Einstein–Dilaton–ModMax Theory Overview

• Einstein–Dilaton–ModMax theory is a four-dimensional model coupling gravity with nonlinear, conformally invariant ModMax electrodynamics and an exponential dilaton field.
• It features a controlled two-parameter deformation via the ModMax parameter γ and dilaton coupling α, bridging Einstein–Maxwell–dilaton and pure Einstein–ModMax models.
• The framework enables exact solution techniques that generate novel static, rotating, and dihole black-hole configurations beyond classical Maxwell theory.

Searching arXiv for papers on Einstein–Dilaton–ModMax theory and related ModMax gravity solutions. Einstein–Dilaton–ModMax theory is a class of four-dimensional gravitating nonlinear electrodynamics models in which Einstein gravity is coupled to the conformally invariant, duality-invariant ModMax gauge sector and to a scalar dilaton through an exponential gauge coupling. In the formulation used for exact black-hole and dihole constructions, the Einstein–dilaton–ModMax (EDMM) action is

$S_{\text{EDMM} =\frac1{16\pi}\int d^4x\,\sqrt{-g}\Big( R -2\,\nabla_a\Phi\nabla^a\Phi + 4\,e^{-2\alpha\Phi}\,\mathcal L^{(MM)}(F,G) \Big),$

with ModMax Lagrangian

$$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$

where $\gamma\ge0$ is the ModMax deformation parameter and $\alpha$ is the dilaton coupling (Bokulić et al., 22 Jul 2025). A broader Einstein-frame realization used in stationary-axisymmetric analyses writes

$$\mathfrak{L} = \sqrt{-g}\bigg( -\,\mathcal R +2\epsilon_0\, (\nabla \phi)^2 + e^{-2 \alpha_0 \phi } \,\mathfrak{L}_{MM} \bigg),$$

with $\epsilon_0=+1$ for an ordinary dilaton and $\epsilon_0=-1$ for a phantom scalar (Bixano et al., 15 Apr 2026). Across these formulations, the central structural feature is that the dilaton multiplies the ModMax sector by the standard exponential factor familiar from Einstein–Maxwell–dilaton theory, while ModMax replaces Maxwell electrodynamics by a nonlinear but conformally invariant and $SO(2)$-duality-invariant theory. In static purely electric or purely magnetic sectors, the nonlinearity collapses to a Maxwell-like system with a constant $\gamma$-dependent rescaling; in genuinely rotating sectors with both invariants active, intrinsically nonlinear branches appear that need not possess a Maxwell analogue (Bokulić et al., 22 Jul 2025).

1. Definition of the theory and basic field content

The EDMM construction begins from nonlinear electrodynamics minimally coupled to gravity with Lagrangian density

$L=\frac{1}{16\pi}\big(R+4L(F,G)\big),$

where the electromagnetic invariants are built from the field strength $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$0 and its dual. For ModMax electrodynamics one chooses

$$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$1

so that the Einstein–ModMax sector is recovered by setting $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$2, while EDMM introduces the dilaton coupling $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$3 and a kinetic term for $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$4 (Bokulić et al., 22 Jul 2025).

In the alternative Einstein-frame conventions used for stationary-axisymmetric rotating solutions, the ModMax sector is written as

$$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$5

with $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$6 and $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$7, and the exponential dilaton factor remains purely multiplicative (Bixano et al., 15 Apr 2026). The same paper explicitly identifies $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$8 with pure Einstein–ModMax, $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$9 with low-energy effective superstring theory, and $\gamma\ge0$0 with Kaluza–Klein theory. A related generalized Ernst-type treatment includes the same Einstein-frame action and further notes the “entanglement relativity” value $\gamma\ge0$1 within its model space (Bixano et al., 27 Mar 2026).

The Maxwell limit is obtained by taking $\gamma\ge0$2, for which ModMax reduces to the linear Maxwell Lagrangian. In the EDMM setting this implies reduction to Einstein–dilaton–Maxwell, while $\gamma\ge0$3 recovers pure Einstein–ModMax (Bokulić et al., 22 Jul 2025). This controlled two-parameter deformation by $\gamma\ge0$4 and $\gamma\ge0$5 is a defining feature of the theory.

