We study the loss of hyperbolicity of perturbation equations for black hole solutions of scalar Gauss-Bonnet gravity. We consider a class of coupling functions allowing for static black hole solutions with arbitrary small masses. For masses below a minimum value, such solutions become unphysical, because the perturbation equations become elliptic; this arguably corresponds to the loss of validity of the effective field theory. We analyse the dependence of this minimum mass on the parameters of the theory, finding that with an appropriate choice of the coupling function, such mass can be chosen arbitrarily small. However, this does not correspond to larger deviations from general relativity, since observable quantities like the black hole scalar charge are bounded by above.

We study the exclusive electroproduction process $e+p\to e'+p'+X$, with $X$ a single-particle final state, in the forward-proton kinematics relevant for the future Electron-Ion Collider (EIC). We develop a unified $2\to 3$ framework that provides the full event kinematics and incorporates pseudoscalar and vector mesons, as well as axion-like particles and vector mediators such as dark photons. It is based on phenomenological amplitudes constrained by existing photo- and electroproduction data and constructed to admit systematic refinement as new measurements become available. To benchmark the framework, we compare its predictions to flux-factorized descriptions based on the equivalent-photon approximation, demonstrating close agreement for total rates and selected single-differential distributions in the near-real regime, while highlighting the role of finite-$Q^{2}$ correlations for multi-differential observables at larger photon virtualities. As a case study, we perform a detailed kinematic analysis of the missing-proton-energy signature, illustrating how the full $2\to 3$ treatment informs forward-proton acceptance and signal selection in realistic EIC configurations.

Coupling tailored electromagnetic fluctuations to materials provides a resource for controlling correlated quantum matter. By structuring the frequency, spatial, and modal distribution of fluctuations through a new generation of cavity quantum materials, vacuum and thermal spectra can shift phase boundaries and stabilize or suppress orders. This review organizes the field around a fluctuation-focused perspective, surveying a practical design toolbox and recent milestones, and outlining theory-experiment challenges in realistic, multimode, beyond-long-wavelength regimes. We highlight photonic observables and map opportunities for equilibrium and driven control across superconducting, magnetic, moire, and topological platforms.

Coronagraph designs which use photonic integrated circuits have the highest theoretical throughput for off-axis signals, and therefore the highest potential exoplanet yield for future high-contrast direct imaging campaigns. Using the rejected starlight, the photonic integrated circuit may also provide simultaneous wavefront sensing, allowing for the correction of non-common-path aberrations. This work considers how a photonic circuit should be configured to maximize its sensitivity to phase aberrations. Two cases are considered: in the first, the photonic circuit is coupled directly to an electric field in a piecewise manner, while in the second, the circuit is coupled to the field via an optical mode sorter. In either case, this work constructs a unitary matrix which can be applied by a photonic circuit to produce maximum sensitivity.

How can one certify that an unknown quantum state possesses magic without resorting to full state tomography? We address this question by introducing two thermodynamic witnesses that rely solely on energy and heat measurements. First, we define the stabilizer ground-state energy as the lowest energy achievable by any stabilizer state, and the stabilizer gap as the separation between this value and the true ground-state energy. Any state whose energy lies below the stabilizer ground-state energy is therefore necessarily nonstabilizer. This leads to a direct witness of magic using only average-energy measurements. To overcome the limitations when direct energy measurements are inconclusive, we further develop a nonlinear witness based on heat exchange with a thermal ancilla. Specifically, we derive fundamental bounds on heat that are satisfied by all stabilizer states; therefore, their violation certifies the presence of magic. We demonstrate the effectiveness of our approach through several examples, ranging from few-body systems where heat exchange reveals nonstabilizerness even when energy measurements alone fail, to the transverse-field Ising chain, where the stabilizer gap becomes maximal at the quantum critical point.

We provide a complete first-principles based discussion of quantum tunnelling out of initial states carrying a conserved Noether charge. Our main result is a simple, unambiguous Euclidean-time prescription for the calculation of tunnelling rates out of such states. By relying on a combination of the direct approach and the steadyon framework for the evaluation of real-time path integrals, our derivation offers full transparency of its underlying assumptions, and is independent of any ad-hoc generalisations. This strategy also offers a simple explanation for the emergence of complex saddle points for such systems, justifying techniques postulated by earlier works. Our analysis furthermore offers the first results for initial states with both a conserved Noether charge and a non-trivial energy. We first illustrate the main conceptual points of our analysis for the simple example of a point particle in two spatial dimensions carrying a conserved angular momentum. Then, we generalise our results to the case of multiple dimensions and an arbitrary conserved Noether charge, providing an easy-to-implement prescription for the calculation of the tunnelling rate. We furthermore apply our results to the example of a complex scalar field subject to a global U(1)-symmetry with associated charge. These results provide a reliable foundation for the calculation of tunnelling rates in applications in finite-density and charge-asymmetric systems.

