Based on M. Hall's theorem we prove a simple result dealing with real numbers which admit exact approximations by rationals.

Superconductivity in sulfur superhydride H$_{3}$S under extreme pressures has been explained theoretically, but it requires a peaked concentration of the electronic density of states (DOS), which has been found in first-principles calculations. The mechanism of this peak formation, though vital for its high transition temperature, has however remained obscure. We address this problem through detailed analysis of the first-principles electronic wave functions. The valence wave functions are shown to be significantly plane-wave-like. From the Fourier-mode analysis of the self-consistent potential and atomic pseudopotentials, we extract the nearly uniform models that accurately reproduce the first-principles band structure with very few parameters. The DOS peak is shown to be the consequence of the hybridization of specific plane waves. Adjacency of Jones' large zone to the plane-wave spherical Fermi surface is posited to be the root cause of the multiple plane-wave hybridization, the DOS peak formation and its proximity to the Fermi level. The present theory resolves the minimal modeling problem of electronic states in H$_{3}$S, as well as establishes a mechanism that may help to boost the transition temperatures in pressure induced superconductors.

Symmetries and conservation laws are studied for a generalized Westervelt equation which is a nonlinear partial differential equation modelling the propagation of sound waves in a compressible medium. This nonlinear wave equation is widely used in nonlinear acoustics and it is especially important in biomedical applications such as ultra-sound imaging in human tissue. Modern methods are applied to uncover point symmetries and conservation laws that can lead to useful developments concerning solutions and their properties. A complete classification of point symmetries is shown for the arbitrary function. Local low-order conservation laws related to net mass of sound waves are obtained by the multiplier method. Two potential systems are derived yielding potential symmetries and nonlocal conservation laws. For the physical case interesting for this equation, travelling wave solutions are studied leading to shock waves.

We study bidirectional one-dimensional (1-D) shallow-water waves within a class of Boussinesq equations, including the integrable Kaup-Boussinesq (KB) equation and a truncated-dispersion variant, which serves as a representative non-integrable model. For these two systems, the normal-form transformation yields an interaction coefficient of the same general structure, differing only through the dispersion relation. We derive this coefficient and numerically confirm that it vanishes on the resonant manifold for the KB equation, as expected in the literature. In contrast, the non-integrable model admits a non-vanishing interaction coefficient, producing a non-trivial wave kinetic equation (WKE), which is the first known in a 1-D shallow-water setting. The resulting WKE is non-homogeneous in nature due to the non-homogeneity of the corresponding dispersion relation; however, approximate Kolomogrov-Zakharov (KZ) solutions can be derived in a novel way under certain approximations. Numerical experiments in two settings validate the kinetic predictions and elucidate the underlying dynamics: (i) in free-evolution cases of the KB equation, despite complete integrability and the invariance of the discrete nonlinear spectrum guaranteed by isospectrality, an initial arbitrary wavenumber spectrum undergoes substantial evolution driven by quasi-resonant triad interactions; (ii) in forced-dissipated cases of the non-integrable equation, we find stationary power-law spectra that agree with the theoretical predictions.

The article Nuclear Fusion \textbf{66}, 016012 (2026) by Richard Fitzpatrick is based on fundamental errors in the physics of the evolution of the poloidal magnetic flux in tokamaks. This paper was inspired by an article $\big<$arxiv.org/abs/2507.05456$\big>$ by Allen Boozer, which was posted on arXiv in various versions. The September 9, 2025 version was submitted to the Physics of Plasmas and flatly rejected until the issues raised in the Nuclear Fusion paper had been addressed. Not only did the Nuclear Fusion paper make a number of fundamental errors in science, it also misrepresented the views clearly stated in the arXiv article and even more explicitly in email exchanges that were repeatedly cited as ``private communication."

