Microscopic Black-Hole Entropy: Claim and Ensemble Contract

A microscopic black-hole entropy claim is a fully specified comparison between a named microscopic observable and a named macroscopic entropy functional. It becomes meaningful only after the state sector, normalized charges, ensemble and inverse transform, moduli chamber, coupling path, scaling limit, and correction order have been fixed on both sides. Agreement at one declared order licenses no stronger equality.

Required background. Black-Hole Thermodynamics at the QFT Interface supplies the macroscopic entropy and ensemble distinctions; Holographic Duality: Claims, Dictionaries, and Regimes supplies the correspondence criteria.

Helpful background. Thermodynamic Limits, Phases, and Ensemble Equivalence explains when ensemble transforms are controlled; The Witten Index, Vacuum Counting, and Its Failure Modes distinguishes an index from a count.

First we distinguish what can be counted, then build and fill the comparison contract, and finally stress-test two common shortcuts.

Four microscopic objects with different meanings

Let $\Gamma$ be a vector in the quantized charge lattice. A chamber $\mathcal C$ is a connected region of asymptotic moduli in which the stable BPS spectrum does not jump across a wall. A BPS count first requires a finite-dimensional internal sector of normalizable bound states, written $\mathcal H^{\mathrm{BPS,int}}_{\Gamma,\mathcal C}$. The center-of-mass volume is divided out, while exterior hair is either removed or retained by an explicitly stated convention.

Absolute degeneracy. Every state has positive weight:

$$d_{\mathrm{abs}}(\Gamma,\mathcal C) =\operatorname{Tr}_{\mathcal H^{\mathrm{BPS,int}}_{\Gamma,\mathcal C}}\mathbf 1, \qquad S_{\mathrm{abs}}=\log d_{\mathrm{abs}}.$$

Protected trace. A supersymmetric index has signs and may need an insertion $\mathcal R$ to absorb fermion zero modes:

$$I_{\mathcal R}(\Gamma,\mathcal C) =\operatorname{Tr}_{\mathcal H^{\mathrm{BPS,int}}_{\Gamma,\mathcal C}} \left[(-1)^F\mathcal R\right].$$

In a common four-dimensional normalization, the even helicity supertrace is

$$B_{2k}(\Gamma,\mathcal C) =\frac{1}{(2k)!} \operatorname{Tr}_{\Gamma,\mathcal C} \left[(-1)^{2J_3}(2J_3)^{2k}\right].$$

For a state that breaks $4k$ supercharges, the $2k$-th insertion can absorb the corresponding fermion zero modes; conventions can differ by an overall sign. The adjacent BPS-index article develops the detailed normalization, chamber dependence, and cancellation tests. Exact equality with an absolute count requires coefficient-level equality after known multiplet and hair factors. Along the declared large-charge trajectory, a leading entropy match needs only

$$\log d_{\mathrm{abs}}-\log\lvert I_{\mathcal R}\rvert =o\!\left(\log d_{\mathrm{abs}}\right), \qquad I_{\mathcal R}\ne0,$$

but that weaker statement still requires a no-exponential-cancellation argument Dabholkar et al. 2011, § 2, eqs. (2.1) and (2.4)–(2.9), pp. 11–14.

Coarse-grained count. A macrospace $\mathcal M$ is specified by a projector $\Pi_{\mathcal M}$ and has

$$N_{\mathcal M}=\operatorname{Tr}\Pi_{\mathcal M}, \qquad S_{\mathrm{cg}}(\mathcal M)=\log N_{\mathcal M}.$$

Away from a discrete BPS sector, charges alone do not define a finite macrospace. A common choice is a regulated energy window,

$$N_{\Delta E}(\Gamma,E;\mathcal C) =\dim\mathcal H_{\Gamma,[E,E+\Delta E];\mathcal C}, \qquad S_{\Delta E}=\log N_{\Delta E}.$$

The window $\Delta E$, finite-volume regulator, boundary conditions, and order of the infinite-volume limit are part of this definition.

Typical-state statistic. A typicality claim specifies a probability measure $\mu_{\mathrm{typ}}$ on a named state space and an observable, such as a subsystem entropy $S_A(\lvert\psi\rangle)$. It asserts concentration of that observable under $\mu_{\mathrm{typ}}$; it is not another name for $\log d_{\mathrm{abs}}$ or $S_{\Delta E}$.