2. Field equations, constitutive structure, and special symmetries

The ModMax constitutive relation can be expressed through the excitation 2-form

$\gamma\ge0$6

which, after variation of the ModMax Lagrangian, takes the explicit form quoted in the exact-solution analysis of Einstein–ModMax backgrounds (Bokulić et al., 22 Jul 2025). For generic nonlinear electrodynamics the stress tensor can be written as

$\gamma\ge0$7

and ModMax’s conformal invariance implies $\gamma\ge0$8, yielding a traceless electromagnetic stress tensor proportional to the Maxwell form (Bokulić et al., 22 Jul 2025).

In the purely magnetic EDMM sector the field equations are

$\gamma\ge0$9

so the dilaton contributes both directly through its kinetic energy and indirectly by modulating the effective gauge coupling (Bokulić et al., 22 Jul 2025). In the purely electric or purely magnetic sectors with $\alpha$0, the gauge equations reduce to Maxwell form with an overall factor $\alpha$1; in this regime ModMax behaves as a rescaled linear theory. This underlies much of the exact static EDMM solution theory.

A broader covariant formulation introduces $\alpha$2, dressed invariants $\alpha$3 and $\alpha$4, and two functions

$\alpha$5

with $\alpha$6. The constitutive tensor is then

$\alpha$7

and the field equations become

$\alpha$8

together with trace-reversed Einstein equations

$\alpha$9

(Bixano et al., 15 Apr 2026).

Three structural properties recur throughout the literature. First, ModMax is conformally invariant, so its electromagnetic stress tensor is traceless. Second, the equations of motion and stress tensor are invariant under $$\mathfrak{L} = \sqrt{-g}\bigg( -\,\mathcal R +2\epsilon_0\, (\nabla \phi)^2 + e^{-2 \alpha_0 \phi } \,\mathfrak{L}_{MM} \bigg),$$0 electric–magnetic duality rotations in the $$\mathfrak{L} = \sqrt{-g}\bigg( -\,\mathcal R +2\epsilon_0\, (\nabla \phi)^2 + e^{-2 \alpha_0 \phi } \,\mathfrak{L}_{MM} \bigg),$$1 plane,

$$\mathfrak{L} = \sqrt{-g}\bigg( -\,\mathcal R +2\epsilon_0\, (\nabla \phi)^2 + e^{-2 \alpha_0 \phi } \,\mathfrak{L}_{MM} \bigg),$$2

Third, once the dilaton factor $$\mathfrak{L} = \sqrt{-g}\bigg( -\,\mathcal R +2\epsilon_0\, (\nabla \phi)^2 + e^{-2 \alpha_0 \phi } \,\mathfrak{L}_{MM} \bigg),$$3 is present, the simple duality story is modified, and full duality invariance is generally not maintained unless the scalar is transformed appropriately (Bokulić et al., 22 Jul 2025). A common misconception is that ModMax nonlinearity always produces qualitatively new static black-hole equations; in fact, the exact analyses show that in the sectors most often used for solution generation, $$\mathfrak{L} = \sqrt{-g}\bigg( -\,\mathcal R +2\epsilon_0\, (\nabla \phi)^2 + e^{-2 \alpha_0 \phi } \,\mathfrak{L}_{MM} \bigg),$$4 and the theory becomes effectively linear. The genuinely nonlinear novelty emerges most clearly in rotating or dyonic branches with independent $$\mathfrak{L} = \sqrt{-g}\bigg( -\,\mathcal R +2\epsilon_0\, (\nabla \phi)^2 + e^{-2 \alpha_0 \phi } \,\mathfrak{L}_{MM} \bigg),$$5 and $$\mathfrak{L} = \sqrt{-g}\bigg( -\,\mathcal R +2\epsilon_0\, (\nabla \phi)^2 + e^{-2 \alpha_0 \phi } \,\mathfrak{L}_{MM} \bigg),$$6 (Bixano et al., 15 Apr 2026).

