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arXiv · 2607.06870

Phase transitions and uberholography of holographic pure-state geometries

Ning Bao·Keiichiro Furuya·Jacob March
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Abstract

We study the error-correcting properties of pure-state holographic geometries, in which mixed boundary subregions are replaced, via the surface/state correspondence, by the Ryu--Takayanagi (RT) geodesic bounding their entanglement wedges. In AdS$_3$/CFT$_2$ we derive a cross-ratio threshold relation $\eta'/\eta = e^{\Delta H/2}$ for the connected/disconnected transition of the entanglement wedge when two holes are punched in such a geometry. The quantity $\Delta H$ is sourced entirely by geodesics ending on RT boundaries. It shifts the standard two-interval threshold $\eta = 1/2$, and we classify when its sign is fixed by the pattern of hole endpoints. Turning to code properties, we show that the recursive hole-punching underlying uberholography cannot start within an RT-boundary, while an untouched asymptotic boundary can still fractalize, and we find numerically that in the configurations we study it does so with the universal fractal dimension $\alpha \approx 0.786$. The resulting upper bounds on price and distance are nevertheless procedure dependent. In the configurations we study, punching holes on the asymptotic boundary while retaining the RT-boundary yields strictly tighter bounds than first tracing out the RT-boundary and then fractalizing.

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Ning Bao, Keiichiro Furuya, Jacob March. 2026-07-08. Phase transitions and uberholography of holographic pure-state geometries. https://arxiv.org/abs/2607.06870

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