Background: The neutron-deficient Hg isotopes provide one of the most prominent examples of nuclear shape coexistence. While the even-even nuclei have been extensively investigated, much less is known about how an unpaired neutron influences the evolution of collectivity in the odd-mass isotopes.

Purpose: To determine lifetimes of low-spin states in ${}^{189}\mathrm{Hg}$ and investigate the evolution of collectivity in the rotational band built on the $13/2^{+}$ isomer through comparison with neighboring shape-coexisting Hg isotopes.

Methods: Excited states in ${}^{189}\mathrm{Hg}$ were populated in the ${}^{160}\mathrm{Gd}({}^{34}\mathrm{S},5n){}^{189}\mathrm{Hg}$ fusion-evaporation reaction at a beam energy of 185 MeV. The reaction channel of interest was selected by requiring coincidence with at least one neutron detected in the neutron wall array, while prompt $\gamma$ rays were measured with the GALILEO spectrometer. Lifetimes of low-lying states above the $13/2^{+}$ isomer were measured with the recoil-distance Doppler-shift method using the GALILEO plunger device. The experimental results were compared with total Routhian surface calculations.

Results: Lifetimes of the $17/2^{+}, 21/2^{+}$, and $25/2^{+}$ states were measured for the first time and the corresponding $B$($E2$) strengths were extracted. The excitation energies and transition strengths closely follow the systematics observed in neighboring odd- and even-mass Hg isotopes, supporting the interpretation of the rotational sequence as the coupling of an $i_{13/2}$ neutron to the weakly oblate structures of the neighboring even-even cores. The extracted $B_{4/2}$-type ratio is significantly below unity, extending the occurrence of anomalous transition-strength ratios to this odd-mass nucleus. Total Routhian surface calculations reproduce the weakly oblate equilibrium deformation but not the observed evolution of the transition strengths.

Conclusions: The present results indicate that the $i_{13/2}$-based rotational band largely inherits the structural complexity of the neighboring shape-coexisting even-even Hg nuclei. They further demonstrate that electromagnetic transition strengths provide stringent constraints on the evolution of the underlying wave functions beyond those accessible from excitation energies or static mean-field descriptions alone.

We propose an extension of the initial-condition model trento for sampling the number of partons inside the nucleons that participate in a heavy-ion collision. This sampling method is based on parton distribution functions and therefore has a natural dependence on the momentum transferred in the collision and the scale being probed during the collision. We examine the resulting distributions and their dependence on the momentum transfer. Additionally, we explore the sensitivity of different observables on the number of partons using the trento framework and the estimators available therein for final-state observables.

In this study we apply the Remler formalism to charmonium production at Super Proton Synchrotron (SPS) and GSI/Facility for Antiproton and Ion Research (FAIR) energies in order to investigate the effects of baryon-rich matter on charmonium production and dissociation in heavy-ion collisions within the parton-hadron-string dynamics. As a first step the Remler formalism is tested in $p+p$ collisions and then applied to $p+A$ collisions in order to extract the nuclear absorption cross section of charmonium, which is then utilized in heavy-ion collisions. We find that the Remler formalism successfully describes charmonium production in heavy-ion collisions at SPS energies when an in-medium heavy quark potential is implemented, in which $J/\psi$ dissociates near $T_c$. Finally, the same formalism is applied to the low CERN/SPS energy and GSI/FAIR energies, where we estimate charmonium production.

Terbium-152, decaying by electron capture and ${\beta }^{+}$ emission to the ground and excited states in ${}^{152}\mathrm{Gd}$, shows potential for use in PET imaging as part of personalized cancer treatments. We report results of decay spectroscopy measurements, providing nuclear data necessary for medical use of this radionuclide. Sources of ${}^{152}\mathrm{Tb}$ were prepared at CERN-ISOLDE by proton-induced spallation of a tantalum target, followed by on-line mass separation. The decay of these sources was measured at ILL Grenoble using the $\gamma$-ray coincidence spectrometer FIPPS and the PN1 electron-$\gamma$ coincidence spectroscopy setup. We present an updated level scheme for the decay, comprising 751 $\gamma$-ray transitions placed to 200 excited states. Angular correlation analysis and conversion-electron spectroscopy provide spin-parity assignments to levels, multipolarity assignments to $\gamma$ rays, and measurements of electric monopole transition strengths. A transition intensity balance of the resulting level scheme provides beta feeding intensities for each level, showing an 8.9% reduction in the positron emission probability and an increase of 4.5% in the $\gamma$-ray energy emitted per decay compared to the existing evaluated data for this decay.

