Emergent-Gravity Claims: Assumptions, Nonconverses, and Status

“Gravity from entanglement” names several results with different logical strength. Information identities are exact within their operator-algebra domains; holographic entropy and modular equalities are controlled expansions; linearized field equations follow only after dictionary and inversion assumptions; crossed-product and nonlinear constructions cover specified models. No established converse says that entanglement structure alone uniquely selects gravity.

Required background. Beyond Linear Order: Conditional Emergent-Equation Claims supplies the nonlinear obstruction. Crossed-Product Gravitational Algebras and Generalized-Entropy Terms supplies the algebraic proposals.

Helpful background. Claim–Evidence Records, Replication, and Retraction Handling supplies evidence discipline. Claim Status, Freshness, and Research Handoffs supplies the volume-wide claim grammar.

First application. Test a representative entanglement-to-gravity argument line by line and label every imported dictionary, locality assumption, and implication direction.

Implication map

Input and domain	Supported conclusion	Missing converse or promotion
Differentiable family of faithful states	entanglement first law	no gravitational dynamics
RT plus stress-tensor dictionary for all vacuum balls	linearized bulk equation under inversion hypotheses	no derivation of RT, dictionary, or nonlinear action
JLMS in a fixed code subspace	leading boundary/bulk relative-entropy equality and modular action	no exact finite-N equality for all states
Positive second relative entropy	positive canonical energy for matched perturbations	no global nonlinear stability theorem
Crossed product with specified modular/area data	semifinite trace entropy matching generalized entropy in stated models	no unique algebra for every gravitational region
Higher information variations plus complete correlator and EFT data	constraints on interaction vertices	no unique UV completion

The right column is scientifically active content: it prevents a valid conditional result from being reported as a stronger, logically different theorem.

Line-by-line test of the ball argument

Consider the statement “the entanglement first law yields Einstein’s equation.” Expanded, the inference is:

1. choose a holographic CFT vacuum and a small state perturbation;
2. use the exact boundary first law for every ball;
3. import the RT entropy formula and stress-tensor/metric normalization;
4. use a covariant gravitational identity for the AdS–Rindler Killing field;
5. assume boundary conditions, gauge control, and injectivity of the ball transform;
6. infer the linearized equation for the chosen bulk action.

Steps 3–5 contain the gravitational dictionary and locality assumptions. The result is powerful because they make the inference quantitative, not because those assumptions disappear.

Nonconverses and alternative realizations

The first law holds in spin systems, free fields, thermal states, and finite-dimensional algebras with no Einstein bulk. Area-like entanglement scaling likewise occurs in many-body states without a gravitational dual. Tensor networks can model reconstruction and geometry while using discrete exact codes that do not satisfy gravitational dynamics.

Conversely, higher-curvature gravities modify both the entropy functional and field equation while retaining related entanglement identities. Field redefinitions can change the off-shell equation’s form without changing observables. These examples rule out uniqueness claims based only on the first two state variations.

Dated frontier boundary

The literature was checked through 10 August 2026. Recent work continues to strengthen restricted algebraic results rather than remove their hypotheses. A July 2026 analysis proves a JLMS condition and vacuum-subtracted HRT statement for localized coherent excitations in linearized AdS gravity, while explicitly assuming the AdS/CFT isometric map, vacuum matching, linearized Einstein equations, and wedge reconstruction Mondal 2026. A December 2025 two-area construction shows that different included area modes can give different Type-II or Type-III region algebras Cao, Faulkner, and Wang 2025. These are advances within declared constructions, not evidence for a unique nonperturbative algebra of all quantum gravity.

Falsifiers for stronger claims

A strong emergence claim must survive at least four tests:

• dictionary removal: if the conclusion disappears when RT or JLMS is not assumed, the result is conditional on holography;
• interaction variation: if different cubic couplings share all tested first- and second-order data, nonlinear uniqueness fails;
• algebra variation: if changing centers or area modes changes the entropy relation, the algebra must be part of the claim;
• sector escape: if heavy states, QES transitions, or finite-N corrections leave the code domain, a sector-wide equality cannot be globalized.

Failure of a stronger claim does not weaken the correctly scoped linearized or algebraic result.

Durable conclusion

Entanglement and modular theory provide exact boundary identities and unusually sharp consistency conditions on semiclassical holographic gravity. Together with a specified dictionary, they can imply linearized equations, match canonical energy, reconstruct wedge evolution, and realize generalized entropy algebraically. They do not by themselves derive the dictionary, choose a unique nonlinear action, establish a converse, or define quantum gravity beyond the controlled sector.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Cao, Xuchen, Thomas Faulkner, and Zhencheng Wang. “Gravitational Algebras with Two Areas.” arXiv preprint (2025). arXiv:2512.04435.
• Faulkner, Thomas, Monica Guica, Thomas Hartman, Robert C. Myers, and Mark Van Raamsdonk. “Gravitation from Entanglement in Holographic CFTs.” Journal of High Energy Physics 2014, 051 (2014). DOI; arXiv:1312.7856.
• Jafferis, Daniel L., Aitor Lewkowycz, Juan Maldacena, and S. Josephine Suh. “Relative Entropy Equals Bulk Relative Entropy.” Journal of High Energy Physics 2016, 004 (2016). DOI; arXiv:1512.06431.
• Mondal, Avinandan. “Holography in the Linearized Quantum Gravity Regime and Modular Crossed Product.” arXiv preprint (2026). arXiv:2607.27337.