{"paper":{"title":"Generalize cross-ratios in n-dimensional Plane-Based Geometric Algebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Cross-ratios extend as a grade-agnostic projective invariant across all object types in n-dimensional PGA.","cross_cats":["cs.CV"],"primary_cat":"cs.CG","authors_text":"Enzo Harquin (LIGM), Pascal Monasse (ENPC), Stephane Breuils (LAMA), Venceslas Biri (LIGM), Vincent Nozick (LIGM)","submitted_at":"2026-05-18T13:43:15Z","abstract_excerpt":"We develop a complete theory of projective cross-ratios in n-dimensional Plane-Based Geometric Algebra (PGA), R(n,0,1), covering geometric objects of every grade: finite and ideal points, hyperplanes, and intermediate flats. For each object type and configuration, we establish an explicit cross-ratio formula, prove that it recovers the appropriate classical invariant, and identify the canonical pairwise measurement operator. A systematic duality analysis further revealed that all eight configurations organize into four dual pairs under the Hodge dual, and that all measurement operators reduce "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The cross-ratio is established as a grade-agnostic projective invariant within PGA, with explicit formulas for each object type and configuration that recover the appropriate classical invariant (signed distance ratios for parallel cases and sine cross-ratios for secant cases).","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the algebraic structure of PGA together with the Hodge dual and commutator operators faithfully extends the classical projective cross-ratio to all grades and all eight configurations without additional constraints or exceptions.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Generalizes projective cross-ratios to every grade of geometric object in n-dimensional PGA with explicit formulas, duality pairs, and recovery of classical invariants.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Cross-ratios extend as a grade-agnostic projective invariant across all object types in n-dimensional PGA.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a86b38d1f6289f163afbb7d0705e4c333dab381f45a3baa3c880daec37dcfbeb"},"source":{"id":"2605.18398","kind":"arxiv","version":1},"verdict":{"id":"1016de1e-8238-412c-b9ec-5de17ad00513","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T23:27:12.213440Z","strongest_claim":"The cross-ratio is established as a grade-agnostic projective invariant within PGA, with explicit formulas for each object type and configuration that recover the appropriate classical invariant (signed distance ratios for parallel cases and sine cross-ratios for secant cases).","one_line_summary":"Generalizes projective cross-ratios to every grade of geometric object in n-dimensional PGA with explicit formulas, duality pairs, and recovery of classical invariants.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the algebraic structure of PGA together with the Hodge dual and commutator operators faithfully extends the classical projective cross-ratio to all grades and all eight configurations without additional constraints or exceptions.","pith_extraction_headline":"Cross-ratios extend as a grade-agnostic projective invariant across all object types in n-dimensional PGA."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.18398/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"cited_work_retraction","ran_at":"2026-05-19T23:52:10.557046Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"citation_quote_validity","ran_at":"2026-05-19T23:50:03.284820Z","status":"completed","version":"0.1.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T23:33:29.748554Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"external_links","ran_at":"2026-05-19T23:31:43.101357Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T23:31:19.773595Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T23:30:57.651944Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T23:21:58.726262Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"70f34af4dc42c56ba0266ca8cecca05b1c29fb4e4039fc0ad134a5816a89e580"},"references":{"count":11,"sample":[{"doi":"","year":2002,"title":"Journal of Mathe matical Imaging and Vision 16(2), 131–154 (2002)","work_id":"a96ff0b8-492e-4a4a-8c8a-9c982b57ffd8","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"URL https://bivector.net/PGA4CS.html","work_id":"39182d43-2ac1-4ee7-8164-b83075f56c49","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"URL https://bivector.net/PGADYN.html","work_id":"070a770d-62c9-4edc-9e6f-3b97c89eb066","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2007,"title":"Morgan Kaufmann ( 2007)","work_id":"b229fa45-2be2-48b7-bce3-b2b27b2c81e4","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.14279/depositonc","year":2011,"title":"Gunn, C.: Geometry, kinematics, and rigid body mechanic s in Cayley-Klein geometries. Ph.D. thesis (2011). DOI 10.14279/depositonc e-3058","work_id":"b236852d-c86f-4b77-aced-b5feeb3e3f23","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":11,"snapshot_sha256":"0e3c4b53302032f89293a14bedb21133a883fc3e586150a45587a3eabd33fb9a","internal_anchors":1},"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"}