Nonlinear Geometrizability of State-Dependent Proto-Area in Approximate Holographic Codes

Abstract: State-dependent proto-area data produced by approximate recovery need not be compatible with a single local bulk metric. Using recovery maps calibrated on the code channel and held fixed along a logical-state family, we derive exact finite-resolution criteria and, near the hyperbolic disk, 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. Finite networks give a polyhedral realization problem with primal and dual certificates, stable reconstruction, and explicit witnesses of nongeometry. In the continuum, the geometric tangent space is the range of the rank-two geodesic X-ray transform. A metric-forced Jacobi equation determines the normal Hessian of the renormalized boundary-length image and yields a gauge-invariant quadratic obstruction. Under a split-regularity hypothesis, nearby geometric data form a local graph; the two-jet criterion itself is unconditional for regular data. Hamiltonian-skewed codes realize both first-order nongeometry and a response whose first obstruction appears only at quadratic order. The compatible metric perturbation is reconstructed modulo boundary-fixing diffeomorphisms.