Corrections, Nonuniform Limits, and Failure Modes

A semiclassical bulk approximation is controlled only after its observable, state, kinematic domain, and order of limits have been fixed. Corrections suppressed at fixed energy and fixed time can become order one when energy scales with a gap, time scales with entropy, or the number of light species scales with $N$. Such loss of uniformity is a failure of the approximation in an enlarged domain; by itself it is not a failure of the proposed duality.

Required background. Observable and Regime Matrix for Quantum Gravity supplies the object–observable–regime distinctions used below, and Large-N Factorization and Classical Bulk Scaling fixes connected-correlator normalization.

Helpful background. EFT Truncation Errors and Breakdown Diagnostics treats generic truncation estimates; Spectral Statistics, Form Factors, and Late-Time Evidence develops late-time observables; Large-N, Loop, and ℏ Approximation Hierarchies separates loop expansions; and Curvature, Coupling, Loop, Derivative, and Secular Hierarchies treats secular breakdown.

Build a dimensionless correction budget

Let $\mathcal A_{\mathrm{ref}}$ be a positive comparison scale with the same physical units as the observable $\mathcal A$. It may be a fixed natural scale or a specified reference profile, but it must not hide parametric growth with $N$, energy, or time. A conservative schematic budget is

$$\begin{aligned} \frac{\lvert\mathcal A-\mathcal A_0\rvert}{\mathcal A_{\mathrm{ref}}} &\lesssim \epsilon_{\mathrm{loop}} +\lvert c_{\mathrm{string}}\rvert\epsilon_{\mathrm{string}}\\ &\quad +\lvert c_{\mathrm{der}}\rvert\epsilon_{\mathrm{der}} +\lvert c_{\mathrm{sec}}\rvert\epsilon_{\mathrm{sec}}\\ &\quad +\lvert c_{\mathrm{np}}\rvert\epsilon_{\mathrm{np}} +\epsilon_{\mathrm{rem}}. \end{aligned}$$

Every $\epsilon_i$ is a nonnegative control magnitude; the displayed inequality is not a signed decomposition. The symbols become useful only after the theory, state, observable, and kinematics have been specified. Here $D=d+1$ is the bulk spacetime dimension, and $Q$ is the largest local dynamical or curvature scale sampled in the bulk rather than an unredshifted boundary frequency. For $d>2$, define the convention-dependent positive factor $\kappa_d$ and the IR-matched Newton coupling $G_L$ by $C_T=\kappa_dL^{d-1}/G_L$; in $d=2$, first make the corresponding conversion from the Virasoro central charge to the declared stress-tensor convention. Also define

$$r_G(Q)\equiv\frac{G_{\mathrm{eff}}(Q)}{G_L}.$$

The factor $N_{\mathrm{eff}}(Q)$ is the diagram-weighted light-species sum, while $b_D(Q)$ contains the remaining dimensionless diagram and regulator dependence.

Representative correction controls and the assumptions that must accompany them

Effect	Typical dimensionless control	What must be checked
Bulk gravitational loops	εloop ∼ |bD(Q)| κd Neff(Q) rG(Q) (QL)d − 1/CT	The diagram coefficient, weighted species sum, running-coupling ratio, dimension, and normalization of CT
Finite string length	εstring ∼ (Qℓs)k	In a top-down example at AdS-scale kinematics this may become λ−p, with theory-dependent k, p > 0
Higher derivatives	εder ∼ (Q/Λder)q	The physical omitted threshold; ΛderL ∼ Δgap only when the declared single-trace gap controls it
Secular growth	εsec ∼ ε0 |F(t/L)|	The small initial coefficient and calculated growth law, not a crossover time inserted by definition
Nonperturbative sectors	εnp ∼ e−Inp |Fnp(t)|	Which action or entropy controls Inp, and whether time or multiplicity compensates the exponential

String-scale effects are encoded in higher-derivative Wilson coefficients after massive string modes are integrated out. Count a given $\alpha'$ correction under either the string entry or the generic derivative entry, not both. Likewise, contributions already absorbed into $G_{\mathrm{eff}}(Q)$ must be excluded from $b_DN_{\mathrm{eff}}$.

