Boundary-Completed Noether-Charge Stability Theorem

• The theorem identifies the missing curvature in the fixed-volume entropy Hessian by combining bulk canonical energy with asymptotic Noether-charge contributions.
• It employs the covariant phase space formalism and fixed-volume constraints to prove the reverse isoperimetric conjecture in stationary asymptotically AdS black holes.
• The result has broader implications for stability analysis and higher-derivative gravity, offering a unified boundary completion mechanism.

Searching arXiv for the specified paper and closely related Noether-charge and boundary-charge works. arXiv search query: (Kumar, 25 Jun 2026) The Boundary-Completed Noether-Charge Stability Theorem is a theorem in stationary asymptotically AdS black-hole thermodynamics that identifies the reverse isoperimetric conjecture with a fixed-volume entropy bound derived from covariant phase space. In the formulation proved in "Reverse Isoperimetric Conjecture as a Noether-Charge Stability Theorem" (Kumar, 25 Jun 2026), the bulk Hollands–Wald canonical energy is not the full fixed-volume entropy Hessian: along exact stationary black-hole families it vanishes, and the missing curvature is supplied by a constrained asymptotic Noether-charge Hessian at infinity. The theorem therefore introduces a boundary-completed canonical energy whose positivity yields entropy concavity on admissible fixed-volume components, while a zero-energy rigidity condition determines the equality sector (Kumar, 25 Jun 2026).

1. Geometric and thermodynamic setting

The theorem is formulated in $d$-dimensional asymptotically AdS gravity, mainly illustrated in $d=4$ but written in a $d$-dimensional covariant framework. AdS boundary conditions are reflecting, with a compact spherical conformal boundary when compactness is assumed. The black holes under consideration are stationary and possess a smooth bifurcate Killing horizon generated by

$\chi^a = t^a + \Omega_i \phi_i^a,$

where $t^a$ is the asymptotic time translation and $\phi_i^a$ are the axial Killing fields; the horizon has surface gravity $\kappa$ and bifurcation surface $B$ (Kumar, 25 Jun 2026).

The thermodynamic ensemble is the fixed-volume ensemble. The cosmological constant is treated as pressure,

$P = - \Lambda /(8\pi G),$

and the thermodynamic volume $V$ is its conjugate in the extended Iyer–Wald first law. Comparisons are made at fixed $d=4$0, fixed $d=4$1, and fixed external couplings $d=4$2. On the stationary asymptotically AdS solution space $d=4$3, with physical phase space $d=4$4 after quotienting diffeomorphisms and gauge transformations trivial at infinity or acting trivially on the charges, the relevant component is

$d=4$5

Admissibility of $d=4$6 is central. The paper requires: every $d=4$7 is connected to the Schwarzschild–AdS reference $d=4$8 by an admissible piecewise smooth stationary path; the boundary-completed Hessian identity holds along such paths; boundary-completed positivity holds on the component; interior zero-energy paths are rigid; and maximizing sequences satisfy a properness condition so they do not escape to non-compact horizons, singular geometries, or disconnected boundary sectors (Kumar, 25 Jun 2026). The fixed-volume constraint is imposed through

$d=4$9

with $d$0 along tangent directions $d$1.

2. Covariant phase space, Wald entropy, and boundary completion

The theorem is built on the Iyer–Wald covariant phase-space formalism. For a diffeomorphism-invariant $d$2-form Lagrangian $d$3,

$d$4

with $d$5 the field equations and $d$6 the symplectic potential $d$7-form. The symplectic current is

$d$8

and the Noether current for $d$9 is

$\chi^a = t^a + \Omega_i \phi_i^a,$0

with $\chi^a = t^a + \Omega_i \phi_i^a,$1 on shell (Kumar, 25 Jun 2026).

For a Cauchy slice $\chi^a = t^a + \Omega_i \phi_i^a,$2, the Hamiltonian variation identity is

$\chi^a = t^a + \Omega_i \phi_i^a,$3

Its second variation on stationary backgrounds decomposes into a bulk canonical-energy term and boundary terms at infinity and at the bifurcation surface. The bulk term is the Hollands–Wald canonical energy,

$\chi^a = t^a + \Omega_i \phi_i^a,$4

for a perturbation $\chi^a = t^a + \Omega_i \phi_i^a,$5 around a stationary solution with horizon generator $\chi^a = t^a + \Omega_i \phi_i^a,$6 (Kumar, 25 Jun 2026).

