AWC Conjecture L: Wedderburn Classification of Left Multiplication Enveloping Algebras of Cayley–Dickson Algebras in Characteristic p and a Refutation of the Cartan Doubling Recursion

Let Fp be the prime field, and let An be the n-th Cayley–Dickson doublingalgebra, of dimension 2n. This paper studies the Wedderburn structure over Fpof the left multiplication enveloping algebra EnvL(An) of An, and examines thefeasibility of a Cartan-matrix recursion along the doubling tower. The classificationresult falls into four branches: when p = 2, An is commutative and associative,EnvL(An) ∼ = An is a local algebra, and the Cartan matrix is [2n]; when p is oddand n = 1, EnvL(A1) is isomorphic to A1, which either splits as Fp × Fp or is thefield Fp2, with Cartan matrices I2 and [1], respectively; when p is odd and n = 2,the left regular representation gives EnvL(A2) ∼ = A2 ∼ = M2(Fp), of dimension 4; when p is odd and n ≥ 3, EnvL(An) = EndFp(An) ∼ = M2n(Fp), and the Cartan matrix is [1]. Along with the classification is a basis-product closure fact: for everyprime p and every n, the ordered product of standard basis elements is still a signedmultiple of some standard basis element and is nonzero; hence statistics supportedon zero basis pairs with positive density are empty at the object level. On thisbasis, a Cartan doubling recursion of the form )︃ Cn+1 = (︃Cn X XT Cn +∆n cannot hold at the matrix-size level: for every n ≥ 1, the size of Cn+1 is always 1, whereas the right-hand side of the recursion is assembled from two copies ofCn and hence has size at least 2; at the same time, permuting rows and columnsdoes not change matrix size, so this excludes dependence on the specific valuesof the cross blocks and correction terms. Exact integer and modular-arithmeticenumerations over finite ranges provide consistency checks for structure constants,block formulas, closure dimensions, and centralizers; the general conclusions areindependently supported by the classification proof and the size inequality.