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A. A. Araújo Filho

Publications and source records attributed to A. A. Araújo Filho.

At least 19 recordsLinked to original sources
arXiv2026

An exact Simpson-Visser family in bumblebee gravity

We construct an exact Simpson-Visser solution in metric bumblebee gravity, supported by a phantom scalar, magnetic nonlinear electrodynamics, and a spacelike configuration of fixed norm. The independent vector equation determines a local Lagrange multiplier and a temporal component that permits a regular ingoing extension across metric horizons. Reconstruction of the matter functions identifies a continuous family of wormholes, null minimum geometries, and black-bounces within one action. In the black hole case, the areal photon sphere and critical timelike orbit radii retain their Schwarzschild values, whereas angular deflection, periapsis advance, and photon orbit instability depend on the bounce and bumblebee parameters. The metric shadow coincides with the Schwarzschild prediction at fixed lapse mass and areal observer position; in addition, the adopted Sgr A* shadow diameter and M87* angular mass comparisons do not determine these parameters separately. The coupled axial equations yield distinct tensor, metric, and transverse bumblebee characteristics. We compute fundamental auxiliary massless scalar and ordinary Maxwell probe resonances by Chebyshev spectral collocation and estimate the tensor and bumblebee resonance scales at leading geometric optics order. Nonnegative probe potentials exclude exponentially growing probe modes. For negative nonminimal coupling and a nonzero temporal case, however, the supporting electromagnetic field develops an angular gradient instability in its extraordinary polarization. The unstable characteristic persists in the coupled principal symbol, while a variational construction establishes growing modes on the fixed geometry and lower bounds on their growth rates.

gr-qc↗
arXiv2026

Fermi Acceleration Mechanisms Beyond Lorentz Symmetry

We construct models for first- and second-order Fermi acceleration of particles, incorporating generic frame transformations, dispersion relations, and conservation laws. Within this framework, we study deformations of Lorentz symmetry via the $κ$-Poincaré algebra in the bicrossproduct and classical bases, which respectively deform and preserve the relativistic dispersion relation. We also examine explicit Lorentz symmetry violation and compare the results with deformed relativity and special relativity. The energy spectra present different shapes when one considers deformation or violation of Lorentz symmetry in superluminal or subluminal scenarios. One of the possible outcomes is an intense decay of the spectrum for higher energies. We also gather the predictions of all the scenarios considered, comparing the spectra produced at the source in the energy domain, which is the variable in which cosmic ray observatories report their measurements.

gr-qc↗
arXiv2026

An exact rotating black hole solution in Palatini Kalb-Ramond gravity

We construct an exact stationary axisymmetric vacuum configuration in metric affine Kalb-Ramond gravity with zero cosmological constant. Independent variation of the metric, connection, and antisymmetric field yields the complete field equations and the algebraic relation between the physical and auxiliary geometries. A Kerr auxiliary metric and a closed corotating vacuum two form of fixed positive norm determine the physical metric. The tensor kernel contains the horizon generator, and the metrics, independent connection, and vacuum field share a smooth extension across the nonextremal physical Killing horizon and the rotation axes at finite radius for unequal deformation factors. We obtain an explicit sufficient domain for a causal exterior and an asymptotically locally flat end at fixed nonpolar angles. The physical Hamiltonian yields the timelike and null equations, equatorial effective potentials, circular orbit properties, radial stability, and perihelion advance. Numerical photon evolution determines the critical curve on an orthonormal observer screen. Its size, shape, and centroid depend on $ε=ξ_2b^2/[2(1+ξ_1b^2)]$ at fixed normalized mass scale and spin, with Kerr recovered at $ε=0$. Geometric observations probe this combination and retain an exact degeneracy between the curvature couplings. A static equatorial comparison with Mercury gives a conditional sensitivity of $2.30\times10^{-11}$ for $|2ε|$ under the stated angular convention and mass setting.

gr-qc↗
arXiv2026

Thermodynamic and statistical properties of a multifractional modified dispersion relation via the grand-canonical ensemble