3. Stationary-axisymmetric reduction and generalized Ernst formalisms

For stationary, axisymmetric spacetimes, EDMM and its scalar generalizations admit a Weyl-type reduction with metric

$$\mathfrak{L} = \sqrt{-g}\bigg( -\,\mathcal R +2\epsilon_0\, (\nabla \phi)^2 + e^{-2 \alpha_0 \phi } \,\mathfrak{L}_{MM} \bigg),$$7

electromagnetic potential $$\mathfrak{L} = \sqrt{-g}\bigg( -\,\mathcal R +2\epsilon_0\, (\nabla \phi)^2 + e^{-2 \alpha_0 \phi } \,\mathfrak{L}_{MM} \bigg),$$8, and scalar field $$\mathfrak{L} = \sqrt{-g}\bigg( -\,\mathcal R +2\epsilon_0\, (\nabla \phi)^2 + e^{-2 \alpha_0 \phi } \,\mathfrak{L}_{MM} \bigg),$$9 (Bixano et al., 27 Mar 2026). This reduction leads to a generalized sigma-model system in the real potentials

$\epsilon_0=+1$0

where $\epsilon_0=+1$1, $\epsilon_0=+1$2 is a magnetic-type potential, $\epsilon_0=+1$3 is a rotational Ernst-like potential, and $\epsilon_0=+1$4 (Bixano et al., 27 Mar 2026).

In static axisymmetric Einstein–ModMax configurations restricted to purely electric or purely magnetic sectors, the reduction simplifies to an Ernst-type system with gravitational potential $\epsilon_0=+1$5 and complex electromagnetic potential $\epsilon_0=+1$6. In the purely magnetic sector the field equations become

$\epsilon_0=+1$7

$\epsilon_0=+1$8

with the purely electric system obtained by replacing $\epsilon_0=+1$9 by $\epsilon_0=-1$0 (Bokulić et al., 22 Jul 2025). These are direct ModMax deformations of the Einstein–Maxwell Ernst equations.

The more general stationary-axisymmetric scalar-coupled formalism introduces complex one-form variables

$\epsilon_0=-1$1

$\epsilon_0=-1$2

which satisfy a compact generalized Ernst system (Bixano et al., 27 Mar 2026). In the special sector $\epsilon_0=-1$3, equivalently $\epsilon_0=-1$4, the functions $\epsilon_0=-1$5 and $\epsilon_0=-1$6 become constants $\epsilon_0=-1$7, and the equations simplify dramatically. The authors describe this as analytically manageable while retaining genuinely nonlinear ModMax features (Bixano et al., 27 Mar 2026).

An important interpretive point follows from the “trivialization” result in the frozen sector. If the scalar is absent or constant, frozen ModMax can be mapped by a constant linear redefinition of electromagnetic potentials to a Maxwell-like system. This suggests that a dynamical scalar field is essential for obtaining genuinely new stationary-axisymmetric ModMax effects in that sector (Bixano et al., 27 Mar 2026). That conclusion aligns with later rotating dilatonic constructions in which the scalar is not an ancillary ingredient but part of the mechanism supporting branches without Maxwell analogue (Bixano et al., 15 Apr 2026).

4. Solution-generating transformations and static exact solutions

A major advance in EDMM was the construction of generalized Harrison transformations that preserve purely electric or purely magnetic sectors and therefore act as solution-generating maps inside the effectively linear $\epsilon_0=-1$8 regime (Bokulić et al., 22 Jul 2025). In Einstein–ModMax the electric Harrison transformation is

$\epsilon_0=-1$9

while the magnetic Harrison transformation is

$SO(2)$0

These transformations can add charges to vacuum seeds, embed solutions in a Melvin–ModMax universe, or do both simultaneously (Bokulić et al., 22 Jul 2025).

For EDMM, the magnetic sector admits a Harrison–dilaton transformation acting on the metric, gauge potential, and dilaton: $SO(2)$1

$SO(2)$2

$SO(2)$3

with

$SO(2)$4

The exponent $SO(2)$5 in the metric scaling is the familiar Einstein–Maxwell–dilaton Harrison–dilaton structure, now dressed by the ModMax factor $SO(2)$6 (Bokulić et al., 22 Jul 2025).

