Quantum Horizon Tadpole and Emergence of a de Sitter Interior

Abstract: In a paper \cite{Chu:2026dhx}, the quantum stability of the fuzzy sphere was established at large but finite $N$. In addition, a tadpole was identified for the scaling fluctuation mode of the matrix geometry. In this paper, we show that the tadpole generates a positive surface tension and tends to contract the sphere if the horizon is left isolated. However, when the horizon is coupled to gravity, the tadpole requires the bulk geometry to adjust according to the junction condition of general relativity. Assuming a static, spherically symmetric and non-singular vacuum interior, pure de Sitter space is selected. The matching also fixes an intriguing large-$N$ soldering relation between the de Sitter time inside and the Schwarzschild time outside. Potential cosmological implications are discussed.