These four objects answer different questions. A calculation must keep the same object from its microscopic definition through every transform and into the macroscopic comparison.

Ensemble transforms preserve the counted observable

For any declared microscopic object $X(\Gamma)$—for example $d_{\mathrm{abs}}$ or $I_{\mathcal R}$—one must first choose which mutually compatible charges are transformed. In an electric–magnetic split $\Gamma=(p^I,q_I)$, a standard mixed polarization fixes the magnetic charges $p^I$ and sums the electric charges:

$$Z_X(p,\phi) =\sum_{q}X(p,q)e^{-\phi^I q_I}.$$

A fixed-charge quantity is recovered only after the inverse transform. For integer-normalized $q_I$, normalize each $\phi^I$ to have imaginary period $2\pi i$; then coefficient extraction is

$$X(p,q) =\int_{\mathcal C_\phi} \left[\prod_{I=1}^{r_e}\frac{\mathrm d\phi^I}{2\pi i}\right] Z_X(p,\phi)e^{\phi^I q_I}.$$

Here $r_e$ is the number of transformed electric charges. The electric–magnetic polarization, contour $\mathcal C_\phi$, periodicities, and treatment of poles are part of the coefficient definition. A change of variables supplies a fixed Jacobian. Any further measure or prefactor in a proposed gravitational integral is additional dynamical data; it is not a freely selectable factor in the inverse of the displayed series.

A microcanonical quantity fixes energy within a declared window and fixes the conserved charges. A canonical partition function transforms energy while keeping specified charges fixed. A grand-canonical partition function also sums over one or more charges with chemical potentials; a mixed ensemble transforms only a chosen subset. Inverting an index generating function returns an index, not an absolute degeneracy.

Define the exponent

$$\Phi(\phi;p,q)=\log Z_X(p,\phi)+\phi^I q_I, \qquad \left.\partial_I\Phi\right|_{\phi_\star}=0.$$

At a single dominant nondegenerate saddle, the Gaussian approximation has the schematic form

$$\log\lvert X(p,q)\rvert =\operatorname{Re}\Phi(\phi_\star;p,q) -\frac12\log\left\lvert \det{}'\!\left(\frac{H_\star^{\perp}}{2\pi}\right) \right\rvert+\cdots .$$

Here $H_\star^{\perp}$ is the Hessian along the steepest-descent directions, and the prime omits zero modes. A separately derived prefactor $M(p,\phi)$ adds $\log\lvert M(p,\phi_\star)\rvert$. Zero modes and negative modes require separate treatment; competing saddles can interfere and must be summed before taking the absolute value. The leading Legendre transform retains only $\operatorname{Re}\Phi(\phi_\star)$, while the determinant and derived prefactors can change logarithmic terms without changing the leading exponential. Black-hole loop calculations likewise have to keep track of zero modes and the chosen ensemble Sen 2013, abstract.

For an energy-only canonical partition function, even the canonical entropy is $S_{\mathrm{can}}=(1-\beta\partial_\beta)\log Z$, not simply $\log Z$. Writing “entropy” next to an untransformed partition function therefore does not identify the fixed-charge quantity.

Macroscopic entropy and the horizon sector

At two-derivative order in Einstein gravity, a regular stationary horizon gives

$$S_{\mathrm{EH}}(\Gamma)=\frac{A_H(\Gamma)}{4G_D}.$$

For a classical diffeomorphism-invariant theory with higher derivatives, the corresponding local horizon functional is the Noether-charge or Wald entropy Wald 1993, theorem and eqs. (25)–(26). For extremal black holes with an AdS₂ near-horizon region, the classical entropy-function construction fixes near-horizon data and returns the Wald entropy at its extremum Sen 2005, § 3, eqs. (3.1)–(3.5).

