What does ``Big Bang" mean? What was the actual origin of these two words? There are many aspects hidden under this name, which are seldom explained. They are discussed here. To frame the analysis, help will be sought from the highly authoritative voices of two exceptional writers: William Shakespeare and Umberto Eco. Both have explored the tension existing between words and the realities they name. And this includes names given to outstanding theorems and spectacular discoveries, too. Stigler's law of eponymy is recalled in this context. These points will be at the heart of the quest here, concerning the concept of ``Big Bang", which only a few people know what it means, actually. Fred Hoyle was the first to pronounce these words, in a BBC radio program, with a meaning that was later called inflation. But listeners were left with the image he was trying to destroy: the explosion of Lemaître's primeval atom (an absolutely wrong concept). Hoyle's Steady State will be carefully compared with inflation cosmology. They are quite different, and yet, in both cases, the possibility of creating matter/energy out of expanding space is rooted in the same fundamental principles: those of General Relativity. As is also, the possibility of having a universe with zero total energy, anticipated by R.C. Tolman, in 1934 already. It will be shown, how to obtain accelerated expansion from negative pressure; how to reconcile energy conservation with matter creation in an expanding universe; and a curious relation between de Sitter spacetime and Steady State cosmology. Concerning the naming issue, it will be remarked that, today, the same label ``Big Bang" is used in very different contexts: (a) the Big Bang Singularity; (b) as the equivalent of cosmic inflation; (c) speaking of the Big Bang cosmological model; (d) to name a very popular TV program; and more.

We provide a lower bound for the embedding capacity of higher-dimensional symplectic ellipsoids, formulated in terms of the Lagrangian capacity of ellipsoids. Our approach relies on examining the Borman--Sheridan class of a Weinstein neighborhood of a suitable monotone Lagrangian torus, using Tonkonog's string topology-based computation of the gravitational descendants of the torus.

We present a multifractal formalism for measures on infinite dimensional metric spaces, in terms of scales instead of dimensions in the classical multifractal analysis. We prove a multifractal formalism with a suitable scaling, called order, for the Wiener measure, which is the probability law of the standard Brownian motion. We also prove the fundamental Frostman Lemma on a large class of Polish spaces, for which the increasing sets lemma holds.

GW200105 is the first gravitational wave detection to show signs of eccentricity, it also is a neutron star - blackhole binary. This raises the need for waveforms that incorporate tidal effects on quasi-elliptic orbits. We tackle the problem of finite size effects within the post-Newtonian framework, including the mass-type quadrupole and octupole, as well as the current-type quadrupole deformations in the adiabatic approximation. The computations are performed at the second-and-a-half relative post-Newtonian order. We first derive the quasi-Keplerian parametrization of the conservative motion; we then express the radial separation and phase with their time derivatives in terms of the orbital frequency, the time eccentricity and the eccentric anomaly. To obtain these as functions of time, we invert the generalized Kepler equation while also discussing the convergence of eccentricity expanded results. We provide those results to the fourteenth order in eccentricity. Finally, we exploit the already known radiation reaction term of the acceleration in order to derive the secular and oscillatory evolutions of the orbital elements. The companion paper contains the derivation of the radiated fluxes and the amplitude modes of the strain. All relevant results are provided in an ancillary file.

The randomized linear combination of unitaries (LCU) method with many applications to early fault-tolerant quantum computing algorithms has been proposed. This quantum algorithm computes the same expectation values as the original, fully coherent LCU algorithm using a shallower quantum circuit with a single ancilla qubit, at the cost of a quadratically larger sampling overhead. In this work, we propose a quantum algorithm intermediate between the original and randomized LCU that manages the trade-off between the sampling overhead and circuit complexity. Our algorithm divides the set of unitary operators into several groups and then randomly samples LCU circuits from these groups to evaluate the target expectation value. Notably, we reveal that across all grouping strategies, the mechanism of the sampling overhead reduction can be solely characterized by a metric we call the reduction factor. Moreover, we analytically prove an underlying monotonicity of the reduction factor in the group size: larger group sizes entail smaller sampling overhead. Finally, our framework enables a more flexible algorithmic design by systematically yielding intermediate implementations of LCU-based algorithms; we provide intermediate implementations of non-Hermitian dynamics simulation, ground-state property estimation, and quantum error detection. Besides, we demonstrate this principle by deriving intermediate trade-off scaling in sample complexity and ancillary space for quantum linear system solver.

