Abstract Warped orbifold compactifications generically admit a warp-anisotropy parameter α characterizing the ratio of internal to external warp exponents. In a recent work, this parameter was shown to be topologically discretized and fixed for a specific nine-dimensional model whose boundary sector supports self-dual Yang–Mills configurations with fractional instanton numbers quantized by a finite torsion exponent. Here we investigate the structure of this discretization mechanism across (d + 1 + q) -dimensional warped orbifold compactifications within the same boundary-sector class, where d is the number of external dimensions, 1 is the orbifold direction, and q is the dimension of a Ricci-flat internal space. We derive the general junction relation α = (1 + dx) / (1 − (q − 1) x) with x = ρ/σ, and show that boundary topological consistency with torsion exponent N restricts x to the discrete set m/ (N − m). The number of physically allowed anisotropy sectors is ⌊ (N − 1) /q⌋ and decreases monotonically with increasing internal dimension q. A sharp uniqueness criterion emerges: when ⌈N/2⌉ ≤ q N, the anisotropy is uniquely determined within this class without any selection principle. When q ≥ N, no physical solution exists. The previously studied case d = q = 4, N = 18 is recovered as one entry in the classification, and the criterion explains why a selection principle was necessary in that case while identifying the regime where it becomes unnecessary.
Mu Sung Lee (Thu,) studied this question.