Necessary, Sufficient, and Heuristic Bulk Criteria

No single familiar large-$N$ signal is sufficient for a local Einstein bulk. Factorization supports weak interactions, a large stress-tensor coefficient supports a Planck hierarchy once a gravitational dictionary is known, spectral sparsity limits the light field content, a higher-spin gap removes a central obstruction to Einstein dynamics, and controlled Mellin growth supports a derivative expansion. Each conclusion is conditional on a theory, sector, observable class, energy window, and precision. The useful result is therefore a map from hypotheses to licensed conclusions, with explicit failures of tempting converses—not a checklist promoted to a theorem.

Required background. Sparse Spectra, Large Gaps, and Semiclassical Bulk Criteria defines light-state sparsity; Higher-Spin Gaps and Einstein-Regime Obstructions isolates the spin-sensitive condition; and Corrections, Nonuniform Limits, and Failure Modes supplies the finite-$N$, finite-gap, and scaled-limit qualifications.

Helpful background. Falsifiers, Negative Results, and Counterexamples gives the general logic of a decisive failure test, while Large-Gap Constraints and CFT-Side Locality Tests develops the boundary-data constraints.

The target determines what is necessary

A condition is necessary only relative to a target conclusion. Here “sufficient” will mean sufficient for reproducing a declared set of CFT correlators by a perturbative bulk theory within a stated energy window and to stated orders in $1/N$ and the derivative expansion. It will not mean existence or uniqueness of an exact quantum-gravity completion. With that distinction in place, three targets should not be conflated:

1. A weakly interacting bulk field sector: connected correlators of normalized single-trace operators are parametrically suppressed and admit a particle-like organization.
2. A perturbative local AdS effective field theory: the selected light single-trace sector has an approximate Fock-space organization, omitted exchanges are separated from it, and its correlators can be reproduced by a derivative expansion below the lowest relevant EFT cutoff.
3. An Einstein regime: in addition, the stress tensor has the correct isolated graviton role and omitted single-trace operators of spin $J>2$ are parametrically heavy enough that higher-spin exchanges do not compete with graviton dynamics in the domain of interest.

A theory can satisfy the first target and fail the other two. Conversely, a bulk with light higher-spin gauge fields is still a bulk, but it is not an Einstein EFT. The word “necessary” without its target silently excludes valid non-Einstein duals.

What each criterion supports—and what it does not

The main implications are best stated with their missing converses.

Factorization. If normalized connected $n$-point functions scale as $N^{2-n}$ in a matrix-like sector, then cubic and higher interactions of the corresponding canonically normalized fields are parametrically weak. This is necessary for the ordinary classical-field limit of that sector. It is not sufficient for dynamical gravity or an interacting local EFT. Generalized-free correlators can have a free-field representation on a fixed AdS background while furnishing neither a standalone local boundary CFT with a stress tensor nor a metric dictionary. Vector models show independently that factorization need not imply an Einstein regime.

Large $C_T$. In a theory already known to have an Einstein-like AdS dual,

$$C_T\propto\frac{L^{d-1}}{G_{d+1}}$$

turns $C_T\gg1$ into a Planck hierarchy. Without the dictionary, a large stress-tensor coefficient is boundary data, not proof of a metric bulk. It says nothing by itself about the string scale, higher-spin gap, or number of light species.

Sparse light single-trace spectrum. A uniformly finite set of light single-trace primaries is compatible with—and, once a dictionary exists, supplies candidate field content for—a bulk description with finitely many elementary fields below a cutoff. If the number grows along the family, that growth must instead be included in the loop-counting species factor. The threshold, spin range, and dependence on $N$ must always be stated. Sparsity alone neither constructs the interactions nor proves locality.

