We report the observation of charge-exchange collisions between trapped calcium monohydride molecular ions ($^{40}$CaH$^+$) and ultracold potassium atoms ($^{39}$K) in a hybrid ion-atom trap. The measured charge-exchange rate coefficient is significantly suppressed relative to the Langevin rate constant for the system. We use $\mathit{ab\ initio}$ quantum-chemical calculations to model the (CaH-K)$^+$ complex in the ground and excited electronic states and to identify possible charge-exchange mechanisms. Our calculations rule out a direct non-radiative charge-exchange reaction and instead point to a radiative mechanism, but do not quantitatively reproduce the measured rate, highlighting the need for a full-dimensional quantum dynamics treatment that includes vibrational motion and intermediate complex formation. Our work demonstrates that cold hybrid ion-atom platforms with molecular ions enable access to richer chemical complexity and collisional dynamics inaccessible in purely atomic systems.

Brittle materials subjected to thermal shocks experience strong temperature gradients that in turn give rise to mechanical stresses that can be large enough to induce fracture. This work presents a complete model for phase-field fracture for coupled thermo-mechanical problems, wherein the bulk material properties, the material strength, and the fracture toughness are specified independently. The capabilities of the model are assessed across a wide span of scenarios in thermo-mechanical fracture, from the propagation of large pre-existing cracks to crack nucleation under spatially uniform states of stress. In particular, we revisit the controlled quenching of glass plates, and demonstrate how the model captures experimentally observed crack patterns across a range of thermal loads. Ceramic disks subjected to infrared radiation are also examined, with the model reproducing both straight cracks in notched specimens and branching in intact specimens. Finally, ceramic pellets subjected to rapid power pulses are examined, with the model explaining experimental transitions from intact to fractured pellets and the important role of material strength. The results demonstrate that the complete phase-field model unifies the treatment of distinct fracture scenarios under thermal shock, surpassing classical approaches and enabling more reliable prediction of brittle fracture in extreme environments.

By modelling the upper layers of the Sun in terms of a two-layer setup where a free-surface exists within the computational domain, we numerically study the interaction between the surface gravity, or the fundamental ($f$) mode, and the magnetic fields. Earlier such works were idealized in the sense that the static magnetic fields were imposed below the photosphere, i.e., the free-surface, to detect signatures of sub-surface magnetic fields and flows on the $f$-mode. In this work, we perform three-dimensional (3D) numerical simulations where the interior fluid below the photosphere is stirred helically at small scales, thus facilitating an $α^2$-dynamo. This allows us to investigate how these self-consistently generated large-scale magnetic fields influence the properties of the $f$-mode. We find that when the magnetic fields saturate near the equipartition values with the turbulent kinetic energy of the flow, the $f$-mode is significantly perturbed. Compared to the non-magnetic case, or the kinematic phase of the dynamo when fields are too weak, we note that the frequencies and the strengths of the $f$-mode are enhanced in presence of saturated magnetic fields, with these effects being larger at larger wavenumbers. This qualitatively confirms the earlier findings from observational and numerical works which reported the $f$-mode strengthening due to strong sub-surface magnetic fields.

Lax-Wendroff methods combined with discontinuous Galerkin/flux reconstruction spatial discretization provide a high-order, single-stage, quadrature-free method for solving hyperbolic conservation laws. In this work, we introduce automatic differentiation (AD) for performing the Cauchy-Kowalewski procedure used in the element-local time average flux computation step (the predictor step) of Lax-Wendroff methods. The application of AD is similar for methods of any order and does not need positivity corrections during the predictor step. This contrasts with the approximate Lax-Wendroff procedure, which requires different finite difference formulas for different orders of the method and positivity corrections in the predictor step for fluxes that can only be computed on admissible states. The method is Jacobian-free and problem-independent, allowing direct application to any physical flux function. Numerical experiments demonstrate the order and positivity preservation of the method. Additionally, performance comparisons indicate that the wall-clock time of automatic differentiation is always on par with the approximate Lax-Wendroff method.