The low-lying excitation observables of even-even nuclei, e.g., the excitation energies of the first $2^{+}$ and $4^{+}$ states, $E\left(2_1^{+}\right)$ and $E\left(4_1^{+}\right)$, and the reduced electric quadrupole transition probability between the ground state and the first $2^{+}$ state, $B(E2;0_1^{+}\to 2_1^{+})$, serve as promising experimental probes for nuclear shell evolution in unstable nuclei far from the $\beta$-stability line. Previous machine learning studies on low-lying nuclear states have typically employed independent single-task models that neglect the intrinsic correlations among $B(E2;0_1^{+}\to 2_1^{+}), E\left(2_1^{+}\right)$, and $E\left(4_1^{+}\right)$, potentially leading to unphysical divergences in practical extrapolation. In addition, the manual selection of hyperparameters introduces subjective tuning bias and model dependence. To overcome these limitations, we propose a multitask Gaussian process (MTGP) approach that jointly predicts the three observables while preserving their mutual physical couplings. The MTGP hyperparameters are automatically optimized via Bayesian optimization. The examination of the established MTGP method demonstrates its significant improvement in predictive robustness over previous single-task approaches, both in terms of the prediction accuracy and the description of inherent correlations among the three observables. We then apply the MTGP model to predict unmeasured $B(E2;0_1^{+}\to 2_1^{+}), E\left(2_1^{+}\right)$, and $E\left(4_1^{+}\right)$ values around the magic shells $N,Z=20$ and 28. Systematic analysis of the MTGP-predicted observables along their respective isotonic and isotopic chains reveals nuanced shell-evolution patterns at $N,Z=20$ and 28: the $N=20$ closure weakens in the neutron-rich $Z=8–12$ region, strengthens at $Z=14–20$, and weakens again for $Z\ge 22$; the $N=28$ closure is quenched at $Z=12,14$, robust at $Z=16–28$, and shows signs of weakening in ${}^{58}\mathrm{Zn}$ and ${}^{60}\mathrm{Ge}$; the $Z=20$ closure persists in ${}^{34,36}\mathrm{Ca}$ but gradually weakens beyond $N=32$; and the $Z=28$ closure remains strong in ${}^{48,50,52}\mathrm{Ni}$ and weakens in neutron-rich nickel isotopes beyond $N=46$. These data-driven results support that the MTGP-based systematics capture the evolution of shell gaps and collectivity, suggesting the MTGP as a potential tool for nuclear structure studies.

This work formulates and implements the phaseless auxiliary-field quantum Monte Carlo method in the Hartree-Fock-Bogoliubov (HFB) manifold for systems with strong pairing correlations. HFB walkers are represented in both the product-state and Thouless forms with respect to a flexible reference vacuum, with explicit expressions for the corresponding normalization factors in terms of the representation matrices. In addition, the stochastic gauge formalism is extended from the Slater determinant manifold to the HFB framework, providing additional flexibility in controlling stochastic fluctuations. Benchmark calculations for the Richardson model demonstrate the accuracy and numerical stability of the method.

Recently, influence of strong fields on nuclear decay processes is progressively becoming a topic of great interest plus the existence of some ambiguities. In this study, an approximate calculation of the Schrödinger equation with reduced $r$ dependence is employed instead of the classical Coulomb wave-function program to provide a description of decay width and decay energy in quasistatic (adiabatic) laser fields. More importantly, going beyond the quasistatic approximation, the strict time evolution of the wave function and the periodic modulation of the decay energy are investigated by solving the time-dependent Schrödinger equation with the split operator method, which captures the dynamical behavior of the $\alpha$ cluster during the decay process. First, our results corroborate earlier reports that the decay width is modified only negligibly by strong laser fields. Likewise, the shift in decay energy is found to be extremely small, which validates the rationality of using fixed experimental decay energies in earlier studies. This modification effect will exist more obviously in a nucleus with the higher proton-neutron asymmetry, the lower decay energy, and the octupole deformation. Second, our results demonstrate that strong fields can indeed produce a periodic modulation of decay energy, with a cycle mean shift on the order of ${10}^{-7}\mathrm{eV}$ at an intensity of ${10}^{24}\mathrm{W}/{\mathrm{cm}}^2$.