We study the thermodynamic and statistical properties of a gas governed by a multifractional modified dispersion relation of the form $ω^{2}=k^{2}+4E_{*}^{-1/2}k^{5/2}$, where $E_{*}$ sets the characteristic scale of the multifractional correction. Working within the grand-canonical ensemble, we derive the modified density of states, the grand potential, the partition function, and the main thermodynamic quantities for both bosonic and fermionic sectors. The deformation changes the available phase-space distribution and produces nonstandard thermal scalings controlled by the ratio $T/E_{*}$. In the infrared regime, the usual relativistic gas behavior is recovered with leading corrections proportional to powers of $(T/E_{*})^{1/2}$. In the ultraviolet regime, the density of states scales as $\varrho(ω)\propto ω^{7/5}$, corresponding to an effective density-of-states dimension $d_{\mathrm{eff}}=12/5$. As a consequence, the Stefan-Boltzmann law is deformed from $u\propto T^{4}$ to $u\propto E_{*}^{3/5}T^{17/5}$, while the equation-of-state parameter approaches $w=5/12$ instead of the standard radiation value $w=1/3$. We also analyze thermal stability, particle number and energy fluctuations, Bose-Einstein condensation, and the degenerate Fermi gas limit. The multifractional correction increases the critical temperature of a conserved bosonic gas and modifies the Fermi energy, pressure, sound speed, and low-temperature heat capacity of degenerate fermions. A direct fit to the COBE/FIRAS monopole spectrum yields $E_{*}>0.36\,\mathrm{MeV}$ at $95\%$ CL, while the conservative GWTC--3 propagation constraint gives $E_{*}>1.18\times10^{19}\,\mathrm{GeV}$ at $90\%$ credibility when the dispersion relation is assumed to be universal.

cond-mat.stat-mech↗
arXiv2026

Higher derivative Hořava-Lifshitz fermions: from superconductivity to atomic spectroscopy

We investigate an isotropic Hořava-Lifshitz fermion theory supplemented by a dimension-six spatial derivative operator, preserving first order time evolution while modifying the ultraviolet dispersion relation. Within the regime of validity of the derivative expansion, we develop three complementary consequences of the $z=2$ sector. First, a Bardeen-Cooper-Schrieffer/Nambu-Jona-Lasinio (BCS/NJL) mean field treatment of fermion pairing yields the particle and antiparticle quasiparticle spectra, the finite temperature gap equation, and the gauge invariant criterion for a stable superconducting phase. The modified dispersion alters both the density of states governing the pairing scale and the electromagnetic vertices entering the Meissner response. Second, using the scalar pole denominator as a minimal spin independent exchange kernel, we derive an exponentially damped and oscillatory massive potential and obtain the corresponding Born scattering amplitudes. The quartic momentum dependence renders both the forward and total cross sections finite and produces a large transfer behavior distinct from Rutherford scattering. Finally, we evaluate the induced shifts of the hydrogenic $1S$, $2S$, and $2P$ levels in closed form, retaining the full dependence on the interaction range. In other words, these latter results gives direct spectroscopic constraints on the product of nuclear and leptonic couplings, with muonic atoms, showing the respective sensitivity to the short distance Hořava-Lifshitz sector.

gr-qc↗
arXiv2026

Light bending and observational bounds in dyonic Kalb-Ramond gravity

We investigate the weak and strong gravitational lensing of a dyonic black hole in Kalb-Ramond gravity. After fixing the asymptotic normalization, we distinguish the effective charge governing the local null trajectories from the global angular identification associated with Lorentz-symmetry breaking. In the weak-deflection regime, we evaluate the Gaussian curvature explicitly and apply the Gauss-Bonnet theorem with a perturbed ray boundary, recovering the complete second order mass contribution. Independent calculations based on the orbit equation, the turning point integral, and Fermat's principle reproduce the same bending angle. In the strong-deflection regime, Tsukamoto's method yields closed expressions for both strong-deflection coefficients, with exact charge dependence. We derive finite-distance lens equations and the associated image positions, magnifications, flux ratios, and differential arrival times, retaining the physical winding condition. Shadow sizes inferred from observations of Sgr~A$^*$ and M87$^*$ yield conditional charge bounds, complemented by an estimate from S2 precession. We further construct combinations of strong-lensing observables that separate the effective charge from the conical parameter within the model: angular separation and relative brightness remove the explicit winding dependence, while a timing ratio incorporating an independent distance estimate isolates the conical deformation.

gr-qc↗
arXiv2026

Charged particles and fields in bumblebee gravity: Periodic orbits and superradiance

