Evidence Programs for Holographic Duality

No single celebrated agreement establishes a holographic duality. A persuasive evidence program asks whether different parts of a specified dictionary survive tests with different observables, calculational routes, approximations, and failure modes. Its conclusion is therefore not “many papers agree.” It is a bounded statement such as: these parameter, operator, and normalization maps work in these protected or asymptotic sectors, subject to these shared premises and unresolved corrections.

Required background. Holographic Duality: Claims, Dictionaries, and Regimes fixes what is being tested, and Exact Statements, Saddle Expansions, and Conditional Derivations fixes the logical status of each comparison.

Helpful background. Exact-Observable Identities as Duality Tests develops protected tests; AGT and Exact-Correspondence Dictionaries: Status and Limits supplies a correspondence-level comparison; Anomaly Polynomials and Inflow supplies structural anomaly tests; and Benchmark Reproduction and Data Provenance supplies numerical verification practice.

Reading path. First build an evidence record, then work through the circular Wilson-loop example. The evidence classes explain why exact can still mean narrow. The AdS₅/CFT₄ matrix and localization ablation execute the method; the final sections state what survives and provide solved exercises.

From a match to an evidence record

Fix a candidate duality by naming the two theories, their global data, a dictionary $\Phi$, and a regime $\mathcal R$. For one observable, a truncated bulk calculation has the form

$$\mathcal A_i^{\mathrm{CFT}}(\theta) = \mathcal A_{i,[k]}^{\mathrm{bulk}}\!\left(\Phi(\theta)\right) +\Delta_{i,[k]}.$$

Here $\theta$ collects the boundary-theory inputs, $\Phi(\theta)$ is their image under the bulk parameter dictionary, and $[k]$ means that the bulk result is retained through order $k$ in its declared expansion. The remainder $\Delta_{i,[k]}$ must be zero by an exact identity, bounded, or assigned a justified scaling such as $O(N^{-2})$ or $O(\lambda^{-3/2})$. If the two calculations have no common parameter domain, the pair supplies endpoints or a forecast—not a quantitative agreement at the same point.

Record each evidence item as

$$E_i=(Q_i,O_i,R_i,M_i,K_i,U_i,F_i,L_i).$$

• Claim $Q_i$. The exact parameter, operator, state, normalization, dynamical, or global-sector map being tested.
• Observable $O_i$. Its definition, normalization, geometry, state or ensemble, and units.
• Regime $R_i$. Values or limits of $N$, coupling, temperature, charges, separation, cutoff, and expansion order.
• Methods $M_i$. The boundary and bulk calculations, including which side is exact, numerical, perturbative, or conjectural.
• Dependency key $K_i$. Shared identities, input data, calibrations, parameter maps, codes, or asymptotic assumptions. Items with the same key form an evidence family.
• Uncertainty $U_i$. Statistical errors, discretization and finite-volume effects, omitted perturbative orders, large-$N$ corrections, and normalization ambiguity.
• Failure rule $F_i$. A result that would falsify the dictionary entry, or merely show that the chosen approximation or numerical method failed.
• Licensed conclusion $L_i$. The strongest claim that follows if the comparison succeeds. This upper bound is the item’s claim ceiling.

A calibration uses an observable to fix a dictionary parameter; a prediction uses the fixed map on held-out data. The same number cannot be counted once as calibration and again as an independent prediction. Independence is graded rather than binary: two results can use different computations yet share the candidate theory pair, the same value of $\lambda$, or one localization identity.

Worked record: the half-BPS circle

Consider the normalized fundamental half-BPS circular Wilson loop in planar $SU(N)$ $\mathcal N=4$ super-Yang–Mills theory. Localization gives the Gaussian-matrix-model result

$$\left\langle W_\circ\right\rangle_{\mathrm{planar}} =\frac{2}{\sqrt\lambda}I_1(\sqrt\lambda),$$

and hence

$$\left\langle W_\circ\right\rangle_{\mathrm{planar}} \sim \sqrt{\frac{2}{\pi}}\,\lambda^{-3/4}e^{\sqrt\lambda}, \qquad \lim_{\lambda\to\infty} \frac{\log\left\langle W_\circ\right\rangle}{\sqrt\lambda}=1.$$

More precisely, the same Bessel asymptotics give

$$\log\left\langle W_\circ\right\rangle =\sqrt\lambda-\frac34\log\lambda +\frac12\log\frac{2}{\pi} -\frac{3}{8\sqrt\lambda}+O(\lambda^{-1}).$$

The classical fundamental-string worldsheet ending on the boundary circle gives the same leading exponential. The matrix-model identity and its strong-coupling asymptotics are derived in Pestun 2012, § 1, eqs. (1.1)–(1.5), pp. 2–3 and Drukker and Gross 2001, § 1, eqs. (1.1)–(1.3), pp. 1–2.

