Paper 42 — The Gauge Sector of the Black Circle: Structural Reduction and Zeta Framework

This paper analyses the gauge sector of the Black Circle — the irr₃ component of the S₄-invariant spinorial boundary condition BC(0,β) derived in Paper 37. Using the modular structure of Paper 41, the projective-injective property of the Steinberg module St and the S₄-equivariance of the Dirac operator reduce the boundary problem from 4×4 to the irr₃ sector alone: the gravitational singlet (irr₁) is free under IIB-II, and the Majorana spinors (St = irr₈ = 4_s) are outside the BC domain by construction.

The Gauss hypergeometric connection matrix K^{irr₃} is computed in closed Γ-function form for all ω ∈ ℂ. There is no Heun obstruction for this sector. Lemma 2.3 closes OP 42.1: the irr₃ sector carries the spinorial phase Φ_s(ω) = √2·2^{iω/2}e^{iω} (not the scalar phase), as a consequence of the S₄-equivariance of the Dirac operator, Schur's lemma, and the geometric Frobenius shift ρ₊^{(1)} = 1/2 + iω/2 at r = r_s.

The leading-order β-dependent QNM condition H_BC^{irr₃}(ω;β) is derived from the Robin boundary condition cos β·F̂₁^{irr₃}(r_s) + sin β·F̂₂^{irr₃}(r_s) = 0. The single remaining gap is the off-diagonal coefficient C₁(ω) of the full 2×2 spinorial connection matrix (OP 18.1, Paper 17). Section 5 provides a structural argument that IIB-II is algebraically preferred over IIB-I, upgrading OP 37.1 to [Conjecture, IIB-II structurally preferred].