We study the large-time asymptotic behavior of a new KdV soliton gas. We first introduce a pure-soliton Riemann--Hilbert(RH) problem with \(2N\) poles and two different types of residue conditions. We show that, as \(N\to\infty\), this discrete problem converges to primitive-potential RH problem introduced by Dyachenko, Zakharov, and Zakharov, and the jump matrix of this soliton gas RH problem has two nonzero reflection coefficients. To analyze the large-time behavior, we apply the Deift--Zhou nonlinear steepest descent method together with an appropriate \(g\)-function mechanism. Through a sequence of transformations, the original RH problem is reduced to explicitly solvable model problems on an associated hyperelliptic Riemann surface. This allows us to derive an explicit leading-order asymptotic formula for the solution in terms of Jacobi elliptic function. The result provides a rigorous asymptotic description of a new KdV soliton gas and extends the available analysis beyond the previously studied case \(r_2\equiv 0\).

This is a continuation of the paper [Commun. Theor. Phys., 77 (2025) 115006] on the nonlinearization of bilinear equations. The sine-Gordon type and nonlinear Schrödinger type bilinear equations are introduced by Jarmo Hietarinta during his search for integrable bilinear equations. In this paper, we provide a formulation to convert these two types of bilinear equations into nonlinear forms. In addition, the nonlinearization related to the equations involving the Hilbert transformations is also considered. Bell polynomials are employed in the nonlinearization and illustrative examples are provided.

This work introduces a biophysical formalism to describe the spatiotemporal evolution of the chemical profile in tissues, with the novelty of modeling tissue compartmentalization and the mechanism by which cells maintain the system far from thermodynamic equilibrium via production and/or degradation of substances. The models were derived from conservation laws, chemical kinetic theory, and geometric constraints, while considering fundamental properties of tissues to connect theoretical modeling with experimental observations. In a morphogenetic context, each morphogen is described by two coupled reaction-diffusion equations, representing intra- and extracellular dynamics, linked through membrane transport processes such as nonlinear, cross, and anomalous diffusion. We explore the models' morphogenetic potential through diffusion-driven instabilities and discuss how natural tissue heterogeneities influence Turing instabilities and self-organized phenomena. The mathematical structure reveals that two-morphogen systems can produce Turing patterns with multiple characteristic length scales, while the system's dimensionality enables chaotic behavior in well-mixed dynamics. Moreover, due to domain coupling, Turing instabilities are allowed for single-morphogen systems. We used Schnakenberg kinetics to demonstrate that Turing patterns arise even when the activator diffuses faster than the inhibitor (d$<$1), thereby expanding the parameter space for pattern formation. Our results suggest that tissue spatial structure has important consequences for Turing instability mechanisms, in some cases weakening the usual conditions for its emergence while widening the possible patterns it can produce. The proposed framework offers a minimal mathematical basis to explore emergent dynamics in biological and synthetic contexts, with potential applications in developmental biology and tissue engineering.

We study the 1D swarmalator model augmented with self-propulsion. Each swarmalator swims along the ring at a speed $v_0\sinθ_i$ fixed by its orientation $θ_i$. Self-propulsion unfolds the static states of the ordinary model into traveling, breathing, split-wave, and chaotic states. Several of these states admit analytic reductions: an exact drifting two-cluster branch with a closed-form stability spectrum, and a four-cluster split-wave ansatz whose active pair reduces, in a constant-orientation approximation, to an Adler equation. Our numerical evidence suggests that the transition to chaos under broad random initial conditions is not caused by local destabilization of the ordered cluster branches, but by basin reorganization among coexisting attractors. The resulting states may serve as qualitative signatures for confined active oscillator arrays.

We study the one-dimensional swarmalator model in which the phase coupling $K_i'$ is drawn from a uniform distribution. Our main result is a static coupling-split cluster, in which the population partitions across the threshold $K'=0$ that separates positively coupled ($K_i'>0$) from negatively coupled ($K_i'<0$) swarmalators, with smaller order parameter $s=μ/γ$ set by the positive-coupling excess. The familiar async, phase-wave, and sync states persist, but each stability boundary feels a different part of the distribution: async the mean same-coordinate response, sync the most negatively coupled particle, and the phase wave the full density through a logarithmic characteristic equation. At a cusp where its Hopf and real-eigenvalue branches meet, the phase-wave dispersion has a double zero -- the spectral signature of a Bogdanov--Takens point -- and simulations nearby show a small-amplitude breathing limit cycle. For supports containing strongly negatively coupled particles the order parameters instead oscillate persistently.

