{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:7DZ4SGXM44VNQAOW54LC4EINTO","short_pith_number":"pith:7DZ4SGXM","canonical_record":{"source":{"id":"2607.01266","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2026-06-29T20:44:20Z","cross_cats_sorted":["cs.LG","math.FA"],"title_canon_sha256":"6daf02e5906b23515c8af2f5d673710be46660a95d370f2de2e5637f1efd322f","abstract_canon_sha256":"c2dc2edf0579083191d90093a034fa71632b4ab98364f5d1c65c26ee09e48eb8"},"schema_version":"1.0"},"canonical_sha256":"f8f3c91aece72ad801d6ef162e110d9ba556f9e4a3dbf2dc6e2b271d19e56a4e","source":{"kind":"arxiv","id":"2607.01266","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.01266","created_at":"2026-07-03T00:16:55Z"},{"alias_kind":"arxiv_version","alias_value":"2607.01266v1","created_at":"2026-07-03T00:16:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.01266","created_at":"2026-07-03T00:16:55Z"},{"alias_kind":"pith_short_12","alias_value":"7DZ4SGXM44VN","created_at":"2026-07-03T00:16:55Z"},{"alias_kind":"pith_short_16","alias_value":"7DZ4SGXM44VNQAOW","created_at":"2026-07-03T00:16:55Z"},{"alias_kind":"pith_short_8","alias_value":"7DZ4SGXM","created_at":"2026-07-03T00:16:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:7DZ4SGXM44VNQAOW54LC4EINTO","target":"record","payload":{"canonical_record":{"source":{"id":"2607.01266","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2026-06-29T20:44:20Z","cross_cats_sorted":["cs.LG","math.FA"],"title_canon_sha256":"6daf02e5906b23515c8af2f5d673710be46660a95d370f2de2e5637f1efd322f","abstract_canon_sha256":"c2dc2edf0579083191d90093a034fa71632b4ab98364f5d1c65c26ee09e48eb8"},"schema_version":"1.0"},"canonical_sha256":"f8f3c91aece72ad801d6ef162e110d9ba556f9e4a3dbf2dc6e2b271d19e56a4e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-03T00:16:55.890652Z","signature_b64":"9IV5JAJTYtxyWMpekMcsjx9e2j+yTaTewWj0jOa3y3MaI1irg3VIWkND6pjdRsvIxUo+7T6t/4/Krhid/KkHDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f8f3c91aece72ad801d6ef162e110d9ba556f9e4a3dbf2dc6e2b271d19e56a4e","last_reissued_at":"2026-07-03T00:16:55.890296Z","signature_status":"signed_v1","first_computed_at":"2026-07-03T00:16:55.890296Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.01266","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":"2026-07-03T00:16:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ARzNEaz0ZazvTbk0pPgpoe3f0JD0RlwaeQb8wYJ8pYJj1xR5cOgI9vlufCiNmTx/ljPqH2gSLxq/tMEVIqPBCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T08:06:06.801967Z"},"content_sha256":"948882afa34316951743b8b51d61d30237648647aec7497161b9a553eb2f67cd","schema_version":"1.0","event_id":"sha256:948882afa34316951743b8b51d61d30237648647aec7497161b9a553eb2f67cd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:7DZ4SGXM44VNQAOW54LC4EINTO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Fast approximation and learning of binary classification tasks in o-minimal structures using ReLU neural networks","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.LG","math.FA"],"primary_cat":"math.LO","authors_text":"Clemens Kinn, Philipp Petersen","submitted_at":"2026-06-29T20:44:20Z","abstract_excerpt":"We study binary classification problems whose decision sets are given by definable sets in o-minimal expansions of the real field. Motivated by cell decomposition of definable sets, we introduce traceable sets as a classical proxy for definable decision regions and analyze their approximation by ReLU neural networks. Under uniform bounds on the number of connected components and suitable $C^m$ extensions for the boundary functions, we prove that characteristic functions of traceable subsets of $[-1/2,1/2]^n$ can be approximated in $L^p$ to accuracy $\\varepsilon>0$ by ReLU neural networks of si"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01266","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/2607.01266/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":"2026-07-03T00:16:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PhFxMvlooxGsk8+q7tY+cvbme/o6iCFJ2iYN/9LY9pbidYo+6LTmbQ6kdhi4W7+5AiFmejTiSRzLtzNdIGNLBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T08:06:06.802354Z"},"content_sha256":"a533bb568dbd1c4556de1c624bf574d7b3b2d13cd8007481ae6ce22c95bdb922","schema_version":"1.0","event_id":"sha256:a533bb568dbd1c4556de1c624bf574d7b3b2d13cd8007481ae6ce22c95bdb922"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7DZ4SGXM44VNQAOW54LC4EINTO/bundle.json","state_url":"https://pith.science