We present a new mechanism for addressing the cosmological constant problem based on global constraints arising from a lapse function in a higher-dimensional gravitational theory. Inspired by Horava-Lifshitz gravity, we consider a 5d spacetime with anisotropic scaling along a compact extra dimension, while preserving Lorentz invariance in four dimensions. In the deep infrared limit, variation with respect to the lapse generates a global constraint on the 4d geometry, closely analogous to that of vacuum energy sequestering. Although the resulting effective gravitational equations differ from standard sequestering, radiative contributions to the Standard Model vacuum energy are nevertheless cancelled at all orders.

We uncover a combinatorial structure governing the differential equations satisfied by wavefunction coefficients of scalar fields with generic masses in de Sitter space. Using an integral representation of the massive mode functions, we express the Feynman integrals underlying cosmological correlators as twisted integrals of rational functions. In this formulation, the integrals belong to a finite set of master integrals obeying a first-order system of differential equations, which can be derived efficiently in the time-integral representation. We show that these equations admit a simple graphical description in terms of graph tubings, which encode the couplings among basis functions and the evolution of singularities. This structure provides an efficient algorithm to derive the differential equations, and a boundary-centric perspective on massive cosmological correlators in which their analytic structure emerges from underlying combinatorial data. As an illustration, we solve the system in the limits of small and large masses.

Motivated UV frameworks generically predict the existence of multiple axion fields. Their interplay gives rise to novel collective phenomena - including level crossings and the formation of string bundles - which modify the predicted mass and couplings of the QCD axion as a solution to both the strong CP problem and the observed dark matter abundance. Among these effects, the domain wall number is determined by the full anomaly structure of the theory: in the single axion case, the absence of long-lived domain walls imposes $E/N \geq 8/3$ as a theoretical bound on the QCD axion photon coupling, assuming the global structure of the Standard Model gauge group is minimal. We show that this bound can be relaxed in the multi-axion framework. Combined with the fact that the QCD axion can become a subdominant dark matter component, this might render multi-axion scenarios experimentally challenging. Nevertheless, a careful analysis of the parameter space reveals that in most regions where the QCD axion evades detection, an axion-like particle remains visible to next-generation experiments. When all signals fall below future projections, we identify the most promising regions of parameter space to probe in an illustrative two-axion setup.

Conformal supergravity provides an effective off-shell formalism to study higher derivative actions. We show that the $D=4$, $\mathcal{N}=2$ theory admits equivariantly closed forms. These may be used to compute closed-form expressions for supersymmetric observables in a general class of supergravity theories with higher derivative couplings, without any need to solve equations of motion. We discuss applications to holography, presenting results for on-shell actions that are conjecturally valid to all orders in the perturbative $1/N$ expansion.

Achieving sufficiently low residual excited-state populations remains a key challenge in superconducting quantum circuits, particularly for protocols operating close to noise limits or requiring repeated qubit initialization. Existing protocols primarily address this challenge through sophisticated control, engineered dissipation, or feedback mechanisms. Here, we demonstrate an alternative approach in which a superconducting qubit is reset using a physically distinct, intrinsically colder phononic bath. Specifically, we interface a transmon with a high-overtone bulk acoustic resonator (HBAR), enabling cooling of the qubit into GHz-frequency modes. Using this approach, we achieve a residual excited-state population of the qubit below $10^{-4}$, representing an improvement of one to two orders of magnitude compared to existing reset schemes. These results highlight the potential of phononic baths as a resource for high-fidelity qubit initialization in superconducting circuits.

