Typicality, Non-BPS Extensions, and Evidence Limits

A typical black-hole microstate is not merely a complicated state or a representative-looking geometry. Typicality is a probability statement: after fixing a state sector, a measure, an observable, a tolerance, and a time or resolution scale, the exceptional fraction must become small in a declared limit. In the D1–D5 system, ordinary Cardy counting already gives entropy-sized non-BPS densities in controlled regimes, and several smooth non-BPS soliton families are known. What remains unproved is that a quantized smooth-geometry sector is entropy-saturating and typical, or that its strong-coupling observables satisfy the required concentration and dynamical tests.

Required background. BPS Indices, Absolute Degeneracies, and Wall Crossing supplies the protection and cancellation ceiling; Microstate Geometries and Fuzzball Proposals supplies explicit solutions and the known-family entropy shortfall; Absorption, Emission, and Dynamical Tests supplies the distinction between an inclusive response and a state-resolved spectrum.

Helpful background. The Eigenstate Thermalization Hypothesis develops the matrix-element ansatz and its limitations. Claim–Evidence Records, Replication, and Retraction Handling and QEC Evidence, Current Disputes, and Status provide comparison standards for frontier claims.

Evidence cutoff: 29 August 2026.

On this page. Follow the typicality contract, concentration test, D1–D5 evidence comparison, current non-BPS constructions, six promotion gates, adversarial tests, or solved exercises.

Typicality requires a sector, measure, observable, and time scale

Four distinctions prevent most overclaims on this subject.

BPS is not the same as extremal

BPS means that a state saturates a supersymmetry bound and belongs to a shortened representation. Extremal means that a black-hole solution lies at its zero-temperature horizon bound. Many familiar BPS black holes are extremal, but the concepts are not equivalent: extremal non-BPS solutions exist, and a non-BPS state need not be nonextremal. This page is primarily concerned with finite-energy-density, non-BPS states in AdS microcanonical windows and with metastable asymptotically flat solitons that model aspects of nonextremal black holes.

Resolve the energy window before counting

Let $H|n\rangle=E_n|n\rangle$. Fix the boundary conditions, compactification, couplings, and every exact charge or symmetry label collected in $\Gamma$. For a window centered at $E$, define

$$\begin{aligned} \mathcal I_W &=\left\{n:\left|E_n-E\right|\leq\frac{\Delta E}{2}, Q_a(n)=\Gamma_a\right\},\\ P_W&=\sum_{n\in\mathcal I_W}|n\rangle\langle n|, \qquad D_W=\operatorname{Tr}P_W, \qquad S_{\rm mc}=\log D_W . \end{aligned}$$

We use $k_B=\hbar=1$. The level spacing should be much smaller than $\Delta E$, while $\Delta E$ should be small on the scale over which the macroscopic parameters change. The identification $S_{\rm mc}=S_{\rm BH}+o(S_{\rm BH})$ is licensed only in a scaling regime where the black-hole saddle dominates the specified sector.

For a proposed subset of states, entropy saturation means more than having many examples. In a declared large-charge or large-$N$ limit $\lambda\to\infty$, it requires

$$\log D_{\rm cand}(\lambda) =S_{\rm BH}(\lambda)+o\!\left(S_{\rm BH}\right).$$

Only after a classical family has been mapped to distinct quantum states or quantized with its gauge, regularity, and normalizability constraints can one form the counting fraction

$$F_{\rm cand}(\lambda)=\frac{D_{\rm cand}(\lambda)}{D_W(\lambda)}.$$

A raw number of solutions or continuous parameters is not a numerator for this ratio.

Measures answer different questions

Three common measures are inequivalent.

• Uniform eigenstate counting assigns weight $1/D_W$ to each energy eigenstate in $\mathcal I_W$. This is the natural measure for weak or strong eigenstate thermalization.
• Haar measure on shell superpositions samples unit vectors in $P_W\mathcal H$. Canonical typicality can make a small subsystem close to the shell average for overwhelmingly many such vectors, but it does not imply that Hamiltonian eigenstates are Haar-random or that typical vectors admit smooth classical geometries Goldstein et al. 2006, pp. 1–2, especially eq. (5).
• A preparation or quantized phase-space measure weights states according to a dynamical protocol or semiclassical quantization. Its support can differ sharply from both of the measures above.