Large higher-spin gap. Parametrically separating omitted single-trace fields with $J>2$ that couple appreciably to the probed light sector is necessary for an Einstein-like regime over a parametrically large energy interval. It is not necessary for a higher-spin bulk. Within the weakly coupled gravitational-EFT and high-energy eikonal assumptions of Camanho et al. 2016, §§2–4 and 5.4, appreciable non-Einstein graviton three-point structures require new higher-spin physics at the corresponding scale. The conclusion depends on spacetime dimension and the exchanged representations; it is not a theorem that every finite higher-derivative coefficient forces the same totally symmetric tower.

Mellin poles and local contact terms. Exchange poles with factorizing residues plus controlled polynomial contact terms support perturbative AdS EFT. Even complete pole or discontinuity data leave polynomial subtraction or contact ambiguities; their coefficients require additional OPE, low-energy, or subtraction input. Finite pole and Taylor data leave a still larger ambiguity because an entire addition—one analytic everywhere in the Mellin variables—can preserve those data while growing faster than any polynomial in an untested direction.

Heemskerk, Penedones, Polchinski, and Sully established a scalar four-point counting result in a broad planar, gapped setup at the first nontrivial order $O(1/N^2)$ Heemskerk et al. 2009, §§2–4. Fitzpatrick and Kaplan formulated broader criteria for a perturbative AdS EFT Fitzpatrick and Kaplan 2013, §§1–3. Neither result is a general all-orders reconstruction theorem. For a pedagogical treatment of large-$N$ expansions and Mellin space, see Bissi, Sinha, and Zhou 2022, §§4 and 6.

Regge and dispersive control. Polynomial behavior in a Euclidean Mellin regime and a Lorentzian Regge bound are related but distinct assumptions; each needs its own kinematic domain and subtraction count. Dispersive methods turn such assumptions into sharp Wilson-coefficient bounds in specified large-gap setups Caron-Huot et al. 2021, §§1–2 and 5. Locality sum rules provide another precise test: for a QFT already formulated on fixed AdS, they constrain one-bulk/two-boundary form factors and can produce a manifestly local representation Levine and Paulos 2024, §§2–4. That observable-specific result does not by itself make the metric dynamical or derive gravity from an arbitrary CFT.

Top-down dictionary. A specified boundary theory, bulk string construction, dictionary, and controlled parameter regime provide an extensively tested duality candidate far stronger than correlated indicators. Even then, a supergravity truncation has finite-$N$, string-scale, Kaluza–Klein, and nonperturbative limits. One example does not prove a universal criterion for all CFTs.

Four theories evaluated on the same map

The comparison below uses “yes” only for the stated large-$N$ sector; it does not erase coupling, spin, or operator-class qualifications. “Single-trace analogue” in a vector model means a singlet bilinear, not a matrix trace.

Theory or data class	Evidence that works	Missing hypothesis or obstruction	Evidence class, uncertainty, and licensed conclusion
Generalized-free scalar data	Exact mean-field factorization and crossing for the chosen correlator; a free scalar representation on fixed AdS may exist	No intrinsic local stress tensor, metric dynamics, $N$ parameter, or controlled finite-$N$ completion need be present	Formal correlator data: mean-field correlators, possibly represented by a free field on fixed AdS; no dynamical metric or complete holographic CFT follows
Critical large-$N$ $O(N)$ vector model	Vector-model factorization, a local stress tensor, and typically $C_T\sim N$	Singlet bilinears include even-spin currents $s=4,6,\ldots$ that are conserved at $N=\infty$ and only weakly broken at finite $N$; the stress tensor remains conserved at $s=2$	Concrete CFT plus a proposed higher-spin dual: a weakly coupled higher-spin organization is supported, while locality beyond tested observables and a bulk finite-$N$ completion remain conditional
Weakly coupled four-dimensional $\mathcal N=4$ $SU(N)$ Yang–Mills theory, $\lambda\ll1$	Matrix factorization, $C_T\sim N^2$, and double-line genus counting	Long towers of low-dimension single-trace operators and nearly conserved higher-spin currents prevent a parametrically large string-scale gap	Perturbation theory in a named local CFT plus large-$N$ counting: string-like topology is supported, but gap control and an Einstein truncation are absent
Strong-coupling $AdS_5\times S^5/\mathcal N=4$ $SU(N)$ dual pair	Explicit dictionary, $C_T\sim N^2$, factorization, and a string gap with $L^4/\ell_s^4=\lambda$	$g_s=\lambda/(4\pi N)$ must be small for perturbative strings; supergravity also needs $\lambda\gg1$ and $EL\ll\lambda^{1/4}$; Kaluza–Klein modes remain at the AdS scale	Top-down duality candidate with extensive checks: perturbative Type IIB string theory, and supergravity within the declared hierarchy; exact finite-$N$ control and a unique nonperturbative formulation are not established by these limits alone