A new geometric shaping technique, referred to as coset shaping, is proposed and analyzed for coded QAM and PAM signaling. This method can be applied to both information and parity bits without introducing additional complexity. It is shown that, as the error-correcting code length and the modulation order grow, the gap to capacity of the proposed shaping scheme can be made arbitrarily small. A Gallager-type bound is provided together with numerical results, including performance comparisons for the shaping scheme combined with short and mid-length binary-coded, as well as nonbinary LDPC-coded QAM signaling

We contribute to the recent line of work on responsibility measures that quantify the contributions of database facts to obtaining a query result. In contrast to existing work which has almost exclusively focused on monotone queries, here we explore how to define responsibility measures for unions of conjunctive queries with negated atoms (UCQ${}^\lnot$). Starting from the question of what constitutes a reasonable notion of explanation or relevance for queries with negated atoms, we propose two approaches, one assigning scores to (positive) database facts and the other also considering negated facts. Our approaches, which are orthogonal to the previously studied score of Reshef et al., can be used to lift previously studied scores for monotone queries, known as drastic Shapley and weighted sums of minimal supports (WSMS), to UCQ$^\lnot$. We investigate the data and combined complexity of the resulting measures, notably showing that the WSMS measures are tractable in data complexity for all UCQ${}^\lnot$ queries and further establishing tractability in combined complexity for suitable classes of conjunctive queries with negation.

This study presents an Initial Data Analysis (IDA) of the German Transplantation Registry (TxReg) data for a better data understanding and to inform future data analyses. The IDA is focusing on data on first-time kidney-only transplantations in adult recipients from deceased donors between 2006 and 2016 and refers to data from 14,954 recipients and 9,964 donors across 25 tables. Investigated aspects include missing data patterns and structure, data consistency, and availability of event time data. Results show that missing data proportions vary widely, with some tables nearly complete while others have over 50% missing values. Missing data patterns are identified using a decision tree approach. An influx and outflux analysis demonstrates that some variables have high potential for imputing missing data, while others were less suitable for imputation. We identified 168 multi-sourced variables that are reported by multiple data providers in parallel leading to discrepancies for some variables but also providing opportunities for missing data imputation. Our findings on event time data demonstrate the importance of carefully selecting the variables used for event time analyses as results will strongly depend on this selection. In summary, our findings highlight the challenges when utilizing the TxReg data for research and provide recommendations for data preprocessing and analysis in future analyses.

We study a honeycomb Kondo lattice model in which Dirac conduction electrons are coupled to a spin-1/2 Kitaev quantum spin liquid. For weak Kondo coupling, the spins fractionalize into Majorana fermions comprising a gapless Dirac mode and three gapped visons. In second order perturbation theory, the Kondo coupling gives rise to local Hubbard repulsions and spin-spin interactions between conduction electrons, as well as a vertex coupling electrons to gapless Majorana fermions. We analyze the resulting low-energy field theory using a perturbative renormalization group (RG) scheme, accounting for additional density-density interactions generated under RG. At criticality, electrons decouple from Majorana fermions but all three electron interactions acquire positive values. An analysis of susceptibility exponents reveals that the fractionalized Fermi liquid becomes unstable towards antiferromagnetic order and that superconductivity is disfavored.

We demonstrate that the thermodynamic friction metric governing dissipation in slowly driven continuous-time Markov chains is equivalent to the commute-time embedding and the resistance distance. This equivalence yields complementary insights: The commute-time embedding demonstrates the intrinsic cost of transporting probability across dynamical bottlenecks, while the resistance distance maps thermodynamic dissipation to Joule heating in an electrical network. We further demonstrate that the linear-response thermodynamic distance is a discrete $L^2$-Wasserstein optimal transport cost evaluated along paths of equilibrium distributions, extending a continuous-state correspondence to discrete networks. This conceptual synthesis of linear-response thermodynamics, random walks on graphs, electrical circuits, and optimal-transport theory connects independently developed geometric frameworks, reduces complex metric calculations to simple circuit algebra, and provides a clear physical picture of dissipation as the energetic cost of routing probability through the state space network.

