Papers

In this paper we will examine if nonlocal quantum field theory will suffer from the fermion doubling pathology. We find that for a nonlocal Dirac theory, that no additional fermion species are introduced. This is provided that the form factor is nonvanishing at every point. The proof follows from the invertibility of the entire-function operator, which implies that the nonlocal Dirac operator has exactly the same kernel and finite-momentum zero set as the original local Dirac operator. We will distinguish this result from the standard Nielsen-Ninomiya theorem, which applies local lattice fermions on a compact Brillouin zone. We provide a general criterion for fermion doubling, a test for genuine and false doubling, and then a test procedure for mathematical and physical fermion doubling. We then will go on to distinguish this result from finite derivative truncations, which can introduce spurious polynomial zeros. We then conclude that fermion doubling is absent in the full continuum nonlocal theory.

We present a geometric derivation of Liénard–Wiechert fields in flat-space and AdS, emphasizing the origin of antipodal matching. In flat-space, the field of a uniformly moving charge is rewritten in coordinates centered on the source timelike geodesic. In this frame the charge is at rest and the solution is Coulombic, so the matching of the leading data at null infinity arises from describing a static field in a non-centered frame. We extend this construction to global AdS, where uniform motion is replaced by motion along a timelike geodesic. Starting from the static Coulomb solution at the center, we reconstruct the field of a freely moving charge in arbitrary global coordinates using embedding-space invariants. The resulting closed-form field obeys exact antipodal covariance in the bulk, and its boundary null-fringe limit reproduces the usual flat-space antipodal matching relation. We also describe an image-charge interpretation: the flat-space Coulomb field is represented after conformal compactification by an image singularity at spatial infinity, while the AdS Coulomb seed may be viewed as a charge together with an opposite image charge in a reflected copy. Together, these perspectives give a unified picture of Coulombic Liénard–Wiechert fields, antipodal matching, and the AdS-to-flat-space limit.

I construct the Type-IIB axion–dilaton wormhole partition function from charge-sector data. In a chosen axion charge, equivalently form-field flux sector, the long-distance saddle calculation supplies a two-end operator term with coefficient matrix \(C^{ij}_\nu\). The labels \(i,j\) label end-insertion operators; the labels \(A,B\) label parent universes. Reduction data \(b\) convert this matrix into scalar coefficients \(W_\nu[b]\). The wormhole partition function in the theta variable is \(Z_{\rm wh}(\theta;b)=\sum_\nu W_\nu[b]\e^{i\nu\theta}\). I analyze properties and constraints this coefficients satisfy: discrete-symmetry covariance, phase, absolute bounds, moment positivity, Cauchy–Schwarz inequalities for the unreduced coefficient matrix, complex-\(\theta\) domains, charge-lattice tails, and the dilute Bessel/Skellam limit. The \(\theta\)-dependence of the wormhole partition function is the Fourier transform of the charge-sector scalar coefficients.

We revisit the old problem of analytically continuing QFTs to fractional dimension $d\in\mathbb C$. We observe that common theories like QCD and QED have branch cuts in the complex $d$ plane. In particular, many operators in their low-energy CFT have OPEs and scaling dimensions that jump as a function of $d$.

In this work, we present the procedure to find series representations of tree-level cosmological correlators using the Kontorovich-Lebedev-Fourier (KLF) space formalism. This framework allows us to trade the in-in nested time integrals for frequency integrals over rational propagators and vertex functions, which encode interactions among quantum fields on a de Sitter background. Because these functions are the key objects to understand in order to perform a diagrammatic computation, we derive their relevant analytic properties by using both their integral representation and series representation in terms of Lauricella functions. For a vertex involving any number of fields, we obtain the location of singularities, the corresponding residues and the large-frequency asymptotic behaviour. Gathering these properties at each frequency integration allows us to compute a tree-level correlator directly, without relying on the differential equations it satisfies. To illustrate this procedure, we provide a complete treatment of the double-exchange diagram. The computation naturally distinguishes the different physical contributions, whether to the background or to the cosmological collider signal. The newly derived result is expressed at most in terms of a double series over hypergeometric functions, which simplifies the analytical expression of the correlator.