We study how time-periodic, spatially localized states are born from the linear spectrum of a \emph{finite} lattice as the nonlinearity is switched on. In earlier work we treated this question for a diatomic chain with on-site nonlinearity and developed a framework that continues a near-edge linear mode in amplitude and controls the resulting perturbation series uniformly in the chain length. The present paper shows that the same framework applies to the more difficult case of Fermi--Pasta--Ulam--Tsingou (FPUT) interaction nonlinearity. The key is a structural relation between the FPUT and on-site nonlinearities, which allows the estimates obtained in the on-site setting to be transferred to the FPUT setting. As before, the analysis yields a quantitative radius of convergence, $\eps=Θ(1/\sqrt{n})$ for a chain of length $2n$, below which the near-edge mode stays extended and above which the orbit localizes and its frequency leaves the band. The diatomic chain is used only as a test case; both the formation mechanism and the method are model-independent and are expected to extend to other short-range nonlinearities and to higher dimensions.

Equilibrium quantum many-body methods differ not only in approximation, but in which information they represent explicitly. We formulate this distinction by fixing an equilibrium specification and viewing every representation as an encoder from admissible states to represented variables. The identity encoder gives a full-state representation, whereas a non-injective encoder gives a reduced representation whose value labels a fiber of compatible states. For a specified task, an exact decoder exists on a state class if and only if the task is constant on the encoder fibers within that class. Variational principles, reconstruction correspondences, functionals, kernels, and closures are different realizations of additional structure used to select, restrict, or approximate the task-relevant content of a fiber when the retained variable alone is insufficient. Static moments and imaginary-time correlation functions are unified as restrictions of a complete equilibrium readout functional to different probe families. Within the same principle, quantum embedding can be viewed as consistency or replacement between global and local descriptions through reduced interface encoders and their conjugate fields.

The Brownian sphere, also known as the Brownian map, is a canonical random metric measure space homeomorphic to the two-dimensional sphere $\mathbf S^2$. It can be interpreted as the uniform measure on surfaces homeomorphic to $\mathbf S^2$, in the sense that it arises as the scaling limit of many natural models of random planar maps chosen uniformly from a given class. It is also equivalent to the $\sqrt{8/3}$-Liouville quantum gravity sphere. We prove that the Brownian sphere is quasisymmetrically rigid, meaning that, almost surely, it has no nontrivial quasisymmetric automorphisms. We also show that two independent Brownian spheres are almost surely not quasisymmetrically equivalent. Our argument also gives a new proof that the conformal structure of the Brownian sphere is almost surely determined by its metric structure.

We review random matrix approaches to chaotic wave scattering and transport in open systems. Starting from the effective non-Hermitian Hamiltonian formulation, we discuss the scattering matrix, reaction matrix, time delays, and complex resonances as complementary probes of open chaotic dynamics. We emphasize universal statistics governed by symmetry, openness, and channel coupling. Topics include the maximum-entropy description of fixed-energy scattering and its applications to quantum transport, energy correlations, resonance and eigenfunction statistics, and selected wave-chaotic phenomena induced by finite absorption. The focus throughout is on non-perturbative methods and universal structures underlying open quantum and wave chaotic systems.

We study the approximation of certain renormalized conserved quantities for the NLS hierarchy with nonzero boundary conditions, in the long-wave regime, by the energies of the KdV hierarchy. We extend this to all $n \in \mathbb{N}$ by proving an approximation result for the transmission coefficient of the Lax operator of the NLS hierarchy, which is a Dirac operator in the nonrelativistic regime, by the transmission coefficient of a Schrödinger operator, which is the Lax operator of the KdV hierarchy. This yields a formal approximation result between the hierarchies, which we quantify using energy methods and previously established well-posedness results.