For an observable functional $X_O$ and target value $X_{O,{\rm th}}$, a typicality claim has the operational form

$$p_O(\epsilon;\lambda) =\mu_\lambda\!\left( \left|X_O-X_{O,{\rm th}}\right|>\epsilon \right) \longrightarrow0 .$$

The statement is incomplete unless it names $\mu_\lambda$, $O$, $\epsilon$, and the limit $\lambda\to\infty$. A result for one operator or one time window does not automatically extend to all simple observables or all times.

ETH is an eigenbasis statement

For a suitably simple, neutral operator $O$ in one resolved chaotic symmetry sector, the standard ETH ansatz is

$$O_{mn} =\overline O(\bar E)\delta_{mn} +e^{-S_{\rm mc}(\bar E)/2} f_O(\bar E,\omega)R_{mn},$$

where

$$\bar E=\frac{E_m+E_n}{2}, \qquad \omega=E_m-E_n.$$

Here $\overline O$ and $f_O$ are smooth on the selected microcanonical scale. After the smooth envelope is removed, $R_{mn}$ is modeled as a zero-mean, locally unit-variance irregular variable with statistics compatible with Hermiticity. This is an ansatz about a distinguished Hamiltonian eigenbasis, not a Haar statement about arbitrary shell superpositions D’Alessio et al. 2016, §4.2, pp. 31–34, especially eqs. (62)–(63).

Its diagonal and off-diagonal parts answer different questions. For a fixed $O$ and tolerance $\epsilon$, define the exceptional eigenstate fraction

$$f_{O,\epsilon}(W) =\frac{1}{D_W} \#\left\{n\in\mathcal I_W: \left|O_{nn}-\overline O_W\right|>\epsilon\right\}, \qquad \overline O_W=\frac{1}{D_W}\sum_{n\in\mathcal I_W}O_{nn}.$$

Weak ETH requires $f_{O,\epsilon}\to0$ for the declared observable class. Strong ETH requires the largest diagonal deviation in the window to vanish. Off-diagonal ETH additionally constrains transitions, relaxation, and late-time fluctuations. Counting states, constructing one geometry, or matching one inclusive response does not test all three statements.

A reproducible concentration test

A finite-dimensional audit can test weak typicality without assuming the full ETH ansatz. Define the diagonal variance

$$\sigma_{O,W}^{2} =\frac{1}{D_W} \sum_{n\in\mathcal I_W} \left|O_{nn}-\overline O_W\right|^{2}.$$

Markov’s inequality applied to the nonnegative squared deviation gives

$$f_{O,\epsilon}(W) \leq\frac{\sigma_{O,W}^{2}}{\epsilon^{2}}.$$

Thus a demonstrated bound $\sigma_{O,W}^{2}\leq e^{-cS_{\rm mc}}$ implies an exponentially small exceptional fraction for fixed $\epsilon$. This is a reproducible weak-typicality result for the chosen diagonal observable, sector, and window. It is not strong ETH, it says nothing about $O_{m\neq n}$, and it does not by itself prove relaxation from a specified initial state.

Two controls show why means and large families are insufficient:

1. If half the states have $X=X_{\rm th}+\Delta$ and half have $X=X_{\rm th}-\Delta$, then the mean is exactly thermal while $p_X(\epsilon)=1$ for every $\epsilon<\Delta$.

2. If $D_{\rm cand}=e^{\alpha S_{\rm BH}}$ with fixed $0<\alpha<1$ while $D_W=e^{S_{\rm BH}}$, then

   $$F_{\rm cand}=e^{-(1-\alpha)S_{\rm BH}}\longrightarrow0.$$

   The candidate family is exponentially large and nevertheless exponentially atypical under uniform eigenstate counting.

D1–D5 evidence versus the non-BPS target

The following table separates evidence that is often compressed into one narrative. The contracts are deliberately different; success in one row does not fill the missing column in another.

