Typicality, Non-BPS Extensions, and Evidence Limits

Protected indices, selected smooth geometries, and low-energy response matches establish important microscopic mechanisms, but typical non-BPS black holes demand more: an entropy-sized state space, a measure, stable strong-coupling dynamics, and observables reproducing thermal behavior for overwhelmingly many states. No single protected result supplies those ingredients.

Required background. BPS Indices, Absolute Degeneracies, and Wall Crossing supplies the protection ceiling; Microstate Geometries and Fuzzball Proposals supplies explicit solutions; Absorption, Emission, and Dynamical Tests supplies response evidence.

Helpful background. Claim–Evidence Records, Replication, and Retraction Handling supplies evidence classification; QEC Evidence, Current Disputes, and Status supplies a comparison with another frontier interpretation.

Evidence cutoff: 25 July 2026.

What typicality would require

Let $\mathcal H_{E,\Gamma}$ be a microcanonical sector with

$$\dim\mathcal H_{E,\Gamma}\sim e^{S_{\rm BH}(E,\Gamma)}.$$

A typical-state claim must specify a measure—usually Haar measure on the energy shell or a physically prepared ensemble—and a class of observables. For a simple operator $O$, an eigenstate-thermalization statement takes the form

$$O_{mn} =O_{\rm th}(\bar E)\delta_{mn} +e^{-S(\bar E)/2} f_O(\bar E,\omega)R_{mn}.$$

Counting states tests $\dim\mathcal H$; a smooth geometry constructs selected vectors or coherent sectors; an absorption calculation tests averaged matrix elements. None alone verifies this equation for a measure-one set of non-BPS states.

First application: compare evidence with the required conclusion

For the D1–D5 system, place three established results side by side:

Evidence	Controlled quantity	Strongest direct conclusion
protected index and Cardy growth	signed BPS trace at large charge	protected entropy asymptotics
smooth capped solutions	selected states with fixed charges	horizonless realization mechanism
greybody agreement	thermal or ensemble-averaged low-energy two-point response	channel-specific dynamical dictionary

A typical nonextremal claim additionally needs the absolute non-BPS degeneracy, the strong-coupling measure, state-by-state or concentration bounds for simple observables, stability for the relevant time, and a finite-$N$ description of late-time discreteness. This comparison makes the missing premises explicit without discounting the results that are controlled.

Current non-BPS extensions

Non-BPS microstate geometries constructed through Q-ball-like excitations show that smooth capped solutions need not be supersymmetric Ganchev, Houppe, and Warner 2022. Microstrata enlarge the nonlinear excitation space in consistent truncations Ganchev et al. 2023, and vector superstrata add further six-dimensional modes Čeplak and Hampton 2024.

These are current primary existence results through the stated cutoff. They do not yet supply a demonstrated entropy-saturating quantization, a measure dominated by smooth geometries, or generic Schwarzschild-like non-BPS evolution. Stability and uplift constraints must be checked family by family.

Adversarial control: promote one success too far

Try each of the following as sole evidence for typicality:

• a large protected index;
• one smooth deep-throat geometry;
• one low-energy absorption channel.

The index can hide cancellations and says nothing direct about non-BPS lifetimes. One geometry proves existence but has zero measure in an exponentially large continuum unless a counting argument says otherwise. One response function can agree because of symmetry and infrared universality while other operators differ. The typicality conclusion fails each test even though the original result survives.

The evidence ceiling is therefore tiered: robust protected counting in special sectors; explicit and increasingly rich capped solutions; nontrivial low-energy dynamical matches; exploratory non-BPS families. A general microscopic description of typical nonextremal black holes, including exact finite-$N$ late-time behavior, remains unestablished at the cutoff.

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

• Čeplak, Nejc, and Shaun D. Hampton. “Vector Superstrata: Part Two.” arXiv:2405.05341 [hep-th] (2024). arXiv.
• Ganchev, Bogdan, Stefano Giusto, Anthony Houppe, Rodolfo Russo, and Nicholas P. Warner. “Microstrata.” arXiv:2307.13021 [hep-th] (2023). arXiv.
• Ganchev, Bogdan, Anthony Houppe, and Nicholas P. Warner. “Q-Balls Meet Fuzzballs: Non-BPS Microstate Geometries.” Journal of High Energy Physics 2022, 1 (2022): 031. DOI. Open PDF.