{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:6QDFC2Y4U7PG646RJLUYHTFPOH","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":"62f80cc2f7e1bf01bdc441db144ff1c58a541f85b00010ed393581da69da6f36","cross_cats_sorted":["cs.AI","cs.LG","stat.ML"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2026-05-17T21:30:13Z","title_canon_sha256":"8e2ad1d8fba0ed6441fbfa822016d407e90e873fd5e608931be87cddd501d7fe"},"schema_version":"1.0","source":{"id":"2605.17660","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.17660","created_at":"2026-05-20T00:04:51Z"},{"alias_kind":"arxiv_version","alias_value":"2605.17660v1","created_at":"2026-05-20T00:04:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.17660","created_at":"2026-05-20T00:04:51Z"},{"alias_kind":"pith_short_12","alias_value":"6QDFC2Y4U7PG","created_at":"2026-05-20T00:04:51Z"},{"alias_kind":"pith_short_16","alias_value":"6QDFC2Y4U7PG646R","created_at":"2026-05-20T00:04:51Z"},{"alias_kind":"pith_short_8","alias_value":"6QDFC2Y4","created_at":"2026-05-20T00:04:51Z"}],"graph_snapshots":[{"event_id":"sha256:504b4c132042cef61d4b89c57119f971ffbaa0179b110940d5cb8e8f175cf416","target":"graph","created_at":"2026-05-20T00:04:51Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"Under the NTK injectivity assumption (linear independence of log-sum-exp functions modulo affine functions), gradient flow in the conditional Wasserstein metric converges to global minima whenever the initial loss is sufficiently small."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The mean-field limit and the NTK injectivity condition remain valid approximations for the finite but large transformers that are actually trained; this premise enters when the convergence theorem is stated after the injectivity characterization."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Develops a mean-field neural PDE model for transformer training, proves forward-pass well-posedness via function-space ODEs, derives conditional Wasserstein gradients, and shows global convergence of gradient flow under an NTK injectivity condition equivalent to linear independence of log-sum-exp fu"},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Gradient flow in the conditional Wasserstein metric converges to global minima for infinitely deep and wide transformers when the initial loss is small enough and the attention NTK is injective."}],"snapshot_sha256":"0ef8e9c083ae82fb032841b05de678609a723a0bae106e4075ad4349d14ce7f2"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"9f10c5a324cfd2b25af36ad09efb6164574e418af8cd7ff37bf1f8d1b6d2aee2"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"name":"doi_title_agreement","ran_at":"2026-05-19T22:31:19.483169Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"cited_work_retraction","ran_at":"2026-05-19T22:22:11.642152Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"doi_compliance","ran_at":"2026-05-19T22:20:57.559089Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"citation_quote_validity","ran_at":"2026-05-19T21:49:44.270375Z","status":"skipped","version":"0.1.0"},{"findings_count":0,"name":"ai_meta_artifact","ran_at":"2026-05-19T21:33:23.541285Z","status":"skipped","version":"1.0.0"},{"findings_count":0,"name":"claim_evidence","ran_at":"2026-05-19T21:21:57.458298Z","status":"completed","version":"1.0.0"}],"endpoint":"/pith/2605.17660/integrity.json","findings":[],"snapshot_sha256":"bac2ec79408c39f98fce8a078365c1009a4553dbe1fa1e5d7a30ef8a506dd882","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Transformers have become the dominant architecture in modern machine learning, yet the theoretical understanding of their training dynamics remains limited. This paper develops a rigorous mathematical framework for analyzing gradient-based training of transformers in the mean-field regime, where both the depth (number of layers) and width (number of attention heads) tend to infinity. While ResNet training can be understood as controlling a neural ODE, transformer training corresponds to controlling a neural PDE, due to the coupling of multiple token distributions through the attention mechanis","authors_text":"Gabriel Peyr\\'e, Maarten V. de Hoop, Rapha\\\"el Barboni, Takashi Furuya","cross_cats":["cs.AI","cs.LG","stat.ML"],"headline":"Gradient flow in the conditional Wasserstein metric converges to global minima for infinitely deep and wide transformers when the initial loss is small enough and the attention NTK is injective.