We prove dynamical Amrein-Berthier uncertainty principles for fractional Schrödinger flows. For the free Hamiltonian $H=(-Δ)^α$ on $L^2(\mathbb{R}^n)$, with $α>\frac{1}{2}$, we show that two--time localization on finite measure sets $E,F$ forces the quantitative estimate \begin{equation*} \|u(t)\|_{L^{2}}\lesssim_{E,F,T,n,α} \|u(0)\|_{L^{2}(E^{c})} + \|u(T)\|_{L^{2}(F^{c})}, \qquad T\neq0,\ t\in \mathbb{R} \end{equation*} for $u(t)=e^{-itH}u(0)$ at every time. The threshold $α>\frac{1}{2}$ is tied to the stationary phase structure of the fractional kernel. If $α\ge1$ the sets can be arbitrary finite measure sets; if $\frac{1}{2}<α<1$ we impose the finiteness of a natural interaction energy \begin{equation*} \textstyle \mathcal{I}_γ(E,F) = \int_{\mathbb{R}^n \times \mathbb{R}^n} \mathbf{1}_{F}(x)|x-y|^{2γ}\mathbf{1}_{E}(y)\,dx\,dy<\infty, \qquad γ= \frac{n(1-α)}{2 α-1} \end{equation*} of the pair $(E,F)$, essentially equivalent to a sufficiently fast joint decay of the measure of the sets at infinity. In particular, compact support at two distinct times is impossible for a nonzero solution. We also prove corresponding results for one dimensional fractional Hamiltonians $(-\partial_x^2+V)^α$ under weighted scattering assumptions, and for higher order Hamiltonians $(-Δ)^m+V$ for suitable classes of decaying potentials $V$.

We discuss a problem posed by A.~Agrachev asking how many times a usual circle in $\mathbb R^n$ should be traversed to admit a deformation by curves with nowhere degenerating Frenet frame. It turns out that the answer depends on a specific topology which we consider. For the literal $C^n$ curve topology, the least number of turns of a plane circle admitting arbitrarily small nondegenerate perturbations is \[ k(2)=1,\qquad k(3)=2,\qquad k(n)=1\quad(n\ge4). \] This jet-level problem is different from the original Frenet-control problem by Agrachev. We show that in the literal interpretation of Agrachev's problem one has a simple spherical Fenchel obstruction in all dimensions $n\ge4$. To retain a nontrivial turn-counting problem, we introduce decorated turn data. In $\R^4$ the datum is a pair $(p,q)$ recording tangent-plane and normal-plane turns; we prove that every nonresonant pair $(p,q)$ with $p,q>0$, $p\ne q$, is accessible by small positive constant Frenet controls. In even dimension $2r$ the analogous datum is a vector $(p_1,\ldots,p_r)$, and every vector with pairwise distinct positive entries is accessible by constant controls. Odd dimensions require genuinely time-dependent openings since constant controls cannot close the base curve.

We prove lower bounds for the Hausdorff dimensions of various natural sets associated with the Liouville quantum gravity (LQG) metric. We prove that the set of 3-star points (i.e., starting points of three disjoint geodesics) has Hausdorff dimension at least two with respect to the LQG metric, which is conjectured to be optimal. Our proof works for a general class of planar length metrics which also includes, e.g., Kendall's Poisson roads metric. We additionally prove a dimension lower bound of one for the set of 2-star points intersected with the boundary and for the metric net intersected with the boundary, as well as a dimension lower bound of two for the intersection of two metric nets. In the particular setting of LQG, we obtain sharper lower bounds for the Hausdorff dimensions of the set of 2-star points and the LQG metric net, with respect to both the Euclidean metric and the LQG metric. Our proofs are primarily topological. The key idea is to express the sets of interest in terms of non-constancy sets of Lipschitz functions.

We survey recent progress in the classification of hyperkähler and Hermitian gravitational instantons (i.e., complete noncompact 4 dimensional Ricci-flat manifolds with quadratic curvature decay), as well as the construction of non-Hermitian gravitational instantons via harmonic maps. We also present a list of open questions related to gravitational instantons.

We apply a deep-neural-network (DNN) ansatz to the symmetrized spectator vector of the subtracted three-body Faddeev integral equation for identical bosons near the Efimov limit. The network is trained by minimizing the residual of the discretized integral equation, while the positive binding scale associated with the three-body energy is treated as a trainable parameter. Deterministic diagonalization of the same discretized kernel is used only as an a posteriori numerical benchmark. As preliminary validation, the neural-solver strategy is tested on the analytically solvable hydrogen radial problem. At unitarity, the DNN reproduces the Efimov ground-state binding scale with a DNN--deterministic deviation of $0.022\%$, while the first excited state is recovered to $0.002\%$. The deterministic solver recovers the universal Efimov scaling ratio $e^{2π/s_0}\simeq 515.03$, and the neural method traces the bound-state branches as a function of the inverse scattering length $1/a$ by continuation from the unitary solution. These results indicate that DNN-based residual minimization can provide a compact and differentiable representation of a renormalized few-body integral-equation solution in a regime governed by discrete scale invariance.