This paper proposes a multistatic radar (MSR) system utilizing a distributed wireless synchronization protocol. The wireless synchronization protocol uses a two-tone waveform exchange for frequency synchronization and a bi-directional waveform exchange for time synchronization, independent of GPS. A Bayesian Cramer-Rao lower bound (BCRLB) framework is developed to quantify the impact of synchronization offsets on joint delay and Doppler estimation, and consequently, on target localization and velocity estimation accuracy. Simulation results derived from the analytical expressions establish the extent to which the residual synchronization offsets degrade the MSR's performance. The performance of the synchronization links primarily depends on the synchronization-link channel and transmit parameters; optimizing these parameters enables the MSR configuration to surpass the monostatic performance and approach the ideal case. Furthermore, the simulated synchronization-link parameters suggest that practical implementation is feasible.

We phenomenologically derive a cosmological model that includes both a cosmological constant term $Λ/3$ and a dissipative driving term $β(2 H^{2} + \dot{H})$ by applying both the first law of thermodynamics and an effective entropy (that is proportional to the Bekenstein--Hawking entropy) to matter creation cosmology. Here $H$, $\dot{H}$, and $β$ are the Hubble parameter, the time derivative of $H$, and a non-negative dimensionless coefficient used for the effective entropy, respectively. The dissipative term is proportional to the Ricci scalar curvature, suggesting that the dynamic creation pressure has the same dependence. We examine the model's background evolution in the late universe and its horizon thermodynamics. The present model supports a transition from a decelerating universe to an accelerating universe when $β<0.5$.The second law of thermodynamics is always satisfied on the horizon, and maximization of entropy is satisfied in the final stage. In addition, we study first-order density perturbations related to structure formation, by applying a neo-Newtonian approach to the present model. We then examine constraints on the present model using three types of observational data and the transitional and thermodynamic constraints and find that a weakly dissipative universe with $Λ$ is likely favored and consistent with our Universe. We also discuss irreversible entropy due to adiabatic particle creation, assuming a holographic-like matter creation cosmology.

We study the scattering behavior of an anisotropic inhomogeneous Lipschitz medium at a fixed wave number, continuing our previous work [SIAM J. Math. Anal., 56(4):4834-4853, 2024] and using free boundary techniques from [arXiv:2506.22328]. Our main results can be categorized into two distinct cases. In the first case, we show that in two dimensions, piecewise $C^{1}$ or convex penetrable obstacles with corners, and in higher dimensions, obstacles with edge points, always induce nontrivial scattering for any incoming wave. In the second case, we prove that piecewise $C^{1}$ obstacles with corners in two dimensions (and with edge points in higher dimensions) with angles $\notinπ\mathbb{Q}$ always produce nontrivial scattering for any incoming wave.

We present computations of individual mode-to-mode energy transfers from direct numerical simulations of homogeneous isotropic turbulence. Unlike previous approaches based on shell-filtered velocity fields, this method distinguishes between the energy exchanged by each pair of modes within a triad. We introduce a potential function based on the energy content of the modes involved and show that it predicts the distribution of intense energy transfers in the vicinity of the sampling mode considered. By performing simulations with forcing applied at intermediate wavenumbers, we demonstrate that the region of most intense transfers is determined by the spectral location of the energy-containing scales rather than by the local or nonlocal character of the triad. Direct energy exchanges with the energy-containing range are suppressed by geometric constraints from the divergence-free condition, but persist as residuals when the sampling mode is close to the energy-containing scales. The comparison with an estimator derived from EDQNM theory shows good agreement and recovers the forward, scale-local nature of energy transfer consistent with the cascade picture.

We demonstrate the implementation of $Λ$-enhanced gray molasses cooling on the D$_2$ line of $^{85}$Rb atoms in an optical tweezer array. This technique yields lower atomic temperatures of 4.0(2) $μ$K compared to red-detuned polarization gradient cooling, and consequently extends the $T_2^*$ coherence time of the hyperfine clock qubit by a factor of 1.5. The method is alignment-free and can be readily implemented on laser beams used for magneto-optical trapping, as it only requires frequency and phase modulation control. Our experimental observations are corroborated by a numerical model based on a semi-classical force approach extended to a four-level system, including two hyperfine states of the upper manifold that are 120 MHz apart.

