Icosahedral Spatial Spacetime Crystallisation and Singularity Resolution in Rotating Black Holes:

The ring singularity of the Kerr solution remains a fundamental challenge for general relativity: the continuous differential-geometric structure that generates the theory breaks down completely at r = 0, θ = π/2 in Boyer–Lindquist coordinates. Every existing approach to singularity resolution introduces discrete structure, whether kinematically (the loop quantum gravity area gap), geometrically (a de Sitter core as a boundary condition), or topologically (a zero-point length as a topological order parameter), yet three questions central to any crystallisation account remain unaddressed in the singularity-resolution literature: what continuous symmetry is spontaneously broken as the discrete structure forms; what field serves as the Landau order parameter, vanishing in the symmetric phase and acquiring a non-zero expectation value in the broken phase; and what mechanism prevents the discrete structure from thermalising back to the continuous Kerr geometry under Hawking radiation. These three open questions are the gap the present paper addresses. We propose that the 2I-modulated regular Kerr model, developed and adversarially verified in the companion black hole investigation (publication pending), constitutes a candidate for the first explicit model of spatial spacetime crystallisation as a singularity resolution mechanism. Working through five bridge steps and five assessment questions, we find that the model satisfies the criteria for spatial crystallisation with well-supported confidence. A two-tier spontaneous symmetry breaking structure operates: at Tier 1, curvature-induced condensation drives the condensate amplitude from zero in the exterior Kerr geometry to a stable minimum at the Planck core (Φ̃eq = 2/3, confirmed by 𝑉 ′(2/3) = 0 and V’ ’(2/3) > 0 from the established effective field theory (EFT)); at Tier 2, the selection of which of the 120 equivalent icosahedral orientations is adopted spontaneously, as the potential is 2I-invariant and all orientations are degenerate. The order parameter is the condensate amplitude 𝐴 = |⟨Φ̃⟩|. The symmetry broken is the continuous azimuthal U(1) of Kerr to the discrete C₅ subgroup at the observable level, and the SU(2) spinor bundle symmetry to the binary icosahedral group 2I at the deeper level. Three independent topological invariants protect the broken phase: integer winding numbers from 𝜋1(𝑈 (1)/𝐶5) = ℤ; non-abelian charges from 𝜋1(SU(2)/2𝐼) = 2𝐼, whose coset space is the Poincaré homology sphere; and the thermodynamic winding number 𝑊 = 0 for the regularisation parameter C> 0, which places the regular geometry in a topologically distinct sector from the singular Kerr geometry (𝑊 = 1). An established anti-thermalisation mechanism exists in the class of Russo and Pohl (2025): dissipation–correlation interplay via the Hawking radiation Lindblad channel, analogous to the quantum continuous time-crystal II phase (mechanism established for 𝑚𝐻 ≫ 𝑇𝐻 from vacuum structure at defect sites). Temporal crystallisation, by contrast, is assessed as speculative and likely in the forced-constraint class rather than the spontaneous symmetry-breaking class.The present model is sharply distinguished from the Ecker–Ecker–Grumiller (2026) critical spacetime crystal, which applies the spacetime crystal concept to the Choptuik critical solution (Choptuik, 1993) at the threshold of gravitational collapse: breaking scaling rather than azimuthal symmetry, with the underlying solution singular rather than regular. Our icosahedral crystal is a stable, topologically protected phase inside every rotating black hole above a critical spin-to-mass ratio, resolving the ring singularity via orientation selection. Three residual computations are addressed: the full Kerr thermodynamic winding number (established analytically), the Goldstone ring-phonon velocity (established: 𝑣𝜑 = 𝑐 at leading order in the long-wavelength EFT), and the Lindblad master equation for the 2I condensate under Hawking dissipation (established: H_eff and {L_k} derived from ℒ_EFT; 30-zero orbit structure computed with adversarial pass; anti-thermalisation established for 𝑚𝐻 ≫ 𝑇𝐻 ; Kac-label coefficient identification speculative, negative orbit result).