We study noncommutative analogues and integrable discretisations of nonlinear Schrödinger (NLS)-type systems associated with reduction groups. In particular, we consider the Ablowitz--Kaup--Newell--Segur (AKNS) system, the Kaup--Newell derivative NLS system, and the Mikhailov--Shabat--Yamilov deformation of the derivative NLS system together with their Darboux--Bäcklund transformations and associated lattice equations. We derive the continuum limits of previously constructed integrable lattice systems and recover the corresponding NLS-type partial differential equations. We then construct a noncommutative deformation of the Mikhailov--Shabat--Yamilov system and show that, unlike the AKNS and Kaup--Newell cases, its Lax representation requires the introduction of nonlocal variables. Furthermore, we derive Darboux--Bäcklund transformations and integrable discretisations for the noncommutative derivative NLS and deformation derivative NLS systems in the form of vertex--bond lattice equations. We also construct explicit solutions for a six-point derivative NLS-type lattice equation and for its noncommutative analogue.

We derive the vertical chromaticity $ξ_y$ of the Fermilab Muon g-2 storage ring in closed analytic form. Expanding the Hamiltonian as a Taylor polynomial in the dynamical variables and integrating the equations of motion order by order, we obtain the vertical second-order aberrations of the homogeneous magnetic dipole ($\mathtt{DI}$) and the combined-function dipole-and-electrostatic-quadrupole element ($\mathtt{DIQ}$) used in the muon $g{-}2$ ring. Composing the per-element maps over the periodic dispersion orbit yields a closed-form expression for the vertical chromaticity $\xichromy$ of the continuous-ring $\mathtt{DIQ360}$ model, in direct functional analogy with the horizontal result of our earlier work on the same ring (Ref.~\refcite{ChromCPO11}). Comparison against COSY INFINITY differential-algebra computation shows agreement at the $10^{-11}$ level across all three ring models ($\mathtt{DIQ360}$ closed form and the modular $\mathtt{DIEQ\_ON}$, $\mathtt{DIEQ}$ via per-element composition) for muon $g{-}2$ electrostatic-quadrupole (ESQ) voltages $\Vesq \in [10, 26]\,\mathrm{kV}$.

We formulate a nonlinear geometric theory of visco-anelasticity that unifies viscoelastic and anelastic responses within a single thermodynamic framework. At each material point, the total deformation gradient is multiplicatively decomposed into elastic, viscous, and anelastic distortions, thereby generalizing the Bilby-Kröner-Lee decomposition to visco-anelasticity. The theory explicitly incorporates the material metric, which encodes the evolving natural configuration of the solid, the transformed structural tensors, and provides a consistent formulation of the constitutive equations, the balance laws, the thermodynamic potentials, and the kinetic equations. The first and second laws of thermodynamics are systematically applied to derive the constitutive and evolution equations without invoking observer invariance. Anisotropy is treated in full generality through structural tensors. As illustrative examples, we specialize the general framework to isotropic and transversely isotropic visco-anelastic solids. Two examples within the class of universal deformations, which admit closed or partially closed form solutions, show how the proposed framework can be used to model the coupled viscous and anelastic response of incompressible anisotropic solids with distributed eigenstrains and the associated residual stresses. This geometric framework unifies nonlinear viscoelasticity and anelasticity by coupling time-dependent and eigenstrain-driven effects within a single, fully consistent geometric formulation. In particular, the proposed framework clarifies the geometric structure of the elastic, viscous, and anelastic distortions and resolves ambiguities associated with intermediate configurations in existing formulations of nonlinear viscoelasticity and viscoplasticity.

The Fermat principle of least time determines the path of a single ray of light given its initial and final positions and the medium electromagnetic properties. However, a single ray is not a measurable physical object. We derive here an upgraded variational principle for the geometric optics limit of Maxwell equations in an arbitrary medium, based on measuring the intensity of the wave on two planes. The principle provides the complete Fresnel rays bundle connecting associated points at the first and second planes. One of the applications of the present theory is to use the reciprocity between Fresnel rays and phase normals to determine the phase of an electromagnetic wave from two intensity measurements.

