When does a state-dependent proto-area define a bulk geometry?

Abstract

An area-like function for each boundary region need not come from a single bulk geometry. Cross-region locality requires the data for all regions to factor through a single boundary-length map. We formulate this common-source condition as a nonlinear sewing problem. For a fixed network chamber, we derive exact primal and dual certificates together with an operator-valued criterion for central area operators. On the hyperbolic disk, the first variation lies in the range of the rank-two geodesic X-ray transform. At second order, extremality no longer removes geodesic displacement, and the forced Jacobi equation determines the normal Hessian of renormalized boundary length. We prove a gauge-invariant necessary and sufficient criterion for a regular proto-area coefficient two-jet to arise from a metric two-jet. At a product reference, we prove an exact orthogonal decomposition for the Bogoliubov--Kubo--Mori (BKM) metric associated with the Petz recovery map. If the traced-out factor is nontrivial, no unrestricted reverse stability bound exists; on an observable tangent space the sharp constant is $(1-α_E)/α_E$, with $α_E$ the minimum retained BKM fraction. Combining the geometric and information-theoretic structures yields BKM--Jacobi matching: the negative BKM information-loss form must equal the Jacobi normal acceleration across the full interval family. We construct two isometric code families with complementary obstructions. Exact regional central area operators on the recoverable algebra can fail to sew into one local graph geometry, while a fixed-central-fiber affine logical path has zero first response and a nonzero quadratic witness. Under source density, sampled-data convergence, and first- and second-derivative consistency, corrected discrete witnesses converge to the continuum obstruction.