The quantum entropy function is a different, quantum construction. In Sen’s Euclidean fixed-charge convention, its defining relation is

$$d_{\mathrm{hor}}(p,q) =\left\langle \exp\!\left[-iq_I\oint \mathrm d\theta\,A_\theta^{(I)}\right] \right\rangle_{\mathrm{AdS}_2;p}^{\mathrm{finite}}.$$

The electric-field boundary condition and Wilson line implement fixed electric charges $q_I$; the magnetic charges $p^I$ are held fixed as flux data. The label “finite” means that the boundary-length divergence has been removed. Its classical limit is $e^{S_{\mathrm{Wald}}}$ Sen 2009, § 1 eq. (1.3), § 3, and § 8. This is the fixed-charge AdS₂ framework; the 2005 entropy function is its classical predecessor, not the same definition.

A single AdS₂ path integral describes a single horizon sector. When the relevant broken-supercharge zero modes and helicity insertion are carried by the exterior sector, a four-dimensional single-center supertrace takes the form

$$B_{2k}^{\mathrm{single}}(\Gamma,\mathcal C) =\sum_{\Gamma_{\mathrm{hor}}+\Gamma_{\mathrm{hair}}=\Gamma} d_{\mathrm{hor}}(\Gamma_{\mathrm{hor}}) B_{2k}^{\mathrm{hair}}(\Gamma_{\mathrm{hair}},\mathcal C).$$

A full asymptotic trace can additionally contain multicenter configurations. The helicity insertion belongs to their relative and exterior modes rather than factorizing into an independent horizon index for each center:

$$B_{2k}^{\mathrm{full}}(\Gamma,\mathcal C) =\sum_{\Gamma_{\mathrm{hair}}+\sum_i\Gamma_i=\Gamma} \left[\prod_i d_{\mathrm{hor}}(\Gamma_i)\right] B_{2k}^{\mathrm{rel+hair}} (\Gamma_{\mathrm{hair}},\{\Gamma_i\};\mathcal C).$$

Thus a horizon answer is compared with a microscopic single-center quantity only after the same hair subtraction and attractor-contour prescription have been imposed. If the microscopic object includes all centers in an asymptotic chamber, the macroscopic side must include that decomposition too Dabholkar et al. 2011, § 2, eqs. (2.4)–(2.9), pp. 12–14, Sen 2009, § 8.

The quantum entropy function is the definition; supersymmetric localization is one proposed evaluation method. Its additional assumptions and current limitations are examined in the localization article Sen 2026, abstract.

The reproducible comparison contract

Before accepting an entropy match, record all ten fields below. Each omission changes the mathematical statement rather than merely shortening its description.

1. Theory and state sector. Give the compactification, spacetime dimension, boundary conditions, supersymmetry, and the Hilbert-space sector being traced.

2. Charge map. State the charge lattice, normalization, induced shifts, angular-momentum convention, and the map between microscopic and gravitational charges.

3. Microscopic object. Name $d_{\mathrm{abs}}$, $I_{\mathcal R}$, $N_{\Delta E}$, or a typicality statistic, including insertions and exterior factors.

4. Ensemble and inversion. Say which variables are fixed or summed, and give the contour, measure, and pole prescription that recover the desired sector.

5. Chamber and decomposition. Fix the asymptotic moduli and say whether single centers, multicenters, and hair are included, removed, or convolved.

6. Protection and coupling path. Identify the protected observable, preserved supercharge and boundary data, conditions keeping the protected sector discrete and separated from the continuum, walls avoided, and the path from the calculable regime to gravity.

7. Scaling trajectory and order of limits. A general large-charge path has

   $$\Gamma_a(\Lambda) =\Lambda^{w_a}\widehat\Gamma_a +o\!\left(\Lambda^{w_a}\right), \qquad w_a\ge0,$$

   so different charges may grow at different rates or remain fixed. State all ratios and take thermodynamic, Cardy, weak-coupling, and large-horizon limits in a declared order.

8. Macroscopic object. Name the single saddle or saddle sum, $S_{\mathrm{EH}}$, $S_{\mathrm{Wald}}$, or the fixed-charge quantum path integral, with the same charges and boundary data.

9. Accuracy and uncertainty. Distinguish leading growth, logarithmic order, constant order, exponentially small corrections, and exact finite-charge equality. Record analytic remainders, regulator or contour dependence, and competing saddles.

10. Falsifier and claim ceiling. State what disagreement would refute the claimed match and what stronger inference remains unlicensed.

