{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:BELFBVAJ53HA3JIWYFW7XIKBAF","short_pith_number":"pith:BELFBVAJ","schema_version":"1.0","canonical_sha256":"091650d409eece0da516c16dfba141014b959148588e3f427e7cb6d8e4fe176f","source":{"kind":"arxiv","id":"2402.02619","version":11},"attestation_state":"computed","paper":{"title":"Understanding Addition and Subtraction in Transformers","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["cs.CL"],"primary_cat":"cs.LG","authors_text":"Clement Neo, Fazl Barez, Philip Quirke","submitted_at":"2024-02-04T21:33:18Z","abstract_excerpt":"We use integer addition and subtraction as a controlled, exactly-solvable testbed for what can be said with confidence about the algorithm a low-loss transformer implements - logically and mechanically. We train small transformers (2-3 layers) from scratch, find the edge cases they fail (long carry and borrow cascades), and enrich the training data with them; most resulting models reach >$99.999% accuracy on 5-15 digit problems in under an hour, solving cascades such as 555555555+444444448=+1000000003. Interpretability analysis of these accurate models - not an a-priori guess - surfaces the st"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2402.02619","kind":"arxiv","version":11},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"cs.LG","submitted_at":"2024-02-04T21:33:18Z","cross_cats_sorted":["cs.CL"],"title_canon_sha256":"e44812d30b6a2d80fcebc207b905389c791044997296044a87cc7a2d31b961ef","abstract_canon_sha256":"1ee7684af9e4ffa476bcd419e6893611b81bfc016ee49914820becfb4b0dc4e1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-24T00:23:02.511358Z","signature_b64":"4hIvCI6hw2PjIp0tgpAA6QVKfz72bZfDYc0SasEPXX31Vtjq3syhqdNreOdTy4+QsMzJpaW1LVLg0Q+BlXerCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"091650d409eece0da516c16dfba141014b959148588e3f427e7cb6d8e4fe176f","last_reissued_at":"2026-07-24T00:23:02.510269Z","signature_status":"signed_v1","first_computed_at":"2026-07-24T00:23:02.510269Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Understanding Addition and Subtraction in Transformers","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["cs.CL"],"primary_cat":"cs.LG","authors_text":"Clement Neo, Fazl Barez, Philip Quirke","submitted_at":"2024-02-04T21:33:18Z","abstract_excerpt":"We use integer addition and subtraction as a controlled, exactly-solvable testbed for what can be said with confidence about the algorithm a low-loss transformer implements - logically and mechanically. We train small transformers (2-3 layers) from scratch, find the edge cases they fail (long carry and borrow cascades), and enrich the training data with them; most resulting models reach >$99.999% accuracy on 5-15 digit problems in under an hour, solving cascades such as 555555555+444444448=+1000000003. Interpretability analysis of these accurate models - not an a-priori guess - surfaces the st"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.02619","kind":"arxiv","version":11},"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/2402.02619/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2402.02619","created_at":"2026-07-24T00:23:02.510781+00:00"},{"alias_kind":"arxiv_version","alias_value":"2402.02619v11","created_at":"2026-07-24T00:23:02.510781+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2402.02619","created_at":"2026-07-24T00:23:02.510781+00:00"},{"alias_kind":"pith_short_12","alias_value":"BELFBVAJ53HA","created_at":"2026-07-24T00:23:02.510781+00:00"},{"alias_kind":"pith_short_16","alias_value":"BELFBVAJ53HA3JIW","created_at":"2026-07-24T00:23:02.510781+00:00"},{"alias_kind":"pith_short_8","alias_value":"BELFBVAJ","created_at":"2026-07-24T00:23:02.510781+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":7,"internal_anchor_count":7,"sample":[{"citing_arxiv_id":"2606.03982","citing_title":"Language Models Compare Quantities Using Number-specific and Unit-specific Heuristics","ref_index":17,"is_internal_anchor":true},{"citing_arxiv_id":"2606.03645","citing_title":"The Shape of Addition: Geometric Structures of Arithmetic in Large Language Models","ref_index":15,"is_internal_anchor":true},{"citing_arxiv_id":"2605.22488","citing_title":"Represented Is Not Computed: A Causal Test of Candidate Algorithmic Intermediates in a Transformer","ref_index":21,"is_internal_anchor":true},{"citing_arxiv_id":"2605.14323","citing_title":"Dynamic Latent Routing","ref_index":33,"is_internal_anchor":true},{"citing_arxiv_id":"2605.05115","citing_title":"Manifold Steering Reveals the Shared Geometry of Neural Network Representation and Behavior","ref_index":139,"is_internal_anchor":true},{"citing_arxiv_id":"2605.01148","citing_title":"Arithmetic in the Wild: Llama uses Base-10 Addition to Reason About Cyclic Concepts","ref_index":144,"is_internal_anchor":true},{"citing_arxiv_id":"2604.15306","citing_title":"Generalization in LLM Problem Solving: The Case of the Shortest Path","ref_index":41,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BELFBVAJ53HA3JIWYFW7XIKBAF","json":"https://pith.science/pith/BELFBVAJ53HA3JIWYFW7XIKBAF.json","graph_json":"https://pith.science/api/pith-number/BELFBVAJ53HA3JIWYFW7XIKBAF/graph.json","events_json":"https://pith.science/api/pith-number/BELFBVAJ53HA3JIWYFW7XIKBAF/events.json","paper":"https://pith.science/paper/BELFBVAJ"},"agent_actions":{"view_html":"https://pith.science/pith/BELFBVAJ53HA3JIWYFW7XIKBAF","download_json":"https://pith.science/pith/BELFBVAJ53HA3JIWYFW7XIKBAF.json","view_paper":"https://pith.science/paper/BELFBVAJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2402.02619&json=true","fetch_graph":"https://pith.science/api/pith-number/BELFBVAJ53HA3JIWYFW7XIKBAF/graph.json","fetch_events":"https://pith.science/api/pith-number/BELFBVAJ53HA3JIWYFW7XIKBAF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BELFBVAJ53HA3JIWYFW7XIKBAF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BELFBVAJ53HA3JIWYFW7XIKBAF/action/storage_attestation","attest_author":"https://pith.science/pith/BELFBVAJ53HA3JIWYFW7XIKBAF/action/author_attestation","sign_citation":"https://pith.science/pith/BELFBVAJ53HA3JIWYFW7XIKBAF/action/citation_signature","submit_replication":"https://pith.science/pith/BELFBVAJ53HA3JIWYFW7XIKBAF/action/replication_record"}},"created_at":"2026-07-24T00:23:02.510781+00:00","updated_at":"2026-07-24T00:23:02.510781+00:00"}