Paper 12 — Boundary Conditions and Spinorial Statistics of the Black Circle Interior

Paper 11 established the complete mass spectrum for the 84 ZD-active scalar modes on the de Sitter Black Circle interior, the SO(1,4)-invariant condensate C₀, and two negative results on spinorial boundary conditions. This paper pursues three objectives and delivers definitive verdicts on the open problems inherited from Paper 11.

(i) Boundary conditions — Objective (i). Option (iii) of Paper 11 (Remark 3.6) is closed as a derived negative result: the sign-parity constraint π_A π_B π_C = −1 does not force the overlap of ker(Lₐ) modes with outgoing 𝒜₃ plane waves to vanish. 𝒜₃ elements are of pure ε-type and lie outside the ZD graph; the sign-parity constraint has no jurisdiction over them. A weaker result is derived (Fano support orthogonality): V_ℓ̄ ⊥ span{eₚ, eₖ, eₚ₊ₖ} by index support — insufficient to close OP 11.1 without IIB-I/II. OP 11.6 sub-question (a) is closed as a derived result: standard CCR are valid for S₈₄; non-associativity of 𝒜₄ governs interaction graph topology, not field-value products. The only currently available path to χ(rₛ) = 0 for spinorial modes remains IIB-I/II (Conjecture).

(ii) QNM frequencies — Objective (ii). A fundamental obstacle is identified. The Frobenius exponents at rₛ are α_in = ±iω/2 (from f′_in(rₛ) = 2) and α_out = ±iω (from f′_out(rₛ) = 1). Standard Wronskian-shooting is invalid for this geometry: the factor 1/2 in α_in/α_out is the reciprocal of the Frobenius algebraic ratio ‖Lₐ‖²_F|ₙ₌₄ / ‖Lₐ‖²_F|ₙ₌₃ = 2 (Paper 7), which propagates into QNM physics as the surface-gravity jump κ_in/κ_out = 2 at rₛ. Preliminary QNM candidates under the Frobenius-continuation ansatz are reported (l = 0 and l = 2); the l = 1 result is identified as spurious. The prediction |ω_BC/ω_Sch| ≈ √2 is not confirmed by this method. Rigorous computation via Leaver-type matched continued fractions is formulated as OP 12.6.

(iii) Spinorial statistics — Objective (iii). GL(2,3) ≅ 2S₄ (the binary octahedral group) is derived: Z(GL(2,3)) = {I, 2I} has order 2, PGL(2,3) ≅ S₄, and the extension by ℤ₂ is the unique non-trivial one (Schur multiplier of S₄ is ℤ₂). GL(2,3) has exactly 8 irreducible representations with dimensions {1, 1, 2, 2, 2, 3, 3, 4}; the unique 4-dimensional irrep 4_s is spinorial (the central element 2I acts as −Id₄). Under the discrete spin-statistics theorem (Conjecture — Assumption 12.1), the 4 spinorial modes obey CAR algebra and generate a Grassmann–Fock space of dimension 2⁴ = 16. The full Black Circle Hilbert space is ℋ_BC = ℱ_sym(ℂ⁸⁰) ⊗ ∧•(ℂ⁴).