D1–D5 evidence streams and the additional premise needed for a typical non-BPS conclusion

Evidence stream	Declared contract and observable	Strongest direct conclusion	Missing promotion
Protected trace	Charges, chamber, helicity insertion, contour, coupling path, and asymptotic order	Protected indexed asymptotics in the stated BPS sector	Absolute non-BPS density, uncancelled measure, and transport of unprotected observables
Ordinary Cardy density	Left and right excitation levels, central charges, thermodynamic/Cardy regime, and boundary conditions	Entropy-sized non-BPS density matching generic BTZ or controlled near-extremal entropy at leading order	A statement that a specified smooth-geometry subset carries that density or dominates a chosen measure
Regular classical configurations	Charges, asymptotics, smoothness, causal conditions, truncation or uplift, and any CFT state map	Existence of the stated classical solutions; quantum states or coherent sectors only where mapped or quantized	Entropy-sized orthogonal-state count, measure, and lifetime or stability in the relevant perturbation sectors
Inclusive response	Ensemble, operator, polarization, partial wave, frequency, resolution, normalization, and coupling regime	A weighted spectral density or averaged transition strength in the stated channel	Distribution across states, individual phases and matrix elements, other operators, and late-time discreteness
Typical non-BPS target	Energy window, exact sector, measure, observable class, coupling, tolerance, resolution, and observation time	Only the observables for which concentration and dynamics pass their declared limits	A broad strong-coupling state map and evidence that every required gate below is passed together

Entropy-sized non-BPS counting is a real positive result

A protected index and an ordinary density of states must not be merged. In a two-dimensional CFT with both chiral sectors excited, ordinary Cardy growth reproduces the two-sided BTZ entropy in its stated regime; this is non-BPS state counting, not a signed trace Strominger 1998, §5, pp. 7–8, especially eqs. (5.1)–(5.3). Near extremality, left- and right-moving D-brane excitations likewise reproduce the black-string entropy under the dilute-gas and thermodynamic assumptions Horowitz and Strominger 1996, §§2–3, pp. 5–7.

This result answers the state-density question in controlled sectors. It does not show that those states are all represented by smooth horizonless geometries. For the counted superstrata family with $N_p,J=O(N)$,

$$S_{\rm strata}\sim N^{3/4}, \qquad S_{\rm BMPV}\sim N,$$

and in the Cardy regime,

$$S_{\rm strata}\sim N^{1/2}N_p^{1/4}, \qquad S_{\rm BMPV}\sim N^{1/2}N_p^{1/2}.$$

The known counted family therefore occupies an exponentially small microcanonical fraction in those scalings Shigemori 2019, §4.6, pp. 25–26, especially eqs. (4.91)–(4.94). This excludes typicality for that family, not for unknown fractional modes, stringy states, or as-yet-unconstructed families.

Coarse probes can look thermal before fine probes do

There are also bounded positive typicality mechanisms. Typical Ramond ground states at large central charge reproduce the massless-BTZ two-point function for sufficiently short separations, while at long times the correlator becomes microstate-dependent Balasubramanian, Kraus, and Shigemori 2005, §§3–4. In the finite-radius two-charge orbifold laboratory, an incoherent beam gives the semiclassical, microstate-independent absorption result when $R_y\Delta E\gg1$; finer energy resolution reveals the particular microstate Das and Mandal 2009, §§1 and 4.

Even within a two-dimensional CFT, the sampling class matters. At finite central charge, typical high-energy descendants have thermal stress-tensor correlators, whereas high-energy primary states need an extreme generalized Gibbs ensemble instead of the ordinary canonical ensemble Datta, Kraus, and Michel 2019, §§3–5. These results demonstrate coarse-grained emergence and concentration for specified observables and measures. None is a theorem that typical non-BPS D1–D5 states are smooth geometries.

What present non-BPS constructions establish

The current landscape is not one monotonic sequence toward a generic black hole. Different families control different boundary conditions, mode sectors, and notions of stability.

JMaRT: existence, a state map, and an instability

The JMaRT solutions are smooth, asymptotically flat, non-supersymmetric D1–D5–P solitons with an explicit relation to special CFT states Jejjala et al. 2005, §§1 and 3–6. They establish that supersymmetry is not necessary for smooth horizonless existence. They are not a stable typical ensemble: the family has an ergoregion and supports outgoing modes with positive growth rate Cardoso et al. 2006, §§III–VI and conclusion.

This is a useful example of why “stability” needs a time scale and perturbation class. With the convention $e^{-i\omega t}$, define

$$\gamma_{\max}=\max_a\!\left(0,\operatorname{Im}\omega_a\right).$$

Negligible linear growth over an intended observation time $t_\ast$ requires $\gamma_{\max}t_\ast\ll1$. Linear mode stability still does not prove nonlinear stability or a quantum lifetime.

