Swampland Distance Conjecture (large-field excursions and emergent towers)

Prove the Swampland Distance Conjecture: In any effective field theory consistently coupled to gravity, demonstrate that a large field-space excursion Δφ ≫ 1 (in Planck units) of a modulus φ necessarily leads to the emergence of an infinite tower of states whose masses are exponentially suppressed according to m(Δφ) = m0 e^{−α Δφ}, where α is an order-one constant.

Background

The paper motivates its dark-dimension scenario using Swampland ideas, emphasizing how large excursions in moduli space should trigger towers of light states. The Swampland Distance Conjecture is cited as a key underpinning for connecting the observed cosmological constant to a low Kaluza–Klein scale. Although the authors employ the conjecture as an assumption for phenomenological analysis, it remains mathematically unproven and thus an explicit conjecture stated within the text.

Establishing this conjecture rigorously would strengthen the theoretical basis for associating dark-energy-scale physics with micron-sized extra dimensions, thereby reinforcing the viability of five-dimensional rotating primordial black holes as dark matter candidates in the dark-dimension framework.