Capstone Comparison Across Named Black-Hole Equations

A named differential equation does not come with a preferred modern method. Schwarzschild, Kerr, extremal Reissner–Nordström, a D3-brane, rotating BTZ, and RN–AdS$_5$ exhibit different combinations of endpoint geometry, angular coupling, singularity type, discrete symmetry, and holographic normalization. The useful comparison is therefore not “which equation is Heun?” but “which calculation returns the selected physical connection coefficient on the same sheet?”

This page closes four loops. Schwarzschild calibrates a continued fraction against the original radial Wronskian. Generic subextremal Kerr compares direct, Painlevé V, and SW/NS charts of one coupled angular–radial zero. Extremal Reissner–Nordström and the D3-brane show that two equations with two irregular ends can support different—and not automatically equivalent—exact structures. Finally, rotating BTZ and RN–AdS$_5$ share one holographic source/response ledger while using hypergeometric and Heun connection engines, respectively.

First reading

Read the comparison standard, the scope table, and the four capstone conclusions. The Painlevé, NS, Stokes-word, and holographic normalization audits may be read on a second pass.

Status ledger

Claim	Status and scope
Singularity class selects the best physical method	False; endpoint flags and extra analytic structure decide
Scalar Schwarzschild continued fraction	Exact boundary divisor after the minimal recurrence line is identified with the outgoing Jost line; its numerical evaluation is independently checked
Generic subextremal Kerr Painlevé V formulation	Exact inverse-map equations on a generic chart and a controlled physical zero set after the wild boundary twist and angular equation are imposed
Generic-chart Kerr confluent-Heun/CFT/NS connection formula	Normalization-controlled on the declared chart; instanton truncation, Matone inversion, logarithm lifts, and resonant limits remain approximation or continuation gates
Kerr–de Sitter PVI reduction	Controlled spectral dictionary; resonant physical angular data and the framed endpoint flag still require a regulated direct check
Extremal Reissner–Nordström condition $V_\gamma=-1$	Connection-derived conditional on the displayed Stokes topology, cycle, sheet, and existence or lateral choice of the Borel sum
D3-brane canonical ODE/IM $Q$	Exact canonical Wronskian and functional relations; its physical endpoint continuation in the chapter’s time convention remains numerically supported rather than proved
Rotating-BTZ noninteger-$\nu$ Gamma quotient	Exact nonlocal quotient and QNM towers on the stated generic chart once $\mathcal N_\Phi$ and the finite scheme are declared; integer $\nu$, singular horizon normalizations, and coincident Gamma loci require limits
RN–AdS$_5$ recurrence quotient	Exact canonical Heun connection data on the stated chart; generic normalization still needs the model action and finite scheme, while integral-gap strata additionally need logarithmic counterterms
One universal tau/period/Voros/Q identity for black holes	Not established and generally false without a normalization-complete, sheetwise dictionary

Compare framed boundary divisors, not method names

For a scalar radial equation in divergence form, freeze two physical endpoint lines and define

$$E_B(\boldsymbol p) = p(r)\Wr_r \left[ y_L^{\mathrm{phys}}, y_R^{\mathrm{phys}} \right].$$

The parameter passport $\boldsymbol p$ includes frequency, separation constant, background parameters, discrete labels, branch choices, and any Stokes-sector data. The weighted Wronskian is independent of the match point. Its zero means that the two selected lines coincide. Rescaling either endpoint vector by a holomorphic nowhere-zero function changes $E_B$ but not its local divisor.

A separated rotating problem has two such tests. In the coordinate order $(A,\omega)$, write

$$\mathcal F(A,\omega) = \begin{pmatrix} E_{\mathrm{ang}}(A,\omega) \\ E_{\mathrm{rad}}(A,\omega) \end{pmatrix}.$$

A QNM is a common zero of both components on one tracked angular and frequency sheet. At a transverse root, the invariant simplicity test is

$$J_{A,\omega} = \det \begin{pmatrix} E_{\mathrm{ang},A} &E_{\mathrm{ang},\omega} \\ E_{\mathrm{rad},A} &E_{\mathrm{rad},\omega} \end{pmatrix} \ne0.$$

Suppose a direct method and a modern bridge produce $D_j^{(1)}$ and $D_j^{(2)}$ for $j\in\{\mathrm{ang},\mathrm{rad}\}$. The strongest practical comparison normally available is

$$D_j^{(a)} = u_j^{(a)}E_j, \qquad u_j^{(a)}\ne0, \qquad a=1,2,$$

on one declared chart. Then roots and local multiplicities agree, although the residual values and derivatives need not. If only a list of coincident frequencies is known, even this unit-equivalence statement has not been shown.

Four grades of agreement

Grade	What has actually been established	What may safely be compared
Normalized identity	Same endpoint vectors, scalar gauge, branches, and leading normalizations	Function values, derivatives, residues, zeros, and multiplicities
Equality up to a unit	$D=uE_B$ with $u$ holomorphic and nonzero	Divisor, multiplicity, and transverse versus nontransverse intersection
Sampled roots and orders	A specified finite set of isolated roots and their orders agrees, without proof that the full local zero germs agree	Evidence for spectral locations and multiplicities only
Numerical coincidence	Finite root lists agree within reported precision	Evidence for a dictionary, not the dictionary itself

An AdS correlator contains one more layer. If the future-ingoing solution has boundary coefficients $(\alpha,\beta)$ in unit-leading source and response bases, then standard scalar quantization gives

$$E_{\mathrm{std}} = W_0\alpha,$$

but the renormalized observable is

$$G_{\mathrm R} = \mathcal M\frac{\beta}{\alpha} +P_{\mathrm{loc}}.$$

The ODE fixes a connection quotient. The bulk action, scalar gauge, boundary powers, counterterms, and finite scheme fix $\mathcal M$ and $P_{\mathrm{loc}}$. On a regular nonresonant scalar chart, a nonzero horizon-ingoing solution cannot have both $\alpha=0$ and $\beta=0$; a simple source zero is therefore a genuine pole. Cancellations can instead occur in coupled systems, singular basis charts, projected observables, or the two-parameter rank-loss problem of pole-skipping.