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2026-05-17T21:30:13Z","title":"Training Infinitely Deep and Wide Transformers"},"references":{"count":53,"internal_anchors":0,"resolved_work":53,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"Transformers learn to imple- ment preconditioned gradient descent for in-context learning","work_id":"42670431-9028-4b66-889a-652920100a66","year":2023},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"What learning algorithm is in-context learning? investigations with linear models","work_id":"f67e4fac-adc7-43d4-b334-b8b9d73e40ac","year":2023},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"A convergence theory for deep learning via over-parameterization","work_id":"00507d70-4223-4a3a-9734-7857dbf05620","year":2019},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"Transport equation and Cauchy problem for non-smooth vector fields","work_id":"89b697c2-19c1-4aea-9f8b-ee3e65fac016","year":2008},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"A user’s guide to optimal transport","work_id":"68f196ac-7a64-491a-a457-3a7adbbd7ede","year":2062}],"snapshot_sha256":"c623850a99f2bf16a3b1bd791d3031a39a3fc798de23afad680d4b9fdfef9187"},"source":{"id":"2605.17660","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-19T22:08:22.341277Z","id":"3204ae40-3192-4bd1-b78a-c06e6933c0b5","model_set":{"reader":"grok-4.3"},"one_line_summary":"Develops a mean-field neural PDE model for transformer training, proves forward-pass well-posedness via function-space ODEs, derives conditional Wasserstein gradients, and shows global convergence of gradient flow under an NTK injectivity condition equivalent to linear independence of log-sum-exp fu","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Gradient flow in the conditional Wasserstein metric converges to global minima for infinitely deep and wide transformers when the initial loss is small enough and the attention NTK is injective.","strongest_claim":"Under the NTK injectivity assumption (linear independence of log-sum-exp functions modulo affine functions), gradient flow in the conditional Wasserstein metric converges to global minima whenever the initial loss is sufficiently small.","weakest_assumption":"The mean-field limit and the NTK injectivity condition remain valid approximations for the finite but large transformers that are actually trained; this premise enters when the convergence theorem is stated after the injectivity characterization."}},"verdict_id":"3204ae40-3192-4bd1-b78a-c06e6933c0b5"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:b5c9f1147d14e2c99cf6b64d8deffa81ef45de9c43647f34ab62e4030ace61dc","target":"record","created_at":"2026-05-20T00:04:51Z","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":"62f80cc2f7e1bf01bdc441db144ff1c58a541f85b00010ed393581da69da6f36","cross_cats_sorted":["cs.AI","cs.LG","stat.ML"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2026-05-17T21:30:13Z","title_canon_sha256":"8e2ad1d8fba0ed6441fbfa822016d407e90e873fd5e608931be87cddd501d7fe"},"schema_version":"1.0","source":{"id":"2605.17660","kind":"arxiv","version":1}},"canonical_sha256":"f406516b1ca7de6f73d14ae983ccaf71ffa71e1ba8407e18af9cc750415bf815","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f406516b1ca7de6f73d14ae983ccaf71ffa71e1ba8407e18af9cc750415bf815","first_computed_at":"2026-05-20T00:04:51.429856Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:04:51.429856Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FGEIH978NuJOSsL0pIxgfoNyopx9bbnRmz/S2VJU2qGSxOaQm6VgM6SYacN0byvKWa+ZlO5DUnsTrSWO0YZPDQ==","signature_status":"signed_v1","signed_at":"2026-05-20T00:04:51.430686Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.17660","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b5c9f1147d14e2c99cf6b64d8deffa81ef45de9c43647f34ab62e4030ace61dc","sha256:504b4c132042cef61d4b89c57119f971ffbaa0179b110940d5cb8e8f175cf416"],"state_sha256":"b74ae105a26a3ad11117e528fb5869a93c37d7e2e2ffb9ef9a62370ad9b6706c"}