Let $\Lambda_{\mathrm{der}}$ be the lowest omitted-state, omitted-tower, or genuine ultraviolet strong-coupling scale controlling the derivative expansion. Separately, $\Lambda_{\mathrm{stop}}\leq\Lambda_{\mathrm{der}}$ is the first scale where the chosen fixed-order calculation loses control: a resummable logarithm or a loop crossover can lower $\Lambda_{\mathrm{stop}}$ without lowering the physical derivative scale.

For fixed $b_D$, fixed weighted species content, fixed $r_G$, fixed $QL$, and matrix-like $C_T\sim N^2$, the loop row reduces to the familiar $N^{-2}$ counting. That shorthand fails if $N_{\mathrm{eff}}(Q)$, $r_G(Q)$, or $QL$ grows with $N$. A raw species count is only a proxy for the diagram-dependent weighted sum. Coefficients can also be large, and mixed terms such as $\epsilon_{\mathrm{loop}}\epsilon_{\mathrm{der}}$ can matter. The lowest physical omitted threshold or genuine strong-coupling scale controls $\Lambda_{\mathrm{der}}$; the earliest independent loss of perturbative control sets the prediction stop.

Bulk Interaction Scaling and Effective Cutoffs derives this normalization and develops the distinction among small corrections, physical thresholds, and fixed-order prediction stops.

Calling a result “leading order” is therefore meaningful only together with simultaneous bounds on every relevant control parameter and on the remainder.

Pointwise control is not uniform control

An asymptotic expansion at fixed $x$ is not automatically uniform on an $N$-dependent domain $D_N$. Define the dimensionless error

$$\varepsilon_N(D_N) \equiv \sup_{x\in D_N} \frac{\lvert\mathcal A_N(x)-\mathcal A_0(x)\rvert} {\mathcal A_{\mathrm{ref}}(x)}.$$

The leading approximation is uniform only if $\varepsilon_N(D_N)\to0$. If

$$\mathcal A_N(x)=\mathcal A_0(x)+N^{-2}f(x)+R_N(x),$$

then it is not enough that $N^{-2}f(x)\to0$ at every fixed $x$. Both

$$\begin{aligned} \sup_{x\in D_N}\frac{\lvert N^{-2}f(x)\rvert}{\mathcal A_{\mathrm{ref}}(x)} &\longrightarrow0,\\ \sup_{x\in D_N}\frac{\lvert R_N(x)\rvert}{\mathcal A_{\mathrm{ref}}(x)} &\longrightarrow0. \end{aligned}$$

must be controlled. To claim a uniform asymptotic expansion through the $N^{-2}$ term, require

$$\sup_{x\in D_N} \frac{\lvert R_N(x)\rvert}{\mathcal A_{\mathrm{ref}}(x)} =o(N^{-2}).$$

If an independently established genus expansion makes $N^{-4}$ the next allowed order, this may be strengthened to a uniform $O(N^{-4})$ bound. A pointwise big-O symbol does not supply either supremum estimate.

Crossover times follow from the actual growth law. If $\epsilon_{\mathrm{sec}}=\epsilon_0e^{\gamma t/L}$, then

$$t_\times\sim\frac{L}{\gamma}\log\frac1{\epsilon_0};$$

for $\epsilon_0=N^{-2}$ this gives $t_\times\sim(2L/\gamma)\log N$. If instead $\epsilon_{\mathrm{sec}}=\epsilon_0(t/L)^r$, then $t_\times\sim L\epsilon_0^{-1/r}$. For $r=1$, initial suppressions $S^{-1}$ and $e^{-S}$ therefore lead to parametrically different scales, $LS$ and $Le^S$.

Near a zero of $\mathcal A_0$, use an absolute error scale or compare with the first nonzero term. Dividing by a symmetry-enforced zero does not diagnose accuracy.