The named-equation scope is deliberately asymmetric

The labels below mean exact chart, controlled, conditional, or not established here. The last phrase is not a claim of impossibility; it means that this book has not supplied the operator-to-boundary dictionary needed to use the method physically.

Named boundary problem	Canonical structure	Physical lines or coupled data
Four-dimensional scalar Schwarzschild	Spherical angular problem; confluent-Heun radial equation for $\omega\ne0$	Fixed $L=\ell(\ell+1)$; future-ingoing horizon and outgoing flat-infinity lines
Generic subextremal Kerr Teukolsky	Two coupled confluent-Heun equations	Two regular angular axes; future-ingoing horizon; outgoing flat infinity; shared accessory
Kerr–de Sitter Teukolsky; explicit scalar laboratory	Two coupled general-Heun equations	Two regular angular axes; event-ingoing and cosmological-outgoing lines; shared accessory
Extremal Reissner–Nordström scalar	Doubly confluent Heun; irregular horizon and infinity	Sectorial future-ingoing horizon and outgoing infinity lines
Extremal D3-brane scalar	Generalized modified Mathieu; two irregular ends	Throat-ingoing and infinity-outgoing lines after the stated continuation audit
Nonextremal rotating BTZ scalar	Gauss hypergeometric	Future-ingoing horizon; slow source and fast response at the AdS boundary
Nonextremal RN–AdS$_5$ scalar	General Heun on the stated generic chart	Future-ingoing horizon; slow source and fast response at the AdS boundary

Named problem	Direct control	Additional representation admitted here
Scalar Schwarzschild	Exact Jaffé–Leaver divisor and independent complex-ray Wronskian check	Original SW/NS period rule is a branchwise proposal with numerical tests, not used as the certificate
Generic subextremal Kerr	Coupled endpoint Wronskians; published simultaneous Leaver recurrences	Framed PV inverse map; generic-chart confluent-block/$SU(2)$, $N_f=3$ NS connection formula
Kerr–de Sitter	Coupled event/cosmological and angular Wronskians	PVI inverse monodromy and $c=1$ blocks, controlled with resonance and composite-lift audits
Extremal Reissner–Nordström	Physical endpoint Wronskian remains primary; direct QNM data provide the numerical benchmark	Ordered exact-WKB Stokes word and Voros condition in a declared chamber, cycle, and Borel prescription
Extremal D3-brane	Exact scalar-gauge Wronskian relation; direct numerical benchmarks	Chamber-specific exact WKB in the primary literature; canonical ODE/IM $Q$, QQ, and TBA with a remaining physical-continuation gate
Rotating BTZ	Exact Euler connection coefficients	Renormalized generic noninteger-$\nu$ Gamma quotient and exact QNM towers once the bulk normalization and finite scheme are declared
RN–AdS$_5$	Exact recurrence connection quotient and pseudospectral benchmark	Holographic quotient after gauge and radial-power conversion; full generic normalization ledger is model dependent

Three absences in this table are important. First, a Heun or quantum Seiberg–Witten curve match does not create the discrete orbit of canonical solutions required for ODE/IM. Second, a tau zero does not select a QNM boundary flag. Third, a canonical source/response ratio without the bulk variational problem is not a fully normalized Green function.

Two developed examples are not promoted to full capstones. Schwarzschild–de Sitter has a complete event-to-cosmological Wronskian once its scalar operator is fixed, but this chapter does not print a dedicated canonical tuple or a modern bridge for it. Extremal Kerr has sectorial endpoint bases and restricted noncorotating scalar exact-WKB results, while corotating topology, generic spin, and the full angular–radial dictionary remain separate problems.

Capstone A: Schwarzschild fixes the direct control

Take a four-dimensional massless scalar with time dependence $\ee^{-\ii\omega t}$. The spherical angular problem fixes $L=\ell(\ell+1)$ before the radial solve. With

$$z=\frac{r}{2M}, \qquad \Omega=2M\omega, \qquad x=\frac{z-1}{z},$$

the event horizon and flat infinity are $x=0$ and $x=1$. The Jaffé ansatz factors out the future-ingoing horizon behavior and the outgoing exponential at infinity. Its remaining series coefficients obey

$$\alpha_na_{n+1} +\beta_na_n +\gamma_na_{n-1}=0,$$

where

$$\alpha_n =(n+1)(n+1-2\ii\Omega),$$ $$\begin{aligned} \beta_n=-\bigl[ &2n^2+(2-8\ii\Omega)n \\ &-8\Omega^2-4\ii\Omega+L+1 \bigr], \end{aligned}$$

and

$$\gamma_n=(n-2\ii\Omega)^2.$$

The seed row selects the horizon series line. On a chart where Pincherle’s hypotheses hold, the minimal large-$n$ line is the outgoing Jost line and the radial boundary divisor is represented by

$$F_{\mathrm{CF}}(\Omega) = \beta_0 - \cfrac{\alpha_0\gamma_1} {\beta_1- \cfrac{\alpha_1\gamma_2} {\beta_2- \cfrac{\alpha_2\gamma_3}{\ddots}}}.$$