The compact formula is not yet an evidence record. The missing fields are:

• Claim and observable. The entry maps this particular normalized half-BPS line operator to a fundamental string with the specified boundary contour. It does not test arbitrary Wilson loops.
• Regime. The boundary formula shown is planar and all-$\lambda$; the classical worldsheet comparison is planar and $\lambda\gg1$. Finite-$N$ and string-loop terms are outside this displayed comparison.
• Conventions. Here $\lambda=g_{\mathrm{YM}}^2N$ and $g_{\mathrm{YM}}^2=4\pi g_s$. The representation, trace normalization, scalar coupling, contour orientation, and subtraction of the regularized worldsheet area are fixed.
• Dependency key. The boundary result belongs to the four-sphere localization and matrix-model family. The bulk result uses the string-tension map $L^2/\alpha'=\sqrt\lambda$ and a semiclassical worldsheet saddle.
• Uncertainty and failure. The classical worldsheet supplies only the leading $\sqrt\lambda$ term. Its fluctuation determinant and measure must account for the logarithm and constant before higher inverse-$\sqrt\lambda$ terms are compared; for this normalized single-trace loop, string handles add $1/N^2$ genus corrections. A disagreement in the coefficient of $\sqrt\lambda$ after fixing the same conventions would challenge the string-tension or line-operator map; a disagreement only in a subleading term could instead diagnose an incomplete fluctuation or measure calculation.
• Claim ceiling. The match supports the half-BPS loop/worldsheet map and the string-tension normalization in this corner. It is not a proof of finite-$N$, all-coupling equivalence or of the full extended-operator dictionary.

This example also shows why “protected” does not mean “uninformative.” Supersymmetry makes the boundary quantity unusually controllable, while the large-$\lambda$ limit still probes a geometric string saddle. Protection strengthens the calculation but narrows the conclusion.

Evidence classes and their ceilings

Protection, method, and evidential role are different axes. A numerical bootstrap bound can use protected input; a protected observable can still be evaluated numerically; and an unprotected result can be only a forecast if its two limits do not overlap.

Evidence classes and the strongest conclusion each class licenses by itself

Class	What makes it valuable	Typical ceiling
Structural	Checks symmetry algebras, Ward identities, anomalies, charge quantization, or global sectors	Supports necessary dictionary structure; shared symmetry alone does not establish equivalent dynamics
Protected	Controls dimensions, indices, partition functions, or selected correlators across coupling	Can give exact but sector-limited identities; an index is a signed trace and need not determine the full BPS degeneracy
Unprotected analytic	Tests interaction-sensitive thermodynamics, amplitudes, correlators, or anomalous dimensions	Supports dynamics only in the overlap of the stated expansions and kinematics
Spectral or integrable	Organizes large sets of planar dimensions or worldsheet charges with high analytic precision	Supports the planar spectral dictionary within the integrability framework, not generic finite-N dynamics
Numerical	Can probe unprotected finite-coupling quantities without a weak-coupling series	Supports the simulated or bounded observable after regulator, solver, and extrapolation errors are controlled
Negative or adversarial	Tests a correction estimate, a purported sufficient criterion, or a held-out dictionary entry	Locates which claim failed; the next page classifies dictionary, regime, bound, and method failures separately

AdS₅/CFT₄ regime checkpoint

The worked program below fixes the boundary theory to $SU(N)$ $\mathcal N=4$ super-Yang–Mills and the bulk candidate to type-IIB string theory on $AdS_5\times S^5$ with $N$ units of five-form flux and corresponding boundary data. The coupling convention is

$$\tau=\frac{\theta}{2\pi}+\frac{4\pi i}{g_{\mathrm{YM}}^2}, \qquad \lambda=g_{\mathrm{YM}}^2N, \qquad g_s=\frac{g_{\mathrm{YM}}^2}{4\pi}, \qquad \frac{L^4}{\alpha'^2}=\lambda.$$