/pith/7DZ4SGXM44VNQAOW54LC4EINTO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7DZ4SGXM44VNQAOW54LC4EINTO/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":"2026-07-05T08:06:06Z","links":{"resolver":"https://pith.science/pith/7DZ4SGXM44VNQAOW54LC4EINTO","bundle":"https://pith.science/pith/7DZ4SGXM44VNQAOW54LC4EINTO/bundle.json","state":"https://pith.science/pith/7DZ4SGXM44VNQAOW54LC4EINTO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7DZ4SGXM44VNQAOW54LC4EINTO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:7DZ4SGXM44VNQAOW54LC4EINTO","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":"c2dc2edf0579083191d90093a034fa71632b4ab98364f5d1c65c26ee09e48eb8","cross_cats_sorted":["cs.LG","math.FA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2026-06-29T20:44:20Z","title_canon_sha256":"6daf02e5906b23515c8af2f5d673710be46660a95d370f2de2e5637f1efd322f"},"schema_version":"1.0","source":{"id":"2607.01266","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.01266","created_at":"2026-07-03T00:16:55Z"},{"alias_kind":"arxiv_version","alias_value":"2607.01266v1","created_at":"2026-07-03T00:16:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.01266","created_at":"2026-07-03T00:16:55Z"},{"alias_kind":"pith_short_12","alias_value":"7DZ4SGXM44VN","created_at":"2026-07-03T00:16:55Z"},{"alias_kind":"pith_short_16","alias_value":"7DZ4SGXM44VNQAOW","created_at":"2026-07-03T00:16:55Z"},{"alias_kind":"pith_short_8","alias_value":"7DZ4SGXM","created_at":"2026-07-03T00:16:55Z"}],"graph_snapshots":[{"event_id":"sha256:a533bb568dbd1c4556de1c624bf574d7b3b2d13cd8007481ae6ce22c95bdb922","target":"graph","created_at":"2026-07-03T00:16:55Z","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/2607.01266/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study binary classification problems whose decision sets are given by definable sets in o-minimal expansions of the real field. Motivated by cell decomposition of definable sets, we introduce traceable sets as a classical proxy for definable decision regions and analyze their approximation by ReLU neural networks. Under uniform bounds on the number of connected components and suitable $C^m$ extensions for the boundary functions, we prove that characteristic functions of traceable subsets of $[-1/2,1/2]^n$ can be approximated in $L^p$ to accuracy $\\varepsilon>0$ by ReLU neural networks of si","authors_text":"Clemens Kinn, Philipp Petersen","cross_cats":["cs.LG","math.FA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2026-06-29T20:44:20Z","title":"Fast approximation and learning of binary classification tasks in o-minimal structures using ReLU neural networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01266","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":"sha256:948882afa34316951743b8b51d61d30237648647aec7497161b9a553eb2f67cd","target":"record","created_at":"2026-07-03T00:16:55Z","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":"c2dc2edf0579083191d90093a034fa71632b4ab98364f5d1c65c26ee09e48eb8","cross_cats_sorted":["cs.LG","math.FA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2026-06-29T20:44:20Z","title_canon_sha256":"6daf02e5906b23515c8af2f5d673710be46660a95d370f2de2e5637f1efd322f"},"schema_version":"1.0","source":{"id":"2607.01266","kind":"arxiv","version":1}},"canonical_sha256":"f8f3c91aece72ad801d6ef162e110d9ba556f9e4a3dbf2dc6e2b271d19e56a4e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f8f3c91aece72ad801d6ef162e110d9ba556f9e4a3dbf2dc6e2b271d19e56a4e","first_computed_at":"2026-07-03T00:16:55.890296Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-03T00:16:55.890296Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9IV5JAJTYtxyWMpekMcsjx9e2j+yTaTewWj0jOa3y3MaI1irg3VIWkND6pjdRsvIxUo+7T6t/4/Krhid/KkHDA==","signature_status":"signed_v1","signed_at":"2026-07-03T00:16:55.890652Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.01266","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:948882afa34316951743b8b51d61d30237648647aec7497161b9a553eb2f67cd","sha256:a533bb568dbd1c4556de1c624bf574d7b3b2d13cd8007481ae6ce22c95bdb922"],"state_sha256":"70648e40d9ba1cb421183a0fe289afa1bdecf036eb621ba27ac6e4dd0189022a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"toVWkgJTW5B+yBehPiMEI/AA80yJ9ofZZUj9UDzSCntPe2hh96quKzhIkRifSif7TVKgzEu8UFhfOJrSWEcFAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-05T08:06:06.804333Z","bundle_sha256":"041184b34ff9fec6fbfe3e61e9ea40b823b51742c88b5472755a87c7080c1d97"}}