Falsifiers, Negative Results, and Counterexamples

A negative result has force only against a proposition precise enough to be wrong. It must name the theory or proposed dual pair, state and sector, observable and dictionary entry, normalization, domain, limit order, admitted corrections, and uncertainty model. Failure of a solver or approximation therefore does not automatically falsify a duality; conversely, one counterexample can decisively refute a genuinely universal claim.

Required background. Holographic Duality: Claims, Dictionaries, and Regimes identifies the claim, while Evidence Programs for Holographic Duality identifies the test and its assumptions.

Helpful background. Duality Checks, Evidence Independence, Status, and Failure Modes gives the general duality setting. EFT Truncation Errors and Breakdown Diagnostics distinguishes expected breakdown from contradiction.

Reading path. First learn what each kind of failure excludes, then use the six-step decision rule. The central application separates a finite-gap failure, an exact inconsistency, and a failed extrapolation. The final sections give an adversarial approximation test, genuine counterexamples, and exercises.

What a failed test can rule out

A negative result is any outcome that fails to support the proposed conclusion; it may be decisive, inconclusive, or merely uninformative. A falsifier is a reproducible contradiction obtained inside the declared domain of a frozen proposition. A counterexample satisfies the premises of a universal or sufficient-condition claim while violating its conclusion. A method failure shows that the calculation or inference procedure is unreliable without yet deciding the physics.

The distinction can be made mathematically. Let $\mathcal A_P(\mathcal D)$ be the set of values allowed by proposition $P$ in domain $\mathcal D$, after every admitted correction, nuisance parameter, and licensed limiting case has been included. An exact result $x_{\mathrm{exact}}$ contradicts $P$ if

$$x_{\mathrm{exact}}\notin\mathcal A_P(\mathcal D).$$

This is a formal contradiction. A stronger, perturbatively robust exclusion has positive distance from the allowed set,

$$d\!\left(x_{\mathrm{exact}},\mathcal A_P(\mathcal D)\right)>0,$$

or equivalently lies outside its closure. The distinction matters for a claim such as $x>0$: the exact result $x=0$ contradicts it even though arbitrarily small positive values approach zero. For a finite-precision observation with confidence set $\mathcal C_{\mathrm{obs}}(1-\alpha)$, the operational exclusion condition is instead

$$\mathcal C_{\mathrm{obs}}(1-\alpha) \cap \mathcal A_P(\mathcal D)=\varnothing.$$

For example, if $P$ allows $x\in[0.90,1.00]$ while the 95% confidence set is $[1.02,1.08]$, the sets are disjoint and $P$ is excluded at that stated confidence level. If a justified calibration uncertainty enlarges the allowed set through $1.04$, the overlap makes the result unresolved. The exclusion is conditional on the confidence-set construction, coverage, likelihood, nuisance model, and systematic-error treatment; it is not a deductive proof that nature lies outside $P$. Test power instead controls what may be inferred from a non-exclusion.

Five failure types and the narrowest conclusion each supports

Failure type	Proposition challenged	What must be controlled	What may survive
Exact-data or dictionary contradiction	Candidate exact data violate consistency, or two quantities claimed to be exact images disagree	Complete data or a rigorous certificate; for a map, also the theory pair, state, sector, normalization, counterterms, and continuation	Other candidate data, dictionary entries, or a narrower sectoral map
Absence of the claimed regime	A necessary condition for the advertised bulk description cannot be met	Scale hierarchy, spectrum, kinematics, phase, and order of limits	The boundary theory and a stringy, higher-spin, or otherwise non-Einstein bulk
Error-bound violation	An in-domain remainder exceeds an independently justified bound	Uniformity of the bound, numerical and experimental uncertainty, and omitted terms	The leading mechanism with weaker precision or additional corrections
Counterexample to sufficiency	The proposed premises hold but their claimed conclusion fails	Every premise, quantifier, and stated scope of the implication	The premises as necessary conditions or heuristic indicators
Computational-method failure	A solver, truncation, saddle choice, or extrapolation does not support its reported answer	Benchmarks, precision and cutoff refinement, alternative implementations, and out-of-fit prediction	The approximation and the physical proposition tested by another method

These rows are not automatically a ladder. A method failure does not propagate to a physical claim, and failure of one dictionary entry does not propagate to an exact dual pair without an additional completeness premise. The chapter overview’s validity-and-failure diagram shows this branching structure.

Freeze the proposition before naming a falsifier

A reproducible classification takes six steps.

