Pseudorandom Dynamics in the SYK Model and Cryptographic Censorship in JT Gravity
We argue that the SYK model provides a conditional realization of Cryptographic Censorship in JT gravity. By using the Weingarten calculus and random matrix universality, we prove that the SYK disorder ensemble is an approximate unitary $k$-design for all $k=\poly(N)$, with deviation controlled by the spectral form factor. We then formulate the planted-SYK hardness conjecture and provide evidence from spectral universality and the low-degree polynomial framework. Under this conjecture, the approximate design becomes a gravitationally pseudorandom unitary. Together with the efficient causal wedge reconstruction in JT gravity, this leads to the conclusion that typical states in the SYK microcanonical window must have event horizons in their bulk duals, with the horizonless fraction doubly exponentially small. We further identify the regularized geodesic length of the maximal interior slice as the explicit distinguishing operator. Its prediction gap grows linearly with time due to the stretching of the black hole interior, linking Cryptographic Censorship to the complexity equals volume conjecture.