{"id":"palomar/PALOMAR-2026-10-16-000013","url":"https://github.com/macgregor-sudo/compact-albert-recognition/tree/40d3b344180d9ddb78aaa2876873377c1e631515","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-16-000013-v1.json","repo":"https://github.com/macgregor-sudo/compact-albert-recognition","tags":["math.RA","math.DG","msc:17C40","msc:17C37","msc:17C30","jordan-algebras","albert-algebra","octonions"],"score":0,"title":"Finite Observations and Recognition of the Compact Albert Algebra","trust":"high","domain":"math.RA","source":"palomar","authors":["Robert MacGregor"],"reviews":0,"subject":"math.RA","summary":"Let J=H_3(O), the real algebra of 3×3 Hermitian matrices over the division octonions, with product a∘b=(ab+ba)/2 and diagonal trace T. A frame p=(p_0,p_1,p_2) consists of trace-one elements satisfying p_i∘p_i=p_i, p_i∘p_j=0 for i≠j, and p_0+p_1+p_2=1. For frames p,q,r, record the 27 pair traces T(p_i∘q_j), T(q_j∘r_k), T(r_k∘p_i), and 27 triple traces T((p_i∘q_j)∘r_k), for i,j,k∈{0,1,2}. How much do these traces determine? Each trace list in some open neighborhood of the data with pair entries 1/3 and triple entries 1/9, in a fixed twenty-dimensional chart, determines exactly two equivalence…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-16","reactions":{},"formalized":true,"formalizes":[{"title":"Unpublished notes on compact Albert reconstruction and finite observations","identifier":"","relationship":"adapts"},{"title":"Rational invariants of three frames in the split Albert algebra: CubicJordan formalization","identifier":"https://github.com/macgregor-sudo/albert-frame-triple-invariants/blob/dd06ef5d423a0e4a5d814d2c221e999b98887ee1/AlbertLean/CubicJordan.lean","relationship":"adapts"}]}