We develop a contact Tulczyjew formalism for dissipative dynamics on skew algebroids. Starting from the Tulczyjew morphism of an skew algebroid, we identify its contact extension in a local line-bundle trivialization. The local representative is obtained by adding to the ordinary Tulczyjew morphism the Euler vector field contribution on $E^*$. This gives an intrinsic explanation of the contact term appearing in the local contact Tulczyjew morphism. For a contact generating object, the construction produces an implicit dissipative dynamics on the contact phase side. In local coordinates, the matching condition gives the Euler-Lagrange-Herglotz equations on the skew algebroid. In the hyperregular case, the corresponding contact Hamiltonian equations are recovered by Legendre transformation. We also develop the discrete counterpart of the construction. After fixing a discrete admissibility relation, a discrete contact generating object defines a discrete contact Tulczyjew relation on the contact phase space. Discrete Herglotz extremals are obtained by matching consecutive contact momenta, with the usual conormal interpretation in the constrained case. In the regular tangent-bundle case, this recovers standard contact variational integrators, while in the singular or skew algebroid setting the same construction remains meaningful as an implicit discrete relation rather than an a priori update map.

The freedom to change Fock basis seems to ensure a minimum amount of locality in lattice theories in the following sense: If $\lbrace (\hat a_i^\dagger\,,\,\hat a_i)\rbrace$ for $i=1,\dots,n$ is a lattice of creation and annihilation operators and if a given Hamiltonian $\hat H$ induces highly non-local dynamics on that lattice, then it will usually be possible to change to a new set of operators $\lbrace (\hat b_i^\dagger\,,\,\hat b_i)\rbrace$ in terms of which the dynamics appear less non-local. We demonstrate this by turning a highly non-local random matrix model into a local, 1D lattice theory where particles can propagate in localized wave packets. More generally, we show that any Hamiltonian can be made to look like such a theory, with the lattice dispersion relation and the non-integrability of the theory depending on the spectrum of $\hat H$. We argue that our results are a step towards quantum mereology for fields.

Under suitable assumptions, geometric objects such as the spacetime metric and curvature tensor can be reconstructed from the algebra of operators of quantized matter fields in the limit of vanishing Newton's constant. In this framework, the full non-linear Einstein equations can be expressed in the language of operator algebras, extending Jacobson's derivation without invoking the area law for Bekenstein-Hawking entropy. These assumptions can then be used as a criterion for determining whether the semiclassical limit of a given quantum theory admits an emergent gravitational description. Going in the other direction, the discrete spectrum of a holographic theory at finite N can be modelled by adding non-perturbative corrections to semiclassical operator algebras. The type III von Neumann algebra that arises in the vanishing Newton's constant limit can be enlarged by adjoining its modular Hamiltonian. A random matrix theory completion of this enlarged algebra, followed by ensemble averaging, results in a type I von Neumann algebra whose minimal projectors approximate those of the underlying microstates. In the case of an eternal black hole, the dimension of the type I algebra equals the Bekenstein-Hawking entropy with universal logarithmic corrections. The complexity of probe operators in the boundary theory provides a diagnostic of the validity of the corresponding bulk semiclassical effective field theory.

We study the pair creation phenomenon for a real scalar field $φ$ in the presence of a surface that undergoes time-dependent deformations, while imposing Dirichlet-like boundary conditions. Including terms up to fourth order in the departure of the surface from an infinite plane, we present results for the angular dependence of the emission rate for the vacuum-to-pair process as a function of the geometry and the dynamics of the surface, as well as of the momenta of the emitted pair. We check the consistency of the leading contribution with previous results obtained from the imaginary part of the effective action, and clarify how the relation between exclusive probabilities and the imaginary part of the effective action is modified at fourth order by the opening of a two-pair channel.

We consider quantization techniques with a particular emphasis on the phase space quantization, as encountered in quasi-probabilistic descriptions and in string theory. We investigate what is lost by adopting the normal ordering prescription, by revisiting fundamental quantization concepts and the correspondence rules that define the well-known quasi-distributions: the Wigner, Q, and P functions, some of which are inherently ordering dependent. The actual quantum state dictates the necessity of employing discrete countable structures or, alternatively, the complex phase space formulation based on coherent states, and the quasi-distributions depict the existence of the phase space dynamics, of the wave translations and the propagating directions.