We investigate the dynamics of charged particles and fields in the background of a charged black hole in bumblebee gravity, aiming to better understand the role played by the Lorentz-violating parameter $l$. We first review the derivation of the charged bumblebee black hole and investigate its main physical and geometrical properties. We then obtain the circular and bound trajectories of charged particles and find that the innermost stable circular orbit increases with $l$. By using a numerical approach, we compute the total absorption cross section for arbitrary values of the scalar wave frequency $ω$. Since the spacetime is not asymptotically flat, we construct the total absorption cross section by taking into account the modified asymptotic momentum and flux. Analytical approximations are derived in the low- and high-frequency regimes. The low-frequency expression displays a transition velocity that separates two distinct asymptotic branches and generalizes the corresponding Reissner-Nordström result. Our numerical results for the cross section agree with the analytical approximations in their corresponding limits, and the absorption parameter space can be divided into three regions: Unbounded absorption, bounded absorption, and bounded superradiance. In particular, superradiance occurs whenever $ω<ω_{c}$, where $ω_{c}$ is the threshold for superradiance. Moreover, the superradiance amplification decreases as we increase $l$, and charged bumblebee black holes with $l < 0$ lead to greater superradiant scattering than in the Reissner-Nordström case. We also note that although increasing $l$ lowers the potential barrier, it reduces the total absorption cross section because of the geometrical factor $1/(1+l)$ present in the cross section formula.

gr-qc↗
arXiv2026

Quantum geometric signatures in neutrino dynamics around a holonomy black hole

We investigate three flavor neutrino phenomena in an effective holonomy corrected Schwarzschild geometry. We derive the weak-deflection angle through second post-Minkowskian order, obtain the semiclassical phases along radial and nonradial trajectories, and formulate the two-image flavor probability by including magnifications, Fermat phases, and wave packet overlap. The cross path phase acquires a logarithmic holonomy contribution and can retain information about the absolute neutrino mass scale. We also characterize flavor mode correlations and trajectory-flavor entanglement through the balance among path predictability, interference visibility, and I-concurrence. Neutrino-antineutrino annihilation outside an effective neutrinosphere is examined, with holonomy entering the integrated power through the radial proper volume measure. For a solar mass lens and $10\,\mathrm{MeV}$ neutrinos, the numerical results show displaced oscillation fringes and a geometry dependent redistribution among the electron, muon, and tau channels. The electron-neutrino trajectory-flavor concurrence reaches approximately $0.23$, with an entropy near $0.10$ bit, while remaining weakly sensitive to the mass ordering in the selected configuration. By contrast, the annihilation power increases monotonically, reaching an increase of approximately $39.3\%$ relative to Schwarzschild spacetime (for a particular configuration, i.e., $λ=2$ and $M/R_ν=1/3$). Furthermore, at fixed source parameters and $M/R_ν=1/3$, an assumed maximum excess of $10\%$ in the integrated annihilation power relative to Schwarzschild yields the conditional bounds $a/(2M)\leq0.2879$ and $|λ|\leq0.6358$.

gr-qc↗
arXiv2026

A general static and spherically symmetric black hole in traceless metric-affine bumblebee gravity

We construct and analyze a general exact static and spherically symmetric black hole solution in traceless metric-affine bumblebee gravity by considering a vacuum expectation value with temporal and radial components. The temporal component remains constant and the auxiliary geometry is Ricci flat, while Lorentz symmetry breaking is transferred to the physical metric through the metric-affine deformation. After introducing the areal radius and normalizing the Killing time at spatial infinity, the geometry is controlled by a single invariant deformation parameter. The resulting spacetime is asymptotically conical, possesses a regular Killing horizon, and retains the causal structure of the Schwarzschild black hole, with a curvature singularity located at the origin. Initially, we examine its thermodynamic properties, gravitational Doppler effect, and tidal forces. Lorentz symmetry breaking modifies the Hawking temperature and the magnitude of the tidal field, although the characteristic ratio between radial stretching and transverse compression is preserved. The thermodynamic-topology analysis reveals no critical point within the admissible black hole sector. Furthermore, massless scalar perturbations are also investigated through WKB and characteristic time domain solution, showing decaying profiles throughout the sampled parameter range and deformation-dependent quasinormal frequencies. Finally, light deflection and perihelion advance are used to establish stringent Solar System bounds on the invariant deformation, which, at leading order, depends on a combination of the temporal and norm sectors of the bumblebee field.