Two-particle correlations play a pivotal role in understanding the space-time characteristics of particle emission in heavy-ion collisions (HIC). A relative emission source typically represents these characteristics and can be obtained from transport-model simulations such as the Boltzmann-Uehling-Uhlenbeck (BUU) transport model. In this paper, we use the BUU transport model to simulate $pp$ sources while simultaneously using HIC correlation data to infer them for the same collisions, combining the Koonin-Pratt equation with Richardson-Lucy deblurring. In comparing the sources, we can isolate and provide a simple parametrization for source tails missing from BUU sources. These tails may be attributed to secondary decays that contribute at least one member of the pair in the source. With a tail added, the BUU can reproduce 80 MeV/u Ar $+$ Sc and Ca $+$ Ca measurements. We investigate the angular dependence of Ca $+$ Ca correlations to infer the angle dependence of slow emissions. Our parametrization for the tail may be employed when no deblurring for source inference is pursued.

We calculate the absorption mean free paths of electron and muon neutrinos in two-flavor color-superconducting (2SC) quark matter in the density and temperature range that is relevant to binary neutron star mergers. We model the strong interaction between quarks using a Nambu–Jona-Lasinio model, performing calculations self-consistently in the mean-field approximation. Since the 2SC gap is large we restrict our analysis to the contribution of unpaired quarks. We find that at low temperatures absorption by a down quark $\nu +d\to u+e^{-}/{\mu }^{-}$ is kinematically not allowed, so absorption by a strange quark, $\nu +s\to u+e^{-}/{\mu }^{-}$, dominates the mean free path. As temperature or neutrino energy rises, the $d$ quark absorption channel becomes active, and the mean free path shrinks. We find that in equilibrated 2SC matter with an electron lepton fraction $Y_{L_e}=0.1$, the neutrinos form a degenerate gas with a mean free path of meters or less, independently of the temperature.

An experimental study of ${}^{258,260}\mathrm{Sg}$ produced in the complete fusion reactions ${}^{52,54}\mathrm{Cr}+{}^{207}\mathrm{Pb}$ was performed using the SHELS velocity filter. The prompt neutron multiplicities in the spontaneous fission of ${}^{258,260}\mathrm{Sg}$ were measured for the first time, yielding average values of $\overline{\nu }{(}^{258}\text{Sg})=4.9\pm 0.4$ and $\overline{\nu }{(}^{260}\text{Sg})=5.03\pm 0.14$. The $\alpha$-decay branching ratio of ${}^{260}\mathrm{Sg}$ was determined to be $b_{\alpha }=0.29\pm 0.03$. In addition, the half-lives were measured as $T_{1/2}{(}^{258}\text{Sg})=2.2_{-1.0}^{+2.4}\mathrm{ms}$ [95% C.L. (confidence limit)] and $T_{1/2}{(}^{260}\text{Sg})=7.0_{-1.7}^{+2.6}\mathrm{ms}$ (95% C.L.).

Prompt $\gamma$-ray production cross sections are important for nuclear reaction and structure calculations and in several applications. In particular, $\gamma$-ray production data and calculations are used in material assay and neutron diagnostics. ${}^{86}\mathrm{Kr}$ is a common, stable fission fragment, and as such can be used to test nuclear reaction models at the edge of stability, in order to improve calculations extending into unstable fission fragments in the region. Partial prompt $\gamma$-ray cross sections were measured, produced in neutron-induced reactions for neutron energies $11\mathrm{MeV}<E_n<80\mathrm{MeV}$ on ${}^{86}\mathrm{Kr}$: three $\gamma$ rays of ${}^{85}\mathrm{Kr}$ and one $\gamma$ ray each in ${}^{82,83,84}\mathrm{Kr}$ and ${}^{83,85}\mathrm{Br}$. A large part of the total cross section for the ${}^{86}\mathrm{Kr}(n,3n){}^{84}\mathrm{Kr}$ reaction is experimentally observed. The results are compared with predictions from calculations using the ${\mathrm{CoH}}_3$ reaction code and are used to check on the consistency of the parameters in the calculations.

