{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:ENI3YBMCCXVWI5QO7P4LTVCXU7","short_pith_number":"pith:ENI3YBMC","canonical_record":{"source":{"id":"2604.14037","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-04-15T16:18:44Z","cross_cats_sorted":["math.AG","math.CO"],"title_canon_sha256":"0d93ca45237b03a9851121d7f713a8571a4b2d3fd4200597bc3fa977536df7b9","abstract_canon_sha256":"791e5f15db5279a21b921aaff0f7c71161ff640d0b0e1ce8251efaff1fc37dda"},"schema_version":"1.0"},"canonical_sha256":"2351bc058215eb64760efbf8b9d457a7cab0e08db66ce3e90551e723667c18d9","source":{"kind":"arxiv","id":"2604.14037","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.14037","created_at":"2026-07-07T02:17:24Z"},{"alias_kind":"arxiv_version","alias_value":"2604.14037v2","created_at":"2026-07-07T02:17:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.14037","created_at":"2026-07-07T02:17:24Z"},{"alias_kind":"pith_short_12","alias_value":"ENI3YBMCCXVW","created_at":"2026-07-07T02:17:24Z"},{"alias_kind":"pith_short_16","alias_value":"ENI3YBMCCXVWI5QO","created_at":"2026-07-07T02:17:24Z"},{"alias_kind":"pith_short_8","alias_value":"ENI3YBMC","created_at":"2026-07-07T02:17:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:ENI3YBMCCXVWI5QO7P4LTVCXU7","target":"record","payload":{"canonical_record":{"source":{"id":"2604.14037","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-04-15T16:18:44Z","cross_cats_sorted":["math.AG","math.CO"],"title_canon_sha256":"0d93ca45237b03a9851121d7f713a8571a4b2d3fd4200597bc3fa977536df7b9","abstract_canon_sha256":"791e5f15db5279a21b921aaff0f7c71161ff640d0b0e1ce8251efaff1fc37dda"},"schema_version":"1.0"},"canonical_sha256":"2351bc058215eb64760efbf8b9d457a7cab0e08db66ce3e90551e723667c18d9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-07T02:17:24.379669Z","signature_b64":"WpZ3SRL+rxDhFLprXjjIZOkZPOCci8J0gvhH7eIpVIZySIqOtYqxDw8Tzkm6dG0DGRlMNU74kq0VqxDuf5SkDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2351bc058215eb64760efbf8b9d457a7cab0e08db66ce3e90551e723667c18d9","last_reissued_at":"2026-07-07T02:17:24.378832Z","signature_status":"signed_v1","first_computed_at":"2026-07-07T02:17:24.378832Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2604.14037","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-07T02:17:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"71HhZ+bbVMxUUgeCvzUhccbGmgC/AJVthLU7eCJ/IJMJLFvPaiNIH0I70GIwsSWsPOmpa/ZhWMgMS+VL1E6+BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T14:17:05.407602Z"},"content_sha256":"ca43db793b2b5bb531379abc5045aabe9acbe195ed278d0e6a19dabe28354159","schema_version":"1.0","event_id":"sha256:ca43db793b2b5bb531379abc5045aabe9acbe195ed278d0e6a19dabe28354159"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:ENI3YBMCCXVWI5QO7P4LTVCXU7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Complete Symmetry Classification of Shallow ReLU Networks","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Exploiting the non-differentiability of ReLU yields a complete classification of symmetries in shallow ReLU networks.","cross_cats":["math.AG","math.CO"],"primary_cat":"cs.LG","authors_text":"Pranavkrishnan Ramakrishnan","submitted_at":"2026-04-15T16:18:44Z","abstract_excerpt":"Parameter space is not function space for neural network architectures. This fact, investigated as early as the 1990s under terms such as ``reverse engineering,\" or ``parameter identifiability\", has led to the natural question of parameter space symmetries\\textemdash the study of distinct parameters in neural architectures which realize the same function. Indeed, the quotient space obtained by identifying parameters giving rise to the same function, called the \\textit{neuromanifold}, has been shown in some cases to have rich geometric properties, impacting optimization dynamics. Thus far, tech"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Here, in contrast, we exploit the non-differentiability of the ReLU activation to provide a complete classification of the symmetries in the shallow case.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the non-differentiability of ReLU is by itself sufficient to obtain a complete and exhaustive classification of all symmetries for arbitrary shallow ReLU networks without further restrictions on width, weight distributions, or additional regularity conditions.