1. Write the proposition with its quantifiers. “All unitary CFTs have property $Q$” and “this family approaches $Q$ as $N\to\infty$” are different claims.
2. Freeze the physical target. Record the theories, state or ensemble, sector, observable, dictionary map, normalization, regulator, and ordered limits.
3. Freeze the validity domain. State the energy and coupling window, retained orders, spectral assumptions, and the phase in which the comparison is licensed.
4. Predeclare uncertainty and the decision rule. Include correlated systematics and theory error; specify what counts as passing, failing, or remaining unresolved.
5. Run controls and an independent check. Refine cutoffs and precision, vary nuisance choices only within their declared ranges, benchmark special limits, and reproduce the result analytically or with independent code where possible.
6. Name the narrowest failed proposition and strongest survivor. Also state the alternative explanation and the next observation that would distinguish it.

If the observable, domain, or tolerance can still move after the result is seen, stop at unresolved discrepancy. The preceding page on Evidence Independence, Circularity, and Double Counting explains how shared assumptions or calibration data can make apparently independent reproductions fail together.

A useful final sentence has the form: “Under assumptions $A$ in domain $\mathcal D$, result $R$ excludes proposition $P$ at precision $\delta$; claim $S$ survives, and test $T$ distinguishes the leading alternative.” It is deliberately more informative than “the model failed.”

Worked application: finite gap, exact inconsistency, or failed extrapolation?

Use a three-part AdS$_5$/CFT$_4$ benchmark to make the classification reproducible. On the Euclidean boundary, take a Hermitian scalar of dimension $\Delta_\phi=2$ normalized by

$$\langle\phi(x)\phi(0)\rangle=\frac{1}{(x^2)^2}.$$

Let $L$ be the AdS radius and $N$ the matrix-rank parameter. The global (+---) convention applies after Lorentzian continuation. The numbers below are analytic or synthetic test inputs—not observations and not evidence against a real duality—and they freeze three propositions:

• $P_{\mathrm{Ein}}$: the normalized discrepancy is $\mathcal E(E)=\lvert\mathcal O_{\mathrm{CFT}}(E)-\mathcal O_{\mathrm{Ein}}(E)\rvert=(E/M_{\mathrm{HS}})^2$, with $M_{\mathrm{HS}}L=20$, and obeys $\mathcal E\leq0.10$ throughout $0\leq EL\leq20$;
• $P_{\mathrm{CFT}}$: the proposed complete identical-scalar data contain only the identity, so $\mathcal G(u,v)=1$, and satisfy exact crossing throughout the Euclidean domain; the adversarial test point is $z=\bar z=1/3$, hence $(u,v)=(1/9,4/9)$;
• $P_{\mathrm{num}}$: the exact synthetic sequence given below is fitted by $X_N=A+B/N^2$ over each declared three-point window, and the intercept estimates the large-$N$ target within $\delta_{\mathrm{fit}}=0.01$.

The analytic fixtures have no sampling uncertainty; displayed rounding is below $10^{-6}$. Their controls are direct substitution, interchange of the two crossing channels, and evaluation at both passing and failing energies. The numerical branch has no measurement noise but does have model-selection uncertainty, tested by the preregistered fit windows and a held-out or subleading-term check. Thus $P_{\mathrm{Ein}}$ is a conditional EFT error claim, $P_{\mathrm{CFT}}$ an exact consistency claim, and $P_{\mathrm{num}}$ a computational inference. A physical application would replace these synthetic inputs with measured or independently calculated quantities and their full covariance.

Finite gap: the Einstein window can lose control

Let $M_{\mathrm{HS}}$ be the bulk mass associated with the lightest single-trace primary of spin $J>2$. In a parametrically large-gap limit,

$$M_{\mathrm{HS}}L=\Delta_{\mathrm{HS}}+O(1).$$

An observable may then receive a schematic higher-spin correction

$$\epsilon_{\mathrm{HS}}(E) \sim c_{\mathcal O} \left(\frac{E}{M_{\mathrm{HS}}}\right)^q, \qquad q>0,$$

where $c_{\mathcal O}$, the exponent, and other admitted terms depend on the observable and EFT operators. When $E/M_{\mathrm{HS}}\ll1$, gap power counting can support a controlled Einstein window. When $E/M_{\mathrm{HS}}=O(1)$, it no longer guarantees suppression; the actual correction can still be small if its coefficient vanishes or is symmetry-suppressed. Data there cannot refute an approximation whose declared domain ended below the gap. They do refute $P_{\mathrm{Ein}}$ if it promised small error through that energy and the discrepancy survives a computed remainder and uncertainty budget. The exact CFT and a bulk completion containing string or higher-spin states may remain fully viable; see Higher-Spin Gaps and Einstein-Regime Obstructions and Corrections, Nonuniform Limits, and Failure.

