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.

Independent correction parameters

For an observable $\mathcal A(E,t)$, a useful schematic expansion is

$$\begin{aligned} \mathcal A =\mathcal A_0 &+\frac{1}{N^2}\mathcal A_{\mathrm{loop}} +\lambda^{-p}\mathcal A_{\mathrm{string}} +\left(\frac{EL}{\Delta_{\mathrm{gap}}}\right)^q \mathcal A_{\mathrm{der}}\\ &+\left(\frac{t}{t_{\mathrm{sec}}}\right)^r \mathcal A_{\mathrm{sec}} +e^{-S}\mathcal A_{\mathrm{disc}} +\cdots . \end{aligned}$$

The displayed powers are placeholders until the theory and observable determine $p,q,r$. More importantly, the terms are not interchangeable:

• $N^{-2}$ commonly counts a closed-string loop or a matrix-theory handle, after operator normalization.
• $\lambda^{-p}$ can count finite string-length effects in examples where a boundary coupling controls $L/\ell_s$.
• $EL/\Delta_{\mathrm{gap}}$ counts derivatives relative to the first omitted heavy single-trace state.
• $t/t_{\mathrm{sec}}$ detects secular accumulation; its coefficient may be parametrically small even though the product is not.
• $e^{-S}$ characterizes effects tied to finite entropy or fine spectral discreteness, but their precise power and observable dependence are not universal.

There is therefore no meaningful statement that an answer is “leading order” without specifying the simultaneous bounds on all relevant parameters. For example, tree-level supergravity can require $N\gg1$, $\lambda\gg1$, $EL\ll\Delta_{\mathrm{gap}}$, and a time window shorter than both secular and discreteness scales.

Uniformity and crossover estimates

An asymptotic expansion at fixed $x$ is not automatically uniform on an $N$-dependent domain. If

$$\mathcal A_N(x)=\mathcal A_0(x)+N^{-2}f(x)+O(N^{-4}),$$

then the leading approximation is controlled on a domain $D_N$ only if

$$\sup_{x\in D_N} \frac{\lvert N^{-2}f(x)\rvert} {\max(1,\lvert\mathcal A_0(x)\rvert)} \longrightarrow0.$$

This criterion catches two common failures. A correction $N^{-2}e^{\gamma t/L}$ becomes order one at

$$t_{\mathrm{sec}}\sim\frac{2L}{\gamma}\log N,$$

while a derivative term $(EL/\Delta_{\mathrm{gap}})^q$ is small at fixed $E$ but not when $E=\kappa\Delta_{\mathrm{gap}}/L$. The crossover estimate is part of the result, not an optional warning.

Zeros of $\mathcal A_0$ require an absolute error or a comparison with the first nonzero term. A tiny correction divided by a symmetry-enforced zero produces a meaningless relative error. Similarly, large numerical coefficients or a number of light species that grows with $N$ can offset the nominal suppression.

Thermal correlators at finite N

Consider a thermal two-point function in a finite-volume boundary theory with discrete spectrum,

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

After subtracting any stationary diagonal component, a smooth large-$N$ saddle may predict decay because densely spaced phases are replaced by a continuum. At every finite $N$, however, $C_N(t)$ is a quasiperiodic sum. Its infinite-time mean square is generically nonzero:

$$\overline{\lvert C_N\rvert^2} = \lim_{T\to\infty}\frac1T \int_0^Tdt\,\lvert C_N(t)\rvert^2 >0,$$

with the surviving terms selected by equal energy differences. In a chaotic microcanonical band the fine-grained scale is often exponentially small in the entropy, and the inverse mean level spacing gives a Heisenberg time of order $e^S$. The exact coefficient and even the relevant entropy depend on the band and normalization.