We present a data-driven inverse construction of the dilaton field in a bottom-up AdS/QCD description of heavy vector quarkonia. Instead of adopting an \emph{ad hoc} analytic ansatz, we use a multilayer perceptron to learn \(Φ'(z)\) as a smooth function of the holographic coordinate, with \(Φ(0)=0\) imposed to ensure ultraviolet consistency. The dilaton and its derivatives obtained by automatic differentiation generate the holographic potential \(U(z)\), and the associated Schrödinger-like equation is discretized and diagonalized to extract the low-lying eigenmodes. Masses and decay constants are then evaluated from the eigenvalues and the near-boundary behavior of the bulk-to-boundary modes. Training on PDG data for charmonium and bottomonium yields a non-quadratic dilaton profile that resolves the longstanding difficulty of simultaneously reproducing both the heavy-quarkonium spectrum and the monotonic suppression of leptonic decay constants with radial excitation. The combined fit achieves RMS deviations of \(1.26\%\) (charmonium) and \(3.32\%\) (bottomonium). This work establishes neural-network reconstruction as a flexible tool for holographic modeling and provides a basis for future extensions incorporating additional channels, lattice constraints, or finite-temperature backgrounds.

In many matching markets--such as athlete recruitment or academic admissions--participants on one side are evaluated by attribute vectors known to the other side, which in turn applies individual \emph{salience vectors} to assign relative importance to these attributes. Since saliences are known to change in practice, a central question arises: how robust is a stable matching to such perturbations? We address several fundamental questions in this context. First, we formalize robustness as a radius within which a stable matching remains immune to blocking pairs under any admissible perturbation of salience vectors (which are assumed to be normalized). Given a stable matching and a radius, we present a polynomial-time algorithm to verify whether the matching is stable within the specified radius. We also give a polynomial-time algorithm for computing the maximum robustness radius of a given stable matching. Further, we design an anytime search algorithm that uses certified lower and upper bounds to approximate the most robust stable matching, and we characterize the robustness-cost relationship through efficiently computable bounds that delineate the achievable tradeoff between robustness and cost. Finally, we show that for each stable matching, the set of salience profiles that preserve its stability factors is a product of low-dimensional polytopes within the simplex. This geometric structure precisely characterizes the polyhedral shape of each robustness region; its volume can then be computed efficiently, with approximate methods available as the dimension grows, thereby linking robustness analysis in matching markets with classical tools from convex geometry.

Tantalum pentoxide (Ta2O5) is a polymorphic wide-bandgap semiconductor with outstanding dielectric properties and widespread use in optical and electronic technologies. Its rich structural diversity, arising from multiple polymorphs accessible under different synthesis conditions, has made Ta2O5 a long-standing subject of interest. However, a unified understanding of the thermodynamic stability and phase transitions of its polymorphs across pressure-temperature (P-T) space has remained elusive. Here, using first-principles calculations, we map the thermodynamic landscape of Ta2O5 and establish a comprehensive P-T phase diagram together with a phase-stability hierarchy. We find that Gamma-Ta2O5 and B-Ta2O5 dominate the phase diagram over a broad range of P-T conditions: Gamma-Ta2O5 is stabilized at low pressures, while B-Ta2O5 becomes thermodynamically favored at higher pressures up to ~ 60 GPa, beyond which Y-Ta2O5 emerges as the most stable phase. Crucially, the zero-point energy (ZPE), one aspect of nuclear quantum effects (NQEs), plays a significant role in determining relative phase stability, contributing substantially to the Gibbs free energy and altering phase boundaries. A re-entrant phase transition between Gamma and B-Ta2O5 is predicted near ~ 2 GPa, revealing unexpected complexity in the phase behavior of this oxide. More generally, we identify a characteristic temperature (T_0), at which zero-point and thermal phonon contributions to the free energy become comparable, and show that T_0 is approximately one-third of the Debye temperature. This relationship provides a simple, physically transparent criterion for assessing the importance of NQEs in phase stability, with implications extending beyond Ta2O5 to a broad class of complex oxides.