For a leading claim, the cleanest statement is a ratio:

$$\lim_{\Lambda\to\infty} \frac{\log\lvert X_{\mathrm{micro}}(\Gamma(\Lambda))\rvert} {S_{\mathrm{macro}}^{(0)}(\Gamma(\Lambda))}=1.$$

A logarithmic claim is stronger. After every known term larger than $\log\Lambda$ has been collected into $S_{\mathrm{local}}(\Lambda)$, it takes the form

$$\log\lvert X_{\mathrm{micro}}\rvert-S_{\mathrm{local}} =a_{\mathrm{micro}}\log\Lambda+O(1),$$

and requires the independently computed macroscopic coefficient $a_{\mathrm{macro}}$ to equal $a_{\mathrm{micro}}$. Neither statement is an exact finite-charge identity.

Worked contract: leading D1–D5–momentum entropy

The canonical five-dimensional example shows what a filled contract looks like without repeating the derivation given in D-Brane Bound States and the Strominger–Vafa Count.

Theory and sector. Use type IIB string theory on $S^1\times K3$. Take the BPS D1–D5 bound-state sector with right movers in their supersymmetric ground state and left-moving momentum along $S^1$.

Charges. Fix integer D1, D5, and momentum charges $(Q_1,Q_5,n)$ and set $J_L^3=0$. Let $Q_5$ be the conserved D5 Page charge and $Q_1$ the conserved D1 Page charge after including the curvature-induced D1 charge of the wrapped D5-branes. In this convention the symmetric-product level and exact central charges are

$$N=Q_1Q_5+1, \qquad c_L=c_R=6N.$$

If $N_1$ instead denotes the number of explicit D1-branes, the map is $Q_1=N_1-Q_5$; confusing these conventions produces more than an order-one error when $Q_5$ grows Strominger and Vafa 1996, § 3, eqs. (3.1)–(3.4).

Microscopic object. At the weak-coupling symmetric-product point, define the K3 elliptic genus

$$\begin{aligned} \chi_N^{\mathrm{int}}(\mathfrak q,y) &=\operatorname{Tr}_{\mathcal H_{\mathrm{RR}}^{\mathrm{int}}} \!\left[(-1)^{F_L+F_R} \mathfrak q^{L_0-c_L/24}y^\ell\right] \\ &=\sum_{n,\ell}c_N^{\mathrm{int}}(n,\ell) \mathfrak q^ny^\ell, \qquad \ell\equiv2J_L^3. \end{aligned}$$

The internal Hilbert space builds in the declared center-of-mass and hair removal. The $(-1)^{F_R}$ insertion cancels contributions from excited right-moving multiplets, leaving the supersymmetric ground-state contribution. K3 is chosen because this elliptic genus is nonzero; the $T^4$ compactification requires a modified or refined trace to absorb its extra zero modes Dijkgraaf et al. 1997, § 1, eqs. (1.2)–(1.4). An absolute degeneracy is not substituted without a sign or no-cancellation analysis.

Ensemble and chamber. At fixed $N$, $n$, and $J_L^3=0$, the protected coefficient is

$$\begin{aligned} I_{\mathrm{BPS}}(Q_1,Q_5,n,J_L^3=0) &=c_N^{\mathrm{int}}(n,0) \\ &=\oint_{\lvert\mathfrak q\rvert=\rho_{\mathfrak q}} \frac{\mathrm d\mathfrak q}{2\pi i\,\mathfrak q^{n+1}} \\ &\quad\times\oint_{\lvert y\rvert=\rho_y} \frac{\mathrm dy}{2\pi i\,y}\, \chi_N^{\mathrm{int}}(\mathfrak q,y). \end{aligned}$$

The circles encircle the origin inside the Fourier-expansion domain. Work in a chamber in which the D1–D5 bound state persists and treat any multicenter sector separately.

Coupling path. Compute at weak string coupling in the brane CFT and transport the protected trace to the strong-coupling black-hole regime along a supersymmetry-preserving path that crosses no stability wall.

Scaling trajectory. The elementary Cardy derivation uses $c_L=6N=6Q_1Q_5+6$ and $n\gg c_L$, together with a macroscopic horizon. One explicit trajectory is

$$Q_1\sim\Lambda, \qquad Q_5\sim\Lambda, \qquad n\sim\Lambda^{2+\epsilon}, \qquad \epsilon>0,$$

for which $n/(Q_1Q_5)\to\infty$. A fixed ray with all three charges proportional to $\Lambda$ would not establish this particular Cardy regime.