The shell structure of nuclei beyond $N=126$ relevant to the third peak at $A\approx 195$ in $r$-process nuclear abundances in investigated. Within the framework of the relativistic Hartree-Bogoliubov theory in deformed space, a comparative study of the response of nuclei with three different model Lagrangians, NL-SV1, SVI-2, and NL3, is made. Ground-state properties of even-even nuclei in the isotopic chains of $Z=58–72$ in the highly neutron-rich region are investigated. A comparison of binding energies with mass formulas shows that in the extreme region of the $r$-process path, the RMF model SVI-2 without nonlinearities of $\sigma$ and $\omega$ mesons shows an excellent agreement with the mass formula HFB-24. It is shown that in all the above Lagrangian models, in the neutron rich region between the shell closures $N=126$ and $N=184$, there is a preponderance of nuclides exhibiting coexistence of well-deformed prolate and oblate shapes in the ground state. A prolate ground state is shown to evolve smoothly to a coexisting oblate ground state via a near degeneracy with successive addition of neutrons. Induced by an onset of deformation for $N\ge 130$, nuclei approaching the $r$-process path are shown to exhibit enhanced stability in many isotopic chains $(Z>60)$. This has the effect of shifting the expected neutron drip line significantly farther into the neutron-rich region especially for the higher $Z$ chains. Consequences of the enhanced stability, the shape co-existence and degeneracy of shapes in nuclei on the $r$-process nucleosynthesis are discussed.

The change of the nucleon density distribution of a heavy nucleus caused by the presence of a second nucleus at a fixed center-to-center distance is studied in the framework of the deformed Woods–Saxon mean field. The single-particle problem of one nucleus has been solved by diagonalization in a cylindrical harmonic-oscillator basis with the external field of the partner nucleus included in the Hamiltonian before diagonalization. The external field consists of the Coulomb potential of the partner charge distribution and of the nuclear part obtained by double folding of the Migdal density-dependent effective nucleon-nucleon interaction, in the same form as used in the dinuclear system approach. The induced density change $\delta \rho (\mathit{r};R)$ and the corresponding changes of the quadrupole moment, of the mean-square radius and of the interaction energy have been calculated for the ${}^{40}\mathrm{Ca}+{}^{208}\mathrm{Pb}$ system at separations $R=15.5–20$ fm. The results are compared with the first-order (linear-response) treatment of the same external field. Because the mean field is anchored at the origin, the multipole moments are shown to require a translationally invariant definition: a displaced but undeformed distribution acquires a quadrupole moment second order in the displacement, and in this problem that kinematic term is comparable with the induced moment itself. Once it is removed, the intrinsic quadrupole response of ${}^{208}\mathrm{Pb}$ agrees with the first-order one to better than half a percent, and remains so over the full range of the switching parameter for the Coulomb field at $R=20$ fm, while that of ${}^{40}\mathrm{Ca}$ retains a genuine higher-order contribution of 12% to $16\%$. The polarization of both partners is dominated by the repulsive Coulomb tidal field, which produces an oblate deformation of the density, while the attractive nuclear field acts in the opposite sense and partially cancels the Coulomb contribution at the smallest separations. At the larger separations the induced octupole deformation is close to $40\%$ of the quadrupole one for ${}^{208}\mathrm{Pb}$ and to $50\%$ for ${}^{40}\mathrm{Ca}$, and it is the more sensitive indicator of the breakdown of the one-center picture. The polarization energy is a dipole quantity, the induced quadrupole deformation contributing only a few percent of it; it is verified by coupling-constant integration and is quoted under several correction prescriptions for the spurious translational component and for the absence of the residual interaction. The sensitivity to the geometry of the folding densities and the limits of validity of the one-center description are established quantitatively.

To improve the prediction accuracy of $\alpha$-decay half-lives in heavy and superheavy nuclei, a physics-guided residual-correction framework combining the effective liquid drop model (ELDM) with machine-learning methods is proposed. The ELDM is first used as the macroscopic baseline for describing the barrier-penetration process, and XGBoost and TabPFN models are then employed to learn the residual deviations between ELDM predictions and experimental data. Several physically motivated descriptors associated with nuclear deformation, Coulomb-decay-energy systematics, and minimum orbital angular momentum are introduced into the residual-learning process. In the global 10-fold cross-validation comparison, TabPFN-term3 achieves the lowest prediction errors, reducing the root-mean-square error and mean absolute error from 0.567 and 0.417 for the ELDM baseline to 0.348 and 0.248, corresponding to improvements of 38.62% and 40.53%, respectively. Regional, parity-separated, and superheavy extrapolation analyses further support the effectiveness and transferability of the residual-correction framework. Overall, the proposed approach effectively reduces systematic deviations not fully described by the macroscopic ELDM baseline while retaining a direct connection to the underlying $\alpha$-decay physics.