Competing saddles create a shrinking crossover region

Nonuniformity also appears when two Euclidean saddles exchange dominance. Suppose, in a regulated finite-volume system,

$$\begin{aligned} Z_N(x) &\simeq e^{-N^2I_1(x)}+e^{-N^2I_2(x)},\\ \frac{Z_2}{Z_1} &=e^{-N^2\Delta I(x)}. \end{aligned}$$

where $\Delta I=I_2-I_1$ is real. If $\Delta I(x)\simeq a(x-x_c)$ near the crossing and one approaches it as $x=x_c+y/N^2$, then

$$\frac{Z_2}{Z_1}\longrightarrow e^{-ay}.$$

Both saddles survive in an $O(N^{-2})$ window even though either saddle is exponentially suppressed at fixed $x\ne x_c$. At finite $N$ the two-term answer is smooth, while the strict limit $-N^{-2}\log Z_N\to\min(I_1,I_2)$ develops the familiar cusp. If the action scales as another power of $N$, the crossover width changes accordingly. The Hawking–Page example is developed in Witten 1998, § 2.3.

Finite-N thermal correlators retain spectral discreteness

Consider a suitably smeared Hermitian operator $O$ in a finite-volume boundary theory with discrete spectrum. Assuming the thermal sums converge,

$$\begin{aligned} C_N(t) &=\frac{1}{Z(\beta)} \sum_{m,n} e^{-\beta E_m}\lvert O_{mn}\rvert^2\\ &\qquad\times e^{\,i(E_m-E_n)t}. \end{aligned}$$

The stationary part includes every zero-frequency matrix element, not merely the diagonal terms:

$$\begin{aligned} C_N^{(0)} &=\frac1{Z(\beta)} \sum_{E_m=E_n}\\ &\qquad\times e^{-\beta E_m}\lvert O_{mn}\rvert^2,\\ \widetilde C_N(t) &=C_N(t)-C_N^{(0)}. \end{aligned}$$

This distinction matters when energy levels are degenerate. Group equal nonzero energy differences into

$$\begin{aligned} A_\omega &\equiv\frac1{Z(\beta)} \sum_{\substack{m,n\\E_m-E_n=\omega}} \\ &\qquad\times e^{-\beta E_m}\lvert O_{mn}\rvert^2, \qquad \omega\ne0. \end{aligned}$$

Then the oscillating correlator and its exact infinite-time mean square take the compact form

$$\begin{aligned} \widetilde C_N(t) &=\sum_{\omega\ne0}A_\omega e^{i\omega t},\\ \overline{\lvert\widetilde C_N\rvert^2} &\equiv\lim_{T\to\infty}\frac1T \int_0^Tdt\,\lvert\widetilde C_N(t)\rvert^2\\ &=\sum_{\omega\ne0}\lvert A_\omega\rvert^2. \end{aligned}$$

It is positive whenever $O$ connects at least one pair of distinct energy levels. Thus a continuum saddle can decay while the exact finite-$N$ answer retains quasiperiodic noise. The noise amplitude depends on the operator, state, symmetries, and degeneracies; no universal power of $e^{-S}$ applies.

In a fixed symmetry sector, let $D_{\mathrm{distinct}}$ count distinct energy values in a band of width $W$. If parametrically large degeneracies are absent and $D_{\mathrm{distinct}}\sim e^{S_{\mathrm{spec}}}$, then the mean distinct-level spacing and Heisenberg time scale as

$$\begin{aligned} \delta E &\sim We^{-S_{\mathrm{spec}}},\\ t_H &\sim\frac{2\pi}{\delta E} \sim\frac{2\pi}{W}e^{S_{\mathrm{spec}}}. \end{aligned}$$

This is the scale on which individual levels become resolvable. A Poincaré recurrence requires many phases to realign to a specified accuracy and is generally far longer and more model dependent; when many frequency differences are rationally independent, estimates can be doubly exponential in the spectral entropy. The Heisenberg and Poincaré scales, and the normalized infinite-time mean square, are distinguished in Barbón and Rabinovici 2003, § 2, especially pp. 4–8.

