Paper 11 — Canonical Quantisation of the Black Circle Interior

This paper initiates canonical quantisation of the 84 ZD-active scalar fields on the static de Sitter interior of the Black Circle — a conjectured horizonless compact object whose interior geometry is governed by the zero-divisor structure of the sedenions 𝒜₄.

Five main results are established.

(1) Boundary conditions. The Klein–Gordon equation for ZD-active fields reduces to a Regge–Wheeler radial equation. The Barrabès–Israel junction analysis at the Schwarzschild radius rₛ gives a derived negative result for ξ=0: only field continuity is imposed — no Neumann condition on ∂ᵣφₖ. The Born Rule theorem alone does not force the Dirichlet boundary condition for spinorial modes; the gap is precisely characterised and three resolution paths are identified.

(2) Complete mass spectrum. Derived via Python and GAP 4.15.1. The ZD pair graph decomposes into 7 isomorphic connected bipartite components (one per Fano class, 12 vertices each), with integer eigenvalues {−4, −2, 0, +2, +4} and multiplicities {7, 14, 42, 14, 7}. The PSL(2,7) permutation character on 84 elements decomposes as χ₁ ⊕ χ₃ ⊕ χ₃′ ⊕ 4χ₆ ⊕ 3χ₇ ⊕ 4χ₈. The 7 tachyonic modes (μ = −4) lie precisely in the irr₇ representation — the same 7-dimensional irrep governing Papers 7 and 10.

(3) De Sitter-invariant condensate and T_μν. The unique condensate C₀ is derived from SO(1,4) transitivity and the Einstein equation: C₀⁴ = 3 / (2048π G³ M² λ). The stress-energy tensor T_μν at the condensate reproduces Λ_eff = 3/rₛ² exactly — the first explicit T_μν for 𝒜₄ matter content.

(4) Breitenlohner–Freedman stability. Level 1 (irr₇, μ = −4) satisfies the BF bound if and only if λGM² < 27π/128, yielding a falsifiable mass threshold M_BF. For M > M_BF the irr₇ sector is BF-unstable. Levels 2–5 are unconditionally stable.

(5) Curvature coupling ξ — derived negative result. The space of PSL(2,7)-invariant symmetric bilinear forms on ℝ⁸⁴ has dimension 28 (GAP 4.15.1): ξ cannot be fixed by symmetry alone.