Entropy is the Iyer–Wald Noether-charge entropy,

$\chi^a = t^a + \Omega_i \phi_i^a,$7

equivalently

$\chi^a = t^a + \Omega_i \phi_i^a,$8

The extended first law is

$\chi^a = t^a + \Omega_i \phi_i^a,$9

with

$t^a$0

In the Euclidean grand-canonical ensemble, $t^a$1 (Kumar, 25 Jun 2026).

The distinctive step is the boundary completion. The theorem defines

$t^a$2

where $t^a$3 is the constrained second variation of asymptotic Noether charges at infinity, Legendre-corrected at fixed $t^a$4. The fixed-volume Hessian identity is

$t^a$5

In short form,

$t^a$6

with $t^a$7 and $t^a$8 (Kumar, 25 Jun 2026).

3. Stability mechanism: positivity, concavity, and rigidity

The theorem’s key observation is that the entropy Hessian at fixed thermodynamic volume is not controlled by the bulk canonical energy alone. Along exact stationary black-hole families, $t^a$9, so

$\phi_i^a$0

The entire curvature of $\phi_i^a$1 along such a stationary family is then provided by the asymptotic boundary block: $\phi_i^a$2 whenever the stationary branch is thermodynamically stable at fixed $\phi_i^a$3 (Kumar, 25 Jun 2026).

The same boundary block may be expressed through the Legendre combination of asymptotic charges. Along a stationary path at fixed $\phi_i^a$4, $\phi_i^a$5, and $\phi_i^a$6,

$\phi_i^a$7

so

$\phi_i^a$8

This is the sense in which the asymptotic charge Hessian supplies the missing entropy curvature (Kumar, 25 Jun 2026).

Bulk positivity enters on the dynamical, charge-free subspace. For compact spherical Einstein–AdS black holes under reflecting AdS boundary conditions and regular horizon boundary conditions, canonical-energy stability gives

$\phi_i^a$9

after quotienting pure gauge, with positivity attributed to Regge–Wheeler–Zerilli/Kodama–Ishibashi master-field positivity. Combining this bulk positivity with positivity of the stationary boundary block yields concavity of entropy along admissible fixed-volume paths: $\kappa$0 With $\kappa$1 at the Schwarzschild–AdS reference, concavity implies the global bound

$\kappa$2

Equality is determined by zero-energy rigidity. If

$\kappa$3

along the interior of an admissible path, the path must project to the equality sector $\kappa$4. This excludes genuinely rotating, distorted, or hairy branches from saturating the inequality; only the static equality-degenerate sector remains (Kumar, 25 Jun 2026).

4. Reverse isoperimetric conjecture and the equality sector

The theorem states that on any admissible fixed-volume component $\kappa$5 of stationary AdS phase space,

$\kappa$6

with equality precisely on the allowed equality sector $\kappa$7. In Einstein gravity, where $\kappa$8, this reproduces the reverse isoperimetric conjecture (Kumar, 25 Jun 2026).

The reverse isoperimetric ratio in $\kappa$9 dimensions is

$B$0

equivalently

$B$1

Hence the entropy bound becomes

$B$2

so Schwarzschild–AdS maximizes entropy at fixed thermodynamic volume (Kumar, 25 Jun 2026).

The equality sector $B$3 contains the Schwarzschild–AdS reference saddle $B$4 and any static equality-degenerate branches at fixed $B$5. The paper explicitly includes static RN–AdS when charge variations are permitted but $B$6 fixes $B$7 and hence $B$8. These cases are treated as part of the equality sector rather than as violations of the entropy bound (Kumar, 25 Jun 2026).

By contrast, rotating Kerr–AdS directions are not part of $B$9. At fixed $P = - \Lambda /(8\pi G),$0 they strictly lower $P = - \Lambda /(8\pi G),$1, starting at quartic order in the rotation parameter $P = - \Lambda /(8\pi G),$2 in $P = - \Lambda /(8\pi G),$3, and their boundary-completed energy is strictly positive for $P = - \Lambda /(8\pi G),$4. The theorem therefore interprets the entropy cost of turning on rotation as the asymptotic charge Hessian at infinity, not as a bulk canonical-energy effect along the stationary family (Kumar, 25 Jun 2026).

