Paper 21 — The Level-2 Sector of the Black Circle

This paper is the twenty-first in a series on zero-divisors in Cayley–Dickson algebras and the Black Circle programme.

Paper 11 of the series established that the Black Circle mass spectrum contains 14 exactly massless modes at Level 2. The coincidence between this multiplicity and the dimension of the exceptional Lie group G₂ raises a natural question: are these modes Goldstone bosons of a spontaneous G₂-symmetry breaking triggered by condensation of the Level-1 tachyon?

This paper answers the question completely and negatively. Three independent obstructions are exhibited: the PSL(2,7)-module structure of the G₂ adjoint representation does not match the actual Level-2 content (representational obstruction); Goldstone's theorem predicts 6 massless modes, not 14 (combinatorial obstruction); and G₂ is not a symmetry of the Black Circle action S₈₄ (structural obstruction). All results are verified with GAP 4.15.1.

A positive structural result accompanies the refutation. Under the SU(3)-path embedding PSL(2,7) ⊂ SU(3) ⊂ G₂, the Level-2 eigenspace carries SU(3) quantum numbers 8 ⊕ 6, which excludes the Standard Model Higgs mechanism as a mass source. The de Sitter symmetry-breaking subgroup S₄ resolves Level-2 into four multiplets; the curvature coupling ξ assigns uniform mass to all 14 modes regardless of multiplet structure, providing a third negative result for the mass-gap problem.

An exact algebraic identity is established: Level-3 = 3 × Level-2 as PSL(2,7)-modules, with the factor 3 fixed by the characteristic polynomial of the ZD adjacency matrix. Four open problems are stated.