The Love numbers of a gravitating body are response coefficients encoding its tidal deformability. In compact binary systems, they appear in the gravitational waveform during the inspiral phase and will be measurable by upcoming gravitational-wave observatories. This review provides a comprehensive and pedagogical account of the theoretical foundations of Love numbers and surveys the most recent advances in the study of tidal effects in compact objects, with particular emphasis on black holes and neutron stars. We begin with a gentle introduction to tidal effects in Newtonian gravity, leading into a discussion of how to robustly define tidal responses in general relativity using the effective field theory and post-Newtonian frameworks. After an overview of the perturbation theory of black holes and neutron stars, we review the computation of Love numbers and dissipative response coefficients in a wide range of settings, including the static, dynamical, and nonlinear tidal responses of Kerr black holes and neutron stars in four-dimensional general relativity. We further discuss the extension of these results to charged black holes and a number of "new physics" scenarios, including higher-dimensional black holes and other black objects, (anti-) de Sitter black holes, supergravity black holes, and theories beyond general relativity. Finally we provide an overview of the tantalizing zoo of hidden symmetries of general relativity that have been uncovered in the attempt to explain the famous vanishing of static black hole Love numbers.

The Milky Way is surrounded by a hot diffuse circumgalactic medium (CGM) with temperatures of millions of degrees. Recent X-ray observations with the eROSITA satellite discovered a significant temperature asymmetry of this hot CGM, with the southern hemisphere being on average hotter than the northern one by a relative difference of $Δ T/T \approx 12\%$, where $T$ is averaged over the entire CGM. In this Letter, we investigate whether the passage of the Magellanic Clouds can be responsible for this asymmetry by means of a hydrodynamical/N-body simulation. In the simulation, the Magellanic Clouds induce a relative motion of the Milky Way's disc of up to 40 km/s. This motion leads to compression of the CGM gas in the southern hemisphere, resulting in an overall temperature increase in that region. We estimate a south-north temperature difference of $Δ T/T \approx 13-20\%$, consistent with the observations. We find that this temperature asymmetry is a recent phenomenon that began ~100 Myr ago.

We propose a novel cosmological scenario in which baryonic neutron stars could plausibly form in the early universe. If baryogenesis initially produces an excessively-large baryon asymmetry, $Y_B \gg 10^{-10},$ the baryonic mass inside the horizon can exceed the minimum neutron star mass before big bang nucleosynthesis (BBN). While this large asymmetry is present, non-relativistic baryons can dominate the universe and enhanced density perturbations on small scales can gravitationally collapse Hubble patches shortly after horizon re-entry. For some initial perturbations, just below the threshold for black hole formation, this collapse will be arrested only by nuclear pressure, possibly resulting in neutron star formation. Afterwards, there must be a large entropy injection to restore the observed baryon asymmetry, $Y_B \sim 10^{-10}$, and preserve the successful predictions of standard BBN. Unlike neutron stars that form from stellar collapse, primordial neutron stars can, in principle, be as light as $\sim 0.1 M_\odot$, limited only by the nuclear equation of state.

Topological stabilizer codes, such as the toric and surface codes, are leading candidates for fault-tolerant quantum computation. While their decodability under stochastic noise has been extensively studied, the effects of coherent errors, which involve quantum interference, remain less explored. In this work, we study the decodability of toric codes on honeycomb and square lattices subject to $X$- and $Z$-type coherent errors generated by the $X$- and $Z$-rotations on each qubit. We establish a duality between these decoding problems and 1+1D monitored dynamics of non-interacting Majorana fermions. This duality shows that the Altland-Zirnbauer symmetry class of the dual Majorana dynamics governs the universal structure of the decodability phase diagram. We show that the honeycomb-lattice toric code (hTC) with $X$-type error is dual to class-DIII dynamics, while the hTC with $Z$-type error and the square-lattice toric code (sTC) with both error types are dual to class-D dynamics. The key distinction arises from time-reversal symmetry. In class DIII, the generic transition out of the decodable phase is dual to a measurement-induced transition between dynamical phases with area-law and logarithmic entanglement scaling. In contrast, in class D, the generic decodability transition corresponds to a transition between two topologically distinct area-law phases. To explore these transitions in microscopic models, we consider hTC and sTC with $X$-type errors as representatives and introduce a minimal two-parameter coherent error model with spatially varying rotation angles. Using analytical and numerical methods, we map out the decodability phase diagrams and characterize the universal behavior of the transitions. We find that the decodability of sTC is more vulnerable to spatially varying coherent errors than uniform ones.