The process of evolution by natural selection leads to phenotypes of increasing fitness. For cellular chemical reaction networks, this means optimising a variety of fitness functions such as robustness, precision, or sensitivity to external stimuli. We argue that these diverse goals can be achieved by a versatile, generic mechanism: coupling chemical reaction networks to reservoirs that are strongly out of equilibrium. Using theory and numerics we show that this mechanism of optimization comes at the price of significant heat dissipation. We compute the heat flux caused by kinetic proofreading in {\it Escherichia coli} and show that it constitutes a significant fraction of the total heat flux experimentally measured in this model organism. We then demonstrate that the degree of optimality achievable saturates, and that Nature appears to operate near saturation despite high energetic costs. We conclude that `life is hot' largely because of the need for a versatile mechanism to optimise a variety of fitness functions.

We construct entanglement surfaces for rotating cylindrical black holes in a double holographic setup, extending previous results to the case of stationary backgrounds. We analyze both the 5d braneworld construction as well as the embedding in 10d type IIB string theory. We couple the rotating cylindrical black hole to a non-gravitating bath, and study island and Hartman-Maldacena surfaces. Properties of island surfaces are characterized by three regimes, bounded by two critical parameters. In addition to the critical value known for the static case, we find that a new one emerges, related to the extremal limit of the rotating black hole. This behaviour is present both for the bottom-up as well as the top-down models, for which we find qualitative agreement.

We present a model featuring sub-GeV vector dark matter by augmenting the Standard Model with a new non-Abelian dark $SU(2)_D$, spontaneously broken by the vacuum expectation values of a scalar doublet and a triplet. Interactions between the dark and visible sectors arise through a dimension-5 non-Abelian \textit{avatar} of kinetic mixing portal, inducing effective magnetic dipole couplings of the dark matter, with the photon and Z boson. The resulting spectrum of the dark gauge bosons naturally exhibits an inverse mass hierarchy between the dark matter and the $Z^\prime$, leading to interesting phenomenology at fixed-target experiments such as LDMX through dark off-shell bremsstrahlung, dark Higgs-strahlung, invisible vector meson decay, and visible decays. We compute the thermal relic abundance across sub-GeV dark matter masses, with regions of freeze-out proceeding via forbidden annihilation into dark sector states or direct annihilation into Standard Model states. Bounds from a broad set of laboratory probes, along with cosmological and astrophysical observations, are incorporated in our analysis. Among them, the most restrictive bounds originate from direct detection experiments, Big Bang Nucleosynthesis, collider searches, and the CMB. Our results demonstrate that a sizeable region of parameter space remains consistent with the observed relic abundance and current experimental constraints, and that fixed-target searches should be considered in tandem with direct detection and cosmological probes for an effective and comprehensive search strategy, especially in the off-shell regime.

We provide a complete characterization of all finite-dimensional quantum systems that saturate the Margolus-Levitin quantum speed limit at arbitrary Uhlmann-Jozsa fidelity. Employing a purification-based approach, we prove that mixed-state saturation occurs precisely when three structural criteria are fulfilled: the state's support is confined to the sum of two energy eigenspaces (the ground level and a single excited level); each eigenvector of the state with nonzero weight is a fixed superposition of one ground- and one excited-state energy eigenvector (determined by the minimizer of the objective function identified by Giovannetti et al.) and all such eigenvectors evolve in mutually orthogonal subspaces. These requirements impose a strict rank bound, ruling out saturation by any faithful state. For quantum bits, we derive a purity-resolved and tight Margolus-Levitin bound that reduces to the pure-state result in the limit of unit purity. Through a time-reversal argument, we further extend the dual Margolus-Levitin quantum speed limit to mixed states and establish the corresponding saturation conditions.

We explore new infrared dualities in $(2+1)$-dimensional quantum field theories involving Majorana fermions. Building on the recently proposed operator-deformation approach to bosonization dualities, we incorporate the bosonization of neutral fermions into the established web of Dirac dualities. Starting from the decomposition of a Dirac fermion into two Majorana components, we derive a hierarchy of dual bosonic descriptions based on orthogonal Chern-Simons-matter theories. This construction leads to novel boson-boson dualities, including an explicit correspondence between an $SO(N)$ theory with two real scalars and a $U(1)$ theory with one complex scalar, and their generalizations to arbitrary numbers of Majorana flavours. We analyze boundary conditions preserving chirality and demonstrate their role in matching gravitational anomalies and chiral central charges. Additional deformations yield orthogonal analogues of Ising-Gross-Neveu and pairing transitions. The resulting framework provides a unified Majorana-based extension of the three-dimensional duality web and clarifies the infrared structure of non-Abelian bosonization.