We develop a quantum kinetic theory of two-dimensional electron gases in which exchange is treated self-consistently at the Hartree-Fock level and enters as a nonlocal, momentum-dependent field in phase space. By starting from the Coulomb Hamiltonian, we derive a Hartree-Fock-Wigner equation for the electronic Wigner function and obtain a closed fluid model with exchange-corrected pressure, force, and current. For a single layer, we show that exchange renormalizes the Fermi velocity and can drive a long-wavelength plasmonic instability at low densities. In coupled layers, the same framework predicts acoustic-optical mode coupling, and an instability forming long-lived charge-imbalance patterns that are not predicted by classical Vlasov and Boltzmann models. Finally, we apply the kinetic model to the Coulomb drag problem and show how exchange substantially enhances the drag resistivity in dilute GaAs double wells, quantitatively matching experimental observations.

We prove that topological isomorphism on procountable groups is not classifiable by countable structures, in the sense of descriptive set theory. In fact, the equivalence relation $\ell_\infty$ expressing that two sequences of reals have a bounded difference is Borel reducible to it. This marks substantial progress on an open problem of Kechris, Nies and Tent (2018): to determine the exact complexity of the isomorphism relation among all non-archimedean Polish groups.

We revisit the matching conditions for self-guided laser pulse propagation in plasma and refine their formulation to maximize the energy of electrons produced via laser wakefield acceleration. Bayesian optimization, combined with particle-in-cell simulations carried out in a quasi-three-dimensional geometry and a Lorentz-boosted frame, is employed. The optimization identifies the maximum electron energy that a self-guided laser wakefield accelerator, driven by a laser of a given energy, can produce, together with the corresponding acceleration distance. Our results further demonstrate that electrons with energies close to the maximum value can be obtained across a relatively wide range of input parameters and without the need for their precise tuning. This provides substantial flexibility for experimental implementation and significantly relaxes the operational constraints associated with self-guided laser wakefield accelerators.

Weak gravitational lensing of the cosmic microwave background (CMB) is a powerful probe of cosmology, providing insight into structure formation and the evolution of the universe. Current and upcoming CMB experiments such as SPT-3G and the Simons Observatory (SO) provide high-resolution, low-noise temperature and polarization maps that are ideal for lensing reconstruction. The global minimum variance (GMV) quadratic estimator for CMB lensing reduces reconstruction noise over the standard quadratic estimator (SQE). In this work, we extend the GMV framework to incorporate the tSZ-deproj and cross-ILC foreground mitigation techniques, which enhance robustness against contamination from astrophysical sources. For a simulation study using SPT-3G Ext-10k and SO Extended configurations at $\ell_{\mathrm{max}}^T = 3500$, the lensing bias at $L < 1000$ is reduced from $\sim4\%$ with standard GMV and SQE to $2\%$ with tSZ-deproj, and to $< 1\%$ with cross-ILC. These methods enable the construction of foreground-cleaned lensing maps suitable for cross-correlation analyses, with direct relevance for current and future surveys.

We determine the 2-local homotopy types of G_2-gauge groups over S^4.