The planar limit takes $N\to\infty$ at fixed $\lambda$. A classical supergravity calculation additionally needs $\lambda\gg1$ to suppress string-scale curvature corrections and $g_s\ll1$ to suppress string loops. The five-dimensional Newton coupling scales as $G_5/L^3\sim N^{-2}$. The D3-brane and AdS₅/CFT₄ regime page owns the full parameter-controlled derivation; the present page asks how much independent evidence those identifications receive.

An exact correspondence claim would concern the fixed finite-$N$ theory and a complete string-theory dictionary, not only its classical-gravity corner. Every row below therefore states the smaller region it actually tests.

An AdS₅/CFT₄ evidence matrix

The overview separates favorable comparisons from forecasts and consistency constraints. A favorable row has two sufficiently specified results that agree to the declared accuracy. A forecast supplies nonoverlapping endpoints. A constraint excludes possibilities but does not select the holographic answer.

AdS₅/CFT₄ evidence items, control regimes, dependency families, and claim ceilings

Item	Observable and comparison	Regime, control, and dependency	Status and ceiling
A. Weyl anomaly	Boundary a and c versus the coefficient of the holographic Weyl anomaly	Leading large N; classical five-dimensional gravity; anomaly family	Favorable structural match of stress-tensor normalization at leading order
B. Protected KK spectrum	Half-BPS dimensions and representations versus Kaluza–Klein masses on the five-sphere	Protected single-trace or single-particle sector at large N; protected-spectrum family	Favorable protected match; no claim about long multiplets or finite-N trace relations
C. Half-BPS circle	Localized Wilson-loop matrix model versus a fundamental-string worldsheet	Planar, strong coupling on the bulk side; localization family	Favorable line-operator and string-tension test in one protected geometry
D. Integrated correlator C	Localized sphere-partition-function derivative versus the independently known R⁴ type-IIB coefficient in stress-tensor-multiplet data	Large N at fixed complex coupling, followed by the flat-space limit; partition-derivative family	Favorable precision check of one selected string-interaction coefficient
E. Integrated correlator H	A fourth mass derivative of the same partition function used to constrain higher-order bulk interaction data	Large-N expansion and finite-N estimates; same partition-derivative family as D	Consistency constraint and dictionary application, not a separately favorable match
F. Planar integrability	The universal large-spin scaling function across perturbative gauge theory, the BES equation, and semiclassical strings	Planar, all-coupling integral equation conditional on integrability; integrability family	Favorable calibration-aware spectral benchmark; no finite-N theorem
G. Thermal entropy	Free plasma and strongly coupled black-brane entropy densities	Large N at opposite coupling endpoints; thermal family	Scaling support and interpolation target, not a same-coupling match
H. Lattice static potential	Polyakov-line potential in a three-colour lattice study versus strong-coupling square-root scaling	One principal 12⁴ lattice, lattice coupling at most 40, no continuum or large-N extrapolation; lattice family	Provisional favorable bare-parameter trend, not precision continuum confirmation
I. Numerical bootstrap	Crossing and positivity bounds on unprotected operator data, compared with large-central-charge supergravity data	Rigorous within stated numerical assumptions; pure-crossing family, or a hybrid family when localization and integrability are imported	Exclusion and compatibility evidence; saturation is an additional conjecture

The labels A–I are local handles for this worked matrix, not universal grades. The details below make the normalizations, uncertainties, and downgrade rules explicit.

Structural and protected comparisons

A — Weyl anomaly. With standard four-dimensional normalization, $SU(N)$ $\mathcal N=4$ SYM has

$$a_{\mathrm{CFT}}=c_{\mathrm{CFT}}=\frac{N^2-1}{4}, \qquad a_{\mathrm{bulk}}^{\mathrm{cl}} =c_{\mathrm{bulk}}^{\mathrm{cl}} =\frac{\pi L^3}{8G_5}=\frac{N^2}{4}.$$

Thus classical gravity reproduces the $O(N^2)$ coefficient while the $-1/4$ term lies beyond the classical approximation. Henningson and Skenderis derive the bulk anomaly and compare it with the large-$N$ field-theory result Henningson and Skenderis 1998, § 3.2, eqs. (20)–(23), pp. 7–8. If $G_5$ is calibrated from this central charge, the same coefficient is no longer a held-out prediction for another stress-tensor observable. A leading coefficient that remained wrong after matching trace conventions and counterterms would falsify this normalization map; the finite $-1/4$ difference does not.