A search is presented for the two-body charmed baryonic decays $\overline{B}_{(s)}^{0}\toΛ_{c}^{+}\overlineΛ_{c}{}^{-}$, using a data sample collected by the LHCb experiment during 2011--2012 and 2015--2018 corresponding to an integrated luminosity of $9\,\mathrm{fb}^{-1}$. The first observation of the $\overline{B}_{s}^{0}\toΛ_{c}^{+}\overlineΛ_{c}{}^{-}$ decay is reported with $6.2σ$ significance, along with $4.3σ$ evidence for the $\overline{B}^{0}\toΛ_{c}^{+}\overlineΛ_{c}{}^{-}$ decay. The branching fractions are measured to be $\mathcal{B}{}(\overline{B}^{0}\toΛ_{c}^{+}\overlineΛ_{c}{}^{-}) = (1.01^{+0.27}_{-0.28} \pm 0.08 \pm 0.15) \times 10^{-5}$ and $\mathcal{B}{}(\overline{B}_{s}^{0}\toΛ_{c}^{+}\overlineΛ_{c}{}^{-}) = (5.0 \pm 1.3 \pm 0.5 \pm 0.8) \times 10^{-5}$, where the first uncertainty is statistical, the second systematic, and the third due to external inputs. These results provide novel experimental inputs for the theoretical framework describing two-body baryonic decays of $B$ mesons via $W$-emission and $W$-exchange mechanisms.

Self-interacting dark matter (SIDM) is an intriguing alternative to the standard cold dark matter (CDM) paradigm, which predicts that dark matter halos typically have large, isothermal cores. Numerical simulations have shown that dynamical friction ceases to operate in cores of (roughly) constant density, a phenomenon known as core stalling. In addition, such cores often are unstable to a dipole instability that gives rise to dynamical buoyancy, causing massive central objects to move outward. Thus far, these manifestations of core dynamics have only been demonstrated in collisionless systems. In this paper, we use idealized N-body simulations to study whether core stalling and dynamical buoyancy operate in SIDM halos. We find that if the self-interactions are sufficiently strong, neither core stalling nor buoyancy are present, and a massive perturber will quickly sink all the way to the center of its host. In doing so, it gravitationally contracts the core, catalyzing a strongly accelerated core collapse. The reason why core dynamics are so different in SIDM halos is that self-interactions drive the halo's distribution function to a featureless exponential, removing any inflections or plateaus that are responsible for the dipole instability and core stalling in the case of CDM. We discuss implications of our finding for constraining the nature of dark matter by using observations of massive objects such as supermassive black holes (SMBHs), globular clusters, and nuclear star clusters in the central regions of galaxies.

The rapid integration of AI writing tools into online platforms raises critical questions about their impact on content production and outcomes. We leverage a unique natural experiment on Change$.$org, a leading social advocacy platform, to causally investigate the effects of an in-platform ''write with AI'' tool. To understand the impact of the AI integration, we collected 1.5 million petitions and employed a difference-in-differences analysis. Our findings reveal that in-platform AI access significantly altered the lexical features of petitions and increased petition homogeneity, but did not improve petition outcomes. We confirmed the results in a separate analysis of repeat petition writers who wrote petitions before and after introduction of the AI tool. The results suggest that while AI writing tools can profoundly reshape online content, their practical utility for improving desired outcomes may be less beneficial than anticipated, and introduce unintended consequences like content homogenization.

We combine existing perturbative results to show that a precise analytic determination of the charm-quark contribution to the hadronic light-by-light (HLbL) part of the muon anomalous magnetic moment is possible. Working in the $\overline{\mathrm{MS}}$ scheme, we include the NLO $\mathcal{O}(α_s)$ correction, which significantly reduces the residual renormalization-scale dependence and the perturbative uncertainty. Our final result is $a_μ^{\mathrm{HLbLc}}=3.65(25)\times 10^{-11}$, in good agreement with recent lattice determinations.