At fixed Hamiltonian $H$, changing the initial state changes the cyclic pair and hence generally the Lanczos basis and spread. For every normalizable polynomial preparation $\lvert\psi_Q\rangle=N_Q^{-1/2}Q(H)\lvert K_0\rangle$, we reconstruct its state-Krylov problem exactly from reference cyclic data. The transfer assumes no integrability and applies on finite or infinite cyclic support. Reweighting the reference spectral measure by $\lvert Q\rvert^2$ gives a finite-band connector for every prepared amplitude and a finite-rank Christoffel-Darboux projection for cumulative probabilities and spread, without rerunning ambient-space Lanczos. Fixed-degree seeds preserve the limiting Jacobi coefficients of asymptotically constant chains. Every Charlier number-state jump obeys $K_r(t)\ge K_0(t)$, with strict inequality for $r\ge1$ away from revivals. At every fixed finite $r$, the Hermite endpoint has exact all-level amplitudes, whereas the continuous-$q$-Hermite chord chain of double-scaled SYK has an all-level root-free re-Lanczos reconstruction. Writing $D_r^{\rm H}$ and $D_r^{\mathrm{ch}}(t)$ for the mean absolute Fock and chord displacements, respectively, the bounds are $K_r\ge D_r^{\rm H}$ and $K_r(t)\ge D_r^{\mathrm{ch}}(t)\ge\lvert\overline n_r(t)-r\rvert$. Thus rebuilt spread bounds physical displacement and signed chord drift. At fixed $0\le q<1$ and time, $K_r(t)-D_r^{\mathrm{ch}}(t)\to0$ as $r\to\infty$, with both approaching the same folded-Bessel limit. In Liouville space, the exact unnormalized gap measure family on an open inverse-temperature interval determines the positive transition-resolved measure, assuming the thermal kernel is known and strictly positive and the requisite exponential moments exist. One cyclic solution can therefore be reused across polynomial seeds while separating physical propagation, basis response, and rebuilt spread.

We analyze the finite-temperature two-point function of the large-$N$ SYK model at intermediate couplings away from the infrared fixed point. Specifically, we examine its analytic structure in the complex time plane, tracking the complex-time singularities over a range of temperatures. The location of the leading singularity lies on the imaginary axis. It controls the short-time dynamics of operator complexity, defining an `effective temperature' for the correlator. The next-to-leading singularity lies outside the thermal strip set by the above effective temperature. It has been argued that this could be interpreted in terms of bouncing null geodesics in the emergent black hole geometry. Both these singularities persist all the way down to zero temperature. We discuss our observations and motivate the related emergent geometry using a kinematic space perspective.

We describe the variable flavor number scheme to three-loop order, which modifies the massless parton densities by single- and two-mass effects and introduces heavy-quark parton distribution functions for charm and bottom. A renormalization group analysis shows the validity of this picture at large scales $Q^2$, where it resembles the non-power-suppressed heavy-flavor corrections completely. We also provide numerical implementations of a series of charged and neutral current Wilson coefficients.

The development of Group Theory in Mathematics as well as its impact in Physics has been spectacular during the twentieth century, and more particularly these last fifty years. If its contribution to Particle Physics deserves a special consideration, its usefulness cannot be neglected in other domains like Statistical Physics for example, and might also be of interest in Theoretical Biology. Some of these aspects will be examined.

We extend the coadjoint orbit approach to bosonization systematized by Delacretaz-Du-Mehta-Son to double-scaled non-relativistic Fermi liquids with central inverted harmonic oscillator potential. In 1+1d, this reproduces the Das-Jevicki collective field theory describing quasiparticle excitations of the $c=1$ matrix model. In 2+1d, this yields an effective field theory for noncritical M-theory, namely the 2+1d Fermi liquid proposed by Hořava and Keeler as a unified framework for characterizing noncritical string vacua. With a suitable gauge choice this theory reduces to a continuous family of 1+1d chiral bosons resembling Das-Jevicki collective fields coupled by additional interactions. Utilizing the underlying integrability of noncritical M-theory, we compute select fermion density correlation functions order-by-order in perturbation theory, and provide evidence that they agree with their Fermi liquid counterparts in a suitable semiclassical limit. This represents a step toward an effective spacetime gravity description of noncritical M-theory, which we expect to shed light on both the landscape of 2d noncritical strings and 3d quantum gravity in general.

