{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:XUZYZ6WJ2IQCODT4FRQFXGTQR4","short_pith_number":"pith:XUZYZ6WJ","canonical_record":{"source":{"id":"2209.04882","kind":"arxiv","version":5},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cond-mat.dis-nn","submitted_at":"2022-09-11T14:44:26Z","cross_cats_sorted":[],"title_canon_sha256":"d1f9b3e6adef6d8605b533e813efd934717469af03d26c5b790773bb4833a638","abstract_canon_sha256":"8330213db51730afcaf134bbdf675213dfe888721262f345d4951c933a8efef9"},"schema_version":"1.0"},"canonical_sha256":"bd338cfac9d220270e7c2c605b9a708f218263fa56baa0ed144720b16e3096a7","source":{"kind":"arxiv","id":"2209.04882","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2209.04882","created_at":"2026-07-05T08:37:03Z"},{"alias_kind":"arxiv_version","alias_value":"2209.04882v5","created_at":"2026-07-05T08:37:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2209.04882","created_at":"2026-07-05T08:37:03Z"},{"alias_kind":"pith_short_12","alias_value":"XUZYZ6WJ2IQC","created_at":"2026-07-05T08:37:03Z"},{"alias_kind":"pith_short_16","alias_value":"XUZYZ6WJ2IQCODT4","created_at":"2026-07-05T08:37:03Z"},{"alias_kind":"pith_short_8","alias_value":"XUZYZ6WJ","created_at":"2026-07-05T08:37:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:XUZYZ6WJ2IQCODT4FRQFXGTQR4","target":"record","payload":{"canonical_record":{"source":{"id":"2209.04882","kind":"arxiv","version":5},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cond-mat.dis-nn","submitted_at":"2022-09-11T14:44:26Z","cross_cats_sorted":[],"title_canon_sha256":"d1f9b3e6adef6d8605b533e813efd934717469af03d26c5b790773bb4833a638","abstract_canon_sha256":"8330213db51730afcaf134bbdf675213dfe888721262f345d4951c933a8efef9"},"schema_version":"1.0"},"canonical_sha256":"bd338cfac9d220270e7c2c605b9a708f218263fa56baa0ed144720b16e3096a7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:37:03.371659Z","signature_b64":"ZqzgtEyJ7sceZQ6en28V50vo+bY3YO9Md2QvoIUqc/W1efJl5kK0c94MGIIECtYhGLTfpwxva+k91cV/Lq9pBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bd338cfac9d220270e7c2c605b9a708f218263fa56baa0ed144720b16e3096a7","last_reissued_at":"2026-07-05T08:37:03.371224Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:37:03.371224Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2209.04882","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T08:37:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tRjZOVGfMyN7hf8zpVEfnzdp8zQr8k/ZLduDxwETTvyP1l9IBsNXUy37SVQapJ6F9DT3P3vRMwyoSywEE6erCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-24T02:39:24.028482Z"},"content_sha256":"84b81b2eaa6a13d443e547d1b87890a0f60a63e33e5ce6dfe83027f9a617bd71","schema_version":"1.0","event_id":"sha256:84b81b2eaa6a13d443e547d1b87890a0f60a63e33e5ce6dfe83027f9a617bd71"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:XUZYZ6WJ2IQCODT4FRQFXGTQR4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A statistical mechanics framework for Bayesian deep neural networks beyond the infinite-width limit","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cond-mat.dis-nn","authors_text":"F. Ginelli, M. Gherardi, M. Pastore, P. Rotondo, R. Pacelli, S. Ariosto","submitted_at":"2022-09-11T14:44:26Z","abstract_excerpt":"Despite the practical success of deep neural networks, a comprehensive theoretical framework that can predict practically relevant scores, such as the test accuracy, from knowledge of the training data is currently lacking. Huge simplifications arise in the infinite-width limit, where the number of units $N_\\ell$ in each hidden layer ($\\ell=1,\\dots, L$, being $L$ the depth of the network) far exceeds the number $P$ of training examples. This idealisation, however, blatantly departs from the reality of deep learning practice. Here, we use the toolset of statistical mechanics to overcome these l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2209.04882","kind":"arxiv","version":5},"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/2209.04882/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":"2026-07-05T08:37:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aAh6VuESzB2Hb4XkChpj3orZNx3ElSGjBpNKenHUeQjiOb1FLQPVdNgUfKROHQdne5ACTbERtnI9yziGdeENBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-24T02:39:24.029118Z"},"content_sha256":"7c2bf9d41b1bf3a569042487e44d7b9534c00fc2e556de748de9e8e232569096","schema_version":"1.0","event_id":"sha256:7c2bf9d41b1bf3a569042487e44d7b9534c00fc2e556de748de9e8e232569096"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XUZYZ6WJ2IQCODT4FRQFXGTQR4/bundle.json","state_url":"