Let $v=V_{K3}/\alpha'^2$. Along the trajectory, choose the compactification moduli and coupling so that

$$\frac{L^2}{\alpha'} \sim g_s\sqrt{\frac{Q_1Q_5}{v}} \longrightarrow\infty, \qquad e^{2\Phi_{\mathrm{hor}}} \sim g_s^2\frac{Q_1}{vQ_5}\ll1.$$

These conditions suppress string-scale curvature and horizon string loops. In the hierarchy $n/(Q_1Q_5)\to\infty$, local control is most transparent in the six- or ten-dimensional uplift; the reduced five-dimensional area gives the same leading entropy.

Macroscopic object. Use the single-center five-dimensional two-derivative black-hole entropy with the same normalized charges.

Claim and uncertainty. The licensed leading statement is

$$\log\lvert I_{\mathrm{BPS}}(Q_1,Q_5,n,0)\rvert =2\pi\sqrt{Q_1Q_5n} +o\!\left(\sqrt{Q_1Q_5n}\right) =S_{\mathrm{BH}}^{(0)} +o\!\left(S_{\mathrm{BH}}^{(0)}\right).$$

This protected coefficient has the same leading Cardy exponent as the BPS-state count in the original Strominger–Vafa construction Strominger and Vafa 1996, §§ 3–4, pp. 6–9. It is a sharpened indexed observable, not literally the finite-charge degeneracy computed there. The result does not establish equality with an absolute degeneracy, an exact finite-charge formula, logarithmic agreement, a generic non-BPS count, or typical-state dynamics.

Evidence status and falsifier. The evidence consists of an independent weak-coupling index asymptotic and strong-coupling area calculation, joined by the stated BPS-protection assumptions. After matching charge normalization and exterior factors, a different coefficient of $\sqrt{Q_1Q_5n}$ would falsify the leading claim. A disagreement only in the coefficient of $\log\Lambda$ would falsify a logarithmic match, not the leading one.

Two adversarial checks

Exponentially large sectors can almost cancel

Let $S_0(\Lambda)\to\infty$, choose $0<\alpha<1$, and let $N_\Lambda$ and $M_\Lambda$ be the nearest positive integers of the same parity to $e^{S_0}$ and $e^{(1-\alpha)S_0}$. For a sign $\sigma=\pm1$, define bosonic and fermionic counts

$$d_B=\frac{N_\Lambda+\sigma M_\Lambda}{2}, \qquad d_F=\frac{N_\Lambda-\sigma M_\Lambda}{2}.$$

Both are nonnegative and individually grow as $e^{S_0}$. Yet

$$d_{\mathrm{abs}}=d_B+d_F=N_\Lambda, \qquad I=d_B-d_F=\sigma M_\Lambda,$$

so

$$\log d_{\mathrm{abs}} =S_0+o(1), \qquad \log\lvert I\rvert =(1-\alpha)S_0+o(1).$$

The two logarithms differ by $\alpha S_0+o(1)$—a leading-order amount—even though the bosonic and fermionic sectors have the same leading exponential. With exact cancellation, $I=0$ and $\log\lvert I\rvert$ is not defined at all. What survives is the existence of exponentially many states; what fails is the promotion of a signed trace to their absolute count.

The inverse transform preserves the leading saddle but changes the logarithm

Consider $n\to\infty$ at fixed $\kappa>0$; more generally, require $\kappa/n\to0$ and $\sqrt{\kappa n}\to\infty$. Take a one-variable generating function with small-$\beta$ behavior

$$Z(\beta)\simeq\exp\!\left(\frac{\kappa}{\beta}\right), \qquad \kappa>0.$$

The fixed-level coefficient is

$$d(n)\simeq \int_{\mathcal C_\beta}\frac{\mathrm d\beta}{2\pi i} \exp\!\left(\beta n+\frac{\kappa}{\beta}\right).$$

Its positive saddle and Hessian are

$$\beta_\star=\sqrt{\frac{\kappa}{n}}, \qquad \Phi(\beta_\star)=2\sqrt{\kappa n}, \qquad \Phi''(\beta_\star)=\frac{2n^{3/2}}{\sqrt\kappa}.$$

The Gaussian factor therefore gives

$$\log d(n) =2\sqrt{\kappa n} -\frac34\log n +\frac14\log\kappa +O(1).$$

The leading exponential agrees with the Legendre saddle, while the inverse-transform determinant supplies a logarithmic term. A derived prefactor, extra chemical potentials, or zero modes can change that coefficient. If $\kappa$ scales along a multi-charge trajectory, the displayed $\tfrac14\log\kappa$ contributes to the total coefficient of $\log\Lambda$. What survives is leading ensemble equivalence at the controlled saddle; what fails is equality of subleading entropies without the complete transform.