The vector-model row is realized concretely by the critical $O(N)$ model and its higher-spin AdS proposal Klebanov and Polyakov 2002, abstract and §§2–3. The $O(1/N)$ breaking of the higher-spin currents in the critical theory is computed in Giombi and Kirilin 2016, §4, especially eq. (4.16). This example defeats the inference

$$\text{factorization}+\text{large central charge} \quad\Longrightarrow\quad \text{Einstein gravity}.$$

The generalized-free row is deliberately limited: its possible free field on fixed AdS is not an interacting theory of dynamical geometry. The matrix row shows that double-line topology is also insufficient. The top-down row refers to the large-$N$, strong ’t Hooft-coupling regime of the $AdS_5\times S^5$ correspondence Maldacena 1998, §§2–4. Even there, the Kaluza–Klein tower means that “pure five-dimensional Einstein gravity with no other light fields” would be an overstatement; a finite field subset requires a consistent truncation.

A recent two-dimensional example sharpens the observable dependence. At symmetric-product orbifold points, selected large-$N$ thermal two-point and heavy–light four-point functions can mimic correlators in a BTZ background even though the bulk regime is tensionless and non-Einstein Belin et al. 2025, §§1, 2.2, and 5. A geometric-looking answer for one observable class is therefore evidence to explain, not a sufficient criterion for semiclassical spacetime.

Removing one hypothesis at a time

Test whether each hypothesis is necessary by removing it while holding the others fixed.

Delete factorization. Without factorization, order-one connected correlators are allowed, so canonical bulk interactions need not be parametrically weak. The classical saddle conclusion is no longer licensed. This is a loss of control, not a counterexample to every possible bulk description; a stronger exclusion is unresolved without a specified theory.

Delete the stress-tensor dictionary. Generalized-free data can retain factorization and an apparently sparse primary list while lacking a local stress tensor with the Ward identities needed to identify a graviton. A metric bulk inference then has no boundary anchor.

Delete light-spectrum sparsity. Without sparsity, an increasing number of elementary bulk fields may lie below the proposed cutoff. The strongest survivor is a possibly weakly coupled bulk with an unbounded light-field sector; whether it admits a useful EFT is model-dependent.

Delete the higher-spin gap. Vector models supply an explicit counterexample to an Einstein inference. The surviving claim is a possible higher-spin bulk.

Delete polynomial boundedness. At the level of formal Mellin data, one can add a crossing-symmetric entire deformation that introduces no poles and vanishes through any prescribed Taylor degree, yet grows faster than any polynomial along a declared Mellin direction. This proves only that finite pole and Taylor data do not imply Mellin polynomial boundedness. Establishing a Lorentzian Regge violation additionally requires a convergent Mellin representation and the appropriate second-sheet continuation of the full correlator. The deformation is not by itself a unitary-CFT counterexample.

Delete finite-$N$ control. Two exact functions can share every coefficient of the genus expansion and differ by $e^{-N}$. Perturbative bulk agreement survives; uniqueness of the exact completion does not. Nonperturbative Exponential Effects and Finite-N Sectors develops this distinction.

Delete a fixed-theory specification. An ensemble-averaged model and a single boundary theory can agree on selected averaged observables. The data then do not decide which interpretation is intended. Without a concrete ensemble/fixed-theory pair, the stronger interpretation is unresolved.