Numerical optimal control (GRAPE) can in principle discover pulse shapes that suppress all coherent gate error to machine precision. But when does that capability actually matter? We present a systematic comparison of Gaussian, DRAG, and GRAPE pulses for single-qubit gates on a three-level transmon model parameterized by IQM Garnet hardware ($T_1 = 37\,μ$s, $T_2 = 9.6\,μ$s,$α/2π= -200$ MHz), with the explicit goal of identifying the regimes where numerical optimization provides genuine practical advantage over analytical methods. Our central finding is that properly calibrated DRAG already operates near the decoherence floor. At 20 ns gate time, GRAPE eliminates all coherent error ($1 - F < 10^{-15}$), but DRAG achieves $1 - F = 4.9 \times 10^{-4}$ in coherent error alone,and $8.4 \times 10^{-4}$ under full decoherence -- only $1.2\times$ above GRAPE's decoherence-limited performance. More surprisingly,DRAG is \emph{more robust} than GRAPE to qubit frequency detuning (minimum fidelity 0.990 vs.\ 0.931 over $\pm 5$ MHz), the dominant calibration uncertainty in charge-noise-limited transmons. GRAPE retains superior amplitude robustness (minimum fidelity 0.994 vs.\ 0.990) and provides the only route to guaranteed zero coherent error, which matters at short gate times ($\lesssim 15$ ns) where perturbative corrections break down. These results lead to concrete calibration guidance: (1) properly calibrated DRAG is sufficient for gate times $\gtrsim 20$ ns on hardware with $T_2/T \gtrsim 500$, (2) GRAPE is necessary at short gate times or when targeting error rates well below the decoherence floor, and (3) robust optimal control incorporating frequency uncertainty should be used when detuning is the dominant noise source. We decompose the full error budget (coherent, $T_1$, $T_2$, control noise) and provide the open-source QubitPulseOpt framework for reproducing all results.

The nucleation and dynamics of vortices in the quasi-two-dimensional rotating dipolar Bose-Einstein condensate are explored by taking into account the Lee-Huang-Yang (LHY) correction to the mean-field (MF) theory. Assuming approximate cancellation of the MF interactions, we focus on the formation of a pure LHY superfluid. The effect of rotational frequency $Ω$ is investigated numerically by determining the corresponding number of stable vortices in the superfluid, together with the respective energy per particle $E$ and chemical potential $μ$. The LHY superfluid provides a deep minimum of $E$ and $μ$, indicating that it is a remarkably robust state of quantum matter. By fixing the LHY interaction strength, an exact single-vortex critical frequency is found, along with the respective chemical potential. A notable feature, observed when creating the LHY superfluid with fewer than five vortices, which is understood as being due to the superfluid's nonlinearity and trapping aspect ratio, is the large frequency ranges admitting the production of two and four vortices, as compared to the small frequency ranges to obtain one and three vortices.

Based on the world's largest $ψ(3686)$ data sample, the BESIII Collaboration recently reported a substructure near the $π^+π^-$ mass threshold in the decay $ψ(3686) \to J/ψπ^+π^-$, challenging the established understanding of the dipion invariant mass spectrum. We propose that this substructure arises directly from dipion correlations. Using a chiral unitary approach, we successfully reproduce the observed anomaly, thereby providing strong evidence of dipion correlation in heavy quarkonium decays. This approach also allows us to predict the corresponding dipion correlation function.

Recessions are periods in which the least productive firms in the economy exit, and as the economy recovers, they are replaced by new and more productive entrants. These cleansing effects improve the average firm productivity. At the same time, recessions induce a loss of varieties. In an economy with Homothetic Single Aggregator technology, we show that their long-run welfare effects trade off these two forces. This trade-off is governed by love-of-variety and the elasticity of substitution in aggregate production. If industry output is aggregated using the standard CES aggregator, recessions do not improve long-run GDP or welfare. If the economy features more love-of-variety than CES, the social planner optimally subsidizes economic activity both in steady state and even more so in recessions to avoid firm exit. We use the model and quasi-exogenous variation in demand to estimate love-of-variety. We find it to be significantly higher than implied by CES aggregation, suggesting that even the long-run effects of recessions are negative. Finally, we quantitatively characterize the optimal policy response both along the transition and in the steady state.

This paper shows that discrete Morse-Bott theory can be developed as a natural extension of R. Forman's discrete Morse theory by improving the definition of the discrete Morse-Bott function introduced by S. Yaptieu. To this end, we demonstrate that the combinatorial structure of critical cells can be extended to critical sets intuitively. Furthermore, we establish the discrete Morse-Bott inequalities, providing a unified view that extends both the discrete Morse inequalities and the continuous Morse-Bott inequalities.