https://pith.science/pith/XUZYZ6WJ2IQCODT4FRQFXGTQR4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XUZYZ6WJ2IQCODT4FRQFXGTQR4/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":"2026-08-24T02:39:24Z","links":{"resolver":"https://pith.science/pith/XUZYZ6WJ2IQCODT4FRQFXGTQR4","bundle":"https://pith.science/pith/XUZYZ6WJ2IQCODT4FRQFXGTQR4/bundle.json","state":"https://pith.science/pith/XUZYZ6WJ2IQCODT4FRQFXGTQR4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XUZYZ6WJ2IQCODT4FRQFXGTQR4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:XUZYZ6WJ2IQCODT4FRQFXGTQR4","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":"8330213db51730afcaf134bbdf675213dfe888721262f345d4951c933a8efef9","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cond-mat.dis-nn","submitted_at":"2022-09-11T14:44:26Z","title_canon_sha256":"d1f9b3e6adef6d8605b533e813efd934717469af03d26c5b790773bb4833a638"},"schema_version":"1.0","source":{"id":"2209.04882","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2209.04882","created_at":"2026-07-05T08:37:03Z"},{"alias_kind":"arxiv_version","alias_value":"2209.04882v5","created_at":"2026-07-05T08:37:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2209.04882","created_at":"2026-07-05T08:37:03Z"},{"alias_kind":"pith_short_12","alias_value":"XUZYZ6WJ2IQC","created_at":"2026-07-05T08:37:03Z"},{"alias_kind":"pith_short_16","alias_value":"XUZYZ6WJ2IQCODT4","created_at":"2026-07-05T08:37:03Z"},{"alias_kind":"pith_short_8","alias_value":"XUZYZ6WJ","created_at":"2026-07-05T08:37:03Z"}],"graph_snapshots":[{"event_id":"sha256:7c2bf9d41b1bf3a569042487e44d7b9534c00fc2e556de748de9e8e232569096","target":"graph","created_at":"2026-07-05T08:37:03Z","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/2209.04882/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Despite the practical success of deep neural networks, a comprehensive theoretical framework that can predict practically relevant scores, such as the test accuracy, from knowledge of the training data is currently lacking. Huge simplifications arise in the infinite-width limit, where the number of units $N_\\ell$ in each hidden layer ($\\ell=1,\\dots, L$, being $L$ the depth of the network) far exceeds the number $P$ of training examples. This idealisation, however, blatantly departs from the reality of deep learning practice. Here, we use the toolset of statistical mechanics to overcome these l","authors_text":"F. Ginelli, M. Gherardi, M. Pastore, P. Rotondo, R. Pacelli, S. Ariosto","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cond-mat.dis-nn","submitted_at":"2022-09-11T14:44:26Z","title":"A statistical mechanics framework for Bayesian deep neural networks beyond the infinite-width limit"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2209.04882","kind":"arxiv","version":5},"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":"sha256:84b81b2eaa6a13d443e547d1b87890a0f60a63e33e5ce6dfe83027f9a617bd71","target":"record","created_at":"2026-07-05T08:37:03Z","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":"8330213db51730afcaf134bbdf675213dfe888721262f345d4951c933a8efef9","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cond-mat.dis-nn","submitted_at":"2022-09-11T14:44:26Z","title_canon_sha256":"d1f9b3e6adef6d8605b533e813efd934717469af03d26c5b790773bb4833a638"},"schema_version":"1.0","source":{"id":"2209.04882","kind":"arxiv","version":5}},"canonical_sha256":"bd338cfac9d220270e7c2c605b9a708f218263fa56baa0ed144720b16e3096a7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bd338cfac9d220270e7c2c605b9a708f218263fa56baa0ed144720b16e3096a7","first_computed_at":"2026-07-05T08:37:03.371224Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:37:03.371224Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZqzgtEyJ7sceZQ6en28V50vo+bY3YO9Md2QvoIUqc/W1efJl5kK0c94MGIIECtYhGLTfpwxva+k91cV/Lq9pBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T08:37:03.371659Z","signed_message":"canonical_sha256_bytes"},"source_id":"2209.04882","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:84b81b2eaa6a13d443e547d1b87890a0f60a63e33e5ce6dfe83027f9a617bd71","sha256:7c2bf9d41b1bf3a569042487e44d7b9534c00fc2e556de748de9e8e232569096"],"state_sha256":"c3b514f82ee1a00800492fd1f3abed9c947d9b3143bd0499b9042f4cc6210558"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lMj/ETcGM910zGnWt0jK8cwWD+I5FCjh7iefBuSvrZvHxb7SHUQtcxaVu3nx/tUfT6ztG1Dig9c0zyKYbPvxBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-24T02:39:24.036761Z","bundle_sha256":"3de81a391b5d6e2dab50dddc8a95235c45fb24421692ccc9b7928e9ff98dc76e"}}