Paper 37 — Spinorial Boundary Conditions in the Black Circle

Paper 37 in the series "Zero-Divisors in Cayley–Dickson Algebras."

We continue the analysis of open problem OP 12.1: does the Dirichlet boundary condition χ(r_s) = 0 hold for the spinorial Level-3 modes of S_{84} in the Black Circle interior, without invoking the IIB-I/II conjectures? Two new approaches are pursued.

Approach (iv) — EF regularity for spinors. The Frobenius analysis of the coupled Dirac system (D1)–(D2) at r = r_s identifies a half-integer shift (+1/2) in the outgoing exponent of F̂_1 relative to the scalar case. This propagates into a modified EF phase Φ_s(ω) = √2 · 2^{iω/2} · e^{iω} ≠ 0 for all ω ∈ ℂ. Hence F̂_1(r_s) = D_in^s(ω) = Φ_s(ω) ≠ 0: EF regularity does not force χ(r_s) = 0. Third [Derived-Negative] result for OP 12.1.

Approach (v) — S_4-invariant boundary conditions. The permutation character of S_4 = Stab_{PSL(2,7)}(ℓ_a) on the four off-line Fano points decomposes as χ_perm = irr_1 ⊕ irr_3 (standard, not sign-twisted irr_{3'}), closing OP 37.2 [Derived] (Python computation, verified). By Schur's Lemma the S_4-invariant boundary conditions form a two-parameter family; χ(r_s) = 0 is admissible but not unique. The irr_1 singlet is identified as the gravitational mode under IIB-II, revealing a tension with IIB-I (OP 37.1). A structural link between irr_3 in the spinorial off-line sector and irr_3 in the bosonic on-line sector (Paper 21) is established.

Open problems formulated: OP 37.1 (IIB-I vs IIB-II tension), OP 37.3 (spinorial QNM spectrum), OP 37.4 (energy-minimising selector), OP 12.1 residual.

Supplementary file: OP_37_2_S4_character_restriction.py (Python computation for Theorem 6.1, deposited as supplementary material). (on Zenodo)