5. Higher-derivative extension, admissible sectors, and known violations

The theorem extends naturally beyond Einstein gravity. In higher-derivative gravity, or in theories with non-minimal couplings, the same mechanism applies with Wald entropy rather than area: $P = - \Lambda /(8\pi G),$5 on admissible positive sectors where $P = - \Lambda /(8\pi G),$6 and rigidity holds. A corresponding entropy-ratio form is

$P = - \Lambda /(8\pi G),$7

(Kumar, 25 Jun 2026).

This extension is conditional rather than automatic. The paper emphasizes that positivity of $P = - \Lambda /(8\pi G),$8 need not hold in the presence of extra massive, ghost, scalarized, or non-minimally coupled modes, because either the bulk canonical energy or the boundary charge Hessian may become indefinite. For that reason the theorem is explicitly an admissible-component result, not an unconditional statement about all AdS black holes in all theories. Verification of local positivity and rigidity is given for Schwarzschild–AdS, with bulk positivity attributed to Regge–Wheeler–Zerilli/Kodama–Ishibashi and stationary boundary positivity to fixed-$P = - \Lambda /(8\pi G),$9 thermodynamics (Kumar, 25 Jun 2026).

Known violations are reinterpreted within this framework rather than treated as failures of the variational mechanism. Super-entropic black holes obtained from ultraspinning AdS limits have non-compact horizons and lie outside the compact admissible sector; they also sit on thermodynamically unstable branches. In the theorem’s language, their reverse-isoperimetric violations reflect failure of compactness or properness and of fixed-volume thermodynamic stability, not failure of the boundary-completed Noether-charge mechanism. Likewise, charged AdS planar black holes with Horndeski axions in Einstein–Horndeski–Maxwell gravity indicate that $V$0 fails on that sector, or that rigidity is modified by extra modes, rather than contradicting the theorem’s logic (Kumar, 25 Jun 2026).

The paper therefore places strong emphasis on scope and limitations. The theorem applies to stationary, asymptotically AdS black holes with smooth bifurcate Killing horizons, integrable and finite charges, stationary Iyer–Wald entropy, and thermodynamic volume defined as the pressure conjugate in the extended first law. Non-compact horizons, disconnected components, and sectors lacking global positivity or rigidity remain outside the admissible domain, and establishing component-level positivity and rigidity in general theories is described as an open problem (Kumar, 25 Jun 2026).

6. Relation to broader Noether-charge and boundary-charge literature

The adjective “boundary-completed” belongs to a broader Noether-charge vocabulary in which bulk variational or Hamiltonian objects become physically meaningful only after the relevant surface terms are included. In Einstein gravity, Padmanabhan showed that the Gibbons–Hawking–York boundary term yields a Noether potential whose horizon charge gives

$V$1

coinciding with the Komar energy and clarifying the role of boundary terms in horizon thermodynamics (Padmanabhan, 2012). In Hamiltonian language, Mouchet formulated improved Noether generators by adding a boundary function $V$2 so that the generator is functionally differentiable and conserved on shell, with the improved energy serving as a Lyapunov functional when positivity holds (Mouchet, 2015). In teleparallel gravity, the Iyer–Wald construction produces the torsion-superpotential charge

$V$3

yielding ADM mass and a black-hole entropy that admits a natural volume-integral representation and is stable under boundary completions and Weyl rescalings (Hammad et al., 2019).

Parallel developments exist in gauge, BRST, and boundary-field-space settings. The BRST Noether 1.5th theorem proves that the BRST Noether current of a gauge-fixed rank-1 BV theory decomposes on shell into a BRST-exact term plus a corner term defining Noether charges, and a charge bracket is introduced that canonically represents the asymptotic symmetry algebra even in the presence of symplectic flux and anomalies (Baulieu et al., 2024). In the field-space connection approach, horizontal symplectic geometry removes spurious local-gauge boundary charges and leaves only charges associated with reducibility parameters, with additive gluing across interfaces (Gomes, 2019). For Fermi fields with torsion and boundaries, conservation of the boundary-completed charge is achieved by boundary conditions that enforce vanishing normal flux $V$4 (Mourad et al., 2020). This suggests a common structural theme: boundary completion is not an auxiliary refinement but the mechanism that identifies which part of a Noether statement survives as a physically meaningful charge, flux, or stability criterion.