The Carleman approach is well-known in the field of deterministic classical dynamics as a method to replace a finite number $d$ of non-linear differential equations by an infinite-dimensional linear system. Here this approach is applied to a system of $d$ stochastic differential equations for $[x_1(t),..,x_d(t)]$ when the forces and the diffusion-matrix elements are polynomials, in order to write the linear system governing the dynamics of the averaged values ${\mathbb E} ( x_1^{n_1}(t) x_2^{n_2}(t) ... x_d^{n_d}(t) )$ labelled by the $d$ integers $(n_1,..,n_d)$. The natural decomposition of the Carleman matrix into blocks associated to the global degree $n=n_1+n_2+..+n_d$ is useful to identify the models that have the simplest spectral decompositions in the bi-orthogonal basis of right and left eigenvectors. This analysis is then applied to models with a single noise per coordinate, that can be either additive or multiplicative or square-root, or with two types of noises per coordinate, with many examples in dimensions $d=1,2$. In $d=1$, the Carleman matrix governing the dynamics of the moments ${\mathbb E} ( x^{n}(t) )$ is diagonal for the Geometric Brownian motion, while it is lower-triangular for the family of Pearson diffusions containing the Ornstein-Uhlenbeck and the Square-Root processes, as well as the Kesten, the Fisher-Snedecor and the Student processes that converge towards steady states with power-law-tails. In dimension $d=2$, the Carleman matrix governing the dynamics of the correlations ${\mathbb E} ( x_1^{n_1}(t) x_2^{n_2}(t) )$ has a natural decomposition into blocks associated to the global degree $n=n_1+n_2$, and we discuss the simplest models where the Carleman matrix is either block-diagonal or block-lower-triangular or block-upper-triangular.

We study the motion of a charged test particle in the spacetime with a spherically symmetric black hole which is immersed in a monopole magnetic field. We show that the radial motion of the charged test particle is governed by completely the same equation as that in the case of no magnetic field. This result implies that the black hole will acquire the electric charge if it is surrounded by the collisionless plasma composed of protons and electrons which obey the Maxwell velocity distribution. The drastically different situation appears in the tangential motions of charged test particles due to the magnetic field. The trajectory of a charged test particle in the black hole with the magnetic field of the order of 10 Gauss near the black hole, is confined on a very thin cone as long as the specific angular momentum of the particle is not much larger than the gravitational radius of the black hole times the speed of light. This result leads to a possibility that a plasma lump can hover over the black hole and is very hot, in the monopole magnetic field.

The compatibility constant plays an important role in evaluating the prediction error of the lasso in high-dimensional settings. However, the computation of the compatibility constant is generally difficult because it is a complicated nonconvex optimization problem. In this study, we present a numerical approach to compute the compatibility constant when the support of true regression coefficients is given. We show that the optimization problem reduces to a quadratic programming (QP) once the signs of the nonzero coefficients are specified. In this case, the compatibility constant can be obtained by solving QPs for all possible sign combinations. We also formulate a mixed-integer QP (MIQP) approach that can be applied when the number of true nonzero coefficients is large. We investigate the finite-sample behavior of the compatibility constant for simulated data under various parameter settings and compare the prediction error with its theoretical upper bound. The behavior of the compatibility constant in finite samples is also investigated through a real data analysis.

Purpose: To create modular solutions for interactive real-time MRI using reconstruction algorithms implemented in BART. Methods: A new protocol for streaming of multidimensional arrays is presented and integrated into BART. The new functionality is demonstrated using examples for cardiac interactive real-time MRI based on radial FLASH, where iterative reconstruction is combined with advanced features such as dynamic coil compression and gradient-delay orrection. We analyze the latency of the reconstruction and measure end-to-end latency of the full imaging process. Results: Reconstruction pipelines with iterative reconstruction and advanced functionality were built in a modular way using scripting. Latency measurements demonstrate latency sufficient for interactive real-time MRI, on the order of 30 ms for BART processing and network transfer time, or 200 ms for end-to-end latency including acquisition, vendor processing, and display. Conclusion: With the new streaming capabilities, real-time reconstruction pipelines can be assembled using BART in a flexible way, enabling rapid prototyping of advanced applications such as interactive real-time MRI.