We derive new 3d $\mathcal{N}=2$ dualities with $\mathrm{U}(N)$ gauge groups involving pairs of two-index tensors interacting through a quartic superpotential, (anti)-fundamentals and possibly an adjoint. A strong hint for their validity follows from T-duality on brane setups with O6 planes and stacks of multiple NS fivebranes. These setups engineer two families of 4d models known in the literature as mezzanine and balcony of type $A_k$ and $D_{k+2}$. In 4d these models admit a generalization of Seiberg duality, tested also at the brane level. We study the reduction of such dualities from both the brane and the field theory perspective and we establish the matching of the three-sphere partition functions, assuming the validity of the 4d superconformal indices. The latter is achieved through a double scaling limit, whose structure is dictated by the brane picture, and it requires the validity of new 3d confining dualities for quivers with two gauge nodes. Moreover we provide a proof of these confining dualities via the tensor deconfinement technique.

We study supersymmetric BPS branes (BPS-B) in supergravity theories. Some of these states are anticipated by BPS black-brane (BPS-BB) solutions of supergravity. In particular, we define and distinguish the cone generated by BPS branes from the subcone of charges that admit BPS black-brane attractor solutions in the infrared limit of the supergravity effective field theory. We denote these cones by $C_{\rm BPS-B}$ and $C_{\rm BPS-BB}$, respectively. We conjecture that, in any supersymmetric theory of quantum gravity, every integrally charged state lying in $C_{\rm BPS-BB}$ is realized by a BPS state in the spectrum. Furthermore, we conjecture and present evidence that when $C_{\rm BPS-B}$ is moduli independent, it can be determined as the cone dual to the $C_{\rm BPS-BB}$ under the electric-magnetic pairing.

The Weil-Petersson (WP) volumes of the (compactified) moduli space of ${N}=1$ supersymmetric Riemann surfaces with Neveu-Schwarz (NS) boundaries are frequently discussed in the literature. Such surfaces can also have marked points called Ramond (R) punctures, where the superconformal structure degenerates. Computing the volumes when these R punctures are included is more challenging for the usual differential and algebraic geometry approaches, and they are therefore less well explored. In particular, the spectral curve describing the inclusion of R punctures is apparently unknown, so far. However, the right random matrix model approach can handle the NS and~R sectors on an equal footing. Such a construction is presented, showing how to use a recently developed technique to readily compute many closed-form formulae for $V^{(2m)}_{g,n}(\{b_i\})$, the WP volumes for genus $g$ with $n$ NS-boundaries of geodesic lengths $b_i$ ($i{=}1,\ldots,n$), and $2m$ R-punctures. Several striking relations between volumes (and subsectors thereof) emerge naturally in this approach. Moreover, the hitherto missing spectral curve is presented, and its use for (re-)deriving the $V^{(2m)}_{g,n}(\{b_i\})$ is demonstrated by using topological recursion.

We continue the tree-level S-matrix bootstrap program for quantum gravity, now in ten-dimensional theories with half-maximal supersymmetry. This setting includes the heterotic string and allows us to test whether the string-like structures found in the maximally supersymmetric bootstrap persist with less supersymmetry and in non-planar gauge sectors. Imposing analyticity, crossing symmetry, unitarity, and Regge boundedness, we constraint weakly coupled UV completions of supergravity and super Yang-Mills. In the gravitational sector, much of the boundary of the allowed region is explained by extremal amplitudes supported on a single linear Regge trajectory, or by convex combinations of such amplitudes. In the gluon sector, the non-planar problem reveals a tension between representation-channel positivity and the trace decomposition of the EFT data, obstructing a direct normalization by $G$ or $g_{\rm YM}$; a coupled gluon/graviton bootstrap may be necessary to directly constrain the relative strength of gauge and gravitational interactions. Overall, our results support the emergence of linear Regge trajectories as a robust feature of the tree-level quantum-gravity bootstrap.

The superstring amplitude's Mellin transform in energy is computed at fixed momentum transfer. In the forward limit, it is shown that this transformed amplitude reduces to the Riemann zeta function, while for $t\neq 0$ it represents a deformation of $ζ$. This object exhibits remarkable mathematical properties as a result of physical attributes of the string amplitude. For integer subtractions, the dispersion relation defined by the Mellin transform yields the effective field theory expansion of the string amplitude order by order in $s$ at finite $t$, for which a new closed-form expression is derived.

We compare realistic $N$-body wave functions obtained from solutions of the Schrödinger equation with Gaussian ansätze constrained to the same rms radius. The microscopic wave functions exhibit significantly broader spatial distributions, revealing pronounced non-Gaussian structures. In addition, we investigate possible production channels for $A=4$ clusters using a phenomenological two-body interaction. This study provides a potential mechanism that may help alleviate the underestimation of $A=4$ cluster yields in theoretical models compared to experimental data.