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A complete classification of symmetries in shallow ReLU networks is achieved by using the non-differentiability of ReLU.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Exploiting the non-differentiability of ReLU yields a complete classification of symmetries in shallow ReLU networks.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"5aec6c035e93c0b6eea9a28df76fb7fcba52662888976bc54424c6e4344a5040"},"source":{"id":"2604.14037","kind":"arxiv","version":2},"verdict":{"id":"3dc572a0-276b-41aa-a8b0-18b4093aa525","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T13:43:55.722623Z","strongest_claim":"Here, in contrast, we exploit the non-differentiability of the ReLU activation to provide a complete classification of the symmetries in the shallow case.","one_line_summary":"A complete classification of symmetries in shallow ReLU networks is achieved by using the non-differentiability of ReLU.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the non-differentiability of ReLU is by itself sufficient to obtain a complete and exhaustive classification of all symmetries for arbitrary shallow ReLU networks without further restrictions on width, weight distributions, or additional regularity conditions.","pith_extraction_headline":"Exploiting the non-differentiability of ReLU yields a complete classification of symmetries in shallow ReLU networks."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.14037/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":"3dc572a0-276b-41aa-a8b0-18b4093aa525"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-07T02:17:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MVSVb7jObgWFGOCqUFt5vy3ToaQZNHdJrinA+qJ99Hx18z/5hPCpdi1pEsBD7perEdoE2FAPA7IsKUG3oZDdAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T14:17:05.409087Z"},"content_sha256":"21020469bebf55fcd0e1907c8d65fd3f2b0bafb7ee01fd570176a09c22a28a61","schema_version":"1.0","event_id":"sha256:21020469bebf55fcd0e1907c8d65fd3f2b0bafb7ee01fd570176a09c22a28a61"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ENI3YBMCCXVWI5QO7P4LTVCXU7/bundle.json","state_url":"https://pith.science/pith/ENI3YBMCCXVWI5QO7P4LTVCXU7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ENI3YBMCCXVWI5QO7P4LTVCXU7/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":"2026-08-20T14:17:05Z","links":{"resolver":"https://pith.science/pith/ENI3YBMCCXVWI5QO7P4LTVCXU7","bundle":"https://pith.science/pith/ENI3YBMCCXVWI5QO7P4LTVCXU7/bundle.json","state":"https://pith.science/pith/ENI3YBMCCXVWI5QO7P4LTVCXU7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ENI3YBMCCXVWI5QO7P4LTVCXU7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:ENI3YBMCCXVWI5QO7P4LTVCXU7","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":"791e5f15db5279a21b921aaff0f7c71161ff640d0b0e1ce8251efaff1fc37dda","cross_cats_sorted":["math.AG","math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-04-15T16:18:44Z","title_canon_sha256":"0d93ca45237b03a9851121d7f713a8571a4b2d3fd4200597bc3fa977536df7b9"},"schema_version":"1.0","source":{"id":"2604.14037","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.14037","created_at":"2026-07-07T02:17:24Z"},{"alias_kind":"arxiv_version","alias_value":"2604.14037v2","created_at":"2026-07-07T02:17:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.14037","created_at":"2026-07-07T02:17:24Z"},{"alias_kind":"pith_short_12","alias_value":"ENI3YBMCCXVW","created_at":"2026-07-07T02:17:24Z"},{"alias_kind":"pith_short_16","alias_value":"ENI3YBMCCXVWI5QO","created_at":"2026-07-07T02:17:24Z"},{"alias_kind":"pith_short_8","alias_value":"ENI3YBMC","created_at":"2026-07-07T02:17:24Z"}],"graph_snapshots":[{"event_id":"sha256:21020469bebf55fcd0e1907c8d65fd3f2b0bafb7ee01fd570176a09c22a28a61","target":"graph","created_at":"2026-07-07T02:17:24Z","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":"Here, in contrast, we exploit the non-differentiability of the ReLU activation to provide a complete classification of the symmetries in the shallow case."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"That the non-differentiability of ReLU is by itself sufficient to obtain a complete and exhaustive classification of all symmetries for arbitrary shallow ReLU networks without further restrictions on width, weight distributions, or additional regularity conditions."