It has been documented that Spread-Error equality and a flat rank histogram are necessary but insufficient for demonstrating ensemble forecast reliability. Nevertheless, these metrics are heavily relied upon, both in the literature and at operational numerical weather prediction centers. In this study, we demonstrate theoretically why the Spread-Error relationship is necessary but insufficient for diagnosing reliability up to second order, even when mean bias is absent or accounted for. Assuming joint normality between ensemble members and the reference truth, we further show with idealized experiments that the same covariance structure responsible for this insufficiency also produces false diagnoses of reliability with the rank histogram and the reliability component of the continuous rank probability score. Under this structure and when the ensemble mean is meaningfully different from climatology, the truth lies among the least (most) extreme members when climatological variance is excessive (deficient) in each member. Importantly, this behavior is also shown to be plausible in operational ensemble weather forecasts. Combining these results with calibration principles from statistical postprocessing leads us to conclude that both perfect dispersion and underdispersion are ill-defined. When diagnostics are misinterpreted as indicating the latter, improper tuning can lead to further deterioration of forecast quality, even while improving Spread-Error and rank histogram behavior. To address these issues, we propose a new reliability diagnostic based on three easily computed statistics, motivated by the structure of the joint distribution of ensemble members and the reference truth up to second order. The diagnostic separates contributions to unreliability originating from climatology and predictability, enabling a more precise and robust characterization of ensemble behavior.

We explore a phenomenological model of dark energy in which space is treated as an elastic brane with uniform tension $T_s$ and supports a longitudinal phonon sector described by three scalar fields $ϕ^I$. At the background level the construction reproduces a perfect fluid whose enthalpy and bulk modulus are controlled by two dimensionless parameters $\varepsilon$ and $κ$, which encode the elastic and viscous response and determine both the effective equation of state $w_{\rm eff}(z)$ and the physical sound speed thriugh $c_s^2 = κ/\varepsilon$. Motivated by the consistency requirements of a phonon effective description, we focus on the stable regime $0 < κ< \varepsilon < 1$, which prevents tachyonic behavior and is natural given that $\varepsilon$ and $κ$ are defined as fractional response coefficients. To connect the model predictions to current observational summaries, we map $w_{\rm eff}(z)$ onto the Chevallier--Polarski--Linder form $w(z) = w_0 + w_a z/(1+z)$ over the redshift range relevant to DESI. We then use the public DESI DR1 compressed Gaussian likelihood in $(w_0,w_a)$ for BAO combined with Pantheon+ Type I a supernovae and Planck 2018, and scan the model parameter space to identify viable regions. We find that the model can reproduce the DESI compressed constraints at the background level, with representative best match values $(w_0,w_a) \simeq (-0.83,-0.74)$ and therefore, a near luminal sound speed $c_s^2 \simeq 0.9618$ as an output of the fit rather than a built in restriction. The covariance weighted distance is found to satisfy $χ^2 \ll 1$ in the $(w_0,w_a)$ plane. The associated phonon mass scale $m_ϕ= H_\star/(2π)$ is ultralight, corresponding to a Compton wavelength of order the cosmological horizon, consistent with interpreting the relevant excitations as infrared collective modes of the brane.

Introduction: We describe the foundation of PETRIC, an image reconstruction challenge to minimise the computational runtime of related algorithms for Positron Emission Tomography (PET). Purpose: Although several similar challenges are well-established in the field of medical imaging, there have been no prior challenges for PET image reconstruction. Methods: Participants are provided with open-source software for implementation of their reconstruction algorithm(s). We define the objective function and reconstruct "gold standard" reference images, and provide metrics for quantifying algorithmic performance. We also received and curated phantom datasets (acquired with different scanners, radionuclides, and phantom types), which we further split into training and evaluation datasets. The automated computational framework of the challenge is released as open-source software. Results: Four teams with nine algorithms in total participated in the challenge. Their contributions made use of various tools from optimisation theory including preconditioning, stochastic gradients, and artificial intelligence. While most of the submitted approaches appear very similar in nature, their specific implementation lead to a range of algorithmic performance. Conclusion: As the first challenge for PET image reconstruction, PETRIC's solid foundations allow researchers to reuse its framework for evaluating new and existing image reconstruction methods on new or existing datasets. Variant versions of the challenge have and will continue to be launched in the future.