This is the first application. If $N\to\infty$ is taken at fixed $t$, the level spacing vanishes before the late-time limit and the saddle can decay to zero. If late time is explored at fixed $N$, discreteness produces a noise floor, a spectral-form-factor plateau, and—at much longer, nonuniversal times—Poincaré recurrences. In terms of a recurrence-sensitive operation,

$$\lim_{N\to\infty}\limsup_{t\to\infty}\lvert C_N(t)\rvert \quad\text{need not equal}\quad \limsup_{t\to\infty}\lim_{N\to\infty}\lvert C_N(t)\rvert.$$

The semiclassical black-hole saddle is not expected to reproduce the exact late-time quasiperiodic signal by itself. This tension and the role of subdominant saddles were emphasized by Maldacena 2003 and Barbon and Rabinovici 2003. Random-matrix spectral correlations offer a quantitative model for ramp-and-plateau behavior in suitable chaotic systems, not a theorem that every holographic theory has the same fine spectrum; see Cotler et al. 2017.

Adversarial scaled limits

Two elementary fixtures expose nonuniformity.

Entropy-scaled time. Suppose an omitted contribution has the form

$$\delta C_S(t)=e^{-S}\frac{t}{L}.$$

For every fixed $t$, it vanishes as $S\to\infty$. Along the sequence $t=\alpha Le^S$, it instead equals $\alpha$. The algebraic example is not a model of an entire correlator; it proves that pointwise exponential suppression alone cannot license a statement uniform up to the Heisenberg scale. A real calculation must determine the actual time dependence.

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

$$\frac{\delta\mathcal A}{\mathcal A_0} =c_q\left(\frac{EL}{\Delta_{\mathrm{gap}}}\right)^q.$$

At fixed $E$ it vanishes when $\Delta_{\mathrm{gap}}\to\infty$. At $E=\kappa\Delta_{\mathrm{gap}}/L$, it is $c_q\kappa^q$, independent of the gap. Choosing $\kappa$ of order one has moved the observable to the cutoff, where the truncated EFT is not predictive.

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 budget

Before using a bulk approximation, record:

1. the normalized boundary observable and the candidate bulk quantity;
2. the state or ensemble and all extensive parameters;
3. the held-fixed variables in the $N$, coupling, gap, energy, entropy, and time limits;
4. the leading retained term and the first omitted term in each expansion;
5. an absolute or relative error norm appropriate near zeros;
6. crossover scales at which each omitted term reaches the tolerance;
7. whether the estimate is pointwise, averaged, or uniform;
8. the alternative description expected after breakdown.

The list prevents a frequent category error: a saddle can fail as an approximation to a fine-grained observable while remaining an excellent approximation to a coarse-grained one.

Evidence ceiling and handoffs

Evidence cutoff: 25 July 2026. Exact finite-volume spectral decompositions establish discreteness, while particular holographic examples and random-matrix models demonstrate concrete late-time mechanisms. These results do not supply a universal recurrence time, a universal nonperturbative completion, or a proof that all large-$N$ CFTs have semiclassical black-hole duals. String corrections belong to the string-regime treatment; detailed high-energy and late-time diagnostics belong to their dedicated chapters; generic EFT error estimation belongs to the renormalization and EFT volume.

Exercises

1. A loop correction scales as $N_{\mathrm{sp}}/N^2$, where $N_{\mathrm{sp}}$ is the number of light species. For which growth $N_{\mathrm{sp}}\sim N^\alpha$ is the loop expansion parametrically suppressed?

   Solution

   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. Why does a nonzero finite-$N$ late-time mean square not contradict pointwise decay of the strict large-$N$ saddle?

   Solution

   The two statements take different limits and use different observables. Replacing a discrete spectrum by a continuum at fixed time can yield decay, while time averaging at fixed finite $N$ retains coincident frequency differences. The saddle estimate is not uniform over arbitrarily late times.

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

• Barbon, José L. F., and Rabinovici, Eliezer. “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.
• Cotler, Jordan S.; Gur-Ari, Guy; Hanada, Masanori; Polchinski, Joseph; Saad, Phil; Shenker, Stephen H.; Stanford, Douglas; Streicher, Alexandre; and Tezuka, Masaki. “Black Holes and Random Matrices.” Journal of High Energy Physics 2017, 118 (2017). doi:10.1007/JHEP05(2017)118.
• 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.