Microstrata: nonlinear existence with an open late-time question

The Q-ball construction produced three-parameter nonextremal microstrata perturbatively and numerically as a proof of concept Ganchev, Houppe, and Warner 2021, §§1 and 9. Later microstrata are smooth non-BPS solitons in a consistent three-dimensional truncation, asymptotic to $AdS_3\times S^3\times T^4$, with dual multi-particle non-BPS supergraviton states and nonlinear binding energies Ganchev et al. 2023, §§1 and 5–7. Consistency of the truncation guarantees a classical uplift of its solutions; the explicit six-dimensional construction of the simplest special locus has now also been given Ramella and Warner 2026, §§1 and 6.

The evidence remains family-specific. Time-dependent microstrata escape some special-locus restrictions, but their perturbation series develops secular terms, some of which are not known to resum and may signal a long-time instability Houppe 2024, §§5–6, pp. 17–21. Neither stationary numerical robustness nor a consistent uplift supplies an entropy-sized quantization or generic nonlinear stability.

New construction tools broaden existence, not typicality

Systematic sigma-model and generalized-Ernst methods now generate static smooth non-BPS solitons with the same mass and charges as specified nonextremal black holes, including a non-BPS analogue of a Gibbons–Hawking center Chakraborty and Heidmann 2025, §§1, 4, and 5. Standard-Kaluza–Klein-asymptotic rotating topological stars provide a discrete tower of smooth non-BPS solutions that lies outside, but can approach, the conserved charges of a highly boosted Kerr black string; the solutions possess a five-dimensional ergoregion, and their dynamical analysis remains future work Heidmann, Pani, and Santos 2026, §§1, 2.5, and 5.

Restricted stability results can also be positive. For a static topological-star family, coupled gravitational and electromagnetic odd modes decay, while spherically symmetric even modes leave a finite stable parameter domain and an unstable domain Bena et al. 2024, §§3–5. This does not transfer automatically to rotating stars, Kaluza–Klein-momentum modes, or nonlinear perturbations.

These are substantial advances in the existence problem. An algebraic solution generator is not a state count, a discrete tower is not automatically entropy-sized, and matching conserved charges is not a proof of typical observables.

Vector superstrata belong in a different evidence row: they enlarge the supersymmetric D1–D5–P solution space and therefore serve as a BPS mode-content comparator, not as a non-BPS extension Čeplak and Hampton 2024, abstract and §§1–2.

The dominant phase depends on the compactification

A 2026 microcanonical analysis emphasizes another hidden variable: the typical phase depends on energy and on the ratio $R_T/R_{\rm AdS}$. For $R_T\gg R_{\rm AdS}$ it argues for an intermediate lattice of localized black holes with $S^5\times S^3$ horizon topology, while explicitly identifying part of that phase diagram as conjectural because the exact solutions are not known Aharony, Frumkin, and Mehl 2026, §§1, 2, and 5.3. A statement about “the typical D1–D5 state” is therefore incomplete without its compactification and energy regime.

Six gates for a typical non-BPS claim

The chapter diagrams show how evidence streams connect; the table below supplies operational pass conditions. The gates are cumulative for the strongest claim, but a narrower claim may need only the relevant subset.

Pass conditions and present status for a typical non-BPS black-hole interpretation