For a large-$N$ family of thermal bands with $\delta E_N\to0$, as in a high-entropy black-hole band, taking $N\to\infty$ at fixed $t/L$ can remove sensitivity to individual levels before the late-time regime is examined. Large $N$ alone does not imply this spectral densification. At each finite $N$, under the convergence conditions that make the smeared spectral sum almost periodic, recurrence-sensitive limits can instead retain the discrete signal. When the required inner limits exist and the resulting quantities are finite, define

$$\begin{aligned} L_{\mathrm{late\ first}} &\equiv \lim_{N\to\infty}\limsup_{t\to\infty} \lvert\widetilde C_N(t)\rvert,\\ L_{\mathrm{planar\ first}} &\equiv \limsup_{t\to\infty}\lim_{N\to\infty} \lvert\widetilde C_N(t)\rvert. \end{aligned}$$

These iterated quantities need not be equal; failure of any required inner or outer limit to exist is another form of nonuniform behavior. The positive mean square above measures persistent time-averaged noise; the limsup is sensitive to rare recurrence peaks, so one quantity does not determine the other.

The dominant semiclassical black-hole saddle does not reproduce the exact quasiperiodic signal by itself. Subdominant saddles can produce an entropy-suppressed non-decaying contribution without reconstructing the detailed recurrence pattern or necessarily its correct magnitude; see Maldacena 2003, § 3, pp. 13–15 and Barbón and Rabinovici 2003, §§ 3–4.

Do not identify this operator-correlator noise with the plateau of the spectral form factor

$$K_\beta(t)=\frac{\lvert Z(\beta+it)\rvert^2}{Z(\beta)^2}.$$

They weight matrix elements and energy pairs differently. A smooth ramp is normally exposed only after a stated time average, energy-window average, spectral smoothing, or ensemble average; a single spectrum is noisy. Random-matrix universality gives a quantitative model in suitable chaotic regimes after the appropriate unfolding or averaging, not a theorem about every large-$N$ CFT. See Cotler et al. 2017, §§ 1, 3, 5, and 8 and Altland and Sonner 2021, §§ 2.1–2.4.

A controlled special case makes the perturbative and nonperturbative roles unusually explicit. In $\mathcal N=1$ JT gravity, a double-scaled analysis assigns the ramp to a perturbative genus term and the plateau to a nonperturbative random-matrix completion; Griguolo et al. 2024, §§ 1 and 5 develops this resurgent ramp–plateau transition. That solvable model is evidence for a mechanism, not a universal finite-$N$ theorem for higher-dimensional CFTs.

Two adversarial scaled limits

Two elementary fixtures expose nonuniformity.

Entropy-scaled time. Two simple omitted contributions are

$$\begin{aligned} \delta C_S^{\mathrm{alg}}(t) &=\frac1S\frac{t}{L},\\ \delta C_S^{\mathrm{exp}}(t) &=e^{-S}\frac{t}{L}. \end{aligned}$$

For every fixed $t$, both vanish as $S\to\infty$. Along $t=\alpha LS$, the first equals $\alpha$; along $t=\alpha Le^S$, the second does too. These algebraic fixtures are not models of an entire correlator. They show why a suppression known only at fixed time cannot license a statement uniform over entropy-growing domains; a real calculation must determine the actual coefficient and time dependence.

Gap-scaled energy. Suppose the first omitted contact term is

$$\frac{\delta\mathcal A}{\mathcal A_{\mathrm{ref}}} =c_q\left(\frac{Q}{\Lambda_{\mathrm{der}}}\right)^q.$$

Consider a family with fixed $q>0$, fixed $QL$, $\Lambda_{\mathrm{der}}L/\Delta_{\mathrm{gap}}\to a>0$, and $c_q$ uniformly bounded. The correction then vanishes at fixed $Q$ as the gap grows. Along $Q=\kappa\Lambda_{\mathrm{der}}$, it becomes $c_q\kappa^q$; it remains parametrically order one when $c_q\to c\ne0$ and $\kappa=O(1)$. This path has moved the observable to the derivative cutoff. If a string, multiparticle, or higher-spin threshold lies lower, that lower scale controls the first breakdown instead.