We consider a scenario in which the Milky Way (MW) and M31 have had a previous pericentric passage, and investigate its compatibility with self-interacting dark matter (SIDM). Using initial conditions sampled from Local Group (LG) analogues in the IllustrisTNG simulation, we perform controlled re-simulations of the MW-M31 orbit, evolving the system under both standard cold dark matter (CDM) and various SIDM cross-sections. We find that the deep baryonic potential of the MW preconditions the halo's thermal structure, establishing an initial negative temperature gradient. This drives SIDM halos to bypass the standard core-formation phase and enter immediate core-collapse, resulting in monotonically increasing central densities. In full orbital simulations, the compact stellar component (disk/bulge) of the MW analog remains robust against tidal disruption for pericenter distances as close as $r_{\rm peri}\lesssim20$ kpc during an encounter at cosmic time $\sim8$ Gyr. The diffuse stellar halo is comparatively more susceptible, facing disruption for $r_{\rm peri}\lesssim100$ kpc. Our results demonstrate a dichotomy in structural evolution: the compact disk/bulge is sensitive to intrinsic SIDM thermodynamics but dynamically robust against the pericenter encounter, whereas the diffuse stellar halo is largely independent of the specific SIDM model but more vulnerable to orbital tidal disruptions.

This note was prepared as a handout for the MAT401 course ``Polynomial equations and fields", taught at the University of Toronto in Spring 2026. It presents a proof of a necessary condition for the solvability of algebraic equations by radicals, based on Galois theory. We begin with a brief overview of the relevant basic results from Galois theory, as covered in MAT401, and use -- without proof -- several standard (and relatively simple) results from the course textbook [1]. The sufficient condition for solvability by radicals, which is based on linear algebra, we will present in the next handout.

This note presents Galois theory for finite fields. It was written as a handout for the MAT401 course ``Polynomial equations and fields'' taught at the University of Toronto in Spring 2026. We use without proofs some basic properties of finite fields and of finite field extensions which we already covered in class. Firstly, we describe an extension $K\subset F$ of a finite field $K$ of a given degree $n$. We show that the set of all intermediate fields for this extension is in one-to-one correspondence with the set of all divisors $k$ of the degree $n$. Then we describe the Galois group of this extension which is the cyclic group of order $n$. The set of subgroups of this group also is in one-to-one correspondence with the set of all divisors $k$ of the degree $n$. It allows us to prove the Galois correspondence for that extension. In the last section, we state basic theorems of Galois theory for arbitrary fields which will be proven later in the course.

We firstly show that due to their resplendency ordered henselian valued fields admit relative field quantifier elimination in the Denef--Pas language expanded by linear orders in the field and residue field sort. Secondly, we deduce from a dimensionality reduction theorem that any set definable over an ordered henselian valued field is a Borel set with respect to the order topology. Our results are contextualised within Shelah's classification conjecture of NIP fields and its connections to the study of definable henselian valuations and the Fundamental Theorem of Statistical Learning.

The rapid rise of artificial intelligence, and in-memory computing has reinvigorated research on scalable, energy-efficient, and reconfigurable photonic hardware. Non-volatile phase-change materials (PCMs) are attractive, as they offer large refractive index contrast, wavelength-scale footprints, and zero static power consumption. However, current PCM-based electrically controlled photonic devices are plagued by high insertion loss and low endurance. One prevalent hypothesis for these material limitations come from electromagnetic scattering in the interface and large programming volumes, respectively. Here, we validate this hypothesis by showing that nano-structuring of PCM minimizes optical loss and enhances the endurance. By tapering both ends of a wide bandgap PCM Sb2Se3 segment on a silicon waveguide, we suppressed the insertion loss by ~94% (resulting in a loss of ~0.1 dB per π phase shift). Through combining tapering and segmentation, we achieved high optical modulation amplitude (~70%), low loss (~0.5 dB per π phase shift), low-voltage (< 5V) actuation, and record high endurance greater than 100 million cycles. This work showcases the substantial advantage of nanopatterning PCMs to attain low loss and high cyclability.

Superradiant phase transitions (SRPTs), characterized by photon condensation and macroscopic matter polarization, are forbidden in equilibrium for homogeneous fields by no-go theorems. Here, we show that Floquet driving can circumvent this constraint in a Landau polariton system consisting of a two-dimensional electron gas coupled to a terahertz cavity in a DC magnetic field. An off-resonant AC magnetic field modulates the cyclotron frequency and light--matter coupling strength while leaving the diamagnetic term unchanged, generating an additional DC coupling contribution. This drives the system across a critical threshold into a superradiant phase, characterized by photon condensation and Landau-level polarization in the ground state of the Floquet Hamiltonian. This quasiequilibrium approach offers a route to SRPTs distinct from driven-dissipative schemes.