Gate	Operational pass condition	Status at the cutoff	Decisive next result
1. Entropy size	$\log D_{\rm cand}=S_{\rm BH}+o(S_{\rm BH})$ in a declared non-BPS window and scaling	Passed by ordinary CFT density in some Cardy/near-extremal sectors; not passed by a quantized smooth non-BPS geometry subset	Gauge-reduced quantization and count of the candidate smooth sector
2. State map and measure	Distinct quantum states, a named measure $\mu$, and $\mu(\mathcal C_{\rm cand})$ or its limiting fraction	Maps exist for selected families; no nonvanishing typical fraction is established	Injective or controlled-many-to-one state map plus measure comparison
3. Coupling and phase transport	Coupling path, operator identification, compactification, and all crossed phase boundaries controlled	Protected quantities transport selectively; generic non-BPS spectra and real-time data do not	Nonprotected strong-coupling calculation or transport theorem
4. Observable concentration	$p_O(\epsilon;\lambda)\to0$ for a declared simple-operator family; off-diagonal data added for dynamical ETH	Positive results exist for selected CFT observables, states, times, and resolutions; no broad smooth-geometry result	State-resolved distribution or rigorous variance bounds at strong coupling
5. Lifetime and stability	Perturbation sector, boundary conditions, $t_\ast$, linear spectrum, nonlinear control, and decay channel declared	Mixed and family-dependent: explicit instabilities, restricted stable domains, and unresolved secular behavior all occur	Channel-complete lifetime or stability analysis in the target scaling
6. Finite-$N$ late times	Level spacing $\delta E$, probe resolution, and $t_H\simeq2\pi/\delta E$ included when recurrence or information-recovery claims are made	Finite-volume CFT discreteness is known; its statewise bulk realization is not controlled for generic non-BPS microstates	Normalized finite-$N$ spectral data through the claimed time regime

The last gate is conclusion-dependent. It is not required for a leading entropy or a pre-Heisenberg-time coarse response; it is required for claims about exact recurrence, plateaus, information recovery, or parametrically late times.

Adversarial promotions

Promote a protected index to typical non-BPS states

Attempted conclusion. A large protected trace proves an entropy-sized set of typical non-BPS black-hole states.

Failure mode. The insertion can create cancellations, the trace lives in a BPS sector, and only protected data are guaranteed to cross the coupling path. Ordinary non-BPS Cardy counting must be supplied separately.

Strongest surviving claim. The declared indexed asymptotics are robust in their chamber and sector.

Decisive next test. Compute the absolute non-BPS density in the same window, then show the relevant observables and measure transport to the strong-coupling phase.

Promote one geometry to a typical family

Attempted conclusion. One smooth deep-throat solution, or a finite-parameter classical family, represents a typical black-hole microstate.

Failure mode. A classical solution is not yet a count of orthogonal quantum states. Every individual ray has zero Haar measure, so invoking “zero measure” for one geometry is uninformative; the missing objects are a state map or quantization and a specified counting or preparation measure.

Strongest surviving claim. The stated field equations, regularity conditions, charges, and any explicit state map establish existence in that family.

Decisive next test. Quantize the reduced phase space, compare its entropy with $S_{\rm BH}$, and measure concentration rather than visual resemblance.

Promote one averaged response to statewise ETH

Attempted conclusion. One greybody or absorption match proves that typical microstates have thermal matrix elements.

Failure mode. The inclusive response is a weighted sum of squared transition matrix elements. It does not determine their phases, their distribution over initial states, other polarizations, or fine spectral lines. The bimodal control above can reproduce the exact mean while every state remains nonthermal at tolerance $\epsilon<\Delta$.

Strongest surviving claim. The normalized spectral density agrees in the declared ensemble, channel, frequency, and resolution regime.

Decisive next test. Resolve the state distribution and variance for a family of simple operators, then test off-diagonal data and the finite-$N$ time window.

Common pitfalls

Calling every non-BPS solution nonextremal. Supersymmetry and extremality are distinct classifications. State both properties separately.

Calling a Haar result ETH. Canonical typicality concerns random shell vectors; ETH concerns matrix elements in the Hamiltonian eigenbasis.

Counting classical parameters as states. Continuous moduli require symplectic reduction and quantization before they contribute to $D_W$.

Equating an average with concentration. A mean response can be thermal while the state distribution is broad or bimodal.

Treating stationary existence as stability. Separate regularity, linear mode stability, nonlinear stability, and quantum decay time.

Exercises

1. Concentration from a variance bound

Suppose $\sigma_{O,W}^{2}\leq e^{-cS_{\rm mc}}$ with $c>0$. Bound the fraction of eigenstates whose diagonal expectation value differs from $\overline O_W$ by more than a fixed $\epsilon>0$. Why is the result not full ETH?

Solution

The concentration inequality gives

$$f_{O,\epsilon}(W) \leq\frac{e^{-cS_{\rm mc}}}{\epsilon^2} \longrightarrow0.$$

This proves weak typicality for one diagonal observable in the selected sector and limit. It does not bound the largest exceptional deviation, so it is not strong ETH; it also contains no information about off-diagonal matrix elements or real-time relaxation.