Disentangling the effects of nucleon transfer via $Q$ value and deformation on fusion has been a longstanding objective. To address this, a comparative analysis of the barrier distribution, which is an alternative probe to fusion, for the ${}^{14}\mathrm{N}+{}^{178}\mathrm{Hf}$ and ${}^{16}\mathrm{O}+{}^{178}\mathrm{Hf}$ systems have been performed. The consistent alteration in deformation by choosing the same target ${}^{178}\mathrm{Hf}$ mitigates its influence, thereby allowing for an explicit investigation of the effect of the $Q$ value of nucleon transfer for these above systems. Consequently, the quasielastic scattering was measured for ${}^{14}\mathrm{N}+{}^{178}\mathrm{Hf}$ at backward angles. The extracted excitation function and barrier distribution were analyzed in light of coupled-channel calculations, which indicated a significant contribution from the one- and two-neutron transfer channels, alongside the rotational degrees of freedom in the highly deformed ${}^{178}\mathrm{Hf}$ nucleus. Furthermore, the coupling strength was extracted using the chi-square minimization method, yielding a value $\approx 0.201$. The extracted coupling strength of ${}^{14}\mathrm{N}+{}^{178}\mathrm{Hf}$ system was compared with that of the two-neutron transfer in the ${}^{16}\mathrm{O}+{}^{178}\mathrm{Hf}$ system, which indicates that the coupling strength for nucleon transfer decreases with an increase in the $Q$ value. This observed behavior might be due to the spectroscopic structures of the ${}^{16}\mathrm{O}$ and ${}^{14}\mathrm{N}$. The present study indicates that the spectroscopic properties play a significant role, along with $Q$ values, in influencing the fusion reaction dynamics.

In this work, we propose a novel approach for probing the neutron skin thickness of gold (Au) by analyzing the event-by-event distribution of ${\pi }^{-}$ and ${\pi }^{+}$ yield differences. This is achieved through SMASH simulations of ultraperipheral $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s_{NN}}=3\mathrm{GeV}$. Our results demonstrate that the mean value of $\mathrm{\Delta }{n}_{\pi }=n_{{\pi }^{-}}-n_{{\pi }^{+}}$, along with the Pearson correlation and mutual information between $({\pi }^{-}+{\pi }^{+})$ and $({\pi }^{-}-{\pi }^{+})$, all scale linearly with the neutron skin thickness. Moreover, the slope of the line connecting two distinct $\mathrm{\Delta }{n}_{\pi }$ values in the event-by-event distribution also exhibits a linear dependence on the neutron skin thickness. The most sensitive $\mathrm{\Delta }{n}_{\pi }$ pairs are identified as $(-2,-1)$, $(-1,1)$, $(0,1)$, and $(0,2)$. These findings establish a new pathway for determining the neutron skin thickness. Finally, by comparing SMASH, JAM, and ultrarelativistic quantum molecular dynamics simulations under identical initial conditions, we observed a certain degree of model dependence in the outcomes, yet the overall trends remained consistent. However, by extracting slopes from multiple $\mathrm{\Delta }{n}_{\pi }$ pairs in experimental event-by-event data and inferring the corresponding neutron skin thickness, one can assess which model better aligns with the true physical value.

We calculate the valence parton distribution function (PDF) of pion within the theoretical approach based on spectral function representation of valence quarks in the pion. In this approach we assume that the soft partonic structure of pion is defined by the valence $q\overline{q}$ cluster, consisting of current quarks, embedded in the residual field of the pion. The valence PDF is calculated using phenomenological light-front wave functions for the $q\overline{q}$ cluster and the residual field. Our result indicates that the peak position of the $x$ weighted valence PDF (xPDF) depends on parameters characterizing the virtuality of the valence quark cluster and the mass of the residual system. Magnitudes of these parameters are obtained by fitting to the height and the peak position of empirical pion xPDF evaluated at starting $Q_0$. They indicate that unlike the nucleon case, very little residual mass is needed to describe existing pion PDFs at intermediate $x$. They also indicate that the $q\overline{q}$ cluster is highly virtual and on average the interacting quark carries almost all of the momentum of the $q\overline{q}$ cluster. This picture is consistent with the dominance of the Feynman mechanism leaving little room for the hard component in the PDF and practically describing $\approx (1-x)$ behavior observed recently at $x\to 1$ limit. Our nontrivial observation is that the height and the peak position of xPDF define the analytic behavior of valence PDF at $x\to 1$ limit.