These transformations yield explicit static exact solutions. In Einstein–ModMax, acting on Schwarzschild reproduces charged black holes with line element

$SO(2)$7

and $SO(2)$8 in the electric case, with an analogous magnetic form. Extremality occurs at $SO(2)$9 or $\gamma$0, showing that $\gamma$1 rescales the charge contribution (Bokulić et al., 22 Jul 2025).

The same solution-generating machinery also constructs Melvin embeddings and balanced multi-center configurations. Its significance lies less in algebraic novelty than in the demonstration that a substantial portion of the Einstein–Maxwell exact-solution technology survives the inclusion of ModMax and a dilaton in the effectively linear magnetic and electric sectors.

5. Extremal EDMM black holes and balanced black diholes

The explicit EDMM black-hole solution extracted from the dihole construction has metric, gauge field, and dilaton

$\gamma$2

$\gamma$3

obtained as the infinite-separation limit of a two-center magnetically charged solution (Bokulić et al., 22 Jul 2025). This is the EDMM analogue of the extremal Einstein–dilaton–Maxwell black hole, with ModMax entering through the $\gamma$4 factor in the gauge potential and the induced rescaling of physical charge.

The horizon is at $\gamma$5, and the Ricci scalar behaves as

$\gamma$6

so for $\gamma$7 the extremal horizon is singular, just as in the standard extremal Einstein–Maxwell–dilaton case (Bokulić et al., 22 Jul 2025). Asymptotically, the geometry is flat, the dilaton vanishes, and the gauge field approaches a monopole form. The magnetic charge is obtained from

$\gamma$8

and is related to the metric parameter by $\gamma$9 (Bokulić et al., 22 Jul 2025).

The corresponding balanced two-center configurations are EDMM black diholes: two extremal black holes with opposite magnetic charges in equilibrium, supported by an external ModMax–dilaton Melvin field. The Bonnor–ModMax–dilaton seed is transformed into the EDMM dihole solution

$L=\frac{1}{16\pi}\big(R+4L(F,G)\big),$0

with corresponding expressions for $L=\frac{1}{16\pi}\big(R+4L(F,G)\big),$1, $L=\frac{1}{16\pi}\big(R+4L(F,G)\big),$2, and $L=\frac{1}{16\pi}\big(R+4L(F,G)\big),$3 given explicitly in the source (Bokulić et al., 22 Jul 2025). The conical singularity can be removed by tuning the external magnetic field to

$L=\frac{1}{16\pi}\big(R+4L(F,G)\big),$4

This balancing condition is the central equilibrium result: the external Melvin-type field exactly compensates the attraction between the oppositely charged extremal black holes (Bokulić et al., 22 Jul 2025).

In the $L=\frac{1}{16\pi}\big(R+4L(F,G)\big),$5 limit these solutions reduce to their Einstein–Maxwell or Einstein–dilaton–Maxwell counterparts, including the dilaton dihole associated in the paper with Emparan–Maldacena–Papadopoulos. In the $L=\frac{1}{16\pi}\big(R+4L(F,G)\big),$6 limit the EDMM dihole becomes the pure Einstein–ModMax dihole (Bokulić et al., 22 Jul 2025). These reductions clarify that EDMM is not a disconnected theory but a deformation of known exact families.

6. Rotating dilatonic ModMax branches and the absence of a Maxwell analogue

Static purely magnetic or purely electric EDMM sectors are not the whole theory. A separate line of work constructs rotating, dyonic, dilatonic ModMax solutions in a genuinely nonlinear sector characterized by

$L=\frac{1}{16\pi}\big(R+4L(F,G)\big),$7

equivalent to a constant dressed ratio $L=\frac{1}{16\pi}\big(R+4L(F,G)\big),$8 (Bixano et al., 15 Apr 2026). In this branch the metric retains the stationary-axisymmetric Weyl form, but both $L=\frac{1}{16\pi}\big(R+4L(F,G)\big),$9 and $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$00 are nonzero and $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$01, producing a nontrivial gravitomagnetic structure.