Thus

$$F_{\mathrm{CF}}=0 \quad\Longleftrightarrow\quad E_{\mathrm{rad}} = \Delta\Wr_r \left[ R_H^{\mathrm{in}}, R_\infty^{\mathrm{out}} \right] =0,$$

after the two recurrence-to-ODE line transfers have been established. This last clause is the physical content. Convergence of an abstract continued fraction alone proves only that two sequence lines align.

For $(\ell,n)=(0,0)$, backward evaluation stabilizes to

$$\begin{aligned} M\omega_{00} ={}& 0.110454939080419685875461506168 \\ &- 0.104895717086880958781739066855\ii. \end{aligned}$$

An independently initialized Riccati integration along a complex ray makes the horizon and outgoing logarithmic derivatives agree at several match points. The recurrence depth and the ODE cutoff, asymptotic order, contour, and integrator tolerance are independent refinement axes. This is why the benchmark is stronger than agreement between two truncations of the same continued fraction.

What the control does not say

At $\Omega=0$, the flat-infinity wave basis collapses and the radial equation demotes to a hypergeometric static problem. On a minimality wall, the physical line must be defined by lateral causal continuation. Moreover, an asymptotically flat Green function can contain a branch-cut contribution from the long-range potential; a list of QNM poles is not a complete time-domain expansion.

The original Schwarzschild SW/NS rule agrees impressively with direct data, but its radial boundary identification was proposed rather than derived from a normalized connection coefficient. It is therefore a useful cross-check, not a replacement for this direct certificate. The later Kerr connection formula below has a stronger status on its own generic chart; that upgrade must not be retroactively applied to every black-hole quantum curve.

Capstone B: Kerr has one spectrum and several charts

For generic subextremal Kerr, the spin-$s$ Teukolsky angular and radial operators are both confluent Heun. They share a separation constant, but their endpoint pairs are different. Keep the physical system primary:

$$E_{\mathrm{ang}}(A,\omega)=0, \qquad E_{\mathrm{rad}}(A,\omega)=0.$$

The first equation aligns the north- and south-regular angular lines. The second aligns the future-ingoing outer-horizon line with the outgoing Jost line. Holding an arbitrary $A$ fixed while solving the radial equation does not produce a Kerr QNM.

Direct recurrence lane

Leaver’s published two continued fractions represent the angular and radial divisors after their minimal solutions have been transferred to the corresponding physical endpoints. This chapter has derived the coupled Kerr Wronskians and the full scalar Schwarzschild recurrence; it cites rather than rederives the general Kerr recurrence coefficients. In either implementation, solve simultaneously for $(A,\omega)$, continue the angular sheet from its spin-weighted spherical seed, and monitor $J_{A,\omega}$. Return every candidate to the original angular and radial Wronskians; recurrence depth is not the only residual.

Painlevé V lane

Each confluent-Heun operator determines a PV inverse problem. In the confluent-Heun normalization fixed on the preceding page, the required stack in sector $j$ is

$$\begin{gathered} \tau_V (\boldsymbol\theta_j;\sigma_j,\eta_j;t_j)=0, \\ t_j \left. \partial_t\log\tau_V (\boldsymbol\theta_{j,-};\sigma_j-1,\eta_j;t) \right|_{t=t_j} -\frac{\theta_{0,j}(\theta_{t,j}-1)}2 =t_jc_j, \qquad j\in\{\mathrm{ang},\mathrm{rad}\}. \end{gathered}$$

Here $\boldsymbol\theta_{j,-}=(\theta_{0,j},\theta_{t,j}-1, \theta_{\star,j}+1)$. The first equation places the scalar reduction on the confluent-Heun slice, and the shifted tau derivative reconstructs the accessory. A third, separately derived triangular connection condition must retain the selected endpoint flag. In the radial chart of the preceding page it fixes the wild twist to $\eta=\eta_0$; the angular chart uses its own regular-axis flag. The displayed accessory shift is fixed by the scalar gauge, not fitted. Only after both boundary gates are imposed and the two sectoral inverse maps share the same $(A,\omega)$ has the PV formulation produced a Kerr candidate.

This lane controls a physical zero set, not automatically an absolutely normalized scattering matrix. It also explains why a bare statement “$\tau_V=0$” is incomplete: the tau zero, accessory equation, wild boundary twist, angular closure, and branch continuation play different logical roles.

Confluent-block and SW/NS lane

For the source convention $s=0,-1,-2$, the generic confluent-Heun connection formula gives a stronger local bridge: after the scalar gauges, irregular-block normalization, $U(1)$ scheme, and parameter map are fixed, a selected connection coefficient is expressed in $SU(2)$, $N_f=3$ NS data. Reserve $a_{\mathrm K}$ for Kerr rotation and $a_{\mathrm{SW}}$ for the quantum A-period.

