Researchers have developed exact finite-resolution criteria to determine when proto-area data, generated by approximate recovery in holographic codes, can be compatible with a single local bulk metric. These holographic codes, which use recovery maps calibrated on the code channel and fixed along a logical-state family, are fundamental for understanding the AdS/CFT correspondence, a conjecture relating theories of gravity in anti-de Sitter (AdS) spaces to conformal quantum field theories (CFT) on their boundaries.

The criteria focus on the necessary and sufficient conditions for a regular “proto-area two-jet” to arise from a “metric two-jet” on a time-reflection-symmetric asymptotically AdS$_3$ slice. In the realm of finite networks, this translates into a polyhedral realization problem that includes primal and dual certificates, stable reconstruction, and explicit witnesses of non-geometry. In the continuum, the geometric tangent space is described as the range of the rank-two geodesic X-ray transform.

A metric-forced Jacobi equation is key to determining the normal Hessian of the renormalized boundary-length image, revealing a gauge-invariant quadratic obstruction. Under a split-regularity hypothesis, nearby geometric data form a local graph, and the two-jet criterion itself is unconditional for regular data. Hamiltonian-skewed codes demonstrate both first-order non-geometry and a response whose first obstruction appears only at quadratic order. This allows for the reconstruction of the compatible metric perturbation, modulo boundary-fixing diffeomorphisms, which has implications for understanding how quantum information is encoded in spacetime geometry.