In the frozen benchmark, $q=2$, $c_{\mathcal O}=1$, and $M_{\mathrm{HS}}L=20$. Direct substitution gives

$$\mathcal E(EL=5)=\frac{1}{16}=0.0625, \qquad \mathcal E(EL=20)=1.$$

The first point satisfies the declared $0.10$ tolerance, while the second violates it inside the advertised endpoint. This rejects the benchmark’s $P_{\mathrm{Ein}}$, not $P_{\mathrm{CFT}}$. The calculation is exactly reproducible because the discrepancy generator was defined as part of the fixture; it does not assert that a real CFT has coefficient one.

Causality makes the distinction concrete. Write the four- and six-derivative graviton-coupling coefficients as $\alpha_2$ and $\alpha_4$. In the weakly coupled, tree-level analysis of Camanho et al. 2016, § 1, pp. 2–4; §§ 3.3–4.4, pp. 20–34; §§ 5.2–5.5, pp. 36–43; § 8, pp. 47–49, causality requires an infinite tower of higher-spin states at parametrically related scales

$$M_{\mathrm{HS}}^2\sim\lvert\alpha_2\rvert^{-1}, \qquad M_{\mathrm{HS}}^4\sim\lvert\alpha_4\rvert^{-1}.$$

A finite collection does not suffice; in string theory, the tower Reggeizes the amplitude and removes the time advance. The result excludes the claimed low-energy regime, not a UV completion that supplies the tower. The large-gap route from CFT data to perturbative bulk locality is developed under explicit assumptions by Heemskerk et al. 2009, §§ 3–4, pp. 10–25.

Exact CFT consistency: a certificate must reach the full claim

The identity-only benchmark already gives an exact contradiction without a solver. Identical-scalar crossing requires

$$v^{\Delta_\phi}\mathcal G(u,v) =u^{\Delta_\phi}\mathcal G(v,u).$$

At the frozen point, the residual is

$$\left(\frac{4}{9}\right)^2 -\left(\frac{1}{9}\right)^2 =\frac{5}{27}\ne0.$$

Interchanging $u$ and $v$ flips the sign, which checks the channel convention. The symmetric point $u=v$ would give zero even for this inconsistent candidate and is therefore not a discriminating control. The nonzero exact residual rejects the benchmark’s $P_{\mathrm{CFT}}$ while saying nothing about complete data sets that contain the required nonidentity operators.

For a general identical-scalar exclusion, absorb the identity contribution into $F_{\mathbf 1}$ and write

$$F_{\mathbf 1} +\sum_{\mathcal O\ne\mathbf 1} \lambda_{\phi\phi\mathcal O}^{2} F_{\Delta,J}=0, \qquad \lambda_{\phi\phi\mathcal O}^{2}\geq0.$$

Suppose a rigorously validated linear functional $\alpha$ satisfies

$$\alpha(F_{\mathbf 1})=1, \qquad \alpha(F_{\Delta,J})\geq0$$

for every operator allowed by the frozen spectrum assumptions. Applying $\alpha$ makes the left side at least one, so the assumed data set cannot solve crossing. This is an exact contradiction of that CFT-data proposition—and hence of every dictionary that requires those data—only when block errors, the continuous positivity ranges, and numerical roundoff are enclosed by a valid certificate. A residual from a truncated OPE sum, or a solver status that changes with derivative order or precision, instead diagnoses truncation or method failure. The crossing and positivity page derives the sum rule; solver certificates and precision and error budgets develop the numerical controls. The separating-functional logic is reviewed in Simmons-Duffin 2017, § 10.4, pp. 54–55, while current algorithms and software are surveyed by Rychkov and Su 2024, §§ II–VII.

Large-N extrapolation: fit instability is not a physical contradiction

Use the transparent, dimensionless calibration sequence

$$X_N=1+\frac{1}{N^2}-\frac{16}{N^4}, \qquad X_\infty=1.$$

These are synthetic exact values, not CFT data. Fit each three-point window by unweighted least squares to the misspecified model $X_N=A+B/N^2$, omitting the $N^{-4}$ term. The only uncertainty here is structural: the preregistered stability tolerance is $\delta_{\mathrm{fit}}=0.01$.

