Warp Anisotropy Is Not a Free Parameter: Fixing α through Junction Conditions and Boundary Topology

Abstract Warped compactifications with multiple warped submanifolds generically admit an anisotropy between external and internal warp factors, which is often treated as a free continuous parameter. We show that this anisotropy is not arbitrary. The argument proceeds through three successive constraints: (i) the bulk Einstein equations on a Ricci-flat internal space establish a continuous family of anisotropic solutions; (ii) Israel junction conditions reduce this family to a single dimensionless ratio of boundary energy densities; and (iii) boundary topological consistency discretizes this ratio. The discretization is traced to the group cohomology $H^{4}(B(\mathbb {Z}_{2}\times \mathbb {Z}_{9}),\mathbb {Z}) \cong \mathbb {Z}_{2}\oplus \mathbb {Z}_{9}$, whose torsion exponent $\mathrm{lcm}(2,9)=18$ controls the fractional instanton lattice on transverse linking four-cycles. The boundary allocation selected under a minimal fractional unit principle yields the ratio $\rho /\sigma =1/17$ and hence $\alpha =3/2$. The result removes an otherwise tunable degree of freedom and significantly increases the predictive content of warped compactifications without introducing additional stabilizing fields.