We establish a rigorous geometric correspondence between thermal partition functions of discrete-spectrum quantum systems with bounded ground energy and the Chern character of "virtual physical sheaf" over spacetime. By interpreting Hamiltonian dynamics as a $U(1)$-equivariant flow on the quantum phase space $\mathbb{CP}^n$ and pushforward to spacetime, we show that the finite-temperature partition function emerges as the integral of the Chern character of "virtual physical sheaf" over spacetime. The construction extends naturally to infinite dimensions through trace class guaranteed by Weyl's aspmtotic law. Using the Grothendieck-Riemann-Roch formalism, we prove pushforward invariance of the Chern character under thermal compactification on arbitrary manifolds, providing a topological foundation for thermal traces in quantum field theory. This framework unifies spectral theory with characteristic class theory, offering a geometric interpretation of partition functions based on operator-algebraic approach.

Fully developed isotropic turbulence exhibits a dual nature: a continuous, scale-invariant energy cascade coexists with discrete, intense vortex filaments. We show that this duality arises from a spontaneously broken SO(3) gauge symmetry. By identifying the specific angular momentum $\mathbf{L} = \mathbf{r}\times\mathbf{u}$ as a non-Abelian gauge connection and the radial velocity $u_r$ as a Higgs field, the turbulent vacuum is described by the SO(3) Georgi-Glashow model. When the radial strain condenses, the symmetry breaks SO(3) $\to$ U(1), generating a topological mass gap $M_W = gv$. This gap partitions the energy into a massless U(1) sector (the solenoidal background) that sustains the Kolmogorov cascade, and a massive SO(3)/U(1) sector that is confined to vortex filaments. Using high-resolution DNS data (JHTDB, $Re_\lambda\approx433$), we empirically verify three key predictions: (i) the energy spectra obey a strict 1:2 equipartition over the inertial range, with a sharp divergence at $M_W \approx 40$; (ii) the radial Higgs field extracted around isolated vortex cores follows the exact BPS monopole profile $H(r)=\coth(r/\eta)-\eta/r$ with $\eta = 0.0093$ domain units and the VEV $v = 0.338$, identifying the ubiquitous "worms" as macroscopic 't Hooft-Polyakov monopoles; (iii) the Wilson loop computed from the velocity field exhibits a clean area law $\langle W_C \rangle \sim e^{-\sigma A}$ with string tension $\sigma = 0.303 \pm 0.009$, directly confirming the confining nature of the turbulent vacuum.

In generalized, also known as $p$-brane Galilei limits, the speed of light $c$ becomes infinite in the directions transverse to a $(p+1)$-dimensional Lorentzian worldvolume. In this paper, we explain that allowing the worldvolume to be Euclidean, and thus time to be transversal, $p$-brane Galilei limits turn into Carrollian ones, that we refer to as "Euclidean $p$-brane Carroll limits", in which $c$ goes to zero in the $p+1$ worldvolume directions. This leads to a unified approach to taking Galilean and Carrollian limits, whose consequences we explore for $p=0$. We show that the spacetime symmetries that arise from the Euclidean 0-brane Carroll limit can be centrally extended to what we will call the Alice algebra, similar to how the Bargmann algebra centrally extends the Galilei symmetries. This gives rise to the novel notion of an Alice particle, and we obtain the Bargmann and Alice particle actions from a unified limit of the action of a relativistic massive particle or tachyon, suitably coupled to a one-form gauge potential. In the presence of a cosmological constant, we find that the Bargmann and Alice particles undergo stable motion for negative and positive cosmological constant, respectively. Finally, we show that the Bargmann and Alice particle actions can be obtained from null reduction of a massless particle action in a relativistic spacetime with one and two times. Our results indicate that in 10 dimensions, the Alice particle is a decoupling limit of a D$0{}^*$-brane in Hull's type IIA${}^*$ theory, similar to how the Bargmann particle is related to a D$0$-brane in type IIA string theory.

Sign-switching dark energy has recently been proposed as a minimal modification of the late-time expansion history aimed at alleviating tensions within the standard cosmological model. In this work, we investigate ECDM, a smooth realisation of this scenario, with the dark energy density gradually transitioning from a negative to a positive value. We develop a consistent formulation of the perturbation equations that remains well behaved even when the dark energy equation-of-state parameter diverges during the transition. We confront the model with a comprehensive set of cosmological observations, including cosmic microwave background measurements from Planck 2018, ACT DR6 and SPT-3G, baryon acoustic oscillation measurements from DESI DR2, Type Ia supernova distances from Pantheon+, and local Hubble constant measurement of SH0ES. The inclusion of perturbations allows us to assess the impact of the model on structure growth and CMB anisotropies, providing a more thorough test of sign-switching dark energy. Our results show that this class of models is fully compatible with current precision cosmological observations while alleviating the Hubble tension and providing a compelling modification of the late-time dynamics of the Universe.