The boundary discontinuity (BD) design is a non-experimental method for identifying causal effects that exploits a thresholding rule based on a bivariate score and a boundary curve. This widely used method generalizes the univariate regression discontinuity design but introduces unique challenges arising from its multidimensional nature. We synthesize over 80 empirical papers that use the BD design, tracing the method's application from its formative stages to its implementation in modern research. We also overview ongoing theoretical and methodological research on identification, estimation, and inference for BD designs employing local polynomial regression, and offer recommendations for practice.

Cosmological determinations of the number of relativistic neutrino species, $N^{ }_{\rm eff}$, are becoming increasingly accurate, and further improvements are expected both from CMB and BBN data. Given this context, we update the evaluation of $N^{ }_{\rm eff}$ and the current entropy density via the momentum-averaged approach. This allows for a numerically fast description of neutrino decoupling, easily portable to an array of new physics scenarios. We revisit all aspects of this approach, including collision terms with full electron mass dependence, finite temperature QED corrections to the equation of state, neutrino oscillations, and the modelling of neutrino ensembles with effective chemical potentials. For integrated observables, our results differ by less than $0.04\%$ from the solution of the momentum-dependent evolution equation. We outline how to extend the approach to BSM settings, and will highlight its power in Part II. To facilitate the practical implementation, we release a Mathematica and Python code within nudec_BSM_v2, easily linkable to BBN codes.

The linear absorption spectrum of excitons in TMDC monolayers under the influence of an in-plane magnetic field is theoretically studied. We demonstrate that in-plane magnetic fields induce a hybridization between spin-bright and spin-dark exciton transitions, resulting in a brightening of spin-dark excitons. We analytically investigate spectral features including resonance energy shifts, broadening and amplitudes ratios. In particular, for a MoSe$_2$ monolayer with radiatively-limited linewidth, we find a complex interplay of dark-bright splitting and linewidth difference of both involved spin-bright and spin-dark excitons.

We continue the study of the two-dimensional dispersive Anderson model (DAM), i.e. the nonlinear Schrödinger equation with multiplicative spatial white noise. For this model, global well-posedness on the periodic domain was established by Visciglia and the second author (2023), and global well-posedness on the full space was established by Debussche, Visciglia, and the authors (2024). We show that, under suitable initial conditions and suitable periodization procedure of the noise, the periodic global dynamics of the DAM converges in spaces of local domains to that of the DAM on the full space as the period goes to infinity. In order to control the growth of the noise and obtain a priori bounds for solutions independent of the periodicity, we introduce periodic weights and construct weighted function spaces on periodic domains. In Appendix, we also discuss the same problem for the parabolic Anderson model.

We consider a Dirichlet series $D(F,G;s)$ attached to two automorphic forms $F$ and $G$ of an orthogonal group of real signature $(2,4)$, involving their Fourier--Jacobi coefficients. When $F$ is a Hecke eigenform and $G$ a lift of a Jacobi-Poincaré series, our main result gives that $D(F,G;s)$ is equal to the standard $L$-function attached to $F$, up to an explicit constant. To establish this, we use a correspondence between binary Hermitian forms and ideals of quaternion algebras, as established by Latimer, together with the fact that the even Clifford algebra of a three-dimensional definite quadratic space can be identified with a quaternion division algebra. Our work should be seen as a generalisation of a work of Kohnen and Skoruppa, whose result corresponds to the case of the orthogonal group of real signature $(2,3)$.

We show that even weak nonreciprocal alignment leads to large-scale structure formation in flocking mixtures. By combining numerical simulations of a binary Vicsek model and the analysis of coarse-grained continuum equations, we demonstrate that nonreciprocity destabilizes the ordered phase formed by mutually aligning or anti-aligning species in a large part of the phase diagram. For aligning populations, this instability results in one species condensing in a single band that travels within a homogeneous liquid of the other species. When interactions are anti-aligning, both species self-assemble into polar clusters with large-scale chaotic dynamics. In both cases, the emergence of structures is accompanied by the demixing of the two species, despite the absence of repulsive interactions. Our theoretical analysis allows us to elucidate the origin of the instability, and show that it is generic to nonreciprocal flocks.