This paper elaborates on the implications of the relationship between the Second and Third Laws and provides a comprehensive formal and historical justification for the logical redundancy of the Nernst heat theorem. By revisiting the Nernst-Einstein debate, the underlying hypotheses that lead to the traditional view of the Third Law as an independent postulate are examined. It is argued that the historical rejection of Nernst's proof -- motivated by Einstein's insistence on the practical non-performability of cycles at absolute zero -- overlooks the fact that a universal Second Law already precludes such cycles, rendering an independent Third Law an unnecessary complexity. Ultimately, the Nernst theorem is shown to be an essential consistency regulator rather than an independent physical discovery.

The paper solves the problem of continuum limit in bosonic thermal coherent-state path integrals. For this purpose, exact discrete versions of the path integral are constructed for three different orderings of the Hamiltonian: normal, anti-normal and symmetric (Weyl order). Subsequently, their different continuum versions are checked on the harmonic oscillator, to choose the symmetric ordering as a possibly correct choice for all polynomial Hamiltonians. Spotted mathematical subtleties in the simple case serve as a clue to the general solution. Finally, a general justification for the symmetric order is provided by deriving the continuum path integral starting from the exact discrete case using a renormalization procedure in the imaginary time frequency domain. While the role of Weyl order has already been found, the paper provides the missing proof of its suitability for every polynomial Hamiltonian and simplifies the previously established construction by referring only to creation and annihilation operators (without position and momentum operators).

Constructing accurate and computationally efficient surrogate models (or emulators) for predicting dynamical system responses is critical in many engineering domains, yet remains challenging due to the strongly nonlinear and high-dimensional mapping from external excitations and system parameters to system responses. This work introduces a novel Function-on-Function Nonlinear AutoRegressive model with eXogenous inputs (F2NARX), which reformulates the recently proposed $\mathcal{F}$-NARX method from a function-on-function regression perspective. The proposed framework substantially improves predictive efficiency while maintaining high accuracy. By combining principal component analysis with Gaussian process regression, F2NARX further enables probabilistic predictions of dynamical responses via the unscented transform in an autoregressive manner. Such probabilistic prediction capabilities further facilitate active learning for first-passage probability evaluation. The effectiveness of the method is demonstrated through case studies of varying complexity. Results show that F2NARX outperforms state-of-the-art NARX model by orders of magnitude in efficiency while achieving higher accuracy in general. Meanwhile, the active learning approach enables accurate estimation of first-passage failure probabilities for dynamical systems using only a small number of training time histories.

We consider image registration as an optimal control problem using an optical flow formulation, i.e., we discuss an optimization problem that is governed by a linear hyperbolic transport equation. Requiring Lipschitz continuity of the vector fields that parametrize the transformation leads to an optimization problem in a non-reflexive Banach space. We introduce relaxations of the optimization problem involving smoothed maximum and minimum functions and appropriate Orlicz spaces. To derive well-posedness results for the relaxed optimization problem, we revisit and establish new existence and uniqueness results for the linear hyperbolic transport equations. We further discuss limit considerations with respect to the relaxation parameter and discretizations.

We consider spherically symmetric Yang-Mills equations with gauge group $SO(d)$ in $d+1$ dimensional Minkowski spacetime. For any given odd $d\geq 11$, we establish existence and uniqueness (modulo reflection symmetry) of exactly $N$ smooth self-similar solutions, where $N$ is the number of zeros of an explicit polynomial $P_m(z)$ of degree $m=(d-5)/2$ in the interval $0<z<1$. The number $N$ can be determined algorithmically by an explicit computation. Our extensive computations for large odd dimensions suggest that $N=3$ for all odd $d\geq 11$. Two of these self-similar solutions admit closed-form expressions: one has been known previously, while the other appears to be new. Our result points toward a relatively simple landscape of possible blowup scenarios for high-dimensional Yang-Mills equations. Beyond its purely mathematical interest, this rigidity of self-similar blowup may also be relevant from a physical perspective, as it constrains the possible ultraviolet dynamics of non-abelian gauge fields in higher-dimensional Yang-Mills theories arising in string-inspired extra-dimensional setups and in holographic models.