We derive a reduction formula for matrix elements on a causal set background. We derive an infinite tower of relations between correlators, akin to the Schwinger-Dyson equations of the continuum. Combining these two results we are able to express matrix elements in three different forms: as a path integral and as two distinct sums of correlators. We sketch the form that our method - which circumvents explicit use of differential equation of motion operators - takes in flat continuum spacetime where it provides an alternative expression for the standard LSZ result.

We study anomalous Wess-Zumino-Witten terms associated with the chiral anomaly in dense QCD matter under magnetic fields and rotation. By introducing electromagnetic, baryon number, and isospin background gauge fields, we write down the topological couplings of neutral mesons for the $N_f=2$ and $N_f=3$ cases. The resulting terms contain characteristic contributions proportional to $\vec{B} \cdot \vec{\nabla} \phi$ and $\vec{\Omega} \cdot \vec{\nabla} \phi$, where $\phi$ denotes $\pi^0$, $\eta$, or $\eta'$. These results are relevant to chiral soliton lattices in dense rotating matter.

Oscillatory features in the primordial power spectrum, potential signatures of new physics in the early universe, are usually searched for using fixed templates. In this work, we perform a template-free search for primordial features using symbolic regression. We find that both Planck and the combined Planck+ACT+SPT-3G datasets independently select an inverse-$k$ oscillation, $\cos(B/k)$ with $B\simeq4\,\mathrm{Mpc}^{-1}$, as the leading low-complexity feature. Comparing this inverse-$k$ template with standard linear and logarithmic oscillating templates, we find that it fits the data best, showing a weak preference for a non-zero amplitude. Our results show that symbolic regression as a powerful machine learning technique can provide an interpretable, model-independent approach to cosmological discovery.

Based on Geroch's process of dropping a system into a black hole from the vicinity of the horizon, we investigate in this paper the influence of deformation on the Bekenstein entropy bound both for (3+1) and (2+1) dimensions in the context of a generalized uncertainty principle (GUP). While providing a coherent framework that sets an upper limit on the entropy across dimensions we show, within a semiclassical treatment, that while a negative GUP deformation yields a universal relaxation of the bound, a positive deformation tightens it. Our results may be interpreted as a response to Planck-scale modifications of the near-horizon redshift.

In this article, we have presented one of the very important observations, regarding the behavior of mutual information of two different sets of subsystems in the after Page time scenario. This provides us with some deep insights about the redistribution of geometrical correlation between the two different sets of subsystems on a Cauchy slice. In our earlier works, we have shown how the saturation of mutual information between two specific subsystems plays a vital role in obtaining the correct Page curve for the eternal black hole. In those works, we have shown that the mutual information between $B_+$ and $B_-$, that is, $I(B_{+}: B_{-})$, vanishes at scrambling time, which leads to the correct Page curve. That means that at scrambling time, there is no correlation between $B_+$ and $B_-$. Remarkably, it is observed that at this particular value of observer's time, the mutual information between $\mathcal{I}$ and $R$, that is, $I(\mathcal{I}:R)$, becomes singular. This indicates that the regions $\mathcal{I}$ and $R$ become maximally entangled. This provides us with a notion of the transfer of geometric correlation between different regions on the Cauchy slice. In this work, we have also provided a way to calculate the tripartite mutual information of regions $\mathcal{I}$,~$R_+$ and $R_-$, that is, $I(\mathcal{I}:R_+:R_-)$ on the Cauchy slice using the earlier results involving the bipartite regions. This is a new result which was missing in the earlier literature.