The exact rotating solution is expressed in spheroidal coordinates through

$$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$02

with scalar and gauge fields built from the same harmonic function $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$03 and its harmonic conjugate $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$04 (Bixano et al., 15 Apr 2026). The key consistency condition is

$$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$05

Since the Maxwell framework in that parametrization corresponds to $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$06, the branch can connect to Maxwell only at the isolated value $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$07. For the physically emphasized values $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$08 and $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$09, one has $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$10, so the branch has no Maxwell analogue (Bixano et al., 15 Apr 2026).

This result addresses another common misconception: that ModMax merely rescales charges in all sectors. The rotating dilatonic branch shows that once both electromagnetic invariants and rotation are active, there exist exact configurations intrinsically tied to nonlinear ModMax structure rather than deformations of Einstein–Maxwell–dilaton solutions (Bixano et al., 15 Apr 2026).

The asymptotic charges of this family include Komar mass, NUT charge, angular momentum, and both EMD-like and ModMax electric charges. The asymptotic values

$$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$11

show that the parameter $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$12 controls the monopole and NUT sector while $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$13 controls rotation (Bixano et al., 15 Apr 2026). Setting $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$14 yields an asymptotically flat, NUT-free configuration with vanishing mass and net charges but nonzero angular momentum, interpreted in the source as a pure mass dipole with ModMax electric and magnetic dipole moments.

In the prolate branch, $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$15 is an event horizon with angular velocity

$$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$16

and for parameter range

$$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$17

the exterior region satisfies the null energy condition and the curvature singularities remain hidden behind the horizon (Bixano et al., 15 Apr 2026). This establishes a physically well-behaved rotating black-hole sector within nonlinear Einstein–dilaton–ModMax theory.

7. Relation to precursor models and broader theoretical context

A precursor to ModMax arises as the $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$18 limit of a duality-invariant deformation of Born–Infeld theory. Its flat-space Lagrangian can be written as

$$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$19

and in the infinite-tension limit reduces to the ModMax Lagrangian (Nastase, 2021). This precursor is duality invariant for all $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$20, but conformal invariance holds only in the $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$21 ModMax limit.

The same work shows that the precursor admits a DBI-like determinant formulation with scalar coupling,

$$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$22

and that BIon, catenoid, and certain knotted solutions survive in this generalized setting (Nastase, 2021). That analysis does not itself include gravity or a dilaton, but it motivates later Einstein–ModMax–scalar and EDMM models by showing that ModMax-type nonlinear electrodynamics can be coupled naturally to scalars and can preserve nontrivial exact solution sectors.

Within the generalized Einstein–ModMax–scalar framework, the scalar can be ordinary or phantom, and the stationary-axisymmetric formalism is broad enough to encompass dilatonic, Kaluza–Klein, low-energy string-inspired, and related couplings (Bixano et al., 27 Mar 2026). The frozen sector $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$23 is especially significant because it preserves nonlinear ModMax structure while making the reduced equations tractable. Two exact rotating families constructed there demonstrate that ModMax plus a dynamical scalar can generate rotating and magnetized geometries whose Maxwell counterparts are static and purely electric (Bixano et al., 27 Mar 2026).

Taken together, these results place Einstein–dilaton–ModMax theory at an intersection of several research programs: nonlinear electrodynamics in curved spacetime, exact solution generation, dilatonic gravity, and duality-symmetric gauge theory. The current literature suggests two complementary regimes. One is the effectively linear $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$24 regime, where Harrison methods, Melvin embeddings, and extremal diholes can be developed almost in parallel with Einstein–Maxwell–dilaton theory. The other is the genuinely nonlinear rotating regime, where the interplay of $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$25, the scalar coupling, and the invariant ratio $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$26 yields exact branches with no Maxwell continuation (Bokulić et al., 22 Jul 2025). A plausible implication is that future progress will depend on extending the stationary-axisymmetric machinery beyond the frozen or $$\mathcal{L}^{(MM)}=\frac14\left(-F\cosh\gamma+\sqrt{F^2+G^2}\,\sinh\gamma\right),$$27 sectors, especially for rotating black holes, NUT-charged configurations, and backgrounds with additional structure such as a cosmological constant.