We propose a Bayesian framework for feedback boundary control of hyperbolic balance laws. The method propagates a probability distribution over feedback parameters using Lyapunov decay estimates as a likelihood. For linear models, it recovers available analytical stability results and extends to nonlinear regimes where theory is limited. Using first-order local Lax-Friedrichs (LLF) discretizations, we validate the approach on the decoupled wave system and the linearized Saint-Venant equations, reproducing known stability intervals and mixed boundary couplings. We then treat nonlinear and stochastic problems, including the nonlinear Saint-Venant system, one- and two-dimensional Burgers equations, Burgers equation with random initial data, and nonconservative perturbations with source terms, and show that the inferred stability domains are robust with respect to the indicator and the prior. Finally, we demonstrate transfer to a second-order semi-discrete LLF scheme and to a two-parameter feedback model for laser powder bed fusion with power regulation.

When a hidden Markov model permits the conditional likelihood of an observation given the hidden process to be zero, all particle simulations from one observation time to the next could produce zeros. If so, the filtering distribution cannot be estimated and the estimated parameter likelihood is zero. The alive particle filter addresses this by simulating a random number of particles for each inter-observation interval, stopping after a target number of non-zero conditional likelihoods. For outlying observations or poor parameter values, a non-zero result can be extremely unlikely, and computational costs prohibitive. We introduce the Frankenfilter, a principled, partially alive particle filter that targets a user-defined amount of success whilst fixing lower and upper bounds on the number of simulations. The Frankenfilter produces unbiased estimators of the likelihood, suitable for pseudo-marginal Metropolis--Hastings (PMMH). We demonstrate that PMMH with the Frankenfilter is more robust to outliers and mis-specified initial parameter values than PMMH using standard particle filters, and is typically at least 2-3 times more efficient. We also provide advice for choosing the amount of success. In the case of n exact observations, this is particularly simple: target n successes.

We study the phase structure of the (3+1)-dimensional cold and dense QCD with the Kogut--Susskind quark in the strong coupling limit using the tensor renormalization group method. The chiral and nuclear transitions are investigated by calculating the chiral condensate and the quark number density as a function of the chemical potential. For a fixed temporal extent $N_τ=8$, we determine the critical quark masses $m_c^χ$ and $m_c^{n}$ for the chiral condensate and the quark number density, respectively, at which the first-order phase transition terminates with the vanishing discontinuity in thermodynamic quantities. We find that both quantities at the same quark mass exhibit a discontinuity at the same chemical potential, and the resulting critical quark masses are consistent with each other. We also compare our results for the critical quark masses with those obtained from the Monte Carlo simulation in the dual formulation and from the mean-field analysis. We further confirm the first-order phase transition at finite quark mass on a $1024^4$ lattice, which is essentially in the thermodynamic limit at zero temperature, as expected from the mean-field analysis.

In this paper we build a bijection between the meet-irreducible elements of the lattice of the compositions of $n$ with parts in $[1,p]$ equipped with the dominance order, and the edges of the $(n,p)$-Turán graph. Using this bijection, we then compute asymptotically the average value of some statistics on those meet-irreducible compositions.

We prove that, if $G$ is a finite almost simple group and $H$ is a maximal subgroup of $G$, then the $10$th term of the derived series of $H$ is perfect. The same is true if $G$ is perfect and $H$ is core-free. The constant $10$ is best possible.