On the angular sheet, with $c=a_{\mathrm K}\omega$, regularity gives, after the regulated resonant endpoint limit,

$$a_{\mathrm{SW,a}} = \ell+\frac12.$$

The angular Matone map turns this into an equation for $A$. In the radial chart used on the preceding page, the unwanted incoming coefficient is

$$C_{\mathrm{in}}^{\mathrm{NS}} = \mathcal N_{\mathrm r} \left[ 1- \exp\!\left( \partial_{a_{\mathrm{SW}}} \mathcal F_{\mathrm{full}} \right) \right], \qquad \mathcal N_{\mathrm r}\ne0.$$

Therefore, away from the declared Gamma, resonance, Matone-branch, and coalescing-basis divisors,

$$C_{\mathrm{in}}^{\mathrm{NS}}=0 \quad\Longleftrightarrow\quad \partial_{a_{\mathrm{SW}}} \mathcal F_{\mathrm{full}} =2\pi\ii k, \qquad k\in\mathbb Z.$$

The complete NS solve is still coupled: use the angular Matone equation and this radial lattice with one common $(A,\omega)$, their respective angular and radial parameter dictionaries, one fixed $U(1)$ scheme, and coherently tracked sectoral logarithm lifts. A finite instanton sum gives an approximant to that system. At an excluded divisor, return to the unfactored connection coefficient rather than interpreting an apparent $0/0$.

The three lanes meet only at the physical divisor

Lane	Intermediate object	Boundary gate	Return test
Direct	Two minimal recurrence lines	Angular and radial continued fractions	Both original Wronskians and $J_{A,\omega}$
PV	Tau, shifted-tau accessory, composite monodromy, wild twist	Framed triangular connection in both sectors	Both original Wronskians on the reconstructed sheet
CFT/NS	Irregular blocks, Matone map, full NS derivative	Selected angular and radial connection entries	Both original Wronskians while instanton order and continuation path vary

Raw continued-fraction residuals, tau functions, and NS derivatives are not expected to have equal numerical values. They can be different coordinates on the same pair of component divisors. Agreement of the coupled zero germ and its local intersection multiplicity is the meaningful comparison.

Kerr–de Sitter supplies the regular-singularity precursor, not a formula to be specialized mechanically. With distinct horizons, both separated equations are general Heun and their inverse problems are PVI. Removing the cosmological scale requires an operator-level confluence

$$(\text{general Heun},\mathrm{PVI}) \longrightarrow (\text{confluent Heun},\mathrm{PV}),$$

in which regular monodromy becomes formal monodromy and Stokes data. The physical angular data are resonant, and a composite trace can represent a union of connection branches. Thus the Kerr–de Sitter PVI expansion is a controlled capstone only after its lifted endpoint flag, regulated resonant limit, coupled angular–radial solve, and direct Wronskian validation have been retained.

Capstone C: two irregular ends do not imply one quantum bridge

Extremal Reissner–Nordström and the extremal D3-brane both lead to second-order equations with two irregular ends. That shared singularity pattern does not identify their canonical solutions, cycles, or functional relations.

Extremal Reissner–Nordström closes an exact-WKB connection word

For a four-dimensional massless scalar at $Q=M$, set $r=M(1+z)$. After a scalar gauge, the radial equation is

$$\left[ -\frac{\dd^2}{\dd z^2} -(M\omega)^2 \left(1+\frac1z\right)^4 +\frac{\ell(\ell+1)}{z^2} \right]\psi=0.$$

On the positive real ray, the physical line is future-ingoing at the irregular horizon $z=0$ and outgoing at irregular infinity. In the declared four-turning-point topology, continuation crosses two Stokes curves and produces

$$E_{\mathrm{rad}}^B \propto V_\gamma^{1/2} +V_\gamma^{-1/2},$$

where

$$V_\gamma = \exp\!\left[ \mathcal S_\vartheta \oint_\gamma S_{\mathrm{odd}}^{\mathrm{trad}}(z)\,\dd z \right].$$

The cycle orientation, square-root sheet, Borel direction, and ordered Stokes word are part of this equation. Since the companion infinity-frame Wronskian is nonzero,

$$E_{\mathrm{rad}}^B=0 \quad\Longleftrightarrow\quad V_\gamma=-1.$$

This is a boundary-derived Voros condition, not a guess based on the genus of the classical curve. Its present claim is conditional on persistence of the stated topology and existence or lateral definition of the selected Borel sum. High-order Borel–Padé agreement is strong numerical evidence, not a proof of those analytic hypotheses.

The D3-brane closes a canonical ODE/IM orbit

For an extremal D3-brane scalar, use

$$r=L\ee^{y/2}, \qquad \omega L=-2\ii\ee^\theta, \qquad P=\frac{\ell+2}{2}.$$

The radial equation becomes generalized modified Mathieu,

$$-\psi'' + \left[ \ee^{2\theta} (\ee^y+\ee^{-y}) +P^2 \right]\psi=0.$$

Canonical subdominant solutions at $y\to\pm\infty$ define

$$Q(\theta,P) = W_y \left[ \psi_{+,0}, \psi_{-,0} \right].$$

The radial Liouville gauge gives the exact identity

$$W_r[\phi_1,\phi_2] = \frac2L W_y[\psi_1,\psi_2].$$

What is not automatic is the endpoint-line identification. In the chapter’s $\ee^{-\ii\omega t}$ convention, the displayed real-$y$ subdominant phases are time reflected relative to the physical outgoing and throat-ingoing phases. If the required sector continuation maps each canonical line to its physical line up to a nonzero scalar, then

$$E_{\mathrm{rad}}^B = g(\theta,P)Q(\theta,P), \qquad g\ne0,$$

and QNMs are zeros of $Q$ on the chosen frequency sheet. Published Leaver comparisons support this zero-set identification; the continuation and the nonzero factor are not displayed in the cited ODE/IM construction.