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"A complete classification of symmetries in shallow ReLU networks is achieved by using the non-differentiability of ReLU."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Exploiting the non-differentiability of ReLU yields a complete classification of symmetries in shallow ReLU networks."}],"snapshot_sha256":"5aec6c035e93c0b6eea9a28df76fb7fcba52662888976bc54424c6e4344a5040"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2604.14037/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Parameter space is not function space for neural network architectures. This fact, investigated as early as the 1990s under terms such as ``reverse engineering,\" or ``parameter identifiability\", has led to the natural question of parameter space symmetries\\textemdash the study of distinct parameters in neural architectures which realize the same function. Indeed, the quotient space obtained by identifying parameters giving rise to the same function, called the \\textit{neuromanifold}, has been shown in some cases to have rich geometric properties, impacting optimization dynamics. Thus far, tech","authors_text":"Pranavkrishnan Ramakrishnan","cross_cats":["math.AG","math.CO"],"headline":"Exploiting the non-differentiability of ReLU yields a complete classification of symmetries in shallow ReLU networks.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-04-15T16:18:44Z","title":"A Complete Symmetry Classification of Shallow ReLU Networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.14037","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-10T13:43:55.722623Z","id":"3dc572a0-276b-41aa-a8b0-18b4093aa525","model_set":{"reader":"grok-4.3"},"one_line_summary":"A complete classification of symmetries in shallow ReLU networks is achieved by using the non-differentiability of ReLU.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Exploiting the non-differentiability of ReLU yields a complete classification of symmetries in shallow ReLU networks.","strongest_claim":"Here, in contrast, we exploit the non-differentiability of the ReLU activation to provide a complete classification of the symmetries in the shallow case.","weakest_assumption":"That the non-differentiability of ReLU is by itself sufficient to obtain a complete and exhaustive classification of all symmetries for arbitrary shallow ReLU networks without further restrictions on width, weight distributions, or additional regularity conditions."}},"verdict_id":"3dc572a0-276b-41aa-a8b0-18b4093aa525"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:ca43db793b2b5bb531379abc5045aabe9acbe195ed278d0e6a19dabe28354159","target":"record","created_at":"2026-07-07T02:17:24Z","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":"791e5f15db5279a21b921aaff0f7c71161ff640d0b0e1ce8251efaff1fc37dda","cross_cats_sorted":["math.AG","math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-04-15T16:18:44Z","title_canon_sha256":"0d93ca45237b03a9851121d7f713a8571a4b2d3fd4200597bc3fa977536df7b9"},"schema_version":"1.0","source":{"id":"2604.14037","kind":"arxiv","version":2}},"canonical_sha256":"2351bc058215eb64760efbf8b9d457a7cab0e08db66ce3e90551e723667c18d9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2351bc058215eb64760efbf8b9d457a7cab0e08db66ce3e90551e723667c18d9","first_computed_at":"2026-07-07T02:17:24.378832Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-07T02:17:24.378832Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WpZ3SRL+rxDhFLprXjjIZOkZPOCci8J0gvhH7eIpVIZySIqOtYqxDw8Tzkm6dG0DGRlMNU74kq0VqxDuf5SkDw==","signature_status":"signed_v1","signed_at":"2026-07-07T02:17:24.379669Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.14037","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ca43db793b2b5bb531379abc5045aabe9acbe195ed278d0e6a19dabe28354159","sha256:21020469bebf55fcd0e1907c8d65fd3f2b0bafb7ee01fd570176a09c22a28a61"],"state_sha256":"04fbb2957645652437cd4cd52a9ffe89deaa56de81e4d6ff43023bf335fdb587"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mPGXWsdsxKO3FC3LSc4/tv2WH8Hv38pe1IPSR1ph3CJgea04lOsw33tUUVhLwQx6raIiVH+/t12OEnADRFX3BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-20T14:17:05.433870Z","bundle_sha256":"20f9d8b147ccd117690103c8be55b5d40de0df3ac6074f2f2c8c35821bd3a75f"}}