Semi-Inclusive Deep Inelastic Scattering (SIDIS) is a key tool for exploring the three-dimensional structure of the nucleon through Transverse Momentum Dependent parton distributions and fragmentation functions. While leading-power contributions to the SIDIS cross-section are well established, next-to-leading (NLP) of order $1/Q$ and next-to-next-to-leading power (NNLP) corrections of order $1/Q^2$ to the hadronic tensor have only recently begun to be systematically investigated. These corrections are essential for the reliable phenomenology and interpretation of modern high-precision data. In recent papers by one of the authors, NNLP corrections to Drell-Yan process were derived using rapidity factorization formalism. In the present work we extend this approach to SIDIS and obtain analytic expressions for the unpolarized structure functions. We derive NNLP corrections that include convolutions of unpolarized distributions, $f_1$, with unpolarized fragmentation functions, $D_1$, and Boer-Mulders functions, $h_1^\perp$, with Collins fragmentation functions, $H_1^\perp$. We compare our results with previous formulations, provide numerical studies, confront our predictions with HERMES and COMPASS measurements, and present predictions for future experiments at Jefferson Lab and the Electron-Ion Collider.

Aliphatic hydrocarbons, such as normal alkanes, constitute a naturally abundant source of carbon atoms. Of special interest is the formation of cyclic and aromatic products from aliphatic reactants. Combining scanning tunneling microscopy and ab initio calculations, we investigate the thermal induced aromatization of linear n octane molecules on the catalytic Pt(111) surface and the reactions of intermolecular homocoupling between them at temperatures above 600 K. The cycloaromatization of individual n octane molecules requires bending the linear adsorbates prior to their dehydrogenation and the formation of an intramolecular C-C bond, yielding adsorbed benzene rings. In addition, the Pt(111) surface catalyzes a homocoupling reaction by initiating the formation of a C-C bond between the dehydrogenated methyl ends of the chemisorbed n octane molecules and then propagating along the carbon backbone in a zipper like fashion. Our findings provide molecular level insight into the heterogeneous catalytic processes underlying the generation of aromatic products and stable on surface polycyclic species.

The recent Belle II evidence for $B^+ \to K^+ ν\barν$, combined with a suppressed branching fraction ratio $R \equiv \mathcal{B}(B^0 \to K^{*0} ν\barν) / \mathcal{B}(B^+ \to K^+ ν\barν)$, necessitates new physics contributing to both left- and right-handed vector operators. We perform a systematic topological classification of one-loop completions that radiatively generate both operators without tree-level mediation, and construct two minimal benchmark scenarios: a scalar-rich model and a fermion-rich model. Evaluating these frameworks under specific benchmark mass schemes and four distinct flavor structures, we find a generic anti-correlation where enhancing one decay channel typically suppresses the other. A notable exception is a decoupled flavor configuration within the scalar-rich model, which yields simultaneous constructive interference, reducing $R$ by $4.22\%$ while satisfying all complementary flavor bounds. Under a universal coupling assumption, the scalar-rich model yields exact cancellation in the charged mode, highlighting the non-trivial interplay between flavor structure and loop-generated Wilson coefficients.

In the 1980's K.S. Brown proved that the Houghton group $H_n$ is of type $\operatorname{F}_{n-1}$ but not $\operatorname{FP}_n$. We show that, provided $n\ge3$, the same conclusion holds for all subgroups $G$ of $H_n$ that are 'large' in the sense that there is an epimorphism $G\twoheadrightarrow\mathbb{Z}^{n-1}$. Our research leads naturally to the study of generalised permutational wreath products in which the base of the wreath product is a direct product of finite groups which are allowed to vary in isomorphism type from one orbit to another. Such generalised wreath products arise naturally amongst the large subgroups of Houghton groups and are accommodated by a generalised Jordan--Wielandt theorem.

Robust output regulation for linear time-varying systems has remained an open problem for decades. By augmenting the classical immersion viewpoint, we propose the trajectory-matching system immersion framework. It reformulates the regulator equation as a forced system, and demonstrates that finding an internal model is equivalent to reproducing the non-decaying output trajectories of this forced system by constructing an unforced one. This perspective yields an exact algebraic boundary for finite-dimensional internal models, termed finite linear parameterization. It further reveals a distinctive obstruction in time-varying systems: even highly structured, finite-dimensional affine parametric uncertainties can generate infinite-dimensional families of non-decaying error-zeroing signals, thereby precluding exact robust regulation via linear finite-dimensional internal models in general. Hence, we develop a comprehensive approximate robust design, which yields a bounded tracking error that can be arbitrarily small, and avoids explicitly solving the regulator equation. Additionally, it recovers exact regulation when the uncertainty influences the system in some specified ways. Overall, these results clarify the intrinsic limitation of exact finite-dimensional robust regulation for uncertain LTV systems, and provide a general, executable framework for constructing an internal model-based design.