The reciprocal potential and its discrete rotations generate neighboring canonical solutions. Wronskian identities close the exact functional relation

$$Q\!\left(\theta+\frac{\ii\pi}{2}\right) Q\!\left(\theta-\frac{\ii\pi}{2}\right) = 1+Q(\theta)^2.$$

No analogous QQ relation follows merely because the extremal Reissner–Nordström equation is also doubly confluent. Conversely, the D3 functional relation does not turn its $Q$ into the Reissner–Nordström Voros symbol. The two problems possess different extra structures:

Problem	Extra structure actually derived	Spectral object	Remaining gate
Extremal Reissner–Nordström scalar	Ordered two-crossing Stokes transport on a chosen WKB curve	$V_\gamma^{1/2}+V_\gamma^{-1/2}$	Stokes topology, cycle, sheet, and Borel sum
D3-brane scalar	Discrete orbit of normalized modified-Mathieu solutions	Canonical Wronskian $Q$ and QQ/TBA hierarchy	Continuation from canonical subdominant sectors to physical endpoint sectors

A chamber-specific D3 exact-WKB condition also exists in the primary literature. Comparing it with ODE/IM requires a separate cycle-to-canonical- solution dictionary; the common D3 operator alone does not make the two analytic completions identical.

Capstone D: holography keeps the ledger and swaps the engine

Rotating BTZ and RN–AdS$_5$ make a particularly clean comparison because their ODEs are not in the same canonical class. What survives the swap is the causal and variational ledger

$$\begin{gathered} \text{future-ingoing horizon line} \longrightarrow (\alpha,\beta), \\ \frac{\beta}{\alpha} \longrightarrow G_{\mathrm R} = \mathcal M\frac{\beta}{\alpha} +P_{\mathrm{loc}}. \end{gathered}$$

The first arrow is a connection problem. The second is holographic renormalization.

Rotating BTZ supplies the exact Gamma engine

For a nonextremal rotating BTZ black hole with AdS radius one, define

$$T_L=\frac{r_+-r_-}{2\pi}, \qquad T_R=\frac{r_++r_-}{2\pi},$$

and, for a scalar of $\nu=\sqrt{1+m^2}$,

$$h=\frac{1+\nu}{2}, \qquad q_L=\frac{\omega-k}{4\pi T_L}, \qquad q_R=\frac{\omega+k}{4\pi T_R}.$$

Euler’s hypergeometric connection formula gives the noninteger-$\nu$ nonlocal quotient. In the normalization fixed on the correlator page,

$$\begin{aligned} G_{\mathrm R}^{\mathrm{BTZ}} ={}& 2\nu\mathcal N_\Phi (r_+^2-r_-^2)^\nu \frac{\Gamma(-\nu)}{\Gamma(\nu)} \\ &\times \frac{ \Gamma(h-\ii q_L) \Gamma(h-\ii q_R) }{ \Gamma(1-h-\ii q_L) \Gamma(1-h-\ii q_R) } +P_{\mathrm{loc}}. \end{aligned}$$

The two generic source-zero towers are

$$\begin{aligned} \omega_n^{(L)} &= k-4\pi\ii T_L(n+h), \\ \omega_n^{(R)} &= -k-4\pi\ii T_R(n+h), \qquad n\in\mathbb Z_{\ge0}. \end{aligned}$$

No recurrence, PVI tau function, or TBA is needed to make this result more exact. Integer $\nu$ requires the logarithmic, renormalized limit; a singular hypergeometric horizon normalization requires a regularized local frame; and intersecting Gamma divisors require a two-parameter pole-skipping limit. The meromorphic function of complex $\omega$ is the displayed Gamma quotient, not a real-axis $|\Gamma|^2$ abbreviation.

RN–AdS₅ supplies a Heun recurrence engine

For a nonextremal five-dimensional Reissner–Nordström–AdS scalar on the generic recurrence chart, set the AdS radius to one and let

$$D_R=r_+^2-r_-^2, \qquad \nu_4=\sqrt{4+m^2}.$$

After the declared Möbius map and scalar gauge, the horizon-ingoing Heun solution has unit-leading boundary expansion

$$\psi_{\mathrm{in}} = C_+\psi_{\mathrm{src}} +C_-\psi_{\mathrm{resp}}.$$

A three-term recurrence computes the canonical connection coefficients. Undoing the common scalar gauge and using $1-z\sim D_R/r^2$ gives

$$\frac{\beta}{\alpha} = D_R^{\nu_4} \frac{C_-}{C_+}.$$

Consequently, on a nonresonant standard-quantization chart,

$$G_{\mathrm R} = 2\nu_4\mathcal N_\Phi D_R^{\nu_4} \frac{C_-}{C_+} +P_{\mathrm{loc}}.$$

The recurrence paper’s independent pseudospectral comparisons validate the Heun connection engine to high precision in its tested regimes. They do not determine an omitted bulk kinetic normalization or a finite counterterm scheme. At $m=0$, $\nu_4=2$ and the boundary is resonant; the generic two-power quotient must be replaced by a logarithmic connection matrix and renormalized canonical momentum.