These failures downgrade a uniform approximation claim. They do not show that the exact boundary theory lacks a bulk description; they show that additional saddles, heavy states, loops, or nonperturbative sectors are needed for the newly requested precision or domain.

A reproducible correction workflow

Before using a bulk approximation, record:

1. the normalized boundary observable and the candidate bulk quantity;
2. the theory, state or ensemble, volume, regulator, and symmetry sector;
3. the variables held fixed in the $N$, coupling, gap, energy, entropy, and time limits;
4. an error norm and tolerance with the correct units, including its treatment near zeros;
5. the leading retained term and first omitted term in every independent expansion;
6. bounds on coefficients and on the remainder over the full claimed domain;
7. each crossover scale, followed by the earliest one reached along the chosen path;
8. every smoothing, smearing, unfolding, time average, energy window, or ensemble average;
9. the degrees of freedom or competing saddles expected after breakdown;
10. at least one scaled path designed to make a nominally small correction order one.

The workflow prevents a category error: a saddle can fail for a fine-grained observable while remaining accurate after coarse graining. Failure of a saddle, loop expansion, or derivative expansion is not failure of the exact boundary description or of an exact duality statement.

Common pitfalls

Treating pointwise big-O notation as a uniform bound. A remainder can be $O(N^{-4})$ at every fixed point while its coefficient grows without bound on $D_N$. State and prove the supremum estimate needed for the advertised domain.

Calling every late-time scale a recurrence time. The secular crossover, Heisenberg time, dip or ramp scale, noise plateau, and Poincaré recurrence answer different questions. Define each from the observable actually being computed.

Equating correlator noise with a spectral-form-factor plateau. The two quantities have different weights and normalizations. Specify the averaging operation before interpreting a ramp or plateau.

Promoting approximation failure to duality failure. First identify which omitted saddles, states, or sectors have become unsuppressed. Only a contradiction in exact quantities would challenge the exact dictionary.

Exercises

1. Test species-enhanced loop counting

At fixed $QL$, fixed $r_G$, and fixed diagram coefficient, suppose the weighted loop sum scales as $N_{\mathrm{eff}}/N^2$ with $N_{\mathrm{eff}}\sim N^\alpha$. For which $\alpha$ is the loop expansion parametrically suppressed?

Solution — species scaling

The correction scales as $N^{\alpha-2}$. It vanishes for $\alpha<2$, remains order one for $\alpha=2$, and grows for $\alpha>2$. Quoting $N^{-2}$ without the species scaling would miss the latter two cases.

2. Locate a secular crossover

Let $\epsilon_{\mathrm{sec}}=N^{-2}e^{\gamma t/L}$ with $\gamma>0$. Evaluate it along $t=aL\log N$ for $a\ge0$, and find the values of $a$ for which the correction vanishes, remains order one, or grows.

Solution — secular crossover

Along this path, $\epsilon_{\mathrm{sec}}=N^{-2+a\gamma}$. It vanishes for $a<2/\gamma$, is order one at $a=2/\gamma$, and grows for $a>2/\gamma$. Fixed-time suppression therefore gives no uniform control beyond the logarithmic crossover.

3. Resolve the two-saddle window

For the two-saddle model, take $\Delta I=a(x-x_c)$ and $x=x_c+y/N^2$. Show that neither saddle can be discarded at fixed $y$.

Solution — two-saddle window

The ratio is $Z_2/Z_1=e^{-N^2\Delta I}=e^{-ay}$. It stays finite and nonzero as $N\to\infty$ at fixed $y$, so both terms contribute throughout the shrinking $O(N^{-2})$ crossover region.

4. Derive the finite-N mean square

Starting from the spectral sum for $C_N(t)$, subtract the complete zero-frequency block and derive the displayed formula for $\overline{\lvert\widetilde C_N\rvert^2}$. Explain why this time average does not determine the recurrence-sensitive limsup.