If no counterexample is known after deleting a hypothesis, the correct entry is unresolved, not sufficient. Absence of a counterexample in a surveyed class is evidence about that survey, not a theorem.

A conditional sufficiency package

For a unitary $d$-dimensional CFT with a unique stress tensor and a declared family of normalized operators, use the following notation:

• $\epsilon_{\mathrm{fact}}$ measures the suppression of normalized three-point couplings. In standard single-parameter matrix examples, $\epsilon_{\mathrm{fact}}\sim C_T^{-1/2}$.
• $Q$ is the local momentum or curvature scale of the bulk process, and $D=d+1$ is the bulk spacetime dimension.
• $b_D(Q)$ is a dimensionless, process-dependent loop coefficient, including the appropriate loop factors, while $N_{\mathrm{eff}}(Q)$ is the weighted number of species that can run in the loop.
• $\Delta_{\mathrm{EFT}}=L\Lambda_{\mathrm{EFT}}$ is the first scale at which the retained field content, weak-coupling expansion, or derivative expansion ceases to be justified. It need not equal any one physical omitted-state gap.

Use the two-derivative Einstein normalization $C_T=\kappa_dL^{d-1}/G_D$, where $\kappa_d>0$ is the convention-dependent factor fixed in Central Charge, Newton Coupling, and the Planck Scale. A process-dependent estimate of its closed-loop expansion parameter is

$$\delta_{\mathrm{loop}}(Q) \sim \lvert b_D(Q)\rvert G_DN_{\mathrm{eff}}(Q)Q^{D-2} =\frac{\lvert b_D(Q)\rvert\kappa_dN_{\mathrm{eff}}(Q)}{C_T} (QL)^{d-1}.$$

The following package is strong evidence for a perturbative local AdS sector. It is meaningful only if dimensions, OPE coefficients, and the relevant correlators also admit a controlled asymptotic expansion in the declared small parameter, with fixed insertion number and uniform remainder control over the stated operator and kinematic window.

1. Large central scale and factorization. Require $C_T\gg1$, $\epsilon_{\mathrm{fact}}\ll1$, and, for fixed $n\ge3$,

   $$\langle\mathcal O_1\cdots\mathcal O_n\rangle_{\mathrm{conn}} =O\!\left(\epsilon_{\mathrm{fact}}^{n-2}\right).$$

2. Loop control in the claimed window. Require $\delta_{\mathrm{loop}}(Q)\ll1$ for every process used in the inference, not merely at $Q\sim L^{-1}$.

3. A controlled light sector. Require an approximate low-energy Fock-space organization generated by a uniformly finite set of light single-trace operators, or include any growth of that set in $N_{\mathrm{eff}}$. State the operator class and the threshold defining “light.”

4. A higher-spin hierarchy for an Einstein target. Require a parametrically large gap for omitted single-trace operators with $J>2$ that couple to the probed sector. Here $\Delta_{\mathrm{HS}}$ denotes a dimension gap; results using a higher-spin twist gap $\tau_{\mathrm{HS}}$ impose a distinct hypothesis.

5. CFT consistency in the selected sector. Require unitarity, crossing, OPE associativity, Ward identities, and compatibility among all correlators used—not consistency of one four-point function in isolation.

6. Locality data with separate growth assumptions. Require appropriately subtracted Mellin amplitudes with a fixed-order polynomial EFT expansion, plus separately stated Lorentzian Regge bounds and subtraction assumptions.

7. A declared domain and order of limits. Work in a window $QL\ll\Delta_{\mathrm{EFT}}$ and state the orders retained in factorization, loops, and derivatives. Approximate Bulk Locality from Spectral and Mellin Data develops the reconstruction side of this requirement.

Vertices are then organized in $\epsilon_{\mathrm{fact}}$, while closed loops are controlled by $\delta_{\mathrm{loop}}$. The latter becomes $O(\epsilon_{\mathrm{fact}}^2)$ only in standard single-parameter families with $b_D$, $N_{\mathrm{eff}}$, and $QL$ held fixed and $C_T^{-1}\sim\epsilon_{\mathrm{fact}}^2$. Derivative errors are powers of $QL/\Delta_{\mathrm{EFT}}$.

