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.
Comments
96 pages, 5 figures, 2 tables. Substantially revised and shortened, title and abstract updated. BKM-Petz geometry, BKM-Jacobi matching, and code examples expanded
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−αE)/αE, with αE 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.
We study a four-qubit product-EPR holographic code whose reconstructing region contains matter and one leg of a geometry bond. Seven deformations compare local, bipartite, bond-stabilizing, and bond-moving operators. Exact spectra show that only matter-controlled bond motion produces a leading quadratic logical-state dependence of the boundary entropy. We select one state-independent physical recovery by optimizing coherent information of the channel Choi state. A separate semidefinite program maximizes entanglement fidelity over all channels. At fixed coupling the physical recovery attains that optimum to numerical precision, while recovered-entropy subtraction leaves a nonzero variational proto-area response. Total stabilizer Renyi magic is also nonzero for deformations with no boundary response, so it does not characterize the effect. Projecting the exact stabilizer-Renyi quadratic form away from pairwise Pauli tangents defines a leading-order boundary witness. Its global convex minimum is positive for the bond-moving deformation and zero for the six comparisons. For the reference-matter-geometry resource partition, however, pairwise tangents span the full projective tangent space at the product-EPR base, so the witness vanishes for every Hermitian deformation. A two-bound extension retains this degeneracy but exhibits a split-geometry structural residual and a logical-state-dependent cut switch. These results establish a finite-code witness, not a universal magic law or continuum gravitational dynamics.
We study how much of the proto-area response can be inferred from the bulk von Neumann entropy. In the perturbative holographic code setting considered by CCKLP and Witten, we keep the recovery frame fixed and analyze the response averaged over the GUE at second order in the encoding perturbation. Near the maximally mixed bulk spectrum, the response is fixed by the entropy through quadratic order. For d1≥3, spectral information first enters at cubic order through the third centered moment, so states with the same entropy can have different responses. We determine the resulting response range on small equal-entropy level sets and obtain its sharp leading D3/2 dependence, including the coefficient. The corresponding minimax error for prediction from entropy alone is one half of this range. For d1=2, the entropy instead determines the response exactly.
Quantum error-correcting codes provide a powerful framework for emergent spacetime, yet existing subsystem erasure-correcting holographic code models describe only quantum fields on a fixed background: in such codes, the entropic area term is state independent and cannot capture gravitational backreaction. We argue that this limitation is intrinsic to exact subsystem complementary recovery and that incorporating backreaction naturally leads to approximate quantum error correction. We introduce a Ryu-Takayanagi-like entropy decomposition for approximate subsystem erasure-correcting codes, defining bulk matter entropy via optimal recovery and a complementary proto-area entropy as the difference between boundary entropy and recoverable bulk entropy. For a broad class of skewed quantum codes obtained by small nonlocal perturbations of exact codes, the proto-area increases monotonically with bulk entropy, closely aligning with the changes in extremal surface areas in gravitational systems. We identify the origin of this response as a form of tripartite non-local magic in the Choi state of the encoding map, which vanishes in stabilizer codes and controls the leading matter-geometry coupling in approximate subsystem erasure-correcting codes.
Take a homogeneous two-point function C(r), define the information distance d(r) = −ln C(r), and read it as the geodesic length between boundary points separated by r of an emergent (d+1)-dimensional bulk. Then the decay class of C fixes the curvature class of that bulk. We prove this by an elementary, self-contained computation. Exponential decay yields a flat bulk; power-law, scale-invariant decay yields a bulk of constant negative curvature—hyperbolic, anti–de Sitter. For the stretched-exponential family C(r) = exp(−(r/ξ)^α) one obtains an explicit closed form, which runs continuously from flat at α = 1 to hyperbolic at α → 0, with a scale-dependent negative curvature in between. Only α enters. No holographic duality is assumed, and Einstein's equations are never used; yet the qualitative MERA/holographic correspondence—criticality with hyperbolic geometry—drops out as an explicit theorem. The signature is simple: R = −d(d+1) at criticality, R = 0 when gapped. It can be measured directly on a lattice.
One of the boldest ideas in modern physics is that space itself is a kind of error-correcting code. In this picture the interior of a region of spacetime is like a message, and the surface surrounding it is an encoded copy that stores the message redundantly, so that losing part of the surface need not lose the interior. A celebrated formula from holography then says something unexpected: the amount of entanglement carried by a boundary region equals the area of a surface reaching into the interior. This paper gives a complete, elementary proof that this area law is not an extra assumption bolted on by hand, but an unavoidable consequence of the code structure itself. The geometry is simply bookkeeping for redundancy. The derivation reduces a statement about gravity to plain facts about entanglement and about the algebra of operations one can perform on the encoded information. It establishes four things, each from scratch. First, in the simplest case it recovers the area law directly: the entanglement of a boundary region is a fixed geometric term plus the entanglement of whatever the interior itself carries. Second, it proves a symmetry that had previously been handled case by case, namely that a pure joint state assigns the same entanglement to a region and to its complement, and it proves this in full generality using a single clean identity about matrices. Third, it extends the area law to the realistic situation in which the code splits into separate sectors, each carrying its own independent area, so that the geometric term is not a single number but a genuine operator averaged over the sectors. Fourth, it proves the structural fact the whole picture rests on: the algebra of logical operations always decomposes cleanly into independent blocks, which is exactly the sector structure the area law needs. That last step comes down to one small analytic fact, that a matrix which is symmetric in the appropriate sense and also nilpotent must be zero, and the paper proves it directly. A companion result closes the reconstruction side of the story, showing that the operations recoverable from a boundary region are precisely those left untouched by the coarse-graining onto sectors, and that the coarse-grained state already reproduces every such measurement. Every step is elementary and self-contained, and the entire logical core has a companion version reconstructed inside a machine proof assistant, checked line by line with no gaps and resting only on the standard axioms of mathematics. The upshot is that a piece of geometry, an area, can be read off from nothing more than how redundantly a code stores its information. None of the underlying mathematics is new; it traces back to established results on holographic codes, to the entropy formula itself, and to the century-old classification of matrix algebras. What is new is the assembly: a short, honest, self-contained line of reasoning that turns a claim about spacetime into a handful of reusable facts about entropy and matrices, with every link exposed and checkable. Where the picture still reaches past what has been proved, in particular the finer tensor structure hidden inside each sector, the paper marks the boundary plainly instead of hiding it.
We compute the complete relative entropy of a faithful quantum family controlled by spatial translations and Lorentz boosts. The prescribed contrast has a minimal three-dimensional positive Hessian realisation. Finite comparisons determine every minimal single-chart embedding up to an affine transformation and fix the potential's first jet, leaving its normal Hessian free. No positive single-chart realisation on a globally convex domain exists in any finite dimension for the entire control plane, whereas every bounded rapidity strip admits one. An exact dephasing channel preserves the energy-momentum law while reducing the contrast to classical rapidity distinguishability. Calibrating a preparation Hamiltonian gives sharp work-susceptibility and finite-work control-area bounds. Within the specified four-parameter preparation family, intrinsic Bogoliubov-Kubo-Mori (BKM) length bounds entropy change; paths outside that family need not obey this bound, even at the same observable law. A genuine Born lift preserves the commutator form with an exact area remainder independent of normal-potential freedom. These results separate finite-contrast rigidity, calibrated energy response, and geometric compatibility without identifying statistical transport with physical dynamics.