Solution — finite-N mean square

Write $\widetilde C_N(t)=\sum_{\omega\ne0}A_\omega e^{i\omega t}$, where $A_\omega=Z^{-1}\sum_{E_m-E_n=\omega}e^{-\beta E_m}\lvert O_{mn}\rvert^2$. In the long-time average of $\lvert\widetilde C_N\rvert^2$, the integral of $e^{i(\omega-\omega')t}$ vanishes unless $\omega=\omega'$. The result is $\sum_{\omega\ne0}\lvert A_\omega\rvert^2$, including all degeneracies of each energy difference. The average measures the total persistent squared amplitude, whereas the limsup can be controlled by rare phase alignments; neither fixes the other without additional spectral information.

5. Compare fixed and gap-scaled energy

Let $\Lambda_{\mathrm{der}}L=a\Delta_{\mathrm{gap}}$ with fixed $a>0$, take fixed $q>0$ and $c_q\to c\ne0$, and consider $\delta\mathcal A/\mathcal A_{\mathrm{ref}}=c_q(Q/\Lambda_{\mathrm{der}})^q$. Compare the large-gap limit at fixed $QL$ with the path $Q=\kappa\Lambda_{\mathrm{der}}$ at fixed $0<\kappa<1$.

Solution — gap-scaled energy

At fixed $QL$,

$$\frac{Q}{\Lambda_{\mathrm{der}}} =\frac{QL}{a\Delta_{\mathrm{gap}}} \longrightarrow0,$$

so the correction vanishes as $\Delta_{\mathrm{gap}}^{-q}$. Along $Q=\kappa\Lambda_{\mathrm{der}}$, the same correction tends to $c\kappa^q$, which is independent of the gap and nonzero. The fixed-energy expansion is therefore not uniform on an energy domain that grows with the cutoff.

Limits of the claim

Evidence cutoff. The literature check for this page extends through 28 August 2026. The finite-volume spectral decomposition is exact under the stated discreteness and convergence assumptions; the saddle and random-matrix mechanisms are example-dependent. They do not provide a universal recurrence time, a universal nonperturbative completion, or a theorem that every large-$N$ CFT has a semiclassical black-hole dual. The next article, Nonperturbative Exponential Effects and Finite-N Sectors, explains what can replace a failed perturbative description. Mutable results and open disputes continue in the Holography and Quantum Gravity research field guide.

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

• Altland, Alexander, and Julian Sonner. “Late Time Physics of Holographic Quantum Chaos.” SciPost Physics 11, 034 (2021). doi:10.21468/SciPostPhys.11.2.034. Open PDF.
• Barbón, José L. F., and Eliezer Rabinovici. “Very Long Time Scales and Black Hole Thermal Equilibrium.” Journal of High Energy Physics 2003, 047 (2003). doi:10.1088/1126-6708/2003/11/047. Open PDF.
• Cotler, Jordan S.; Guy Gur-Ari; Masanori Hanada; Joseph Polchinski; Phil Saad; Stephen H. Shenker; Douglas Stanford; Alexandre Streicher; and Masaki Tezuka. “Black Holes and Random Matrices.” Journal of High Energy Physics 2017, 118 (2017). doi:10.1007/JHEP05(2017)118. Open PDF.
• Griguolo, Luca; Jacopo Papalini; Lorenzo Russo; and Domenico Seminara. “The Resurgence of the Plateau in Supersymmetric $\mathcal N=1$ Jackiw–Teitelboim Gravity.” Journal of High Energy Physics 2024, 168 (2024). doi:10.1007/JHEP06(2024)168. Open PDF.
• Maldacena, Juan. “Eternal Black Holes in Anti-de Sitter.” Journal of High Energy Physics 2003, 021 (2003). doi:10.1088/1126-6708/2003/04/021. Open PDF.
• Witten, Edward. “Anti-de Sitter Space, Thermal Phase Transition, and Confinement in Gauge Theories.” Advances in Theoretical and Mathematical Physics 2 (1998): 505–532. doi:10.4310/ATMP.1998.v2.n3.a3. Open PDF.