{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:IZV5ET6TRJ72FKQ2NMCBPCV26Q","short_pith_number":"pith:IZV5ET6T","schema_version":"1.0","canonical_sha256":"466bd24fd38a7fa2aa1a6b04178abaf4200aa17469e795fa9f18575132ed96cf","source":{"kind":"arxiv","id":"2507.11688","version":4},"attestation_state":"computed","paper":{"title":"Composing Linear Layers from Irreducibles","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Daisuke Yamada, Travis Pence, Vikas Singh","submitted_at":"2025-07-15T19:46:00Z","abstract_excerpt":"Contemporary large models often exhibit behaviors suggesting the presence of low-level primitives that compose into modules with richer functionality, but these fundamental building blocks remain poorly understood. We investigate this compositional structure in linear layers by asking: can we identify/synthesize linear transformations from a minimal set of geometric primitives? Using Clifford algebra, we show that linear layers can be expressed as compositions of bivectors -- geometric objects encoding oriented planes -- and introduce a differentiable algorithm that decomposes them into produc"},"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":"2507.11688","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2025-07-15T19:46:00Z","cross_cats_sorted":[],"title_canon_sha256":"bb9be324b5886e9aa1dc9ea1c97865c9051b936559f979708e791df5d74689e2","abstract_canon_sha256":"0272133ef5d136b979686a3a461a1c362d6dad48281f356792f1efb2d3ffbf33"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-11T00:08:10.780307Z","signature_b64":"1xOgKR4BwgI9nMpGeiyp1nSTzWswD2tK0nYHBs5bNw8vSX3FLzoadD8xnfIODH6CG0TB/mEWqmp303krMINbBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"466bd24fd38a7fa2aa1a6b04178abaf4200aa17469e795fa9f18575132ed96cf","last_reissued_at":"2026-06-11T00:08:10.779430Z","signature_status":"signed_v1","first_computed_at":"2026-06-11T00:08:10.779430Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Composing Linear Layers from Irreducibles","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Daisuke Yamada, Travis Pence, Vikas Singh","submitted_at":"2025-07-15T19:46:00Z","abstract_excerpt":"Contemporary large models often exhibit behaviors suggesting the presence of low-level primitives that compose into modules with richer functionality, but these fundamental building blocks remain poorly understood. We investigate this compositional structure in linear layers by asking: can we identify/synthesize linear transformations from a minimal set of geometric primitives? Using Clifford algebra, we show that linear layers can be expressed as compositions of bivectors -- geometric objects encoding oriented planes -- and introduce a differentiable algorithm that decomposes them into produc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.11688","kind":"arxiv","version":4},"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/2507.11688/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":"2507.11688","created_at":"2026-06-11T00:08:10.779593+00:00"},{"alias_kind":"arxiv_version","alias_value":"2507.11688v4","created_at":"2026-06-11T00:08:10.779593+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.11688","created_at":"2026-06-11T00:08:10.779593+00:00"},{"alias_kind":"pith_short_12","alias_value":"IZV5ET6TRJ72","created_at":"2026-06-11T00:08:10.779593+00:00"},{"alias_kind":"pith_short_16","alias_value":"IZV5ET6TRJ72FKQ2","created_at":"2026-06-11T00:08:10.779593+00:00"},{"alias_kind":"pith_short_8","alias_value":"IZV5ET6T","created_at":"2026-06-11T00:08:10.779593+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/IZV5ET6TRJ72FKQ2NMCBPCV26Q","json":"https://pith.science/pith/IZV5ET6TRJ72FKQ2NMCBPCV26Q.json","graph_json":"https://pith.science/api/pith-number/IZV5ET6TRJ72FKQ2NMCBPCV26Q/graph.json","events_json":"https://pith.science/api/pith-number/IZV5ET6TRJ72FKQ2NMCBPCV26Q/events.json","paper":"https://pith.science/paper/IZV5ET6T"},"agent_actions":{"view_html":"https://pith.science/pith/IZV5ET6TRJ72FKQ2NMCBPCV26Q","download_json":"https://pith.science/pith/IZV5ET6TRJ72FKQ2NMCBPCV26Q.json","view_paper":"https://pith.science/paper/IZV5ET6T","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2507.11688&json=true","fetch_graph":"https://pith.science/api/pith-number/IZV5ET6TRJ72FKQ2NMCBPCV26Q/graph.json","fetch_events":"https://pith.science/api/pith-number/IZV5ET6TRJ72FKQ2NMCBPCV26Q/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IZV5ET6TRJ72FKQ2NMCBPCV26Q/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IZV5ET6TRJ72FKQ2NMCBPCV26Q/action/storage_attestation","attest_author":"https://pith.science/pith/IZV5ET6TRJ72FKQ2NMCBPCV26Q/action/author_attestation","sign_citation":"https://pith.science/pith/IZV5ET6TRJ72FKQ2NMCBPCV26Q/action/citation_signature","submit_replication":"https://pith.science/pith/IZV5ET6TRJ72FKQ2NMCBPCV26Q/action/replication_record"}},"created_at":"2026-06-11T00:08:10.779593+00:00","updated_at":"2026-06-11T00:08:10.779593+00:00"}