gr-qc↗
arXiv2026

Neutrino oscillations in a Kalb-Ramond black hole background

The analysis examines how neutrinos behave when their trajectories unfold around a black hole sourced by a Kalb-Ramond field, where spontaneous Lorentz symmetry breaking reshapes the surrounding geometry. Instead of following the conventional order, the study focuses first on the observable consequences: alterations in the neutrino-antineutrino annihilation energy output, shifts in the oscillation phase accumulated along the path, and distortions in flavor conversion probabilities induced by gravitational lensing. These features are then tied to the Lorentz-violating spacetime structure, which governs the propagation of the neutrinos. Numerical simulations are carried out for both two- and three-flavor descriptions, with normal and inverted mass orderings.

gr-qc↗
arXiv2026

Exploring a phantom Dirac-Born-Infeld regular black hole via particle emission, wave scattering and geodesics

We examine particle creation, evaporation, scalar wave absorption and scattering, and geodesic motion in the asymptotically flat regular black hole supported by a phantom Dirac-Born-Infeld field. For massless bosonic and fermionic fields, the Bogoliubov transformations yield thermal spectra whose Hawking temperature decreases as the regular core becomes more prominent. Energy-conserving tunneling recovers the same temperature in the low-energy limit, whereas recoil and the DBI contribution introduce nonthermal corrections and suppress particle emission. In the high-frequency regime, the enlargement of the cross section does not compensate for the reduction in temperature, resulting in a lower luminosity and a longer evaporation time. A numerical partial wave analysis shows that the total scalar absorption increases with the regular core scale, approaches the horizon area at low frequencies, and oscillates around an enlarged geometric capture limit at high frequencies. The scattering phase shifts modify the multipolar amplitudes nonuniformly and displace the interference fringes toward larger angles. Furthermore, both null and timelike trajectories experience stronger deflection as the regular core contribution increases.

gr-qc↗
arXiv2026

Black Hole Gravitational Phenomena in Higher-Order Curvature-Scalar Gravity

This work aims to explore the gravitational consequences of a recently proposed black hole solution presented in the literature [Phys. Dark Univ. 50 (2025) 102061]. We initiate our analyzes by taking into account the horizon structure, focusing on both the event and Cauchy horizons. Subsequently, we examine the quasinormal modes by considering all types of perturbations -- scalar, vector, tensor, and spinorial. To strengthen these results, we also compute the time-domain for each perturbation. Next, we turn to the study of optical properties of the black hole. In particular, we investigate null geodesics, the photon sphere and its stability, as well as the corresponding black hole shadows. Following this, we analyze gravitational lensing phenomena in two regimes: the weak-field limit, utilizing the Gauss-Bonnet theorem, and the strong deflection limit, employing Tsukamoto's approach. In addition, we confront the lensing observables with Event Horizon Telescope (EHT) data for $Sgr A^{*}$ and $M87^{*}$. Finally, constraints on the parameter $ξ$ -- which is introduced by higher-order curvature-scalar gravity, thereby differing from the Schwarzschild solution -- are estimated using Solar System measurements such as the precession of Mercury's orbit, gravitational light bending, and time delay (or Shapiro effect).

gr-qc↗
arXiv2026

How does nonmetricity shape quantum emission from rotating bumblebee black holes?

We investigate how nonmetricity affects particle creation and evaporation in rotating bumblebee black holes by comparing metric and metric-affine solutions. We normalize the stationary Killing vectors at spatial infinity before defining physical frequencies, angular velocities, and temperatures. This procedure changes the static comparison and gives the leading calibration $\ell=3X/4$. In the metric solution, $\ell$ shifts the horizons, stationary limit surfaces, and extremal boundary, increases the normalized horizon angular velocity, and suppresses the Hawking temperature. Nevertheless, the surface gravity remains uniform and the scalar wave equation separable. We derive the tunneling factors, quantum occupation numbers, radial potential, and an analytical lower bound for the axisymmetric greybody factor. In the metric-affine geometry, nonmetricity preserves the Kerr coordinate locations of the horizons and stationary limit surfaces, but changes the physical horizon area, normalized angular velocity, and meridional sector. The component $g_{rθ}$ couples angular channels, whereas the local surface gravity depends on latitude when $aX\neq0$. Consequently, the generic rotating configuration admits neither a single global Hawking temperature nor an independent channel-by-channel emission spectrum. In the static limit, under the same Stefan-Boltzmann prescription, both Lorentz-violating geometries have lower luminosities and longer lifetimes, with the metric-affine black hole radiating less and evaporating more slowly. A local slow-rotation expansion preserves this tendency but does not establish a global rotating evaporation hierarchy. Existing weak-field constraints limit fractional corrections to static quantum-emission observables to below $1.3\times10^{-11}$.