B — protected Kaluza–Klein spectrum. At fixed $k\ge2$ as $N\to\infty$, half-BPS single-trace primaries in the $[0,k,0]$ representation have protected dimension $\Delta=k$. Their scalar Kaluza–Klein partners obey

$$m^2L^2=k(k-4), \qquad \Delta(\Delta-4)=m^2L^2,$$

with the standard supersymmetric AdS boundary condition selecting $\Delta=k$; this qualification matters at and near the Breitenlohner–Freedman window. The complete linearized type-IIB Kaluza–Klein spectrum on $S^5$ was organized by Kim, Romans, and van Nieuwenhuizen 1985, Tables III–VI, pp. 397–399, and the chiral-primary map and protected three-point comparison were developed by Lee et al. 1998, §§ 2–4, pp. 3–12. This tests masses, dimensions, $R$-symmetry representations, and selected normalizations in a protected large-$N$ sector. It does not determine the unprotected string spectrum, and finite-$N$ trace relations must be handled separately.

An index is weaker than a protected spectrum: it is a signed trace, so short multiplets can cancel. Index agreement can strongly test charges and shortening data, but it cannot by itself establish equality of all BPS degeneracies Kinney et al. 2007, §§ 2.5 and 4.1–4.2, pp. 12–23.

Localization-derived precision tests

C — the half-BPS circle is the worked record above. D and E begin from the mass-deformed four-sphere partition function $Z_N(m,\tau,\bar\tau)$. Set the sphere radius to $R_{S^4}=1$, so $m$ denotes the dimensionless source $mR_{S^4}$. Two quantities related by supersymmetric Ward identities to integrated stress-tensor-multiplet four-point functions are

$$\mathcal C_N =(\operatorname{Im}\tau)^2 \left.\partial_\tau\partial_{\bar\tau}\partial_m^2\log Z_N\right|_{m=0}, \qquad \mathcal H_N =\left.\partial_m^4\log Z_N\right|_{m=0}.$$

These definitions, their modular properties, and the different present levels of control over $\mathcal C_N$ and $\mathcal H_N$ are stated by Alday et al. 2023, abstract and §§ 1–2. The modular comparison is organized at large $N$ with $\tau$ fixed—so $\lambda\propto N$—and then related to the flat-space type-IIB limit. This is more specific than an unspecified “large-$N$, strong-coupling” regime.

For D, the localization result supplies a genuine check of the independently known $R^4$ coefficient in the stress-tensor-multiplet correlator Binder et al. 2019, §§ 2–4. For E, $\mathcal H_N$ and related integrated constraints are used together with imported flat-space string data to constrain $D^4R^4$ and higher-loop contact terms Chester et al. 2020, § 1.1 and §§ 2–3 and to extend the analysis far beyond the planar term Chester and Pufu 2021, §§ 1, 2.4, and 3. E is therefore a valuable consistency constraint and dictionary application, not a second independently favorable comparison.

The external operators are protected; much of the extracted OPE and interaction data is not. This makes the program dynamically richer than a dimension match, but its outputs remain dependent on the localized $Z_N$ input, Ward identities, and, where used, the flat-space string amplitude. Results that infer bulk coefficients by assuming the holographic map should not be relabeled as independent proofs of that same map.

