{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:NP5QPYWFCFAMJ3UGF6ZZPWUPZ7","short_pith_number":"pith:NP5QPYWF","canonical_record":{"source":{"id":"2606.23235","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-06-22T12:21:37Z","cross_cats_sorted":["math.DS","stat.ML"],"title_canon_sha256":"70c95f366bc444c3140906f2e26a3f213a302d3baf98223c1672b470f41f35c5","abstract_canon_sha256":"eaa4b69ed2eb455472c3cbabaf1493147aaff731fa9ed14c770e52476325b0c2"},"schema_version":"1.0"},"canonical_sha256":"6bfb07e2c51140c4ee862fb397da8fcfece5491c3b0e03ef0a431c0103eddb7e","source":{"kind":"arxiv","id":"2606.23235","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.23235","created_at":"2026-06-23T03:14:14Z"},{"alias_kind":"arxiv_version","alias_value":"2606.23235v1","created_at":"2026-06-23T03:14:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.23235","created_at":"2026-06-23T03:14:14Z"},{"alias_kind":"pith_short_12","alias_value":"NP5QPYWFCFAM","created_at":"2026-06-23T03:14:14Z"},{"alias_kind":"pith_short_16","alias_value":"NP5QPYWFCFAMJ3UG","created_at":"2026-06-23T03:14:14Z"},{"alias_kind":"pith_short_8","alias_value":"NP5QPYWF","created_at":"2026-06-23T03:14:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:NP5QPYWFCFAMJ3UGF6ZZPWUPZ7","target":"record","payload":{"canonical_record":{"source":{"id":"2606.23235","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-06-22T12:21:37Z","cross_cats_sorted":["math.DS","stat.ML"],"title_canon_sha256":"70c95f366bc444c3140906f2e26a3f213a302d3baf98223c1672b470f41f35c5","abstract_canon_sha256":"eaa4b69ed2eb455472c3cbabaf1493147aaff731fa9ed14c770e52476325b0c2"},"schema_version":"1.0"},"canonical_sha256":"6bfb07e2c51140c4ee862fb397da8fcfece5491c3b0e03ef0a431c0103eddb7e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-23T03:14:14.520338Z","signature_b64":"9/RNZwBSyfLIGhnQMIlaf6NDYOdvNB9p6w6xRN6JaJL22UIGkSa0fYowtQRkfE4PXo17WGggRhqW/mSFprtwDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6bfb07e2c51140c4ee862fb397da8fcfece5491c3b0e03ef0a431c0103eddb7e","last_reissued_at":"2026-06-23T03:14:14.519940Z","signature_status":"signed_v1","first_computed_at":"2026-06-23T03:14:14.519940Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.23235","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":"2026-06-23T03:14:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"j8FKSeh098EXO+A8BZ334I9e/7SjbRXu+GmOuGvVvXq719yNEFr6sIBKfSbJ6NUCWvAq87mHpEN0+6IyoQY/Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T19:13:09.190656Z"},"content_sha256":"21509625216bbbb1ed00c860050f7d1771418eab74436b7f9fcec13dfae4a580","schema_version":"1.0","event_id":"sha256:21509625216bbbb1ed00c860050f7d1771418eab74436b7f9fcec13dfae4a580"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:NP5QPYWFCFAMJ3UGF6ZZPWUPZ7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A First-Order Mean Field Control Analysis of Transformer Layers under Cross-Entropy Training","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","stat.ML"],"primary_cat":"math.OC","authors_text":"Cheng Huan, Hongwei Yuan","submitted_at":"2026-06-22T12:21:37Z","abstract_excerpt":"We study Transformer-type residual layers under cross-entropy training through a continuous-depth mean field control viewpoint. Depth is treated as time, layer parameters as controls, and the residual Transformer recursion as an explicit Euler scheme for a controlled hidden-state flow. For fixed controls, we prove an $O(\\varepsilon)$ pathwise approximation of finite-depth trajectories by the continuous flow and combine this with high-probability sampling bounds for the empirical cross-entropy risk. We formulate the limiting population problem as a first-order transport control problem for the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.23235","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.23235/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":"2026-06-23T03:14:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ih/yYmSCKlcvC6PRPx4NbkapMA5FrsLdihgSgeDI1iComNKMO9Vm3XUeylmv0OvS0JiLjMTeBJAh7Ps1GklyAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T19:13:09.191023Z"},"content_sha256":"85b6db840be037aa17bc0c6625e96ee4255769a33f205601600d4251b50c9e00","schema_version":"1.0","event_id":"sha256:85b6db840be037aa17bc0c6625e96ee4255769a33f205601600d4251b50c9e00"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NP5QPYWFCFAMJ3UGF6ZZPWUPZ7/bundle.json","state_url":"https://pith.