Beyond Linear Order: Conditional Emergent-Equation Claims

The entanglement first law fixes a linear variation, and second-order relative entropy fixes a quadratic canonical-energy form. Neither result automatically supplies nonlinear gravitational equations. Extending the inference requires higher state variations, interacting bulk data, corrected entropy functionals, locality, gauge closure, and a sufficiently rich family of regions and states.

Required background. Entanglement First Law and Conditional Linearized Field-Equation Inference supplies the linear argument. Canonical Energy and Second-Order Relative Entropy supplies the quadratic identity.

Helpful background. Exchange Witten Diagrams and Conformal Blocks supplies interaction data. Validity, Unitarity, and Breakdown supplies the gravitational-EFT ceiling.

First application. Carry the ball-region argument to second order and identify which terms are fixed by relative entropy and which require independent bulk dynamics.

What first and second order know

For a state family $\rho(\lambda)$ about $\sigma$,

$$S(\rho(\lambda)\Vert\sigma) =\frac{\lambda^2}{2}\mathcal I_2 +\frac{\lambda^3}{6}\mathcal I_3+\cdots.$$

The vanishing linear term is the first law. In the standard AdS ball argument, equality for all balls plus the RT and stress-tensor dictionaries implies the linearized field equation. $\mathcal I_2$ matches canonical energy when the bulk perturbation already obeys the linearized equations and the correct surface terms are included.

At cubic and higher order, new data appear: CFT three- and higher-point functions, bulk interaction vertices, nonlinear constraint terms, QES displacement, bulk entanglement variations, and loop corrections. These are not fixed by the two-point information metric. Two bulk effective actions can share the same linearized spectrum and quadratic form while differing in their cubic couplings.

Second-order ball calculation

Expand the bulk metric as

$$g(\lambda)=g_{\mathrm{AdS}}+\lambda h^{(1)} +\lambda^2 h^{(2)}+O(\lambda^3).$$

At first order, the integral identity constrains $E^{(1)}[h^{(1)}]$. At second order, the equation has the structure

$$E^{(1)}[h^{(2)}] +E^{(2)}[h^{(1)},h^{(1)}] =8\pi G_N\,T^{(2)}.$$

Relative entropy determines an integrated canonical-energy combination built from $h^{(1)}$. To isolate the local equation for $h^{(2)}$, one must know the nonlinear entropy/charge identity, invert the family of region integrals, fix gauge and boundary data, and specify the interaction tensor $E^{(2)}$. The quadratic information quantity alone does not choose $E^{(2)}$.

Independent inputs needed for closure

A nonlinear emergence claim must state at least:

• the boundary operator algebra and all correlators needed at the claimed order;
• a local bulk field content and derivative expansion;
• the entropy functional, including higher-curvature and bulk-entropy terms;
• a covariant phase-space identity valid at that order;
• constraints and gauge transformations that close without anomaly;
• the state/region family over which the integral transform is injective;
• the $1/N$, coupling, derivative, and amplitude ordering.

Jacobson’s entanglement-equilibrium argument provides a distinct local-small-ball route under assumptions about vacuum entanglement, fixed volume, and an entropy density Jacobson 2016. It is not a derivation from the holographic ball first law alone and has separate higher-curvature qualifications.

Adversarial controls

Change a cubic coupling. Hold the linearized spectrum and two-point normalization fixed while changing an allowed three-point structure. First-law and Fisher data remain unchanged, but nonlinear equations differ.

Add a field redefinition. The printed equation changes while observables do not. An emergence claim framed around one off-shell equation rather than invariant correlators can mistake coordinates on theory space for dynamics.

Restrict the state family. An integral identity on a small subset can leave local components unconstrained. One must establish injectivity rather than assume it.

Omit quantum entropy. At the same order where bulk loops contribute to the field equation, bulk entropy and renormalization can contribute to the QES relation.

Evidence ceiling

Higher relative-entropy variations and covariant identities can constrain nonlinear bulk interactions when supplemented by complete CFT correlator data, a local EFT ansatz, matched entropy corrections, and an injective region family. First-law or canonical-energy equality alone does not derive a unique nonlinear action, its UV completion, or exact finite-N dynamics.

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

• 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.
• Jacobson, Ted. “Entanglement Equilibrium and the Einstein Equation.” Physical Review Letters 116, 201101 (2016). DOI; arXiv:1505.04753.
• Lashkari, Nima, and Mark Van Raamsdonk. “Canonical Energy Is Quantum Fisher Information.” Journal of High Energy Physics 2016, 153 (2016). DOI; arXiv:1508.00897.