2. Weak ETH does not imply strong ETH

In a sequence of windows with dimension $D$, let $O_{nn}=0$ for $D-1$ states and $O_{nn}=1$ for one state. Show that weak typicality holds for every fixed $0<\epsilon<1$, while strong ETH fails.

Solution

The shell mean is $\overline O_W=1/D$. For fixed $0<\epsilon<1$, only the exceptional state violates the tolerance once $D$ is large, so

$$f_{O,\epsilon}=\frac{1}{D}\longrightarrow0.$$

However,

$$\max_n\left|O_{nn}-\overline O_W\right| =1-\frac{1}{D}\longrightarrow1,$$

so the largest deviation does not vanish. Weak ETH allows a vanishing fraction of rare states; strong ETH does not.

3. An exponentially large but atypical family

Let $D_W=e^{S_{\rm BH}}$ and $D_{\rm cand}=e^{\alpha S_{\rm BH}}$ with fixed $0<\alpha<1$. Compute the candidate fraction. What does the result exclude, and what does it leave open?

Solution

The uniform eigenstate-counting fraction is

$$F_{\rm cand} =\frac{e^{\alpha S_{\rm BH}}}{e^{S_{\rm BH}}} =e^{-(1-\alpha)S_{\rm BH}} \longrightarrow0.$$

The counted family is not typical under that measure even though it is exponentially large. The result does not exclude uncounted modes, stringy states, a different candidate family, or a different preparation measure.

4. A thermal mean with no typical state

Half of an ensemble has $X=X_{\rm th}+\Delta$ and half has $X=X_{\rm th}-\Delta$, with $\Delta>0$. Compute the mean, variance, and exceptional fraction for $0<\epsilon<\Delta$.

Solution

The two deviations cancel in the mean:

$$\langle X\rangle=X_{\rm th}.$$

Every state differs from the mean by $\Delta$, so

$$\operatorname{Var}(X)=\Delta^2, \qquad p_X(\epsilon)=1 \quad(0<\epsilon<\Delta).$$

An exact ensemble-average match therefore does not establish concentration or statewise thermality.

5. Audit a mixed evidence package

Suppose a theory supplies (i) a protected index with black-hole growth, (ii) one mapped smooth non-BPS solution, and (iii) one averaged scalar absorption match. State the strongest conclusion from each item and list the minimum missing inputs for a typical non-BPS claim.

Solution

The index establishes the protected signed asymptotics in its stated sector and chamber. The mapped solution establishes a regular classical configuration and its proposed or demonstrated quantum identification. The absorption match establishes the normalized inclusive spectral density in its stated channel and resolution regime.

The promotion still needs an absolute entropy-sized non-BPS count for the candidate subset, a named measure and state map, strong-coupling control, concentration for a declared observable family, and a lifetime or stability analysis over the claimed time. Finite-$N$ level spacing and late-time data must be added if the conclusion includes recurrences or information recovery.

What the synthesis establishes

As of the cutoff, ordinary CFT/Cardy counting supplies entropy-sized non-BPS state densities in controlled $AdS_3$ and near-extremal regimes, and several inequivalent smooth non-BPS soliton families exist. Their evidence includes explicit state maps in special cases, numerical nonlinear solutions, systematic solution generators, and both positive and negative stability results in restricted perturbation sectors.

No construction has yet quantized a generic smooth non-BPS geometry phase space to reproduce the full black-hole entropy, shown that such geometries occupy a nonvanishing fraction of a specified microcanonical measure, or established broad strong-coupling concentration together with the required lifetime and finite-$N$ late-time dynamics for typical nonextremal black-hole eigenstates. That is the present evidence ceiling—not the stronger and incorrect claim that non-BPS state counting or smooth non-BPS solutions are absent.

The conclusion should be revisited if an entropy-saturating smooth-sector quantization, a measure comparison, a channel-complete stability or lifetime theorem, strong-coupling concentration bounds, or a revised dominant-phase analysis appears.

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.

For broader context, continue to the Holography and Quantum Gravity research guide. Ensemble and factorization questions continue in Wormholes, Gravitational Path Integrals, and Ensembles; exact unitarity and recovery questions continue in Black-Hole Information, Islands, and Interiors.

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