The claim ladder and stopping rule

• Leading asymptotic agreement compares only the dominant growth along the declared charge trajectory.
• Logarithmic agreement additionally matches every term larger than $\log\Lambda$, the Gaussian transform, nonzero-mode determinants, zero-mode measures, and ensemble.
• Constant or nonperturbative agreement requires regulator, contour, saddle, and ultraviolet data not fixed by the leading result.
• Exact finite-charge equality identifies the same integer or trace coefficient after all shifts, hair, multicenters, and chamber data are included.

Stop at the highest rung independently computed on both sides. Protection of one trace or success at one rung does not skip the intervening tests.

Common pitfalls

“The protected trace is large, so it is the degeneracy.” For a simple $\pm1$ index, $\lvert I\rvert\le d_{\mathrm{abs}}$. A helicity or refined trace has nonunit weights and yields a degeneracy bound only after those weights and the allowed spins are controlled. In either case, equality of leading growth requires independent control of cancellations and multiplet factors.

“The ensembles agree at the saddle, so their logarithms agree.” The saddle fixes the leading Legendre transform. Hessians, measures, zero modes, and the number of transformed variables enter at logarithmic order.

“Fixed total charges select one black hole.” The same total charge can be carried by exterior modes or several centers. The chamber and decomposition must be specified.

“The area match is exact.” A leading $A_H/(4G)$ comparison omits higher-derivative terms, loops, inverse-transform effects, and finite-charge shifts unless they are separately computed.

“Many states imply typical black-hole behavior.” A count supplies the size of a state set. Typicality also needs a measure and concentration for a stated observable class.

Exercises

1. Quantify a cancellation

In the integer construction above, take $\alpha=1/3$. Find the leading growth of $d_{\mathrm{abs}}$, $\lvert I\rvert$, and $\lvert I\rvert/d_{\mathrm{abs}}$. Which entropy claim survives?

Solution

The absolute count obeys $d_{\mathrm{abs}}\sim e^{S_0}$, while $\lvert I\rvert\sim e^{2S_0/3}$. Hence

$$\frac{\lvert I\rvert}{d_{\mathrm{abs}}} \sim e^{-S_0/3}\longrightarrow0.$$

Both quantities are exponentially large, but their entropy coefficients differ: $\log d_{\mathrm{abs}}\sim S_0$ and $\log\lvert I\rvert\sim2S_0/3$. Only the separately stated indexed and absolute growth claims survive; the index cannot be promoted to the degeneracy.

2. Check the weighted Cardy trajectory

Let $Q_1=q_1\Lambda$, $Q_5=q_5\Lambda$, and $n=\widehat n\Lambda^{2+\epsilon}$ with fixed positive $q_1$, $q_5$, $\widehat n$, and $\epsilon>0$. Determine the large-$\Lambda$ behavior of $n/c_L$ and $S_0=2\pi\sqrt{Q_1Q_5n}$. Why is this more informative than writing $\Gamma=\Lambda\widehat\Gamma$?

Solution

Since $c_L\sim6q_1q_5\Lambda^2$,

$$\frac{n}{c_L} \sim\frac{\widehat n}{6q_1q_5}\Lambda^\epsilon \longrightarrow\infty,$$

so the elementary Cardy hierarchy is explicit. The leading entropy scales as

$$S_0\sim2\pi\sqrt{q_1q_5\widehat n}\, \Lambda^{2+\epsilon/2}.$$

A fixed ray would scale $Q_1$, $Q_5$, and $n$ with the same power, giving $n/c_L\to0$ rather than the stated Cardy regime. Weighted scaling records the actual order of limits.

