Paper 19 — The Majorana Sector of the Black Circle

Paper 19 of the Cayley–Dickson Zero-Divisor Series.

The spinorial sector 4_s of the Black Circle interior carries the unique 4-dimensional irrep of GL(2,3) ≅ 2S₄, with Frobenius–Schur indicator FS(4_s) = +1 [Paper 18, Derived]. The remaining obstacle to the Majorana condition was Osterwalder–Schrader axiom OS2 on the compact de Sitter interior.

This paper closes the chain:

(1) OS2 fails structurally [Derived, negative]: the Black Circle interior is compact with no Euclidean half-space. OP 18.7 closed.

(2) The condensate C₀ preserves SO(1,4) and contributes only a mass renormalisation to 4_s; the sector propagates as a free massive scalar on dS₄ [Derived, cond. IIB-I/II].

(3) Candidate (a) ≡ Bunch-Davies vacuum: the algebraic boundary condition of Paper 17 (C₁ = C₂ = 0 at r = 0) and the canonical QFT vacuum of de Sitter coincide because both impose regularity at the same smooth geometric point [Derived, cond. IIB-I/II, Allen 1985].

(4) The Bros–Epstein–Moschella PCT involution Θ_PCT exists on the Bunch-Davies Hilbert space with Θ²_PCT = +1 [Derived, BEM 1998].

(5) Schur collapse: since 4_s is irreducible and FS(4_s) = +1, the GL(2,3) and BEM real structures coincide and C_GL = Θ_PCT [Derived].

(6) Imaginary-frequency bound states (l ≥ 1) are expelled from the Bunch-Davies Hilbert space by the BEM analyticity condition; the BEM norm is positive-definite [Derived]. For l = 0 the spectrum is [m²_{4_s}, +∞) exactly via Pöschl–Teller reduction [Derived].

Main theorem [Derived, cond. IIB-I/II + FS(4_s) = +1]: the four 4_s modes are Majorana. OP 12.7 closed.

Numerical verification: Rayleigh quotients for l = 1, 2 bound states confirmed < at 0.03% (Python/SciPy/mpmath, AMD Threadripper 2950X).