Paper 36 — Three Algebraic Closures in the Cayley–Dickson Zero-Divisor Series

We close three open problems (OP 29.1, OP 34.1, OP 35.1) arising in Papers 29–35 of the series on zero-divisors in Cayley–Dickson algebras.

(A) [Derived-Negative] We prove that no compact Lie group G satisfies adj(G)|_{PSL(2,7)} ≅ irr_6 ⊕ irr_8 (OP 29.1). The proof proceeds by a dimension argument (dim G = 14 is forced), the exclusion of G_2 via its PSL(2,7)-character (Paper 21), and an order obstruction showing that every homomorphism PSL(2,7) → SU(2) is trivial. This definitively rules out a Goldstone interpretation of the Level-2 masslessness of the Black Circle: the spectral sombrero arises from Z(G_{84}) = C_2 and the bipartite structure of ZD*(A_4), not from spontaneous symmetry breaking.

(B) [Derived, conditional on action matrices] We compute H^2(PSL(2,7), (Z/2Z)^4) ≅ (Z/2Z)^2 as an F_2-vector space (OP 35.1, Part B). The computation uses the HAP free resolution of Z over Z[PSL(2,7)] (ranks 1, 5, 13, 24 in degrees 0–3) and the action of PSL(2,7) on the normal subgroup N = (Z/2Z)^4 of G_{84} = Aut(ZD*(A_4)) by conjugation. The cochain ranks are 4, 20, 52, 96; the coboundary ranks are 3, 16, 34. There are exactly 4 extension classes (3 non-trivial); the extension class [G_{84}] is one of them. The identification of [G_{84}] with the lo-bit projection of the octonionic 2-cocycle c_{oct} (Conjecture 6.1 of Paper 35) remains open.

(C) [Derived/Open] We give an algebraic derivation that the normalised residue s_2 = 14 at the H-pole of the ZD zeta function Z(x) equals dim G_2 = dim Aut(O) (OP 34.1). The numerical identity follows from the Weil factorisation Z(x) = 336x^4 · Z_Weil(x), the residue formula R_k · dim(A_k)^4 = 16·(-1)^k·s_k (Paper 33), and standard Lie theory (dim G_2 = 14). The G_2-equivariant identification of the Weil weight space at x_2 with adj(G_2) — establishing why the Weil mechanism selects dim Aut(A_3) rather than another 14-dimensional structure — remains open.

All GAP computations use GAP 4.15.1 with the CTblLib and HAP packages. GAP scripts are deposited as supplementary files. (on Zenodo)