Parametric AdS scale-separation conjecture

Fix $d\ge 3$. A lower-dimensional AdS factor is a Lorentzian $d$-manifold with constant sectional curvature $-L_{\mathrm{A}\mathrm{d}\mathrm{S}}^{-2}<0$, so $R_{\mu \nu \rho \sigma }=-L_{\mathrm{A}\mathrm{d}\mathrm{S}}^{-2}(g_{\mu \rho }{g}_{\nu \sigma }-g_{\mu \sigma }{g}_{\nu \rho })$; its global quotient, topology and boundary conditions are fixed across each family. For each compactification, supply the higher-dimensional metric ansatz, any warp factor, and the Weyl rescaling to the lower-dimensional Einstein metric $g_E$ used for $L_{\mathrm{A}\mathrm{d}\mathrm{S}}$. Also supply the internal Hilbert space $\mathcal{H}$ (including its measure), and the nonnegative self-adjoint mass-squared operator $A_{\mathrm{K}\mathrm{K}}$ with its domain and boundary conditions, obtained from the stated linearized higher-dimensional field equations and kinetic normalization for the chosen Kaluza–Klein (KK) tower. Require compact resolvent, a zero mode and a least positive eigenvalue ${\mu }_1>0$, measured in the mass units of $g_E$, and define $M_{\mathrm{K}\mathrm{K}}:=\sqrt{{\mu }_1}$. This defines the tower scale used in this question; the chosen field sector and any lighter omitted towers must be stated. For example, for a minimally coupled massless scalar on an unwarped product $g_{d+k}=g_E\oplus {\gamma }_E$ with connected compact smooth internal Riemannian manifold $(K,{\gamma }_E)$ without boundary and of dimension $k\ge 1$, take $\mathcal{H}=L^2(K,d{\mathrm{v}\mathrm{o}\mathrm{l}}_{{\gamma }_E})$ and $A_{\mathrm{K}\mathrm{K}}=-{\mathrm{\Delta }}_{K,{\gamma }_E}$ with domain $H^2(K)$. Here $H^2(K)$ is the space of functions with square-integrable weak derivatives through order two. The metric ${\gamma }_E$ includes the constant normalization that puts the external factor in the declared Einstein frame. For a warped, singular or boundary compactification, neither the unweighted scalar Laplacian nor its unconverted eigenvalues may be substituted for the supplied mass operator. “Controlled” means that the moduli, tadpole/charge constraints, localized-source profiles and backreaction, and omitted curvature, string-loop and higher-derivative corrections obey stated bounds throughout the family. For a specified class of these controlled AdS compactifications, is there no parametric family with $M_{\mathrm{K}\mathrm{K}}{L}_{\mathrm{A}\mathrm{d}\mathrm{S}}\to \mathrm{\infty }$? This is a separate conjecture from the ADC; the allowed singular and non-supersymmetric classes must be stated.

ADC abbreviates the AdS distance conjecture. A parametric family and its control bounds must be stated; in particular, specify moduli stabilization, integrated flux/source charge cancellation, source backreaction, and the expansion controlling omitted corrections. $M_{\mathrm{K}\mathrm{K}}{L}_{\mathrm{A}\mathrm{d}\mathrm{S}}$ is dimensionless and compares compactification scales; it is not the ADC's tower-mass versus $\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }$ scaling.

Scalar, gauge and theory-family data for swampland conjectures

Lüst, Palti, and Vafa, pp. 1–5

Status review dated 2025-09-05: the bounded record is Lüst–Palti–Vafa, PDF pp. 1–5, and Lehnert, §II.A, PDF pp. 6–11. Lehnert states that its reviewed literature was published before 2025 (PDF p. 2, note 8); its discussion distinguishes scale separation from AdS distance and reports examples/counterexamples for related strong forms. These sources do not prove a universal no-parametric-family claim for every controlled class, whose definition and allowed singular/non-supersymmetric cases remain formulation-dependent. This item makes no status claim about literature after that cutoff and does not claim an exhaustive search.