The molecular states in coupled channel system of $B_s^0N- B\Lambda- B^*\Lambda- B\Sigma- B^*\Sigma$ are investigated within one-boson-exchange model that includes $S$-$D$ wave mixing and a tunable short-range $\delta(r)$ term. Bound, resonant, and virtual states are searched by analytically continuing the $S$ matrix in the complex energy plane. In the single-channel analysis, the $ B\Sigma$ and $ B^*\Sigma$ systems with $J^P=1/2^-$ and $3/2^-$ are found to be attractive and form three bound states with reasonable cutoff region, corresponding to $1/2^-( B\Sigma)$, $1/2^-( B^*\Sigma)$, and $3/2^-( B^*\Sigma)$ . When coupled-channel dynamics is included, these states evolve into near-threshold poles that produce enhancements in the $B_s^0N$, $ B\Lambda$, and $ B^*\Lambda$ invariant mass spectra. The pole below the $ B^*\Sigma$ threshold in the $J^P=1/2^-$ system depends strongly on the treatment of the short-range $\delta(r)$ term and becomes a virtual state when this contribution is removed, while the other two resonant poles and their line shapes are only moderately affected. For representative parameter region, the predicted masses of these states lie in the $6.44-6.52$ GeV region and have widths of a few to several MeV. The pole coupling and partial-width analyses indicate that the poles are generated mainly by the $ B\Sigma$ and $ B^*\Sigma$ channels interaction, and their observable signals are expected mainly in the lower open channels $B_s^0N$, $ B\Lambda$, and $ B^*\Lambda$. These results support the existence of near-threshold molecular pentaquark $P_{ \bar b s}$ and provide useful guidance in future experimental searches.

Recently, Calkins and Pate calculated the multi-particle operator product expansions (OPEs) of single-particle celestial operators with two-particle celestial operators in pure Einstein gravity. We apply their construction to the graviton, gravitinos, graviphotons, graviphotinos and scalars in ${\cal N}=8$ supergravity. By performing various contour integrals on these multi-particle OPEs, we obtain ninety-five (anti)commutators for the single-particle contributions from the celestial soft current algebra (derived from the single-particle OPEs) in a two-dimensional boundary. Moreover, these ninety-five multi-particle OPEs provide nontrivial relations between the corresponding celestial amplitudes. The triple-collinear limits between these bosonic and fermionic particles from the four-dimensional bulk are also described as a double check. Finally, we implicitly propose both the multi-particle OPEs of single-particle celestial operators with $(N-1)$-particle celestial operators and the corresponding (anti)commutators for the modes of celestial operators that have linear terms (or single-particle exchange terms due to the sum of $N$-particle factorization channels from amplitudes) on the right-hand sides.

We revisit the species scale in quantum gravity from the viewpoint of effective field theory (EFT). Two common parametric characterizations relate it to the breakdown of perturbative gravity and to the suppression scale of higher-derivative gravitational operators. We examine their correspondence through local threshold corrections from infinite Kaluza-Klein (KK) towers. In four dimensions, we compute the four-derivative one-loop action of minimally coupled scalar and Dirac towers with KK spectra in graviton, radion, and photon backgrounds. Feynman diagrams determine the single-mode coefficients, while analytic summation with full threshold functions tests the logarithmic coefficient inferred from renormalization-group evolution. The leading contribution scales with the effective number of KK levels, reproducing the standard species-scale relation under the stated matching assumptions, but its coefficient depends on the cutoff profile and belongs to EFT matching data. In contrast, the subleading curvature-squared thresholds contain logarithms of the KK mass gap whose spin-dependent coefficients remain stable under changes of the cutoff profile within the class studied here.

Modified theories of gravity based on the symmetric teleparallel framework have recently attracted considerable attention as viable alternatives to General Relativity. In this context, f(Q) gravity, in which the gravitational interaction is encoded in the nonmetricity scalar Q, provides a consistent geometrical formulation that differs from the standard curvature-based description. In this work, we investigate the structure of neutron stars within a family of f(Q) gravity models by employing realistic equations of state, namely FPS, SLy, ENG and MPA1. Using the covariant formulation of f(Q) gravity, we derive the corresponding Tolman-Oppenheimer-Volkoff equations and apply them to model compact stellar configurations. Numerical integration of the field equations provides the mass-radius relations and the maximum masses supported by each equation of state, enabling a direct comparison with current observational constraints. Furthermore, we analyze the behavior of the nonmetricity scalar both inside and outside the stellar object, providing additional insight into the gravitational structure of compact stars in this framework.

Lie-Poisson electrodynamics (LPE) is a non-Abelian and nonlinear deformation of usual electrodynamics, where the gauge algebra is defined through a Lie-algebra-type Poisson bracket on space-time. We focus on the geometric approach to LPE in the absence of charged matter. We establish a non-trivial field redefinition which, under mild technical assumptions, maps the LPE dynamics to that of Maxwell theory. Using this map, for any symmetry of the Maxwell action, we construct generators of LPE symmetries and the corresponding conserved currents. In particular, we obtain deformed Poincaré transformations. We also outline a natural quantization prescription for LPE based on our field redefinition.