science/pith/NP5QPYWFCFAMJ3UGF6ZZPWUPZ7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NP5QPYWFCFAMJ3UGF6ZZPWUPZ7/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":"2026-06-27T19:13:09Z","links":{"resolver":"https://pith.science/pith/NP5QPYWFCFAMJ3UGF6ZZPWUPZ7","bundle":"https://pith.science/pith/NP5QPYWFCFAMJ3UGF6ZZPWUPZ7/bundle.json","state":"https://pith.science/pith/NP5QPYWFCFAMJ3UGF6ZZPWUPZ7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NP5QPYWFCFAMJ3UGF6ZZPWUPZ7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:NP5QPYWFCFAMJ3UGF6ZZPWUPZ7","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":"eaa4b69ed2eb455472c3cbabaf1493147aaff731fa9ed14c770e52476325b0c2","cross_cats_sorted":["math.DS","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-06-22T12:21:37Z","title_canon_sha256":"70c95f366bc444c3140906f2e26a3f213a302d3baf98223c1672b470f41f35c5"},"schema_version":"1.0","source":{"id":"2606.23235","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.23235","created_at":"2026-06-23T03:14:14Z"},{"alias_kind":"arxiv_version","alias_value":"2606.23235v1","created_at":"2026-06-23T03:14:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.23235","created_at":"2026-06-23T03:14:14Z"},{"alias_kind":"pith_short_12","alias_value":"NP5QPYWFCFAM","created_at":"2026-06-23T03:14:14Z"},{"alias_kind":"pith_short_16","alias_value":"NP5QPYWFCFAMJ3UG","created_at":"2026-06-23T03:14:14Z"},{"alias_kind":"pith_short_8","alias_value":"NP5QPYWF","created_at":"2026-06-23T03:14:14Z"}],"graph_snapshots":[{"event_id":"sha256:85b6db840be037aa17bc0c6625e96ee4255769a33f205601600d4251b50c9e00","target":"graph","created_at":"2026-06-23T03:14:14Z","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/2606.23235/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study Transformer-type residual layers under cross-entropy training through a continuous-depth mean field control viewpoint. Depth is treated as time, layer parameters as controls, and the residual Transformer recursion as an explicit Euler scheme for a controlled hidden-state flow. For fixed controls, we prove an $O(\\varepsilon)$ pathwise approximation of finite-depth trajectories by the continuous flow and combine this with high-probability sampling bounds for the empirical cross-entropy risk. We formulate the limiting population problem as a first-order transport control problem for the ","authors_text":"Cheng Huan, Hongwei Yuan","cross_cats":["math.DS","stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-06-22T12:21:37Z","title":"A First-Order Mean Field Control Analysis of Transformer Layers under Cross-Entropy Training"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.23235","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":"sha256:21509625216bbbb1ed00c860050f7d1771418eab74436b7f9fcec13dfae4a580","target":"record","created_at":"2026-06-23T03:14:14Z","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":"eaa4b69ed2eb455472c3cbabaf1493147aaff731fa9ed14c770e52476325b0c2","cross_cats_sorted":["math.DS","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-06-22T12:21:37Z","title_canon_sha256":"70c95f366bc444c3140906f2e26a3f213a302d3baf98223c1672b470f41f35c5"},"schema_version":"1.0","source":{"id":"2606.23235","kind":"arxiv","version":1}},"canonical_sha256":"6bfb07e2c51140c4ee862fb397da8fcfece5491c3b0e03ef0a431c0103eddb7e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6bfb07e2c51140c4ee862fb397da8fcfece5491c3b0e03ef0a431c0103eddb7e","first_computed_at":"2026-06-23T03:14:14.519940Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-23T03:14:14.519940Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9/RNZwBSyfLIGhnQMIlaf6NDYOdvNB9p6w6xRN6JaJL22UIGkSa0fYowtQRkfE4PXo17WGggRhqW/mSFprtwDQ==","signature_status":"signed_v1","signed_at":"2026-06-23T03:14:14.520338Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.23235","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:21509625216bbbb1ed00c860050f7d1771418eab74436b7f9fcec13dfae4a580","sha256:85b6db840be037aa17bc0c6625e96ee4255769a33f205601600d4251b50c9e00"],"state_sha256":"9d7c49079a9e4ccbc70dacd6110306d27fd925051a0f00b3dac6663e61531c13"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rxgUzYRwhr56qZQV0xCsdsNGFsxdf7gcE5GkCIdHMt2LhwfHKBpfEdLMm77K4DsCDeW4otk1hiOosmJyu6hzAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T19:13:09.192974Z","bundle_sha256":"f3fe6aabf218df698c4501480fdf7d443bb6fc954174339280c79ac5aedea8af"}}