Planar spectra and thermal dynamics

F — planar integrability. At one loop, the planar scalar dilatation operator becomes an integrable $SO(6)$ spin chain Minahan and Zarembo 2003, §§ 1–3, while the classical $AdS_5\times S^5$ string sigma model carries an infinite tower of nonlocal conserved charges Bena, Polchinski, and Roiban 2004, §§ 1–4. A concrete common observable is the planar scaling function $f(\lambda)$, defined by the large-spin behavior

$$\Delta(S)-S=f(\lambda)\log S+O(S^0).$$

The Beisert–Eden–Staudacher integral equation reproduces the weak-coupling series and the leading semiclassical-string behavior of this same function Beisert, Eden, and Staudacher 2007, § 5, pp. 17–19. The strong-coupling behavior helped calibrate the dressing-phase construction, so it is not wholly held out. A sharper test is agreement with the independently evaluated four-loop weak-coupling coefficient Bern et al. 2007, §§ IV–V. The quantum spectral curve then supplies a compact all-coupling formulation of the wider planar spectral problem Gromov et al. 2014, eqs. (1)–(6) and discussion. Together these calibration-aware comparisons give detailed support for the planar spectral dictionary. The machinery assumes planar integrability; quantum-spectral-curve numerics are therefore not an independent finite-$N$ simulation or a proof that the full duality is integrable.

G — thermal entropy. At leading large $N$, the weak-coupling and supergravity endpoints are

$$S_{\mathrm{free}}=\frac{2\pi^2}{3}N^2V_3T^3, \qquad S_{\mathrm{strong}}=\frac{\pi^2}{2}N^2V_3T^3 \left[1+O(\lambda^{-3/2})\right],$$

so

$$\frac{S_{\mathrm{strong}}}{S_{\mathrm{free}}} \longrightarrow \frac34.$$

These fixed-temperature formulas and the leading strong-coupling correction are derived by Gubser, Klebanov, and Tseytlin 1998, § 1, eqs. (4)–(8), pp. 2–3. The result matches the nontrivial $N^2V_3T^3$ scaling and gives two limits of one putative function of $\lambda$. It is not equality at a common coupling, and the factor $3/4$ is not evidence for a controlled interpolation through intermediate coupling. This row is therefore an unprotected forecast and scaling check, not one of the favorable equality rows counted below.

Numerical evidence and numerical constraints

H — lattice static potential. A lattice study of the twisted four-dimensional theory used three colours, a principal $12^4$ lattice, and bare lattice couplings through $\lambda_{\mathrm{lat}}=40$. It found a Coulombic Polyakov-line potential whose fitted coefficient followed an approximately $\sqrt{\lambda_{\mathrm{lat}}}$ dependence and lay within roughly ten percent of a holographic curve expressed using that lattice coupling Catterall, Giedt, and Toga 2023, §§ 3–4, pp. 4–8. The calculation probes an unprotected quantity with a method genuinely different from localization and planar spectral equations. Its ceiling is nevertheless low: the study did not demonstrate a nonperturbative match between $\lambda_{\mathrm{lat}}$ and the continuum coupling, or continuum, infinite-volume, and large-$N$ extrapolations. Smearing, coupling renormalization, the lattice regulator, and the treatment of the abelian sector remain systematic controls. Call this an exploratory bare-parameter trend, not a continuum-normalized ten-percent confirmation.

I — numerical bootstrap. Crossing, unitarity, and superconformal symmetry bound unprotected operator dimensions as a function of central charge. At large central charge, the extremal spectrum found in an early numerical study was compatible with supergravity, while identifying the extremal solution with $\mathcal N=4$ SYM was explicitly conjectural Beem, Rastelli, and van Rees 2013, pp. 1–4. Compatibility with an allowed region is not equality, and saturation of a bound is not automatic.

Modern analyses can be substantially sharper because they combine bootstrap equations with localization, integrability, perturbation theory, or holographic input. That gain in precision also creates dependence. A 2025 energy-correlator analysis explicitly combines all five kinds of input Dempsey et al. 2025, abstract and § 1. Such bounds are valuable constraints and precision syntheses, but they cannot be counted as methodologically independent of the inputs they consume.

Remove the localization family

The adversarial test is now concrete. There are six favorable rows: A, B, C, D, F, and H. E is a consistency application, G is a forecast, and I is a constraint.

Dependence can be resolved at two granularities. The strict identity family contains D and E because both descend from derivatives of the same mass-deformed $Z_N(m,\tau,\bar\tau)$. C uses a Wilson-loop insertion and does not descend from that derivative identity, so it survives the strict removal. A broader four-sphere localization family groups C with D and E because all three rely on the same localization framework and overlapping matrix-model parameter map. That broader grouping is a deliberately conservative stress test, not a claim that the observables or identities are identical.

