15:46Z"},{"alias_kind":"pith_short_12","alias_value":"5JJMCT4S4NAZ","created_at":"2026-07-05T08:15:46Z"},{"alias_kind":"pith_short_16","alias_value":"5JJMCT4S4NAZJ4EH","created_at":"2026-07-05T08:15:46Z"},{"alias_kind":"pith_short_8","alias_value":"5JJMCT4S","created_at":"2026-07-05T08:15:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:5JJMCT4S4NAZJ4EHAJQH6PVMVO","target":"record","payload":{"canonical_record":{"source":{"id":"2403.18415","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2024-03-27T10:06:33Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"3c8d779fd42370a32dadc72440c3460ac7ef8ffe390b366222385230139cb5d1","abstract_canon_sha256":"a0da32a816443d26c41a78b81908a519ee9e4a15e0d8091d31736884a827a6db"},"schema_version":"1.0"},"canonical_sha256":"ea52c14f92e34194f08702607f3eacabb774ff89490706553257002fe167eb0a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:15:46.750257Z","signature_b64":"vKXQB4DW1/4wLdXGGuaie2cWm1Lta2AGwj1dy6RMjruYkBHexfBmu6vwe7o7iAYe8SJMpw0cogYtsU2ARhO+CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ea52c14f92e34194f08702607f3eacabb774ff89490706553257002fe167eb0a","last_reissued_at":"2026-07-05T08:15:46.749703Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:15:46.749703Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2403.18415","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T08:15:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rkh7Nyabfn3eKkZTRPgOfEdiQiHVwkUCCSvTK3S3Vcadd2xeVAcGQu9vRcuUYKnG7SaG26bZTFbjdh/51fClCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T20:16:28.425142Z"},"content_sha256":"38d6e4727de6a22af4488c76be679d805af23d88b1cde38369bd488b2bc526d7","schema_version":"1.0","event_id":"sha256:38d6e4727de6a22af4488c76be679d805af23d88b1cde38369bd488b2bc526d7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:5JJMCT4S4NAZJ4EHAJQH6PVMVO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Topos of Transformer Networks","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"cs.LG","authors_text":"Mattia Jacopo Villani, Peter McBurney","submitted_at":"2024-03-27T10:06:33Z","abstract_excerpt":"The transformer neural network has significantly out-shined all other neural network architectures as the engine behind large language models. We provide a theoretical analysis of the expressivity of the transformer architecture through the lens of topos theory. From this viewpoint, we show that many common neural network architectures, such as the convolutional, recurrent and graph convolutional networks, can be embedded in a pretopos of piecewise-linear functions, but that the transformer necessarily lives in its topos completion. In particular, this suggests that the two network families in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.18415","kind":"arxiv","version":3},"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/2403.18415/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":"2026-07-05T08:15:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"L0WhD//5srvYgJUBCt9LBih9qSvLr/a4JvrNR2yvbjPy6nkXVl4ana17MOorLF0PK9VASf4QBUtZEysdtlqNBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T20:16:28.426187Z"},"content_sha256":"2bc6bede14da522b8d4a3209876096f98dcd90865fbc4dbd95076dd173db6ef4","schema_version":"1.0","event_id":"sha256:2bc6bede14da522b8d4a3209876096f98dcd90865fbc4dbd95076dd173db6ef4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5JJMCT4S4NAZJ4EHAJQH6PVMVO/bundle.json","state_url":"https://pith.science/pith/5JJMCT4S4NAZJ4EHAJQH6PVMVO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5JJMCT4S4NAZJ4EHAJQH6PVMVO/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":"2026-08-15T20:16:28Z","links":{"resolver":"https://pith.science/pith/5JJMCT4S4NAZJ4EHAJQH6PVMVO","bundle":"https://pith.science/pith/5JJMCT4S4NAZJ4EHAJQH6PVMVO/bundle.json","state":"https://pith.science/pith/5JJMCT4S4NAZJ4EHAJQH6PVMVO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5JJMCT4S4NAZJ4EHAJQH6PVMVO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:5JJMCT4S4NAZJ4EHAJQH6PVMVO","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":"a0da32a816443d26c41a78b81908a519ee9e4a15e0d8091d31736884a827a6db","cross_cats_sorted":["math.CT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2024-03-27T10:06:33Z","title_canon_sha256":"3c8d779fd42370a32dadc72440c3460ac7ef8ffe390b366222385230139cb5d1"},"schema_version":"1.0","source":{"id":"2403.18415","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2403.18415","created_at":"2026-07-05T08:15:46Z"},{"alias_kind":"arxiv_version","alias_value":"2403.18415v3","created_at":"2026-07-05T08:15:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2403.18415","created_at":"2026-07-05T08:15:46Z"},{"alias_kind":"pith_short_12","alias_value":"5JJMCT4S4NAZ","created_at":"2026-07-05T08:15:46Z"},{"alias_kind":"pith_short_16","alias_value":"5JJMCT4S4NAZJ4EH","created_at":"2026-07-05T08:15:46Z"},{"alias_kind":"pith_short_8","alias_value":"5JJMCT4S","created_at":"2026-07-05T08:15:46Z"}],"graph_snapshots":[{"event_id":"sha256:2bc6bede14da522b8d4a3209876096f98dcd90865fbc4dbd95076dd173db6ef4","target":"graph","created_at":"2026-07-05T08:15:46Z","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/2403.18415/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The transformer neural network has significantly out-shined all other neural network architectures as the engine behind large language models. We provide a theoretical analysis of the expressivity of the transformer architecture through the lens of topos theory. From this viewpoint, we show that many common neural network architectures, such as the convolutional, recurrent and graph convolutional networks, can be embedded in a pretopos of piecewise-linear functions, but that the transformer necessarily lives in its topos completion. In particular, this suggests that the two network families in","authors_text":"Mattia Jacopo Villani, Peter McBurney","cross_cats":["math.CT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2024-03-27T10:06:33Z","title":"The Topos of Transformer Networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.18415","kind":"arxiv","version":3},"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":"sha256:38d6e4727de6a22af4488c76be679d805af23d88b1cde38369bd488b2bc526d7","target":"record","created_at":"2026-07-05T08:15:46Z","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":"a0da32a816443d26c41a78b81908a519ee9e4a15e0d8091d31736884a827a6db","cross_cats_sorted":["math.CT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2024-03-27T10:06:33Z","title_canon_sha256":"3c8d779fd42370a32dadc72440c3460ac7ef8ffe390b366222385230139cb5d1"},"schema_version":"1.0","source":{"id":"2403.18415","kind":"arxiv","version":3}},"canonical_sha256":"ea52c14f92e34194f08702607f3eacabb774ff89490706553257002fe167eb0a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ea52c14f92e34194f08702607f3eacabb774ff89490706553257002fe167eb0a","first_computed_at":"2026-07-05T08:15:46.749703Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:15:46.749703Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vKXQB4DW1/4wLdXGGuaie2cWm1Lta2AGwj1dy6RMjruYkBHexfBmu6vwe7o7iAYe8SJMpw0cogYtsU2ARhO+CA==","signature_status":"signed_v1","signed_at":"2026-07-05T08:15:46.750257Z","signed_message":"canonical_sha256_bytes"},"source_id":"2403.18415","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:38d6e4727de6a22af4488c76be679d805af23d88b1cde38369bd488b2bc526d7","sha256:2bc6bede14da522b8d4a3209876096f98dcd90865fbc4dbd95076dd173db6ef4"],"state_sha256":"002f9fe7c3985f967fd5b877181b36853ccd2d667db95017ca4a0527c8c25bb4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Zx11nwbWr/P+Dyr0wOuMw/0yiLSD4XEW8nXVuDg/RbfNsInLf2YHOyQgF8P19QWtxpH9ofLmToRE8SYGNEzVDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-15T20:16:28.431573Z","bundle_sha256":"82234cdc18107a0bbea0ad4dd0aac93a2495dc4e05d1dd71632000e4624655fa"}}