Net primary productivity (NPP) forms the basis of biological carbon pump, but its estimates in high-latitude regions remain highly uncertain despite its disproportional importance for the global carbon sink. Optical satellites are limited by cloud cover, low irradiance, and shallow light penetration, with uncertainties further exacerbated by the lack of in situ validations and regional model tuning for NPP measurements. This study compared two satellite-based models, a global (VGPM) and a regionally tuned (BIO) NPP model, with a time series of in situ NPP. Using a high-frequency, depth-resolved moored profiler in the subpolar Northwest Atlantic (56°N) in 2016, in situ NPP was estimated by daily bio-optical profiles and prior measurement of photosynthesis-irradiance (P-I) parameters. Our findings indicated that satellite-derived estimates of depth-integrated NPP were overestimated by a factor of 2.5 to 4. However, the reasons for the discrepancies varied between the VGPM and BIO model. VGPM used global photosynthetic parameters with a simplified depth assumption, leading to an unrealistic vertical structure for depth-integrated NPP, despite its surface values were lower than in situ estimates. A major phytoplankton bloom in June-July was missed by VGPM, likely due to the use of non-regionally calibrated OCI Chl-a, which led to an underestimation of biomass. In contrast, the BIO model used regionally tuned POLY4 Chl-a products, and the differences in the assignment of P-I parameters accounted for the remaining discrepancies. This study showed the possibility to reach good agreement between satellite and in situ NPPs if the challenge of P-I assignment can be overcome. We recommend further studies to investigate discrepancies of NPP estimates in high-latitude regions, focusing on data sources and model choices, as well as improving regional model calibration to enhance NPP accuracy.

Automated black-box testing of APIs typically relies on interface specifications that define available operations and data schemas, but offer limited or no behavioural semantics. This semantic gap amplifies the test-oracle problem and limits the generation of effective, stateful call sequences. We introduce IcePick, a framework that achieves systematic state-space coverage for API testing by leveraging model checking. IcePick uses TLA+ to formally model API state evolution, employs the TLC model checker to exhaustively explore reachable states, and generates test sequences that provably cover the behavioural model. To mitigate state-space explosion and improve sequence extraction, we introduce a coverage-guided breadth-first traversal of the TLC state-space graph. To address oracle limitations beyond HTTP status codes, we propose Glacier, a first-order logic contract language that enriches API specifications with executable semantic contracts, enabling automated behavioural verification during test execution. We evaluate IcePick on EvoMaster Benchmark systems, demonstrating that model-checking-guided exploration achieves complete state coverage and reveals faults in multi-operation interactions. We also analyse scalability to characterise practical limits and applicability requirements. Overall, IcePick provides reproducible test suites with strong coverage guarantees for critical API-based systems.

Muon-electron universality is tested in $B^0 \to K^{*0} \ \ell^+ \ell^-$ decays, in the dilepton-invariant-mass region above the $ψ(2S)$ resonance. The analysis uses beauty mesons produced in proton-proton collisions recorded by the LHCb detector at center-of-mass energies of 7, 8, and 13 $\text{TeV}$, corresponding to an integrated luminosity of 9 $\text{fb}^{-1}$. The ratio of branching fractions between the muon and electron channels, $R_{K^{*0}}$, is measured to be $1.08\,^{+0.14}_{-0.12}\text{(stat)} \ \pm 0.07\text{(syst)}$ for a dilepton-invariant-mass squared above 14.0 $\text{GeV}^{2}/\text{c}^{4}$, consistent with the Standard Model prediction. This result represents the most precise measurement of $R_{K^{*0}}$ in this region and the first such measurement performed at a hadron collider.

We study differentially private data release, where a database is accessed through successive, possibly adaptive queries and mechanisms. Existing composition theorems and privacy filters combine worst case per-round privacy parameters, leaving room for more refined accounting based on realised leakage, which we term realisation-level accounting. We propose a realisation-level filtering approach to determine stopping times for data releases, and design one such filter. Despite technical challenges arising from conditioning on realisations and stopping time, we prove that the filter guarantees $(ε, δ)$-differential privacy, with $ε$ and $δ$ chosen by the data handler. Through numerical evidence, we demonstrate that realisation-level filtering provides a path to better utility beyond mechanism-level methods. Furthermore, our proposed filter applies to arbitrary mechanisms, including those that are badly behaved under Rényi differential privacy.