Audit the physical boundary powers

For the radial unknown $R_{\omega,\ell}$ of Barragán Amado, Carneiro da Cunha, and Pallante, the leading large-$r$ terms of their raw radial equation give, for $R\sim r^\lambda$,

$$\lambda(\lambda+4)-m^2=0.$$

Thus the powers are $r^{\Delta-4}$ and $r^{-\Delta}$, with $\Delta(\Delta-4)=m^2$. Their printed boundary-condition equation instead uses $r^{2-\Delta}$, which is inconsistent with this indicial equation and with their gauge $R\propto(z-1)^{\Delta/2}F$. The fast standard-quantization line for this radial unknown is $r^{-\Delta}$; derive boundary powers from the operator before importing a connection dictionary.

The same general-Heun equation can also be encoded in a PVI inverse problem. That route requires the tau collision equation, the shifted accessory equation, and a triangular composite-monodromy condition selecting the future-horizon/source-free line. It is useful for controlled small-black-hole or low-temperature expansions, but a tau zero alone does not return $C_-/C_+$. For the planar RN–AdS$_5$ black brane, a separate low-temperature double scaling $T,\omega\to0$ at fixed $\omega/T$ yields a Heun-to-confluent-Heun Green function including the Fefferman–Graham and holographic prefactor in the authors’ chosen probe normalization, up to local scheme terms. Its control parameter and planar geometry are part of the result; it is not an arbitrary-temperature formula for the spherical black hole used above.

The engine swap can now be summarized without erasing the difference:

Layer	Rotating BTZ	RN–AdS$_5$
Canonical connection engine	Hypergeometric Euler formula	General-Heun recurrence; alternatively framed PVI data
ODE output	Explicit $(\alpha,\beta)$ Gamma coefficients	Canonical $C_\pm$, then physical $D_R^{\nu_4}C_-/C_+$
Generic QNM gate	Source Gamma coefficient vanishes	$C_+=0$
Observable completion	Explicit once $\mathcal N_\Phi$ and the finite scheme are declared	Model action and finite scheme are required generically; integral-gap charts additionally need logarithmic counterterms
Exceptional chart	Integer $\nu$, singular horizon frame, pole-skipping, extremality	Integer exponent gap, recurrence divisors, boundary of the sufficient $\lvert t\rvert>1$ recurrence chart, extremality

The common holographic ledger does not identify the two ODEs. It tells us which output the two different connection engines must deliver.

A comparison certificate closes every loop

For any new claimed black-hole correspondence, record the following before placing it in a method matrix.

1. Physical operator. Print the original separated operator, Fourier convention, field variable, radial interval, angular sheet, and parameter range. An operator known only up to a scalar-gauge or accessory shift is not yet a physical passport.
2. Endpoint flags. Derive horizon behavior in regular coordinates, remote boundary behavior on its causal sheet or AdS domain, and both angular regular lines. Record leading normalizations and Stokes sectors.
3. Primary boundary function. Define the weighted Wronskian, selected connection entry, or source determinant before introducing tau functions, periods, or $Q$-functions.
4. Canonical crosswalk. Give the coordinate map, scalar gauge, local exponent lifts, accessory relation, branches, and exceptional divisors.
5. Method gate. State the additional data actually used: framed monodromy for PVI/PV, a $U(1)$ scheme and quantum cycles for NS, a Stokes graph and Borel direction for exact WKB, or a discrete canonical-solution orbit and divisor strip for ODE/IM/TBA.
6. Nonzero-factor or divisor proof. Establish $D_{\mathrm{method}}=uE_B$ with $u\ne0$, or state explicitly that only a weaker divisor or numerical comparison is known.
7. Coupled closure. In rotating problems solve the angular and radial equations at the same $(A,\omega)$ and test the coupled Jacobian. Do not substitute an untracked “nearest” angular eigenvalue.
8. Return to the ODE. Re-evaluate the original endpoint Wronskians while changing match point, precision, recurrence depth, instanton order, Borel–Padé order, continuation path, or TBA window as applicable.
9. Multiplicity audit. Use an argument-principle count around an isolated scalar zero. For matrix source maps, also determine kernel dimension, partial multiplicities, and the crossing matrix or Jordan-chain data; a Jacobian test alone certifies only a transverse simple intersection.
10. Observable audit. For holography, restore the action normalization, physical radial powers, counterterms, contact scheme, and response numerator before interpreting poles and residues.
11. Exceptional-stratum rebuild. Reconstruct local frames at resonance, extremality, synchronous frequency, threshold, recurrence-coefficient zeros, Stokes-graph mutation, or angular branching. A formula divided by the vanishing quantity cannot validate its own limit.

Passing this certificate does not make two methods identical. It shows that they solve the same framed boundary problem in an overlap domain.

Common pitfalls

Ranking methods by the equation’s name. “Confluent Heun” identifies a singularity class, not a preferred global representation. Schwarzschild is best calibrated directly, Kerr admits a normalization-controlled NS chart, and a generic confluent-Heun black-hole equation may admit neither.

Comparing intermediate functions pointwise. A continued-fraction residual, a tau function, and an NS derivative have different normalization units and analytic divisors. Compare the reconstructed physical boundary functions, their zeros, and multiplicities.

Solving one half of a rotating problem. A radial zero at arbitrary $A$ is a curve in $(A,\omega)$, not a QNM. Close the angular equation on a tracked sheet and retain the full system when the angular graph branches.

