{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:36L3KIOKQFOSSFBFP22KVIZATN","short_pith_number":"pith:36L3KIOK","canonical_record":{"source":{"id":"2607.01311","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-07-01T17:26:19Z","cross_cats_sorted":["stat.ML"],"title_canon_sha256":"4bfb1c42aa6a654f568405dadfcf2fdcbae9679dfc7829b5f986a9d3bdf1ac3c","abstract_canon_sha256":"613a0fdc23d802233e8f13f49721b2754690556befd72c2f547a2fdea4e71b93"},"schema_version":"1.0"},"canonical_sha256":"df97b521ca815d2914257eb4aaa3209b70793c77cd2925b5379c81a32e7bea40","source":{"kind":"arxiv","id":"2607.01311","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.01311","created_at":"2026-07-03T00:16:57Z"},{"alias_kind":"arxiv_version","alias_value":"2607.01311v1","created_at":"2026-07-03T00:16:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.01311","created_at":"2026-07-03T00:16:57Z"},{"alias_kind":"pith_short_12","alias_value":"36L3KIOKQFOS","created_at":"2026-07-03T00:16:57Z"},{"alias_kind":"pith_short_16","alias_value":"36L3KIOKQFOSSFBF","created_at":"2026-07-03T00:16:57Z"},{"alias_kind":"pith_short_8","alias_value":"36L3KIOK","created_at":"2026-07-03T00:16:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:36L3KIOKQFOSSFBFP22KVIZATN","target":"record","payload":{"canonical_record":{"source":{"id":"2607.01311","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-07-01T17:26:19Z","cross_cats_sorted":["stat.ML"],"title_canon_sha256":"4bfb1c42aa6a654f568405dadfcf2fdcbae9679dfc7829b5f986a9d3bdf1ac3c","abstract_canon_sha256":"613a0fdc23d802233e8f13f49721b2754690556befd72c2f547a2fdea4e71b93"},"schema_version":"1.0"},"canonical_sha256":"df97b521ca815d2914257eb4aaa3209b70793c77cd2925b5379c81a32e7bea40","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-03T00:16:57.037202Z","signature_b64":"5P/gHc+vRJ3MpzJ92Vk2hOeWycGuZWC67wgIj9gY3OBob9DaHqalPDfhAtBRoPcrxUMc/aDTy7l82JU+/ctCAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"df97b521ca815d2914257eb4aaa3209b70793c77cd2925b5379c81a32e7bea40","last_reissued_at":"2026-07-03T00:16:57.036859Z","signature_status":"signed_v1","first_computed_at":"2026-07-03T00:16:57.036859Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.01311","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":"2026-07-03T00:16:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ptC+rMNaWcfsvT/oEP7AGDBc46jZbIsWE2qcvNpi1GATNHet6XHF6mLttG0TIaVKdvLpiFr2oEzK/cwDjEMNBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T14:47:25.564052Z"},"content_sha256":"afa6548561b7f5330a24f0395ea51eecc7696c6439896e4efdd34b3100d0ad0f","schema_version":"1.0","event_id":"sha256:afa6548561b7f5330a24f0395ea51eecc7696c6439896e4efdd34b3100d0ad0f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:36L3KIOKQFOSSFBFP22KVIZATN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"From Approximation to Emergence: A Theory of Deep Learning","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.ML"],"primary_cat":"cs.LG","authors_text":"Zhilin Zhao","submitted_at":"2026-07-01T17:26:19Z","abstract_excerpt":"Deep learning has outgrown any single mathematical explanation. From Approximation to Emergence develops a unified, proof-oriented account of modern deep learning theory, tracing a path from the classical foundations of approximation, optimization, and generalization to the contemporary mechanisms of overparameterization, robustness, generative modeling, transformers, in-context learning, scaling laws, interpretability, alignment, and emergence. Rather than presenting isolated results, the book organizes a broad literature into a coherent research narrative: each theory is examined through the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01311","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/2607.01311/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":"2026-07-03T00:16:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+GNUlvoVhxpGLAAqaAWYv+tV/OQuIPmqOKIYOzgUQLAcU8bsKROO49NuNb19rQ7oS57XBnwstWa3ouh7ok2oDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T14:47:25.564437Z"},"content_sha256":"e7bd0ceb3dfa67e7f101504beb00411a466bd50925a1edf10a21de37ccca427b","schema_version":"1.0","event_id":"sha256:e7bd0ceb3dfa67e7f101504beb00411a466bd50925a1edf10a21de37ccca427b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/36L3KIOKQFOSSFBFP22KVIZATN/bundle.json","state_url":"https://pith