Strict same-identity removal and broader four-sphere-localization stress test

Removal rule	Before removal	Removed	After removal	Interpretation
Strict partition-derivative identity	6 favorable rows; 6 granular favorable families	D, plus consistency row E	5 favorable rows and 5 favorable families: A, B, C, F, H	This is the manifest-required same-identity ablation; the Wilson circle survives
Broad four-sphere-localization framework	6 favorable rows; 5 broad favorable families after C and D are grouped	C and D, plus consistency row E	4 favorable rows and 4 broad families: A, B, F, H	This stronger stress test removes all favorable boundary results rooted in four-sphere localization
Other context	Thermal forecast G and bootstrap constraint I	No pure-crossing or thermal input	G and pure-crossing portions of I remain	Any hybrid bootstrap result must lose localization-derived inputs under either applicable removal

After the strict removal, the surviving circle still tests a protected line operator and the string-tension map. After the broad removal, the four surviving favorable families probe leading anomaly normalization, protected single-particle data, planar spectral dynamics, and an exploratory lattice trend. They still share the proposed theory pair and portions of its parameter dictionary. Effective independence can therefore be lower than the displayed count, and no scalar count replaces a dependency graph. These counts are sensitivity summaries, not probabilities or Bayes factors. Evidence Independence, Circularity, and Double Counting develops that graph, calibration check, and alternative-model comparison.

The broad removal also changes the breadth of the surviving conclusion. It eliminates the sharpest all-coupling boundary input and the selected integrated-correlator constraints. The remaining program still has structural, protected, spectral, and numerical variety, but much less direct control over finite-coupling interaction data.

What the program licenses

Taken together, the rows provide broad, multi-channel support for specified entries of the $SU(N)$ $\mathcal N=4$ SYM/type-IIB dictionary. The strongest well-supported regions are protected data and large-$N$, planar, or strong-coupling limits. The program does not establish a theorem of exact finite-$N$, all-coupling equivalence, a complete global and extended-operator dictionary, or independent confirmation of every precision correlator.

A useful evidence program also states what would change that conclusion:

• a leading anomaly or protected-spectrum mismatch after all convention and global-form choices are aligned would challenge a core dictionary entry;
• a failure confined to a known $1/N$, inverse-coupling, finite-volume, or discretization correction downgrades the approximation, not automatically the exact duality;
• a continuum- and large-$N$-controlled numerical observable that disagrees with the mapped string prediction outside the combined uncertainty would supply genuinely adverse dynamical evidence;
• failure of a common localization theorem or shared $Z_N$ identity invalidates every result that actually descends from that root; failure of one insertion, Ward identity, normalization, or bulk comparison need not invalidate its siblings;
• an exact, complete boundary and bulk calculation of the same normalized observable that disagrees would directly falsify that dictionary entry.

Falsifiers, Negative Results, and Counterexamples classifies those outcomes. Claim Status, Freshness, and Research Handoffs explains how to update the verdict without rewriting stable exposition.

Evidence cutoff. 27 August 2026. The matrix includes foundational sources, the 2023 lattice result, 2023–2026 precision-correlator and hybrid-bootstrap developments, and the 2024 overview of precision holography Schäfer-Nameki 2024, § 4.1. The 2026 heavy-operator analysis illustrates that the localization-derived sector continues to expand, but it remains part of a correlated method family Aprile, Dorigoni, and Treilis 2026, abstract and § 1. Reassess this page when a continuum and large-$N$ lattice extrapolation, a revised localization identity, an integrability failure, a new finite-coupling benchmark, a reproducibility failure, or contrary observable result changes a row or its dependency key. Dated assessments and contrary literature belong in the Holography and Quantum Gravity Research guide.

Common pitfalls

Counting papers instead of roots. Ten plots derived from one localized partition function are ten outputs but not ten independent calculational foundations. Record both rows and dependency families.

Calling an endpoint a match. The thermal factor $3/4$ compares weak- and strong-coupling limits. Without controlled overlap, it is a scaling check and an interpolation target.

Confusing protected with complete. An exact anomaly, index, or half-BPS observable can test a narrow dictionary entry exquisitely while saying little about generic long multiplets or real-time dynamics.