Calling a quantum-curve match a spectral proof. Matching normal-form operators or classical curves closes only the operator gate. The physical endpoint flag, analytic completion, cycle or Stokes data, and nonzero normalization factor remain to be derived.

Calling a canonical AdS quotient a correlator. The ratio $C_-/C_+$ can locate generic source zeros. Residues and the full analytic background also depend on the radial gauge, action normalization, counterterms, and finite scheme.

Taking confluence by parameter substitution. Kerr–de Sitter to Kerr, nonextremal to extremal horizons, and finite-temperature to zero-temperature response all reorganize local frames and global analytic data. Derive the scaled operator and its endpoint bases before taking the limit of a formula.

Exercises

1. Prove unit covariance of the coupled Kerr test

Let $\widetilde E_{\mathrm{ang}}=uE_{\mathrm{ang}}$ and $\widetilde E_{\mathrm{rad}}=vE_{\mathrm{rad}}$, where $u$ and $v$ are holomorphic and nonzero near an isolated common zero. Show that the coupled zero germ, its local intersection multiplicity, and its transversality are unchanged.

Solution

The two zero sets are unchanged because multiplication by a unit creates no zero or pole. At a common zero, terms involving derivatives of $u$ or $v$ multiply $E_{\mathrm{ang}}$ or $E_{\mathrm{rad}}$ and vanish. Hence

$$\widetilde J_{A,\omega} = uvJ_{A,\omega}.$$

Since $uv\ne0$, one Jacobian vanishes exactly when the other does. Local intersection multiplicity is likewise invariant under multiplication of the two defining germs by units.

2. Recover the Schwarzschild continued-fraction gate

Starting from the minimal ratio

$$\frac{m_n}{m_{n-1}} = -\cfrac{\gamma_n} {\beta_n- \cfrac{\alpha_n\gamma_{n+1}} {\beta_{n+1}-\ddots}},$$

insert $n=1$ into the seed row $\beta_0m_0+\alpha_0m_1=0$ and recover $F_{\mathrm{CF}}=0$. What additional statement makes this a QNM condition?

Solution

Divide the homogeneous seed by $m_0\ne0$ on the ratio chart and substitute the expression for $m_1/m_0$. This gives

$$\beta_0 - \cfrac{\alpha_0\gamma_1} {\beta_1- \cfrac{\alpha_1\gamma_2} {\beta_2-\ddots}} =0.$$

Pincherle’s theorem identifies a minimal sequence. The further physical statement is that the Jaffé synthesis maps the seed sequence line to the future-ingoing horizon solution and the minimal large-order line to the causally continued outgoing solution at infinity. Without both transfers, the equation is only a recurrence-line condition.

3. Derive the extremal Reissner–Nordström Voros zero

Assume a fundamental infinity frame and

$$\Psi_-^{\mathrm I} = -\ii V_\gamma^{1/2}\Psi_+^{\mathrm{III}} + \left( V_\gamma^{1/2}+V_\gamma^{-1/2} \right) \Psi_-^{\mathrm{III}}.$$

Take a Wronskian with the outgoing vector and derive the QNM condition.

Solution

Antisymmetry kills the outgoing–outgoing term, leaving

$$\begin{aligned} W[\Psi_-^{\mathrm I},\Psi_+^{\mathrm{III}}] ={}& \left( V_\gamma^{1/2}+V_\gamma^{-1/2} \right) \\ &\times W[\Psi_-^{\mathrm{III}},\Psi_+^{\mathrm{III}}]. \end{aligned}$$

The last factor is nonzero because the infinity pair is fundamental. The boundary Wronskian therefore vanishes exactly when $V_\gamma^{1/2}+V_\gamma^{-1/2}=0$, or $V_\gamma=-1$. A logarithmic version still requires a chosen lift of $\log V_\gamma$.

4. Explain why the D3 QQ relation does not transfer to extremal RN

Both equations have two irregular ends. List the additional structure used to derive the D3 QQ relation and explain why singularity class alone is insufficient.

Solution

The D3 derivation uses the reciprocal modified-Mathieu potential, a discrete rotation of its spectral and coordinate variables, canonically normalized subdominant solutions in the rotated sectors, and Plücker identities among their Wronskians. Those data close a finite functional orbit and fix the constant term in the QQ relation. A generic doubly confluent equation has no such symmetry or normalized orbit. Extremal RN instead supplies a particular Stokes graph and a Voros cycle; those ingredients derive $V_\gamma=-1$ but not the D3 functional equation.

5. Recover both rotating-BTZ towers

The BTZ source coefficient is proportional to

$$\alpha_z = \frac{\Gamma(c)\Gamma(\nu)} {\Gamma(h-\ii q_L)\Gamma(h-\ii q_R)}.$$

Use the poles of the denominator Gamma functions to obtain the generic QNM frequencies.

Solution

Because $1/\Gamma(z)$ vanishes at $z=-n$, the two source-zero conditions are

$$h-\ii q_L=-n, \qquad h-\ii q_R=-n.$$

Substitute the definitions of $q_L$ and $q_R$ to find

$$\omega_n^{(L)} =k-4\pi\ii T_L(n+h),$$

and

$$\omega_n^{(R)} =-k-4\pi\ii T_R(n+h).$$

This reasoning assumes that the numerator does not simultaneously vanish or become singular. Coincident loci require the full two-parameter limit.