.science/pith/36L3KIOKQFOSSFBFP22KVIZATN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/36L3KIOKQFOSSFBFP22KVIZATN/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":"2026-07-06T14:47:25Z","links":{"resolver":"https://pith.science/pith/36L3KIOKQFOSSFBFP22KVIZATN","bundle":"https://pith.science/pith/36L3KIOKQFOSSFBFP22KVIZATN/bundle.json","state":"https://pith.science/pith/36L3KIOKQFOSSFBFP22KVIZATN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/36L3KIOKQFOSSFBFP22KVIZATN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:36L3KIOKQFOSSFBFP22KVIZATN","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":"613a0fdc23d802233e8f13f49721b2754690556befd72c2f547a2fdea4e71b93","cross_cats_sorted":["stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-07-01T17:26:19Z","title_canon_sha256":"4bfb1c42aa6a654f568405dadfcf2fdcbae9679dfc7829b5f986a9d3bdf1ac3c"},"schema_version":"1.0","source":{"id":"2607.01311","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.01311","created_at":"2026-07-03T00:16:57Z"},{"alias_kind":"arxiv_version","alias_value":"2607.01311v1","created_at":"2026-07-03T00:16:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.01311","created_at":"2026-07-03T00:16:57Z"},{"alias_kind":"pith_short_12","alias_value":"36L3KIOKQFOS","created_at":"2026-07-03T00:16:57Z"},{"alias_kind":"pith_short_16","alias_value":"36L3KIOKQFOSSFBF","created_at":"2026-07-03T00:16:57Z"},{"alias_kind":"pith_short_8","alias_value":"36L3KIOK","created_at":"2026-07-03T00:16:57Z"}],"graph_snapshots":[{"event_id":"sha256:e7bd0ceb3dfa67e7f101504beb00411a466bd50925a1edf10a21de37ccca427b","target":"graph","created_at":"2026-07-03T00:16:57Z","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/2607.01311/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Deep learning has outgrown any single mathematical explanation. From Approximation to Emergence develops a unified, proof-oriented account of modern deep learning theory, tracing a path from the classical foundations of approximation, optimization, and generalization to the contemporary mechanisms of overparameterization, robustness, generative modeling, transformers, in-context learning, scaling laws, interpretability, alignment, and emergence. Rather than presenting isolated results, the book organizes a broad literature into a coherent research narrative: each theory is examined through the","authors_text":"Zhilin Zhao","cross_cats":["stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-07-01T17:26:19Z","title":"From Approximation to Emergence: A Theory of Deep Learning"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01311","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:afa6548561b7f5330a24f0395ea51eecc7696c6439896e4efdd34b3100d0ad0f","target":"record","created_at":"2026-07-03T00:16:57Z","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":"613a0fdc23d802233e8f13f49721b2754690556befd72c2f547a2fdea4e71b93","cross_cats_sorted":["stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-07-01T17:26:19Z","title_canon_sha256":"4bfb1c42aa6a654f568405dadfcf2fdcbae9679dfc7829b5f986a9d3bdf1ac3c"},"schema_version":"1.0","source":{"id":"2607.01311","kind":"arxiv","version":1}},"canonical_sha256":"df97b521ca815d2914257eb4aaa3209b70793c77cd2925b5379c81a32e7bea40","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"df97b521ca815d2914257eb4aaa3209b70793c77cd2925b5379c81a32e7bea40","first_computed_at":"2026-07-03T00:16:57.036859Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-03T00:16:57.036859Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5P/gHc+vRJ3MpzJ92Vk2hOeWycGuZWC67wgIj9gY3OBob9DaHqalPDfhAtBRoPcrxUMc/aDTy7l82JU+/ctCAw==","signature_status":"signed_v1","signed_at":"2026-07-03T00:16:57.037202Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.01311","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:afa6548561b7f5330a24f0395ea51eecc7696c6439896e4efdd34b3100d0ad0f","sha256:e7bd0ceb3dfa67e7f101504beb00411a466bd50925a1edf10a21de37ccca427b"],"state_sha256":"ac9704fee2ce6ffa97fc533eddecc0006275eae2e32da405f39f82d5f3cc6d41"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7ZGZjT+gzGoWtjzZ3TtuK27THtXC3suhZwKE+liv/2KIG+MzrbJIPXKXS1AVEm+aCwuXAXZFsdB5sV/MeJZbAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-06T14:47:25.566441Z","bundle_sha256":"a8905484ec85917563629a8285859a42789449d9acbe57fce08032ff8a9315e9"}}