Counting calibration as prediction. If the central charge fixes $G_5$, reproducing that same central charge is calibration. Test a held-out observable with the calibrated value.

Treating a bound as a measurement. A numerical bootstrap allowed region establishes exclusion and compatibility. Identifying the theory with a boundary point or assuming saturation requires separate evidence.

Promoting exploratory numerics. Agreement on one lattice size is encouraging, not a continuum result. List the extrapolations and systematics that remain.

Exercises

1. Class and ceiling

Classify the following as structural, protected, unprotected analytic, spectral or integrable, numerical, or negative evidence: (a) equality of a Weyl-anomaly coefficient at leading $N^2$; (b) a half-BPS dimension; (c) a finite-temperature entropy calculation; (d) a quantum-spectral-curve value for a planar anomalous dimension; (e) a lattice static potential. State the claim ceiling of each.

Solution

(a) is structural and supports the leading stress-tensor normalization map, not full dynamics. (b) is protected and supports the corresponding operator/state and mass/dimension entries, not long multiplets. (c) is unprotected analytic; it supports thermodynamic dynamics only where the coupling and large-$N$ approximations overlap—opposite endpoints alone give a forecast. (d) is spectral or integrable and supports the planar spectral dictionary conditional on integrability. (e) is numerical and supports only the simulated observable after finite-spacing, volume, rank, and statistical systematics have been controlled.

2. Repeat the ablation

An evidence list contains favorable rows A, B, C, D, F, and H, plus consistency row E. D and E are derivatives of the same mass-deformed partition function; C uses a Wilson-loop insertion in the broader four-sphere localization framework. Recompute the favorable rows and families after (i) strict same-identity removal and (ii) broad framework removal. Why are the family counts not Bayes factors?

Solution

Initially there are six favorable rows. At strict resolution they form six favorable families. Removing the shared partition-derivative identity deletes favorable row D and consistency row E; five favorable rows and five favorable families remain: A, B, C, F, and H. At broad resolution, C and D count as one four-sphere-localization family, so the six favorable rows form five broad families. Removing that framework deletes favorable rows C and D and consistency row E; four favorable rows and four broad families remain: A, B, F, and H. A family count records a chosen causal grouping but supplies neither likelihoods nor prior probabilities, and all survivors still share the candidate duality and parameter map. It is a sensitivity diagnostic, not a posterior odds ratio.

3. Interpret the thermal ratio

Using

$$S_{\mathrm{free}}=\frac{2\pi^2}{3}N^2V_3T^3, \qquad S_{\mathrm{strong}}=\frac{\pi^2}{2}N^2V_3T^3,$$

compute $S_{\mathrm{strong}}/S_{\mathrm{free}}$. What does the result test, and what does it not test?

Solution

The common factors cancel:

$$\frac{S_{\mathrm{strong}}}{S_{\mathrm{free}}} =\frac{1/2}{2/3}=\frac34.$$

Both endpoints exhibit the expected $N^2V_3T^3$ scaling, and the strong-coupling value supplies a nontrivial target for the same thermal observable. The ratio is not an equality at one coupling and does not determine the function connecting weak and strong coupling.

4. Dictionary failure or approximation failure?

Suppose a finite-$N$ lattice value at $\lambda=20$ disagrees with the classical-supergravity prediction by 18 percent. The lattice calculation has one spacing and one volume, and the bulk calculation omits $1/N$ and inverse-coupling corrections. Does this falsify the duality? What additional comparison would be decisive?

Solution

No. The two results have unresolved lattice, finite-volume, finite-$N$, and inverse-coupling errors, so the residual cannot yet be assigned to the dictionary. One should extrapolate the lattice observable toward the continuum and infinite-volume limits at several $N$, compute or bound the leading bulk corrections in the same normalization, and compare in an overlap regime. A statistically and systematically significant disagreement outside the combined error after those controls would challenge the observable map. An exact disagreement between complete finite-$N$ calculations on both sides would be stronger still.

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

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• Dempsey, Ross, Robin Karlsson, Silviu S. Pufu, Zahra Zahraee, and Alexander Zhiboedov. 2025. “Conformal Collider Bootstrap in $\mathcal N=4$ SYM.” Preprint arXiv:2512.10796. Open PDF.
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