A neglected subleading term moves the fitted large-N intercept

Values of N	Fitted intercept A	Absolute error from 1	Result at tolerance 0.01
2, 3, 4	1.310248	0.310248	Fails
4, 6, 8	1.019391	0.019391	Fails
6, 8, 12	1.003799	0.003799	Passes this check

The middle and late intercepts differ by $0.015592$, already larger than the declared tolerance. Because the exact generating rule is known, the omitted $N^{-4}$ term is the diagnosis. In real data one must compare plausible exponents and subleading terms, propagate covariance, withhold points for prediction, and seek an independent calculation. Until those checks stabilize, only $P_{\mathrm{num}}$ fails; no exact finite-$N$ theory or duality has been tested. A real large-central-charge modular-bootstrap analysis uses correspondingly careful language: its asymptotic gap is reported as numerical evidence from extrapolation, with the explicit caveat that a stronger limiting bound cannot be ruled out Afkhami-Jeddi, Hartman, and Tajdini 2019, § 1, pp. 2–3; § 5.2, pp. 14–16.

The same unfavorable outcome has different logical force at each layer

Observation	Classification	Narrowest rejected proposition	Strongest survivor and decisive follow-up
No higher-spin gap large enough for the promised energy window	Absence of the claimed regime	The advertised Einstein window	Exact CFT and non-Einstein bulk; identify the light tower and retest
In-domain discrepancy exceeds an independently proved EFT remainder	Error-bound violation	The stated precision guarantee	The leading EFT mechanism; include omitted terms or narrow the domain
Validated functional has a strictly positive identity value and is nonnegative on every allowed nonidentity block	Exact-data contradiction	The proposed CFT data plus their spectrum assumptions	Other boundary data or theories; relax one explicit assumption and recertify
Crossing residual moves under cutoff or precision refinement	Computational-method failure	The reported numerical conclusion	The CFT proposition; obtain a stable, independently checked certificate
Large-N intercept moves beyond its fit tolerance	Computational-method failure	The chosen extrapolation model and error bar	All exact finite-N data; add subleading terms and predict held-out points

Push a controlled approximation past its boundary

An adversarial validity test should cross a known boundary on purpose. Consider the exactly solvable response

$$F(\epsilon)=\frac{1}{1-\epsilon} =1+\epsilon+R_2(\epsilon), \qquad R_2(\epsilon)=\frac{\epsilon^2}{1-\epsilon}.$$

For $0\leq\epsilon\leq\epsilon_*=1/2$, the independently proved uniform contract

$$\lvert R_2(\epsilon)\rvert\leq2\epsilon^2$$

is valid. If a measured residual $r=\lvert F_{\mathrm{obs}}-1-\epsilon\rvert$ has total absolute uncertainty $\delta r$, then

$$r+\delta r\leq K\epsilon^2 \quad\Longrightarrow\quad\text{compatible},$$

whereas

$$r-\delta r>K\epsilon^2 \quad\Longrightarrow\quad\text{in-domain violation}.$$

The gap between these two conditions is unresolved. This interval rule avoids converting a central-value fluctuation into a false failure.

An in-domain false bound fails; an out-of-domain mismatch does not falsify the exact response

Advertised contract	Epsilon	Exact remainder	Advertised band	Classification
K = 2 through epsilon = 0.5	0.40	0.2667	0.3200	Compatible in-domain
K = 2 through epsilon = 0.5	0.60	0.9000	0.7200	Outside the licensed domain; expected breakdown
K = 1 through epsilon = 0.5	0.40	0.2667	0.1600	Advertised error guarantee is false

The second row is the required stop rule: once $\epsilon>\epsilon_*$, even an exactly reproduced excess cannot be promoted to a contradiction of the underlying response. The third row is different because the false bound was claimed inside its domain. A fitted envelope would not suffice as an independent proof of $K$; the guarantee must come from a theorem, convergent representation, interval calculation, or another derivation not calibrated on the same test points.

For the finite-gap QFT record above, the analogous control parameter is $\epsilon=E/M_{\mathrm{HS}}$ and $F$ would be the normalized observable reconstructed from the boundary correlator. A real EFT calculation must independently derive its own $K$, $\epsilon_*$, operator content, and uncertainty budget. The rational response is a unit test of the decision rule; it does not manufacture a remainder theorem for the QFT observable.

