Finite N, Horizons, State Dependence, and Reconstruction Limits

Large-$N$ bulk reconstruction is an asymptotic construction, not an exact finite-$N$ isomorphism between local bulk QFT and the full CFT operator algebra. At finite $N$, the boundary spectrum is discrete on a compact spatial manifold, black holes have finite entropy, perturbative mode expansions develop nonperturbative ambiguities, and a smooth horizon can demand state- or code-dependent interior operators. A meaningful claim must state the boundary region, algebra, gravitational dressing, state/code sector, order in $1/N$ and bulk couplings, time range, and error norm. We use Lorentzian global AdS, a compact boundary sphere, and thermal correlators with the standard real-time $i0$ prescription.

Required background. Interacting and dressed reconstruction supplies the perturbative expansion, and causal wedges delimit what causal propagation alone establishes.

Helpful background. Real-time horizon validity tracks contour and saddle assumptions, while finite-size scrambling diagnostics separates genuine information loss from finite-size or symmetry effects.

Where the large-N construction is controlled

Let $P_{\mathrm{code}}$ project onto states obtained from a semiclassical reference state by a bounded number of low-energy insertions. A perturbative reconstruction has the form

$$\Phi_{\mathrm{rec}} =\Phi^{(0)}+\frac{1}{N}\Phi^{(1)}+\cdots +\frac{1}{N^K}\Phi^{(K)},$$

with a statement such as

$$\left\lVert P_{\mathrm{code}} (\Phi_{\mathrm{rec}}-\Phi_{\mathrm{EFT}})P_{\mathrm{code}}\right\rVert \leq \varepsilon_K.$$

The projector is part of the result. Outside its range, high energies change the geometry, mode completeness can fail, and the same boundary polynomial need not approximate the intended relational observable. Likewise, an $O(N^{-K-1})$ remainder is not a bound on effects of order $e^{-cN^p}$; the asymptotic series does not determine those sectors.

Near a horizon, fixed asymptotic time can correspond to a large local boost. Corrections suppressed at early time may grow like

$$\varepsilon_{\mathrm{eff}}(t)\sim \frac{1}{N^p}e^{\lambda_L t}$$

until perturbation theory fails near a scrambling scale $t_*\sim(p/\lambda_L)\log N$. This schematic estimate is a diagnostic, not a universal law: $p$, $\lambda_L$, the relevant norm, and the operator ordering depend on the model. A smooth classical horizon does not grant uniform control at arbitrary boost or time.

A finite-N spectral calculation

Consider a CFT energy window of dimension $D=e^S$ at infinite temperature and an operator with

$$\frac{1}{D}\operatorname{Tr}(O^\dagger O)=1, \qquad O_{nn}=0.$$

Its exact correlator is

$$C(t)=\frac1D\operatorname{Tr}\!\left[O(t)O^\dagger(0)\right] =\frac1D\sum_{m,n}|O_{mn}|^2e^{i(E_m-E_n)t}.$$

For nondegenerate energy gaps, the infinite-time mean square is

$$\overline{|C(t)|^2} =\frac{1}{D^2}\sum_{m,n}|O_{mn}|^4.$$

In an ETH-like toy window with $|O_{mn}|^2\sim D^{-1}$, the normalization is satisfied and

$$\overline{|C(t)|^2}\sim D^{-2}=e^{-2S}, \qquad |C|_{\mathrm{rms}}\sim e^{-S}.$$

The exact finite system therefore has quasiperiodic late-time noise even if the leading black-hole saddle predicts exponential decay to zero. The precise exponent and any nonzero time average depend on the observable, symmetry sectors, gap degeneracies, ensemble, and diagonal elements; the robust point is that the signal is nonperturbatively small in the entropy. Resolving this toy feature requires absolute error $\varepsilon\ll e^{-S}$ and times comparable to inverse many-body level spacings, typically exponential in $S$. No fixed order in $1/N$, with $S\sim N^p$, supplies that accuracy. Maldacena’s analysis of two-sided AdS black-hole correlators makes the tension between semiclassical decay and finite-entropy recurrences explicit (Maldacena 2003, §5).

This is the first application: a large-$N$ reconstructed correlator can be excellent at fixed time and every computed perturbative order while being incapable of resolving an exact finite-$N$ spectral feature. “Small correction” is incomplete until its scale is compared with the target signal.

State dependence is a domain statement

An exterior bulk operator can often be represented state-independently over a broad perturbative sector. A behind-horizon mode is more delicate: its semiclassical commutation relations and entanglement with exterior modes refer to a chosen background state. A state-dependent proposal assigns a family

$$|\Psi\rangle\longmapsto \widetilde O_{\Psi,\mathcal H_{\mathrm{code}}},$$

rather than one linear operator valid on the entire CFT Hilbert space. Such a construction can be consistent on a sufficiently small code subspace, but then superpositions, overlaps between code subspaces, and the allowed measurement algebra must be checked explicitly. Papadodimas and Raju proposed mirror operators of this kind for equilibrium black-hole states (Papadodimas and Raju 2013, §§3–5); Harlow analyzed consistency conditions and limitations of extending that prescription (Harlow 2014, §§2–4). These references establish a proposal and critique, not a universal nonperturbative interior dictionary.

State dependence also appears more mundanely when a reconstruction kernel is built using a state-specific background metric. That dependence should not be confused with nonlinear quantum evolution: within a fixed declared code subspace, the represented operator must still act linearly.

The four-demand adversarial test

Demand simultaneously:

1. one state-independent boundary operator on the full finite-dimensional black-hole Hilbert space;
2. exact local-QFT commutators for independent interior and exterior modes;
3. a smooth-horizon entanglement pattern for every microstate; and
4. validity at arbitrarily late times and exponentially small errors.

The demands omit the assumptions that made effective bulk QFT possible. Exact independent oscillator algebras require unbounded Fock spaces, whereas the black-hole sector has dimension $e^S$. Exact interior/exterior factorization conflicts with gravitational constraints and finite entropy. A universal smooth-horizon entanglement pattern conflicts with allowing all vectors in the microcanonical space, and perturbation theory cannot certify exponential precision.

At least one demand must be weakened: restrict the code subspace and time range; accept approximate commutators in a stated norm; use a state- or background-dependent interior representation with compatibility conditions; or supply a genuinely nonperturbative dictionary. The test does not by itself select among these alternatives. It exposes which missing premise an interior claim needs.

Controlled limits and handoff

The order of limits matters. Taking $N\to\infty$ first at fixed time yields a continuous semiclassical spectrum and suppresses recurrences. Taking $t\to\infty$ at fixed $N$ probes discreteness. Increasing the code energy or number of insertions at fixed $N$ eventually invalidates the background and operator expansion. Near a horizon, holding a small asymptotic perturbation fixed while increasing the boost can also leave the code sector.

Reconstruction error norms turns these qualifications into explicit bounds. Later chapters may represent an operator redundantly through quantum error correction, but any such representation remains relative to an algebra, code sector, and accuracy; it does not create an exact state-independent local interior algebra at finite $N$.

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

• Harlow, D. (2014). “Aspects of the Papadodimas–Raju proposal for the black hole interior.” Journal of High Energy Physics 2014(11), 055. DOI.
• Maldacena, J. (2003). “Eternal black holes in anti-de Sitter.” Journal of High Energy Physics 2003(4), 021. DOI.
• Papadodimas, K., and Raju, S. (2013). “An infalling observer in AdS/CFT.” Journal of High Energy Physics 2013(10), 212. DOI.