We investigate how the compactification of some spatial dimensions in Rindler spacetime affects the vacuum expectation values (VEVs) of the field squared and the energy-momentum tensor for a charged scalar field prepared in the Fulling-Rindler vacuum state. A toral compactification is considered with quasi-periodicity conditions on the field operator along the compact dimensions. The phases of these conditions are interpreted in terms of the magnetic flux enclosed by the compact dimensions. For a general number of spatial dimensions, the components of the VEVs are explicitly separated, corresponding to the expectation values in the Minkowski vacuum. As a limiting case, the VEVs are retrieved in Rindler spacetime with trivial topology. We demonstrate that, for non-zero phases in the periodicity conditions, the vacuum energy-momentum tensor exhibits off-diagonal components with indices in the compact subspace. Near the Rindler horizon, the leading terms in the asymptotic expansions of the field squared and the diagonal components of the energy-momentum tensor coincide with the corresponding VEVs in Rindler spacetime with trivial topology. The off-diagonal components of the energy-momentum tensor vanish on the Rindler horizon. For small accelerations, the difference between the VEVs in the Fulling-Rindler and Minkowski vacua is exponentially small. The exception to this is a massless field with zero phases in the periodicity conditions. In this special case, the difference decays according to a power law as a function of acceleration. As an application, we consider the VEVs near the horizon of cylindrical black holes and topological black holes with toroidal horizons.

We study exclusive electroproduction of dijets in the collinear factorization framework as a probe of generalized parton distributions (GPDs). For quark dijet production, we extend previous analyses by including contributions from helicity GPDs and by assessing an additional leading-order electromagnetic channel governed by elastic nucleon form factors. Furthermore, we investigate exclusive gluon dijet production. We compare our prediction to HERA data and provide projections for measurements at the future Electron-Ion Collider.

We present an analysis of one-loop partition functions for scalars in AdS$_{d+1}$ obeying the Neumann boundary condition and explore phases of scalar field theories in several dimensions at zero and finite temperature. The partition function computation involves an analytic continuation from $\Delta_+$ corresponding to the Dirichlet boundary condition to $\Delta_-$ corresponding to Neumann boundary condition. We show that this can be implemented by deformations of the contour for integral over eigenvalue ($\lambda$) of the Laplace operator as compared to the integral over ${\mathbb R}$ in [arXiv:2201.09043] for the Dirichlet boundary condition. We further contrast these phases with those appearing for the case of scalars obeying the Dirichlet boundary condition and corroborate the occurrence of these phases by studying the long range behaviour of correlators.

We study the energy-momentum tensor of stationary rotating topological solitons in a (2 + 1)-dimensional $\mathbb{C}P^2$ nonlinear sigma model with a stabilizing potential term. We evaluate the distributions of the corresponding shear forces and pressure and study the stability criteria for these solutions. Our results suggest that these solitons become classically unstable for some range of values of the parameters of the system.

We construct an effective field theory description of relativistic superfluids with nonzero angular momentum density. At first order in the derivative expansion, the effective action contains a term with a quantized coefficient, which encodes the Berry phase for the angular momentum of the superfluid condensate. From this single term we derive a number of physical effects, including the relativistic Mermin-Ho relation, an anomalous Ettingshausen effect (exchangeable for an anomalous Hall effect), and an anomalous Hall viscosity.

Determining the local quantum source near a spacelike singularity is essential for assessing semiclassical backreaction inside black holes. By standard point splitting, we provide, to our knowledge, the first complete numerical determination of the renormalized stress-energy tensor (RSET) of a massless scalar field at arbitrary curvature coupling in the four-dimensional Schwarzschild interior. Calculations in the Unruh and Hartle–Hawking states reach $r/M\simeq10^{-4}$, yielding a common leading $r^{-6}$ coefficient tensor and coupling-dependent anisotropic pressure ratios. Uniform estimates of the complete mode sums then establish a rigidity theorem: at each fixed coupling, any Hadamard-state change on the common interior evolution domain contributes only subleading stress. The leading tensor is therefore insensitive to time and directional dependence of the quantum environment, in contrast with state-dependent leading Cauchy-horizon coefficients in asymptotically flat Reissner–Nordström spacetime. Its directional null projections define positive, mixed-sign, and negative regimes in coupling and direction, specifying the local mean focusing source; at conformal coupling, all leading projections are positive.