Invariant Discrete Spherical Lattices with Irreducible Defect: A Mod-24 Cyclic Quotient Approach

This preprint presents a novel discrete lattice discretization of spherical surfaces that achieves high rotational invariance under the cyclic group \(\mathbb{Z}_{24}\), supports volumetric extension into radial shells, maintains locality of operations, and ensures arithmetic closure under both vector addition (with radial overflow) and modular multiplication on diagonal arcs. The lattice is constructed as a \(\mathbb{Z}_{24}\)-quotient of a toroidal base discretization, with points parameterized by polar angle \(\phi \in [0, \pi/2]\), azimuthal angle \(\psi \in [0, 3\pi/2]\), auxiliary parameter \(t \in [0,1]\), and radial shell index \(r\). Residues modulo 24 are partitioned into three types (units, even non-units, multiples of 3) to guarantee multiplicative closure. Points are embedded in \(\mathbb{R}^3\) using a unit radial vector \(R\) and orthogonal displacements along positive and negative polar axes \(P\) and \(N\). A central result is the identification of an irreducible defect: a quarter-hemisphere wedge (azimuthal fraction \(f = 1/4\), deficit angle \(\delta = \pi/2\), integrated curvature \(\pi/2\)) that cannot be closed without violating rotational invariance, arithmetic closure, point uniqueness, or locality. This arises from the intrinsic asymmetry imposed by the prime factors 2 and 3 in \(m = 24\) (generalized to \(m = 6k\), \(k \geq 1\)), limiting the maximal invariant azimuthal span to \(3/4\) of the circle. Irreducibility is proven by contradiction for both the specific \(m=24\) case (via small equatorial ring examples showing overlap/collapse on attempted closure) and the general case (seam-crossing operations induce type mergers or non-local overflows). The lattice approximates a flat metric to \(O(1/N^2)\) error in the thermodynamic limit, with curvature bounded by discrete Gauss-Bonnet theorem incorporating cone-manifold corrections for the defect. Virtual closure of the defect is achieved via Hermitian boosts in the adjoint representation of SU(2), preserving Haar measure and positivity of plaquette terms. Potential applications include regularization in quantum field theory (where the defect may enforce topological lower bounds on spectral quantities, e.g., mass gaps or Casimir-like energies) and advances in discrete differential geometry, computational meshing, and topological modeling. This work stands as an independent contribution to discrete geometry, motivated by long-term exploration of natural spherical discretizations. Feedback and scrutiny are welcome.