6. Separate the RN–AdS₅ spectral and correlator claims

Suppose a recurrence computes $C_-/C_+$ accurately. Which claims can be made without knowing $\mathcal N_\Phi$ and the finite counterterm scheme, and which cannot?

Solution

After the radial-gauge and boundary-power conversion is known, generic zeros of $C_+$ locate standard-quantization QNMs. On a regular nonresonant scalar chart, $C_-$ cannot vanish there as well without making the nonzero ingoing solution identically zero. The quotient also gives relative frequency dependence of the nonlocal response. Without $\mathcal N_\Phi$, one cannot normalize the correlator or its residues. Without counterterms, one cannot state the full analytic background or the scheme-dependent zero set of the complete correlator. At an integral exponent gap, even the generic quotient must first be replaced by the logarithmic renormalized construction.

References

• E. W. Leaver, “An Analytic Representation for the Quasi-Normal Modes of Kerr Black Holes”, Proceedings of the Royal Society of London A 402 (1985), 285–298. Derives the simultaneous Kerr angular and radial continued fractions and the Schwarzschild recurrence used as the direct control.
• S. A. Teukolsky, “Perturbations of a Rotating Black Hole. I. Fundamental Equations for Gravitational, Electromagnetic, and Neutrino-Field Perturbations”, Astrophysical Journal 185 (1973), 635–647. Supplies the separated spin-$s$ Kerr operators and their shared separation data.
• F. Novaes, C. I. S. Marinho, M. Lencsés, and M. Casals, “Kerr–de Sitter Quasinormal Modes via Accessory Parameter Expansion”, JHEP 05 (2019), 033. Develops the coupled Heun/PVI dictionary and controlled near-Nariai and small-rotation expansions.
• B. Carneiro da Cunha and J. P. Cavalcante, “Teukolsky Master Equation and Painlevé Transcendents: Numerics and Extremal Limit”, Physical Review D 104 (2021), 084051. Constructs the coupled Kerr PV inverse maps, boundary twist, numerical validation, and extremal degenerations.
• G. Bonelli, C. Iossa, D. Panea Lichtig, and A. Tanzini, “Exact Solution of Kerr Black Hole Perturbations via CFT$_2$ and Instanton Counting: Greybody Factor, Quasinormal Modes, and Love Numbers”, Physical Review D 105 (2022), 044047. Gives the generic-chart confluent-Heun connection coefficients and Kerr $SU(2)$, $N_f=3$ dictionaries.
• G. Aminov, A. Grassi, and Y. Hatsuda, “Black Hole Quasinormal Modes and Seiberg–Witten Theory”, Annales Henri Poincaré 23 (2022), 1951–1977. Proposes branchwise quantum-period rules for Schwarzschild and Kerr and tests them numerically.
• Y. Hatsuda and T. Shiga, “Exact WKB and Quantum Periods for Extremal Black Hole Quasinormal Modes”, arXiv:2605.01321 (2026). Derives the displayed extremal Reissner–Nordström Stokes word and performs high-order Borel–Padé tests; selected extremal Kerr sectors are treated separately.
• K. Imaizumi, “Quasi-Normal Modes for the D3-Branes and Exact WKB Analysis”, Physics Letters B 834 (2022), 137450. Gives a chamber-specific D3 exact-WKB connection condition and its mutation across Stokes-graph changes.
• D. Fioravanti and D. Gregori, “New Method for Exact Results on Quasinormal Modes of Black Holes”, Physical Review D 112 (2025), 125020. Develops the canonical D3 $Q$, functional relations, TBA, and numerical comparisons used in the ODE/IM audit.
• M. Bianchi, D. Consoli, A. Grillo, and J. F. Morales, “QNMs of Branes, BHs and Fuzzballs from Quantum SW Geometries”, Physics Letters B 824 (2022), 136837. Supplies the D3 physical radial equation and independent direct, WKB, and gauge-theory benchmarks.
• D. T. Son and A. O. Starinets, “Minkowski-Space Correlators in AdS/CFT Correspondence: Recipe and Applications”, JHEP 09 (2002), 042. Establishes the future-ingoing real-time prescription and the exact BTZ thermal correlator.
• D. Birmingham, I. Sachs, and S. N. Solodukhin, “Conformal Field Theory Interpretation of Black Hole Quasi-Normal Modes”, Physical Review Letters 88 (2002), 151301. Matches the rotating-BTZ QNM towers to thermal CFT poles.
• J. Barragán Amado, B. Carneiro da Cunha, and E. Pallante, “QNMs of Scalar Fields on Small Reissner–Nordström–AdS$_5$ Black Holes”, Physical Review D 105 (2022), 044028. Develops the RN–AdS$_5$ Heun/PVI inverse problem and controlled analytic expansions; its boundary powers require the operator-level correction noted above.
• J. Ren and Z. Yu, “Holographic Thermal Correlators from Recursions”, JHEP 06 (2025), 183. Computes RN–AdS$_5$ and charged-dilatonic Heun connection quotients by recurrence and benchmarks their poles pseudospectrally.
• P. Arnaudo and B. Withers, “Exact Low-Temperature Green’s Functions in AdS/CFT: From the Heun Equation to the Confluent Heun Equation”, Physical Review D 111 (2025), L121903. Gives the planar RN–AdS$_5$ black-brane correlator, including the Fefferman–Graham and holographic prefactor in the chosen probe normalization up to local scheme terms, in the controlled low-temperature double scaling.