Calling this package “sufficient” still requires a narrow conclusion. Under the hypotheses of perturbative reconstruction results, it can be sufficient for representing a declared set of correlators by local AdS fields to a stated order. It is not a general theorem guaranteeing a unique exact quantum-gravity Hilbert space, all topologies, a fixed-theory interpretation, or a nonperturbative completion.

Common pitfalls

Confusing a local field on fixed AdS with dynamical gravity. Generalized-free data may have the former representation without a boundary stress tensor or a fluctuating metric. Always state which target is being inferred.

Quoting a gap without its sector. A tower that is decoupled from the selected correlators need not spoil their EFT, while a tower that couples appreciably can. Specify spin, representation, couplings, and how the gap scales along the family.

Treating perturbative agreement as an exact definition. Agreement to every algebraic order in $1/N$ can still miss exponentially small sectors. Exact-completion claims need independent finite-$N$ and nonperturbative information.

Exercises

Diagnose a higher-spin obstruction. A CFT family has $C_T\sim N^2$ and factorization, but currents at every even spin $s=4,6,\ldots$ remain conserved and couple nontrivially to the probed sector. What is the strongest bulk statement supported by these facts?

Solution

They support a weakly interacting large-$N$ organization compatible with light higher-spin bulk fields. They do not support an Einstein EFT with a parametrically separated higher-spin scale. A specific higher-spin bulk interpretation still requires dictionary and locality evidence. If the tower were completely decoupled from the selected sector, this conclusion would apply to the full theory but would not by itself exclude an Einstein-like EFT for those selected correlators.

Compare evidence levels. Why is a top-down dual pair stronger evidence than satisfying the boundary indicators above, yet still insufficient for a universal criterion?

Solution

A top-down construction specifies both candidate descriptions, the dictionary, charges, and tunable regimes, so it supplies a controlled and extensively tested realization rather than an inference from correlated indicators alone. It remains one class of examples with its own compactification and corrections; it does not prove that every CFT satisfying similar indicators has the same kind of bulk or a unique completion.

Test a sparse generalized-free model. Start with generalized-free scalar four-point data and declare that no additional formal single-particle primaries lie below a large dimension. Which entries of the sufficiency package remain unproved?

Solution

Mean-field factorization and the declared spectral gap hold only at the level of the chosen correlator data. Those data may represent a free scalar on fixed AdS, but a local boundary stress tensor with the appropriate Ward identities, a complete unitary CFT operator algebra, metric dynamics, a finite-$N$ completion, and controlled Mellin or Regge behavior for the full sector remain unproved. The example therefore cannot license a local Einstein bulk.

Remove polynomial and Regge control. Suppose all exchange poles and their residues are known and a Mellin amplitude agrees with a local EFT through Taylor degree $K$. Explain why this does not fix a local completion, and state what a rapidly growing entire deformation does not prove.

Solution

Complete pole data still permit crossing-symmetric polynomial contact terms. Finite Taylor data permit still more: multiply a crossing-symmetric entire function by enough powers of centered Mellin invariants to make its Taylor series begin above degree $K$. Such a term can grow faster than any polynomial and therefore violate the assumed EFT bound. It proves that the finite data do not imply polynomial boundedness; without a convergent Mellin representation, Lorentzian continuation, unitarity, and a complete CFT, it is not a physical counterexample by itself.

Check a top-down scaling path. In the standard $AdS_5\times S^5$ conventions, take $\lambda=N^{1/2}$ as $N\to\infty$. Do perturbative strings and a low-energy supergravity window become parametrically controlled?