Counterexamples test sufficiency and universality

Large-$N$ factorization and large central charge do not suffice for an Einstein bulk. At $N=\infty$, the singlet sector of the $O(N)$ vector model contains conserved currents of every even spin, so there is no large higher-spin gap. This is a counterexample to the proposed sufficiency of “large $N$ plus factorization,” not to holography. Klebanov and Polyakov’s original proposal pairs the model with a non-Einstein higher-spin bulk; the boundary spectrum alone supplies the counterexample here, without requiring a claim about the full proposal’s current status Klebanov and Polyakov 2002, §§ 1–2, pp. 1–5. The detailed comparison belongs to Necessary, Sufficient, and Heuristic Bulk Criteria.

A counterexample can also defeat a sharp observable conjecture without threatening its successful limiting regime. For the thermal state of a top-down four-dimensional $\mathcal N=2$ $Sp(N)$ gauge theory, the controlled hierarchy

$$1\ll\lambda\ll N\ll\lambda^{3/2}$$

makes the negative curvature-squared correction dominate the positive finite-coupling correction:

$$\frac{\eta}{s} =\frac{1}{4\pi} \left[ 1-\frac{1}{2N} +O(N^{-2}) +O(\lambda^{-3/2}) \right].$$

The ratio is therefore below the proposed universal bound $\eta/s\geq1/(4\pi)$ at sufficiently large finite $N$ within this hierarchy Kats and Petrov 2009, § 1, eqs. (1.3)–(1.6), pp. 1–3. What fails is universality of the bound. The leading Einstein-gravity result and the holographic dictionary used to calculate the correction are not falsified.

What survives and what should be tested next

Every accepted negative result should end with a scoped downgrade and a next test.

• Method failure: preserve the physical proposition; benchmark, refine, or replace the method.
• Approximation failure: preserve the underlying model; add the omitted orders or narrow the domain.
• Dictionary-entry contradiction: reject that map under the frozen conventions; test an independent observable and inspect global or sector data.
• Absent Einstein regime: preserve a possible non-Einstein bulk; search for the light tower or nonlocal scale responsible.
• Counterexample to sufficiency: weaken the criterion to necessary or heuristic only if that weaker statement is separately supported.
• Exact-pair contradiction: exclude the specified pair only after the quantities and map are exact; other dualities do not inherit the failure.

The chapter overview also provides the compact claim-domain comparison. “Holography” and “quantum gravity” are research frameworks rather than single universally quantified predictions, so no solver residual, failed EFT window, or isolated counterexample can falsify either umbrella term.

Common pitfalls

Moving the domain after seeing the answer. A validity window narrowed only after a mismatch makes the test non-discriminating. Predeclare the domain and treat a justified later change as a new proposition.

Calling a truncated crossing residual exact. A finite OPE truncation lacks the omitted positive tail unless it is rigorously bounded. Instability under derivative order, precision, or spectrum discretization is a method diagnostic, not a CFT contradiction.

Treating expected breakdown as contrary evidence. A perturbative excess beyond its licensed coupling, a large-$N$ fit at small $N$, or a saddle expansion across a phase change says no more than the approximation promised.

Turning non-observation into universal absence. A null search excludes only the parameter region in which its sensitivity and power were quantified. Unseen sectors outside that region remain undecided.

Hiding shared systematics in several small error bars. Correlated normalization, calibration, regulator, or model errors do not become independent by appearing in different calculations. Profile or bound them jointly before claiming reproduction.

Exercises

A calculation claims that, in a fixed state, sector, and normalization, an Einstein EFT satisfies

$$\lvert\mathcal O_{\mathrm{CFT}}(E) -\mathcal O_{\mathrm{Ein}}(E)\rvert\leq0.05$$

for $0\leq EL\leq10$. Two independently implemented, cutoff-stable calculations find a residual $0.09$ with certified total absolute error at most $0.01$ at $EL=8$, and $0.20$ with certified total absolute error at most $0.02$ at $EL=12$. Give an end-to-end classification: target, domain, uncertainty rule, controls, narrowest rejected proposition, strongest survivor, leading alternative, and next discriminating test.

Solution: write the complete scoped result

The frozen target is the displayed observable comparison in the specified state, sector, and normalization; its domain ends at $EL=10$. At $EL=8$, the conservative lower residual is $0.09-0.01=0.08>0.05$, and the independent, cutoff-stable implementations supply the stated controls. This is an in-domain error-bound violation of the advertised Einstein-EFT precision. The $EL=12$ result lies outside the contract and has no falsifying force.

