Holography at finite N: breakdown of bulk reconstruction for subregions

A bstract Within AdS/CFT, focusing on the AdS-Rindler wedge, we show that when N is large but finite, correlation functions of reconstructed bulk operators grow exponentially with bulk momentum, overwhelming the usual 1 /N suppression. The growth starts when the smeared operator’s ultraviolet scale goes beyond a critical value $$ {\Lambda}_{\mathrm{crit}}=\frac{2}{\pi}\ln N $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mtext>crit</mml:mtext> </mml:msub> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> </mml:mfrac> <mml:mo>ln</mml:mo> <mml:mi>N</mml:mi> </mml:math> , which is far below the Planck scale. Above this logarithmic threshold, the large N expansion ceases to be reliable, and the would-be bulk operators cannot be consistently defined as observables in the full quantum gravity theory. Since the AdS-Rindler wedge describes the near-horizon region of black holes, this result implies a sharp ln N cutoff for reconstructing bulk operators across horizons. This has a direct impact on whether and how information from the black hole interior is encoded — a central question in the black hole information paradox.