3. Change an integrand prefactor

Suppose the generating function in the inverse-transform example is multiplied by $\beta^b$, where $b$ is fixed and $b=O(1)$ as $n\to\infty$. Determine the added coefficient of $\log n$ at the same saddle.

Solution

At $\beta_\star=\sqrt{\kappa/n}$,

$$\log\beta_\star^b =\frac{b}{2}\log\kappa-\frac{b}{2}\log n.$$

The coefficient of $\log n$ changes from $-3/4$ to $-3/4-b/2$. Evaluating the prefactor at the old saddle is sufficient at logarithmic accuracy: its induced saddle displacement changes only the $O(1)$ term. The leading term $2\sqrt{\kappa n}$ is unchanged. This is why a derived integrand prefactor can be invisible at leading order but indispensable for a logarithmic comparison.

4. Repair an underspecified claim

A calculation is summarized only as $\log Z=S_{\mathrm{BH}}$. List the missing declarations and rewrite the strongest defensible leading statement.

Solution

The symbol $Z$ does not say whether the microscopic object is an absolute count, protected trace, or generating function. One must name that object; normalize the charges; give the inverse transform and ensemble; fix the energy window when relevant; specify the chamber, hair, and center decomposition; state the protected coupling path; declare the large-charge trajectory and order of limits; name the macroscopic entropy functional and saddle; and state the requested accuracy.

If these data identify a nonzero protected coefficient $I(\Gamma(\Lambda))$ and a leading macroscopic entropy $S_{\mathrm{macro}}^{(0)}(\Gamma(\Lambda))$, the strongest generic leading statement is

$$\lim_{\Lambda\to\infty} \frac{\log\lvert I(\Gamma(\Lambda))\rvert} {S_{\mathrm{macro}}^{(0)}(\Gamma(\Lambda))}=1.$$

This does not assert equality of $Z$ with a fixed-charge coefficient, identify an index with an absolute degeneracy, or match logarithmic and finite-charge corrections.

What the contract licenses

A successful microscopic entropy comparison identifies the same normalized charges, microscopic observable, ensemble, chamber or center decomposition, coupling path, scaling trajectory, macroscopic functional, and correction order. It licenses equality only at the highest order for which those data and their uncertainties have been checked.

Continue to BPS Indices, Absolute Degeneracies, and Wall Crossing for cancellations and chambers; D-Brane Bound States and the Strominger–Vafa Count for the Cardy and area derivations; Higher-Derivative and Quantum Entropy Corrections for subleading terms; and Typicality, Non-BPS Extensions, and Evidence Limits for claims that leave the protected sector.

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

• Dabholkar, Atish, João Gomes, Sameer Murthy, and Ashoke Sen. “Supersymmetric Index from Black Hole Entropy.” Journal of High Energy Physics 2011, 4 (2011): 034. DOI. Open PDF.
• Dijkgraaf, Robbert, Gregory Moore, Erik Verlinde, and Herman Verlinde. “Elliptic Genera of Symmetric Products and Second Quantized Strings.” Communications in Mathematical Physics 185, 197–209 (1997). DOI. Open PDF.
• Sen, Ashoke. “Black Hole Entropy Function and the Attractor Mechanism in Higher Derivative Gravity.” Journal of High Energy Physics 2005, 9 (2005): 038. DOI. Open PDF.
• Sen, Ashoke. “Quantum Entropy Function from AdS₂/CFT₁ Correspondence.” International Journal of Modern Physics A 24, 4225–4244 (2009). DOI. Open PDF.
• Sen, Ashoke. “Logarithmic Corrections to Schwarzschild and Other Non-Extremal Black Hole Entropy in Different Dimensions.” Journal of High Energy Physics 2013, 4 (2013): 156. DOI. Open PDF.
• Sen, Ashoke. “Revisiting Localization for BPS Black Hole Entropy.” Journal of High Energy Physics 2026, 8 (2026): 174. DOI. Open PDF.
• Strominger, Andrew, and Cumrun Vafa. “Microscopic Origin of the Bekenstein–Hawking Entropy.” Physics Letters B 379, 99–104 (1996). DOI. Open PDF.
• Wald, Robert M. “Black Hole Entropy Is the Noether Charge.” Physical Review D 48, R3427–R3431 (1993). DOI. Open PDF.