We investigate the quantum statistical properties of the confining string connecting a static fermion-antifermion pair in the massive Schwinger model. By analyzing the reduced density matrix of the subsystem located in between the fermion and antifermion, we demonstrate that as the interfermion separation approaches the string-breaking distance, the overlap between the microscopic density matrix and an effective thermal density matrix exhibits a pronounced, narrow peak, approaching unity at the onset of string breaking. This behavior reveals that the confining flux tube evolves toward a genuinely thermal state as the separation between the charges grows, even in the absence of an external heat bath. In other words, one cannot tell whether a reduced state of the subsystem arises from a surrounding heat bath or from entanglement with the rest of the system. The entanglement spectrum near the critical string-breaking distance exhibits a rapid transition from the dominance of a single state describing the confining electric string towards a strongly entangled state containing virtual fermion-antifermion pairs. Our findings establish a quantitative link between confinement, entanglement, and emergent thermality, and suggest that string breaking corresponds to a microscopic thermalization transition within the flux tube.

Deforming the algebra of constraint is a well-known approach to effective loop quantum cosmology. More generally, it is a consistent way to modify gravity from the Hamiltonian perspective. In this framework, the Hamiltonian (scalar) constraint is usually the only one to be holonomy corrected. As a heuristic hypothesis, we consider the possibility to also correct the diffeomorphism and Gauss constraints. It is shown that it is impossible to correct the diffeomorphism constraint without correcting the Gauss one, while maintaining a first-class algebra. However, if all constraints are corrected, the algebra can be closed. The resulting differential equations to be fulfilled by the corrections (of the background and of the perturbations) are derived.

The generic Mott transition in one-dimensional quantum systems can be described by the sine-Gordon model with a tilt via bosonization. Because the configuration space of the sine-Gordon model separates into distinct topological sectors, standard local Monte Carlo schemes are limited to very small system sizes. To overcome this limitation, we introduce the smooth worm (SmoWo) Monte Carlo algorithm which enlarges the configuration space to allow smooth transitions between topological sectors. The method combines worm updates with event-chain Monte Carlo moves. We explicitly prove its validity and quantify its performance. Thanks to the substantial acceleration achieved by the SmoWo algorithm, we are able to simulate large system sizes, providing a precise picture of the different phases and critical behaviour of the sine-Gordon model.

In this paper, we present a robust entanglement-assisted synchronization framework for indoor optical wireless systems that explicitly captures the coupling between spatial beam geometry and temporal synchronization accuracy. Unlike conventional approaches that treat beam steering and timing estimation independently, a unified spatio temporal model is developed that links user position uncertainty to the Cramer Rao lower bound of the synchronization error. The framework incorporates key physical impairments, including multipath dispersion, non Gaussian detector jitter, and spatially correlated localization errors. Through analytical modeling and extensive simulations, we show that the proposed system exhibits graceful performance degradation under heavy tailed positioning uncertainty and remains stable in the presence of multipath induced bias. Using realistic single photon detector parameters, the results indicate that synchronization accuracy below $10$ picoseconds can be maintained across a wide range of operating conditions. This level of precision provides a scalable foundation for quantum enabled indoor wireless networks.

Proximity effect in thin-film superconductor (S)/magnet heterostructures with different types of magnets including ferromagnets, antiferromagnets and altermagnets is widely considered in the framework of an effective model, where the heterostructure is replaced by a homogeneous superconductor in the presence of a homogeneous exchange field of a corresponding type. Here we study the extent to which such a model is actually applicable to ballistic thin-film superconductor/magnetic heterostructures. In particular, a comparative analysis of thin-film superconductor/magnetic metal and superconductor/magnetic insulator heterostructures is performed. Metallic and insulating ferromagnets (FM, FI) and altermagnets (AM, AI) are considered. It is shown that in the S/FI and S/AI heterostructures the the proximity effect creates a well-defined spin splitting of the electronic spectra in the S layer. Thus, they are well described by the effective model. At the same time, the proximity effect in S/FM and S/AM heterostructures also creates a spin splitting of the spectra of the S layer, but it has a chaotic spectral and spatial distribution and unpredictable amplitude and, in general, cannot be detected via the spin splitting of the superconducting density of states. Thus, the effective model is not applicable to such heterostructures. Nevertheless, we demonstrate that they support well-pronounced triplet correlations and, thus, can be used for spintronics applications.