Many experimental methods in physics require understanding radiation from single particles into nontrivial electromagnetic mode structures. Such characterization is critical for Cyclotron Radiation Emission Spectroscopy (CRES), an advancing new measurement technique that has the potential to greatly benefit fundamental physics measurements. In CRES, charged particles emit cyclotron radiation at frequencies that provide their energy measurement. As a notable example, the Project 8 experiment aims to kinematically infer the neutrino mass by measuring the energies of electrons emitted in tritium beta decay using CRES. In near-term realizations of Project 8, resonant cylindrical cavities will be used for CRES readout, in a configuration with a magnetic field oriented along the symmetry axis, and electrons following helical cyclotron trajectories confined to the cavity interior. The physics of electromagnetic radiation in these environments is complicated, since it involves both the motion of the emitting particle and the mode structure imposed by the cavity. In this work, we derive and validate an analytic model for how an oscillating, trapped electron radiates into cavity modes, and the power and frequency content of the radiation that can be read out from these events. These results can be used to guide the design of cavities for future CRES and other experiments.

Quantifying causal connections within fireballs resulting from high energy nucleus-nucleus collisions is pertinent to assessing the viability of several proposed mechanisms that directly influence particle production and correlations. Fireball causal connections have previously been studied in the context of $1+1$ dimensional Bjorken flow. We expand into $3+1$ dimensions using Gubser flow, which includes transverse expansion. Our findings suggest that the volume of these causally connected fireballs are on the order of 10–100 ${\mathrm{fm}}^3$ for observables dependent on the formation of the light quark condensates, but could be larger for other observables.

We investigate the internal structure of two exotic tetraquark candidates, ${\chi }_{c1}\left(3872\right)$ and $T_{cc}^{+}\left(3875\right)$, whose masses lie within $1\text{MeV}$ of the $D^0{\overline{D}}^{*0}$ and $D^0{D}^{*+}$ thresholds, respectively. Using the effective-range expansion (ERE) supplemented by resonance compositeness relations on the second Riemann sheet, we extract scattering lengths $a$ and effective ranges $r$ directly from the experimental masses and widths and quantify the $D{D}^{*}$ molecular content through partial compositeness coefficients $X_j$. A two-channel framework couples each state's inelastic detection channel ($J/\psi \pi \pi$; $D^0{D}^0{\pi }^{+}$) to the near-threshold molecular channel. Full Monte Carlo uncertainty propagation with $N=50000$ Gaussian samples yields asymmetric $68\%$ confidence intervals for all observables. We find large negative scattering lengths $a=-8.{49}_{-1.05}^{+0.93}$ fm for ${\chi }_{c1}\left(3872\right)$ and $a=-14.{57}_{-1.66}^{+1.70}$ fm for $T_{cc}^{+}$, both far exceeding the natural hadronic scale $\approx 1/m_{\pi }$ and dominant molecular components $X_2>0.95$ under full compositeness. The results are stable for total compositeness $X\in [0.5,1.0]$, where $1-X$ represents an upper bound on any compact/CDD-pole contribution. In the present study, we go beyond the pole-position Castillejo-Dalitz-Dyson (CDD) criterion by explicitly constructing the inverse amplitude as an ERE background plus a scanned CDD pole term, finding no region of physically reasonable bare-state parameter space that reproduces the pole while leaving a natural background; we fully propagate the loop-function subtraction constant $a\left({\mu }_g\right)$ as a Monte Carlo nuisance parameter jointly with the experimental mass and width, confirming it contributes a negligible ($<0.01\%$) additional uncertainty; and we test robustness against an assumed correlation between $M_R$ and ${\mathrm{\Gamma }}_R$, finding all results stable over a broad range of correlation coefficients. We demonstrate analytically and numerically the scheme-independence of the compositeness coefficients, provide explicit derivations of all ERE formulas from the pole condition, clarify the analytic continuation procedure for closed channels, and compare our results quantitatively with independent ERE, effective-field theory, and other relevant studies. The $Z_c\left(3900\right)$ case is treated separately to show the breakdown of the two-channel approximation for above-threshold states. Graphical illustrations of the Monte Carlo distributions are included.