gr-qc↗
arXiv2026

A rotating black hole in a Hernquist dark matter halo: horizon geometry, thermodynamics, and quantum emission

We investigate the geometrical, thermodynamic, and quantum emission properties of a rotating black hole immersed in a Hernquist dark matter halo. Starting from a static black hole spacetime surrounded by a Hernquist distribution, we construct its rotating counterpart through the noncomplexification formulation of the Newman-Janis algorithm and analyze the modifications induced by the independent halo parameters $ρ$ and $r_s$ and the rotation parameter $a$. The horizon structure is determined from the roots of the radial function $Δ(r)$, while the stationary limit surfaces and the corresponding ergoregions are obtained from the condition $g_{tt}=0$. We show that the Hernquist contribution displaces the outer event horizon toward larger radii and modifies the size of the ergoregion, whereas rotation controls the oblateness of the horizon and the strength of frame dragging. We further derive the surface gravity, Hawking temperature, Bekenstein-Hawking entropy, and heat capacity. The quantum tunneling rate is obtained from the Hamilton-Jacobi method, leading to the corresponding occupation number and a thermal estimate of the particle creation density. Finally, we estimate the Hawking luminosity and evaporation timescales within a Stefan-Boltzmann approximation. All standard Kerr and Schwarzschild results are recovered in the appropriate limiting cases.

gr-qc↗
arXiv2026

Gravitational wave propagation in Hořava-Lifshitz gravity

We investigate the generation and propagation of gravitational waves in the leading parity-even infrared truncation of Hořava-Lifshitz gravity, characterized by the modified tensor dispersion relation $ω^{2}=k^{2}+αk^{4}$. Working in the transverse-traceless sector, we show that the higher-spatial-derivative correction preserves the conventional plus and cross polarizations and introduces neither polarization mixing, helicity splitting, nor gravitational birefringence. We construct the retarded Green function of the modified wave operator and derive the radiation-zone waveform to first order in $α$. The resulting signal exhibits a frequency-dependent amplitude renormalization together with a dispersive propagation phase that accumulates over the source-observer distance. We apply the formalism to a binary black hole system in a quasi-circular orbit and obtain the polarization waveforms for an arbitrary observation direction. We further derive the corresponding energy flux, total luminosity, and adiabatic chirp evolution. In terms of the observed gravitational wave frequency $f$, the leading corrections satisfy $Δh_{A}/h_{A}^{\mathrm{GR}}=-8π^{2}αf^{2}$ and $ΔP/P_{\mathrm{GR}} =Δ\dot{f}/\dot{f}_{\mathrm{GR}} =-16π^{2}αf^{2}$, while the accumulated generation phase has the frequency dependence of a relative third post-Newtonian contribution. By mapping the Hořava-Lifshitz coefficient to the LIGO-Virgo-KAGRA modified-dispersion parametrization, we obtain $-6.2\times10^{2}\,\mathrm{eV}^{-2} <α< 1.9\times10^{2}\,\mathrm{eV}^{-2}$ at $90\%$ credibility from the GWTC-4.0 posterior.

gr-qc↗
arXiv2026

Gauge-covariant Raychaudhuri dynamics for spin-nondegenerate Lorentz-violating congruences