Predicting polymer glass transition temperatures (Tg) with first-principles fidelity has long remained out of reach, as cooling multi-thousand-atom systems over a broad temperature range at acceptable rates exceeds the computational limits of ab initio molecular dynamics (AIMD). Here we employ a hybrid scheme that merges AIMD with accelerated on-the-fly (OTF) machine-learned force-field (MLFF) construction, enabling Tg prediction at quantum-mechanical accuracy with near-classical computational cost. The OTF protocol to construct MLFFs adaptively triggers first-principles calculations only when newly encountered configurations lie outside the current model's domain of confidence, allowing robust, parameter-free MLFFs to be built from merely 1000 AIMD-sampled configurations per polymer. These MLFFs are then utilized to perform long-time cooling simulations on amorphous supercells containing several thousand atoms. Applied across twelve polymers spanning aromatic, aliphatic, heteroatomic, and branched chemistries, the method yields predictions in excellent accord with experiment while reducing computational cost by approximately six orders of magnitude relative to AIMD. This work establishes a new paradigm for predictive polymer modeling, demonstrating that OTF-MLFFs provide a generalizable, accurate, and scalable route to simulating the thermophysical behavior of complex disordered materials at near quantum-mechanical fidelity.

We investigate the holographic mixed-state entanglement measures in the Einstein-Maxwell-Scalar (EMS) theory. Several quantities are computed, including the holographic entanglement entropy (HEE), mutual information (MI), entanglement wedge cross-section (EWCS), and butterfly velocity ($v_B$). Our findings demonstrate that these measures can effectively diagnose phase transitions. Notably, EWCS and MI, as mixed-state entanglement measures, exhibit behavior opposite to that of the HEE. Additionally, we study the butterfly velocity, a dynamic quantum information measure, and observe that it behaves differently from the static quantum information measures. We analyze the butterfly velocity and find that its non-monotonic behavior arises from the competition between two contributions in its expression, which the analytic structure suggests may be correlated with distinct physical interpretations. Moreover, we examine the scaling behavior of the holographic entanglement measures and find that all the critical exponents are equal to $1$, which is twice that of the scalar field. We also explore the inequality between EWCS and MI, noting that the growth rate of MI consistently exceeds that of EWCS during phase transitions. These features are expected to be universal across thermodynamic phase transitions, with the inequalities becoming more significant as one moves away from the critical point.

In this paper, we present a new spectral deferred corrections (SDC) method to solve semi-explicit differential-algebraic equations (DAEs) with the ability to be parallelized. The new scheme restricts numerical integration to differential equations. In Y. Xia et al. (2007), it was shown that each correction elevates the order of the solution by one. We show that this carries over to the new SDC scheme. The derivation of the method combines the approach of SDC and the idea to enforce the algebraic constraints without numerical integration as shown in the $\varepsilon$-embedding method by E. Hairer and G. Wanner (1996). Keeping the algebraic equations as an implicit condition of the system allows an efficient solve of semi-explicit DAEs with high-accuracy. The proposed scheme is compared with other DAE methods. We demonstrate that the proposed SDC scheme is competitive with Runge-Kutta methods for DAEs in terms of accuracy and its parallelized versions are very efficient compared to their associated sequential SDC variants.

We consider higher-point generalizations of the "octagon" large-charge four-point function in planar N=4 super Yang-Mills theory. These n-point polygon correlators are defined as ten-dimensional null limits of generating functions of n-point correlators of protected scalar operators with arbitrary R-charge, and are conjecturally dual to massive scattering amplitudes of W-bosons on the Coulomb branch of the theory. We present a twistor-based direct computation of the polygon loop integrands, and determine the two-loop integrands for any number of external operators. The result is expressed as a linear combination of conformal integrals, with coefficients that are rational functions of the spacetime distances and operator polarizations. We obtain an exact match with a one-loop integrability computation, and study the near-massless limit of the pentagon function.