A complete report is: “Under the stated map, state, sector, and normalization for $0\leq EL\leq10$, the reproduced $EL=8$ residual excludes the $0.05$ Einstein-EFT error claim at the reported uncertainty; the exact CFT, a holographic relation, and a non-Einstein bulk survive. A missing higher-derivative contribution or light state is the leading alternative, and adding the first omitted operator while independently measuring the higher-spin gap is the next discriminating test.”

For the fixture $q=2$, $c_{\mathcal O}=1$, and $M_{\mathrm{HS}}L=20$, evaluate the schematic correction at $EL=5$ and $EL=20$. Classify both results.

Solution: test the finite-gap window

The estimates are $(5/20)^2=1/16$ and $(20/20)^2=1$. In the frozen fixture, the first is compatible with the $0.10$ tolerance and the second is an in-domain error-bound violation that rejects $P_{\mathrm{Ein}}$. In a real theory, reaching $E/M_{\mathrm{HS}}=O(1)$ only removes parametric suppression; the coefficient and full error budget still determine the correction. Neither result contradicts exact CFT consistency or a bulk completion with the higher-spin tower restored.

A numerical crossing residual changes strongly as derivative order and arithmetic precision are raised. Two independently interval-validated functionals later reproduce a strict positivity certificate over the complete assumed spectrum. Classify the two stages.

Solution: separate solver behavior from an exact certificate

The moving residual is computational-method failure, so the proposed CFT data remain undecided. The independently validated separating functionals exclude the frozen spectrum hypothesis. They do not exclude CFTs outside those assumptions or prove that any surviving point is realized.

For the exact response $F(\epsilon)=1/(1-\epsilon)$, verify the rows at $\epsilon=0.40$ and $0.60$ for the contract $K=2$, $\epsilon_*=0.50$.

Solution: test the remainder bound

At $0.40$, $R_2=0.16/0.60=0.2667$ and $2\epsilon^2=0.3200$, so the in-domain bound passes. At $0.60$, $R_2=0.36/0.40=0.9000$ exceeds $0.7200$, but the point is outside the declared domain. This is expected breakdown, not a failure of the exact response.

A large-$N$ theory factorizes but has an unbounded light higher-spin tower. Separately, a top-down finite-$N$ theory gives $\eta/s<1/(4\pi)$. What does each counterexample reject?

Solution: identify the quantified conclusion

The higher-spin example rejects large-$N$ factorization as sufficient for an Einstein bulk; a higher-spin holographic description may survive. The viscosity example rejects the proposed universal lower bound; it preserves both the leading Einstein value and the holographic calculation that reveals the correction.

Current boundary and handoffs

The benchmark cases above classify established logical possibilities; they do not adjudicate every live claim. Domain-specific counterexamples remain with their scientific chapters, theorem-level countermodels belong to Mathematical QFT, and new negative results and responses belong in Holography and Quantum Gravity Research. The next page, Claim Status, Freshness, and Research Handoffs, explains how a scoped result changes a living claim assessment.

Evidence cutoff. The literature check for this page is current through 27 August 2026. A material method revision, failed reproduction, or changed data release should trigger an earlier reassessment of the affected claim.

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

• Afkhami-Jeddi, Nima, Thomas Hartman, and Amirhossein Tajdini. 2019. “Fast Conformal Bootstrap and Constraints on 3d Gravity.” Journal of High Energy Physics 05, 087. DOI; Open PDF.
• Camanho, Xian O., José D. Edelstein, Juan Maldacena, and Alexander Zhiboedov. 2016. “Causality Constraints on Corrections to the Graviton Three-Point Coupling.” Journal of High Energy Physics 02, 020. DOI; Open PDF.
• Heemskerk, Idse, João Penedones, Joseph Polchinski, and James Sully. 2009. “Holography from Conformal Field Theory.” Journal of High Energy Physics 10, 079. DOI; Open PDF.
• Kats, Yevgeny, and Pavel Petrov. 2009. “Effect of Curvature Squared Corrections in AdS on the Viscosity of the Dual Gauge Theory.” Journal of High Energy Physics 01, 044. DOI; Open PDF.
• Klebanov, Igor R., and Alexander M. Polyakov. 2002. “AdS Dual of the Critical O(N) Vector Model.” Physics Letters B 550, 213–219. DOI; Open PDF.
• Rychkov, Slava, and Ning Su. 2024. “New Developments in the Numerical Conformal Bootstrap.” Reviews of Modern Physics 96, 045004. DOI; Open PDF.
• Simmons-Duffin, David. 2017. “TASI Lectures on the Conformal Bootstrap.” In New Frontiers in Fields and Strings, 1–74. Singapore: World Scientific. DOI; Open PDF.