{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:HMGHJP35Z6XTSX6IBSFJS3XL7S","short_pith_number":"pith:HMGHJP35","schema_version":"1.0","canonical_sha256":"3b0c74bf7dcfaf395fc80c8a996eebfca5e3011bd8776058ec47b506b4aa1341","source":{"kind":"arxiv","id":"2502.00873","version":1},"attestation_state":"computed","paper":{"title":"Language Models Use Trigonometry to Do Addition","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.CL","cs.LG"],"primary_cat":"cs.AI","authors_text":"Max Tegmark, Subhash Kantamneni","submitted_at":"2025-02-02T18:55:26Z","abstract_excerpt":"Mathematical reasoning is an increasingly important indicator of large language model (LLM) capabilities, yet we lack understanding of how LLMs process even simple mathematical tasks. To address this, we reverse engineer how three mid-sized LLMs compute addition. We first discover that numbers are represented in these LLMs as a generalized helix, which is strongly causally implicated for the tasks of addition and subtraction, and is also causally relevant for integer division, multiplication, and modular arithmetic. We then propose that LLMs compute addition by manipulating this generalized he"},"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":"2502.00873","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.AI","submitted_at":"2025-02-02T18:55:26Z","cross_cats_sorted":["cs.CL","cs.LG"],"title_canon_sha256":"f5f72a8de5340181ebcc687a58fe9b96ceb8b07ce2852fc59fc995c9f1a88dfb","abstract_canon_sha256":"25d80e4d817ab9e355e72e9510affc477b324f361de0b624218726fac41f2324"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:08:39.436086Z","signature_b64":"OAcItP/tIQ+7jXHlpZKUtKuPDAGqpXRvdXf8q7f8wtSMRYljdGl58Eqt8x+cAH4FcVgj2U/T1zqcWLSDDtR/Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3b0c74bf7dcfaf395fc80c8a996eebfca5e3011bd8776058ec47b506b4aa1341","last_reissued_at":"2026-07-05T10:08:39.435630Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:08:39.435630Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Language Models Use Trigonometry to Do Addition","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.CL","cs.LG"],"primary_cat":"cs.AI","authors_text":"Max Tegmark, Subhash Kantamneni","submitted_at":"2025-02-02T18:55:26Z","abstract_excerpt":"Mathematical reasoning is an increasingly important indicator of large language model (LLM) capabilities, yet we lack understanding of how LLMs process even simple mathematical tasks. To address this, we reverse engineer how three mid-sized LLMs compute addition. We first discover that numbers are represented in these LLMs as a generalized helix, which is strongly causally implicated for the tasks of addition and subtraction, and is also causally relevant for integer division, multiplication, and modular arithmetic. We then propose that LLMs compute addition by manipulating this generalized he"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.00873","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/2502.00873/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":"2502.00873","created_at":"2026-07-05T10:08:39.435688+00:00"},{"alias_kind":"arxiv_version","alias_value":"2502.00873v1","created_at":"2026-07-05T10:08:39.435688+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2502.00873","created_at":"2026-07-05T10:08:39.435688+00:00"},{"alias_kind":"pith_short_12","alias_value":"HMGHJP35Z6XT","created_at":"2026-07-05T10:08:39.435688+00:00"},{"alias_kind":"pith_short_16","alias_value":"HMGHJP35Z6XTSX6I","created_at":"2026-07-05T10:08:39.435688+00:00"},{"alias_kind":"pith_short_8","alias_value":"HMGHJP35","created_at":"2026-07-05T10:08:39.435688+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":10,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.25234","citing_title":"Structuring Sparsity: Block-Sparse Featurizers Capture Visual Concept Manifolds","ref_index":65,"is_internal_anchor":false},{"citing_arxiv_id":"2606.09607","citing_title":"Closure-Validated Circuit Discovery in Attention Heads: Co-activation Proposes, Ablation Disposes","ref_index":6,"is_internal_anchor":false},{"citing_arxiv_id":"2606.03330","citing_title":"FLIPS: Instance-Fingerprinting for LLMs via Pseudo-random Sequences","ref_index":39,"is_internal_anchor":false},{"citing_arxiv_id":"2606.05194","citing_title":"Temporal Preference Concepts and their Functions in a Large Language Model","ref_index":56,"is_internal_anchor":false},{"citing_arxiv_id":"2605.24942","citing_title":"Riemannian-Manifold Steering: Geometry-Aware Generative Autoencoders for Label-Free Steering","ref_index":5,"is_internal_anchor":false},{"citing_arxiv_id":"2606.29522","citing_title":"Do Models Read What They Write? Causal Registers in Scratchpad Reasoning","ref_index":3,"is_internal_anchor":false},{"citing_arxiv_id":"2605.29126","citing_title":"When and How Long? The Readout-Mediator Angle in Temporal Reasoning","ref_index":4,"is_internal_anchor":false},{"citing_arxiv_id":"2605.09967","citing_title":"Tensor Product Representation Probes Reveal Shared Structure Across Linear Directions","ref_index":10,"is_internal_anchor":false},{"citing_arxiv_id":"2605.00847","citing_title":"H-Probes: Extracting Hierarchical Structures From Latent Representations of Language Models","ref_index":12,"is_internal_anchor":false},{"citing_arxiv_id":"2604.20817","citing_title":"Convergent Evolution: How Different Language Models Learn Similar Number Representations","ref_index":12,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/HMGHJP35Z6XTSX6IBSFJS3XL7S","json":"https://pith.science/pith/HMGHJP35Z6XTSX6IBSFJS3XL7S.json","graph_json":"https://pith.science/api/pith-number/HMGHJP35Z6XTSX6IBSFJS3XL7S/graph.json","events_json":"https://pith.science/api/pith-number/HMGHJP35Z6XTSX6IBSFJS3XL7S/events.json","paper":"https://pith.science/paper/HMGHJP35"},"agent_actions":{"view_html":"https://pith.science/pith/HMGHJP35Z6XTSX6IBSFJS3XL7S","download_json":"https://pith.science/pith/HMGHJP35Z6XTSX6IBSFJS3XL7S.json","view_paper":"https://pith.science/paper/HMGHJP35","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2502.00873&json=true","fetch_graph":"https://pith.science/api/pith-number/HMGHJP35Z6XTSX6IBSFJS3XL7S/graph.json","fetch_events":"https://pith.science/api/pith-number/HMGHJP35Z6XTSX6IBSFJS3XL7S/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HMGHJP35Z6XTSX6IBSFJS3XL7S/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HMGHJP35Z6XTSX6IBSFJS3XL7S/action/storage_attestation","attest_author":"https://pith.science/pith/HMGHJP35Z6XTSX6IBSFJS3XL7S/action/author_attestation","sign_citation":"https://pith.science/pith/HMGHJP35Z6XTSX6IBSFJS3XL7S/action/citation_signature","submit_replication":"https://pith.science/pith/HMGHJP35Z6XTSX6IBSFJS3XL7S/action/replication_record"}},"created_at":"2026-07-05T10:08:39.435688+00:00","updated_at":"2026-07-05T10:08:39.435688+00:00"}