We study the interplay of information and prior (mis)perceptions in a Phelps-Aigner-Cain-type model of statistical discrimination in the labor market. We decompose the effect on average pay of an increase in how informative observables are about workers' skills into a non-negative instrumental component, reflecting increased surplus due to better matching of workers with tasks, and a perception-correcting component capturing how extra information diminishes the importance of prior misperceptions about the distribution of skills in the worker population. We sign the perception-correcting term: it is non-negative (non-positive) if the population was ex-ante under-perceived (over-perceived). We then consider the implications for pay gaps between equally-skilled populations that differ in information, perceptions, or both, and identify conditions under which improving information narrows pay gaps.

Learning, inference, memory, and emergence in biological and artificial systems are often described using disparate theoretical frameworks. Here we develop a cognitive field theory in which cognition is described as a collective nonequilibrium phenomenon governed by the geometry and collective spectrum of a learned cognitive manifold. Starting from a stochastic cognitive-field equation on an adaptive Riemannian manifold, we derive an effective cognitive field theory incorporating nonlocal memory kernels and retarded self-energy feedback. The learned cognitive geometry generates a complex collective spectrum characterized by the time-scale density of states $ρ(λ,ω)$, whose relaxation and circulation sectors govern memory persistence and temporal coherence. Integrating out latent slow collective modes produces non-Markovian memory feedback that renormalizes the cognitive forgetting gap $r_{\rm cog}$, enhances collective susceptibility, and drives the system toward a protected near-critical regime characterized by long-time contextual persistence and scale-free temporal organization. The observable cognitive field emerges as a macroscopic order parameter, $ϕ=Ae^{iψ}$, whose amplitude encodes collective cognitive organization and whose phase encodes temporal coherence across distributed collective modes. Within this framework, learning organizes cognitive geometry, cognitive geometry generates a collective spectrum, and the resulting memory feedback stabilizes a memory-dressed cognitive field. The theory provides a unified dynamical description of learning, memory, inference, selfhood, and emergent intelligence in terms of the infrared organization of collective cognitive dynamics.

In this paper, we consider the Dirichlet problem for a class of prescribed Hessian quotient type curvature equations in Minkowski space. For non-convex domains, we prove the existence theorem by establishing the \emph{a priori} estimates without subsolution assumption and Serrin-type condition.

Single-photon polymerization ensures rapid photopolymerization of centimeter-scale structures with features on the order of tens of micrometers, whereas 2PP provides sub-micrometer features at sub-millimeter scales. Existing hybrid approaches combining these techniques typically rely on stitched or layer-by-layer fabrication and often require separate printing platforms, making mesoscale manufacturing time-consuming. Here, we introduce a hybrid unified 3D printer that leverages the complementary strengths of both printing mechanisms to bridge this scale resolution-fabrication time gap. We propose integrating 2PP for high-resolution, localized spatial control with single-photon TVAM for enabling rapid, high-throughput 3D fabrication. In this approach, TVAM first forms millimeter-scale volumetric structures attached on a glass rod, via overprinting, which is then accessible, on the same platform, for subsequent high-resolution 2PP. Without needing to change the photoresin or introducing intermediate post-processing steps, we proceed to demonstrate finely printed structures via 2PP, fabricated both inside (embedded within) and on the surface of the millimeter-scale 3D objects printed with TVAM. Here, TVAM contributes in two distinct ways: by generating a pre-polymerized volume that facilitates subsequent 2PP, and by directly driving layer-less volumetric polymerization in designated regions within seconds. We experimentally demonstrate that this dual-mode strategy provides a mesoscale approach spanning four orders of magnitude in scale for rapid fabrication of millimeter-scale structures featuring 830 nm details. For applications such as micro-optics, biomedical scaffolds and tissue engineering, tens-of-micrometer features are sufficient across the majority of the volume, with higher resolution confined to localized functional regions.