Solution

Yes, within the specified sector and energy range. The string coupling scales as $g_s=\lambda/(4\pi N)=1/(4\pi N^{1/2})\to0$, while the string gap in AdS units grows as $L/\ell_s=\lambda^{1/4}=N^{1/8}$. Thus string loops are suppressed and supergravity is valid for energies satisfying $EL\ll N^{1/8}$. Kaluza–Klein modes remain at the AdS scale, so a finite five-dimensional field subset still requires a consistent truncation.

Limits of the conclusion

Evidence cutoff: 28 August 2026. Known top-down examples, perturbative reconstruction results, and modern locality bounds make the combined criteria scientifically powerful. Vector models, generalized-free data, weakly coupled matrix theories, and symmetric-product orbifolds refute several tempting converses. Formal rapidly growing Mellin deformations expose underdetermination but are not automatically unitary-CFT counterexamples. No general classification identified at this cutoff turns the package into a unique nonperturbative bulk theorem for arbitrary CFTs.

Large-$N$ saddle, matrix/vector, factorization, and master-field methods remain with Nonperturbative Dynamics, while CFT constraints and data remain with Large-Gap Constraints and CFT-Side Locality Tests. Higher-Spin and Vector-Model Holography develops the counterexamples to an Einstein inference. Mutable examples and sufficiency assessments continue in the Holography and Quantum Gravity research field guide; theorem-level existence or reconstruction statements belong to Mathematical QFT.

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

• Belin, Alexandre; Bintanja, Suzanne; Castro, Alejandra; and Knop, Waltraut. “Symmetric Product Orbifold Universality and the Mirage of an Emergent Spacetime.” Journal of High Energy Physics 2025, 190 (2025). doi:10.1007/JHEP05(2025)190. Open PDF.
• Bissi, Agnese; Sinha, Aninda; and Zhou, Xinan. “Selected Topics in Analytic Conformal Bootstrap: A Guided Journey.” Physics Reports 991, 1–89 (2022). doi:10.1016/j.physrep.2022.09.004. Open PDF.
• Camanho, Xian O.; Edelstein, José D.; Maldacena, Juan; and Zhiboedov, Alexander. “Causality Constraints on Corrections to the Graviton Three-Point Coupling.” Journal of High Energy Physics 2016, 020 (2016). doi:10.1007/JHEP02(2016)020.
• Caron-Huot, Simon; Mazáč, Dalimil; Rastelli, Leonardo; and Simmons-Duffin, David. “AdS Bulk Locality from Sharp CFT Bounds.” Journal of High Energy Physics 2021, 164 (2021). doi:10.1007/JHEP11(2021)164. Open PDF.
• Fitzpatrick, A. Liam, and Kaplan, Jared. “AdS Field Theory from Conformal Field Theory.” Journal of High Energy Physics 2013, 054 (2013). doi:10.1007/JHEP02(2013)054. Open PDF.
• Giombi, Simone, and Kirilin, Vladimir. “Anomalous Dimensions in CFT with Weakly Broken Higher Spin Symmetry.” Journal of High Energy Physics 2016, 068 (2016). doi:10.1007/JHEP11(2016)068. Open PDF.
• Heemskerk, Idse; Penedones, João; Polchinski, Joseph; and Sully, James. “Holography from Conformal Field Theory.” Journal of High Energy Physics 2009, 079 (2009). doi:10.1088/1126-6708/2009/10/079. Open PDF.
• Klebanov, Igor R., and Polyakov, Alexander M. “AdS Dual of the Critical $O(N)$ Vector Model.” Physics Letters B 550, 213–219 (2002). doi:10.1016/S0370-2693(02)02980-5.
• Levine, Nat, and Paulos, Miguel F. “Bootstrapping Bulk Locality. Part I: Sum Rules for AdS Form Factors.” Journal of High Energy Physics 2024, 049 (2024). doi:10.1007/JHEP01(2024)049. Open PDF.
• Maldacena, Juan M. “The Large $N$ Limit of Superconformal Field Theories and Supergravity.” Advances in Theoretical and Mathematical Physics 2, 231–252 (1998). doi:10.1023/A:1026654312961. Open PDF.