We present the strategy used to identify and mitigate potential sources of angular systematics in the \textit{Euclid} spectroscopic galaxy survey, and we quantify their impact on galaxy clustering measurements and cosmological parameter estimation. We first surveyed the \textit{Euclid} processing pipeline to identify all evident, potential sources of systematics, and classified them into two broad classes: angular systematics, which modulate the galaxy number density across the sky, and catastrophic redshift errors, which lead to interlopers in the galaxy sample. We then used simulated spectroscopic surveys to test our ability to mitigate angular systematics by constructing a random catalogue that represents the `visibility mask' of the survey; this is a dense set of intrinsically unclustered objects, subject to the same selection effects as the data catalogue. The construction of this random catalogue relies on a detection model, which gives the probability of reliably measuring the galaxy redshift as a function of the signal-to-noise ratio (S/N) of its emission lines. We demonstrate that, in the ideal case of a perfect knowledge of the visibility mask, the galaxy power spectrum in the presence of systematics is recovered, to within sub-per cent accuracy, by convolving a theory power spectrum with a window function obtained from the random catalogue itself. In the case of only approximate knowledge of the visibility mask, we test the stability of power spectrum measurements and cosmological parameter posteriors by using perturbed versions of the random catalogue. We find that significant effects are limited to very large scales, and parameter estimation remains robust; the most impacting effects are connected to the calibration of the detection model.

We study two cross-linked polymer systems in the strong stretching regime. The first consists of two polymers sharing one endpoint, with the other two endpoints coupled by a harmonic potential. Within the weakly bending approximation, we analyze the tensile elastic response for freely jointed or wormlike chains; for the latter, the approximation applies either at large tension or at moderate tension with large persistence length (rodlike limit). We obtain analytic expressions for the force--extension relation and for the longitudinal and transverse mismatch of the cross-linked endpoints. In the thermodynamic limit, the cross-link does not affect the tensile elasticity, but it significantly suppresses transverse fluctuations, effectively forming a loop structure. The second system is a polymer necklace in the thermodynamic limit, composed of two strongly stretched polymers interconnected by a regular sequence of reversible cross-links. Using an analogy with a two-dimensional system of concatenated Gaussian loops ("Gaussian slinky"), we calculate the mean fraction of cross-linked sites as a function of the tensile force and find weak and strong binding regimes connected by a crossover. For shallow binding potential wells (compared with $k_{\rm{B}}T$), we employ a continuum description and exploit the mapping between directed polymers and a two-dimensional quantum particle to determine the crossover behavior and the mean transverse separation between the two polymer chains.

Transformations of covariates are widely used in applied statistics to improve interpretability and to satisfy assumptions required for valid inference. More broadly, feature engineering encompasses a wider set of practices aimed at enhancing predictive performance, and is typically performed as part of a data pre-processing step. In contrast, this paper integrates a substantial component of the feature engineering process directly into the modelling stage. This is achieved by introducing a novel general framework for embedding interpretable covariate transformations within multi-parameter Generalised Additive Models (GAMs). Our framework accommodates any sufficiently differentiable scalar-valued transformation of potentially high-dimensional and complex covariates. These transformations are treated as integral model components, with their parameters estimated jointly with regression coefficients via maximum a posteriori (MAP) methods, and joint uncertainty quantified via approximate Bayesian techniques. Smoothing parameters are selected in an empirical Bayes framework using a Laplace approximation to the marginal likelihood, supported by efficient computation based on implicit differentiation methods. We demonstrate the flexibility and practical value of the proposed methodology through applications to forecasting electricity net-demand in Great Britain and to modelling house prices in London. Methods for building and fitting GAMs with nested transformations are provided by the gamFactory R package, available at https://github.com/mfasiolo/gamFactory, while the code for reproducing the results in this paper is available at https://doi.org/10.5281/zenodo.19239350.