We investigate the Raychaudhuri dynamics of charged spin-nondegenerate Lorentz-violating particle congruences under minimal electromagnetic coupling. The coupling is introduced through the gauge-covariant momentum $P_μ=π_μ-qA_μ$, so that the branch dispersion relation keeps its free functional form, while the electromagnetic field enters through the evolution of $P_μ$. For a generic branch $\mathcal D^{(\pm)}(P)$, the tangent $k^μ_{(\pm)}$ and the momentum Hessian $M^{μν}_{(\pm)}$ determine the covariant acceleration, $a^μ_{(\pm)}=-qM^{μν}_{(\pm)}F_{νρ}k^ρ_{(\pm)}$. As a consequence, the Raychaudhuri equation acquires the branch-dependent electromagnetic source $-q\nabla_μ\!\left(M^{μν}_{(\pm)}F_{νρ}k^ρ_{(\pm)}\right)$. We apply this construction to the $b_μ$, $H_{μν}$, and $d_{μν}$ sectors, obtaining the corresponding branch tangents, Hessians, accelerations, and focusing equations. In flat spacetime, the electromagnetic field modifies the expansion through the divergence of the effective branch force. Therefore, uniform fields may bend the trajectories, whereas local focusing requires field gradients or, in the magnetic case, a coupling to an already deformed congruence. We also develop the analogous description for semiclassical quasiparticle beams, where the band Hessian plays the role of an effective electromagnetic response tensor. For anisotropic parabolic, Dirac-like, and Weyl-type dispersions, the same geometric structure relates electromagnetic textures to beam focusing. In two-branch systems, the opposite Hessians of the branches can produce focusing in one congruence and defocusing in the other, giving a quasiparticle realization of branch-dependent birefringence.

gr-qc↗
arXiv2026

Shadows and lensing signatures of a rotating black hole in a Hernquist dark matter halo

We investigate the optical properties of a rotating black hole immersed in a Hernquist dark matter halo. The spacetime is generated from a static Hernquist black hole through the noncomplexification version of the Newman-Janis procedure, yielding a Kerr-like geometry whose halo contribution is encoded in the radial function $Δ(r)$ \cite{AraujoFilho:2026hernquist}. We derive the null geodesic equations, effective potentials, radial acceleration, and representative three-dimensional photon trajectories around the event horizon and ergoregion. Using the separability of the Hamilton-Jacobi equation, we obtain the critical impact parameters of unstable spherical photon orbits and construct the shadow contours for a distant observer. The rotation parameter mainly shifts and distorts the shadow, whereas the Hernquist halo enlarges the photon capture region and increases the apparent shadow size. Comparing the area-equivalent shadow diameter with the Event Horizon Telescope measurements of Sgr A$^\ast$ and M87$^\ast$, we constrain the dimensionless halo parameter $\hatρ=M^2ρ$. The strongest restriction comes from Sgr A$^\ast$, giving $\hatρ\sim(2.7-3.8)\times10^{-3}$ at $1σ$ and $\hatρ\sim(4.1-5.2)\times10^{-3}$ at $2σ$. We also analyze strong- and weak-field gravitational lensing. In the strong-field regime, the halo shifts the unstable photon orbit and critical impact parameter, controlling the logarithmic deflection angle and the position of relativistic images. In the weak-field regime, the halo contributes already to the leading bending angle and enhances deviations from Kerr as $ρ$ grows. From the Einstein ring of ESO325-G004, we further obtain $0\leq\hatρ\lesssim0.00939$ at $1σ$ and $0\leq\hatρ\lesssim0.01963$ at $2σ$.

gr-qc↗
arXiv2026

Neutrino dynamics in a non-commutative spacetime

We investigate neutrino dynamics in a spherically symmetric non-commutative black hole spacetime generated by a Lorentzian smeared matter distribution. We work with the leading-order Lorentzian-smeared metric, in which the non-commutative parameter $Θ$ enters explicitly through a $\sqrtΘ$ correction to the Schwarzschild geometry. Within this background, we analyze the neutrino-antineutrino annihilation process, the oscillation phase acquired by massive neutrinos along geodesics, and the flavor-transition probability in the presence of weak gravitational lensing. We show that non-commutativity monotonically suppresses the relativistic energy deposition rate relative to the Schwarzschild case, while the Newtonian behavior is recovered at large radii. In the oscillation sector, the weak-field phase receives explicit $\sqrtΘ$ corrections, which modify the distance of closest approach and reduce the accumulated phase with respect to the Schwarzschild limit. For lensed trajectories, the macroscopic transition probabilities remain close to the Schwarzschild prediction; however, a high-precision residual analysis reveals a coherent, non-vanishing oscillatory correction driven by the non-commutative geometry. The dependence on the absolute neutrino mass scale enters only through multi path interference terms and remains subdominant compared with the $Θ$-induced phase shift. The full three-flavor residual analysis confirms that the same non-commutative oscillatory pattern persists beyond the effective two-flavor approximation.

gr-qc↗