We investigate the thermodynamics of the Schwarzschild-AdS black hole in the framework of \(κ\)-deformed non-commutative geometry by constructing an effective \(κ\)-deformed Schwarzschild-AdS metric from the \(κ\)-deformed Newtonian potential. In the extended phase space, we derive a modified first law and the corresponding Smarr relation by treating the \(κ\)-deformation parameter as an additional thermodynamic variable and identifying its conjugate potential. Our analysis shows that \(κ\)-deformation induces critical behaviour and phase transitions in an uncharged Schwarzschild-AdS black hole, with a critical ratio \(P_c v_c/T_c \simeq 0.370\) that is independent of the deformation parameter and close to the Van der Waals value. Interestingly, the \(G - T\) curve exhibits a peculiar double-loop structure, deviating from the standard swallow-tail behaviour associated with a first-order phase transition.

We study the large-scale behavior of solutions to the Allen-Cahn reaction-diffusion equation with Gaussian initial data. We consider the case of short-range dependence in the associated supercritical regime with spatial dimension $d \ge 3$. Under diffusive rescaling, the non-linearity formally vanishes on large scales in this case. Accordingly, we prove a central limit theorem for the rescaled solution, more precisely, that it converges to the solution of the heat equation started from a white noise. These initial conditions for the limit depend non-trivially both on the source of randomness and on the non-linearity. Our proof uses estimates obtained by a combination of comparison principles and Malliavin calculus, initiated by Castillo and Dunlap in arXiv:2509.06260 in the critical case. However, the result there is not a fluctuation result but rather an $L^2_\mathbf{P}$ comparison to a McKean-Vlasov problem with Gaussian solutions. Hence the mechanism behind the Gaussianity of the limit differs, and the proof requires new ideas that should be further applicable to other supercritical problems.

We analyze a reputational bargaining game in which a central player negotiates simultaneously with two peripheral players. Each player is either rational or a commitment type who never concedes and insists on a fixed share, and concessions are publicly observed. The central player's type is global, so actions in one dispute update beliefs in the other and generate reputational spillovers. The game admits a unique equilibrium, enabling a sharp comparison with the bilateral benchmark of Abreu and Gul (2000). Spillovers are payoff-relevant if and only if a peripheral is uniquely the most reputable player initially. In that case, spillovers overturn the bilateral prediction that toughness pays: the central player is never strictly better off and can be strictly worse off; the strongest peripheral loses; and the weakest peripheral can benefit, especially when the center's higher-stakes dispute is with the other peripheral.

The inverse energy cascade in turbulent Taylor-Couette flow is studied in line with the results of the large eddy simulation. The simulation results show that the inverse energy cascade first occurs within the core region of the flow channel of the Taylor-Couette flow at higher Reynolds number. It is uncovered that this phenomenon is induced by the pulsed zero shear stress resulting from the singularities of the Navier-Stokes equation. In the core area between the two cylinders, the shear stress is nearly zero at higher Reynolds number. The turbulence generated there has high turbulent energy due to discontinuity of the tangential velocity. Since the energy transfer between the fluid layers is inhibited due to the low shear stress, the turbulent energy cannot be transferred along the radial direction, and small-scale vortices with high turbulent energy are produced. These small-scale vortices are located with the large-scale vortices and cannot be dissipated owing to low shear stress. A peak in the energy spectrum at middle frequency (or wave number) is formed due to the concentration of the small-scale vortices. As the number of the singular points of the Navier-Stokes equation increases with the increasing Reynolds number, the region with zero shear stress expands along the radial direction, intensifying nonlinear instability and energy accumulation. This, in turn, leads to more prominent peaks in the energy spectrum, resulting in a more pronounced inverse energy cascade.

This study aims to enhance the bidirectional authentication capability of ML-KEM (Module-Lattice-Based Key-Encapsulation Mechanism) by proposing the post-quantum cryptography-based (PQC-based) bidirectional authentication key exchange protocol. Furthermore, it introduces dual-usage certificates combining PQC-based DSA (Digital Signature Algorithm) and PQC-based KEM, which include composite schemes, catalyst schemes, and chameleon schemes. These dual-usage certificates utilize the PQC-based DSA public key and PQC-based KEM public key within the certificate to meet the requirements for bidirectional authentication and encryption, enabling the negotiation of a shared secret key. During the experimental phase, the study validates and compares key exchange message lengths and computation times under different certificate configurations. Finally, instant messaging is presented as an industry application to demonstrate the practical implementation of the proposed protocol.