Let $S$ be a finite set of primes and write $\mathbb{Z}_S$ for the set of those non-zero integers whose prime divisors belong to $S$. Hajdu proved that the abc conjecture implies that the number of terms of any arithmetic progression in $H_S=\{\eta x^l\mid \eta\in \mathbb{Z}_S, x,l\in \mathbb{Z},\ \textnormal{with}\ x>0 \ \textnormal{and} \ l\geq 2 \}$ is bounded. Moreover, if $k\geq 3$ and the exponents of the powers are all $\geq 4$, then the number of such progressions are finite. We consider other sets and prove similar statements for these sets.

Beginning in the 1980s, Mukai introduced birational models of some moduli spaces of curves and some moduli spaces of K3 surfaces. They are defined as geometric invariant theory quotients. Little is known about the boundaries of these spaces. We describe an approach to efficiently evaluate certain invariant polynomials associated to these GIT problems. This allows us to show that several singular curves and surfaces are GIT semistable in Mukai's models. In the appendix, we give a combinatorial formula for an $\operatorname{SL}_n$-invariant in terms of the Gelfand-Tsetlin basis.

We analyze a two-dimensional Bose-Fermi mixture at zero temperature in the presence of a tunable Bose-Fermi attraction. We adopt a diagrammatic T-matrix approach and study the behavior of several thermodynamic quantities for the two species as functions of density, mass ratio, and coupling strength. These include the chemical potentials, the boson momentum distribution function, the condensate density, and Tan's contact parameter. We analytically demonstrate that the present T-matrix formalism recovers the correct second-order perturbative expansion of the chemical potentials in the weak-coupling regime, and test it numerically. The near-universal behavior already found in prior work for the mass-balanced case is confirmed for different masses and becomes even more accurate when the boson mass is large. The mass imbalance emerges as an additional control parameter that qualitatively affects the bosonic momentum distribution. In particular, we found that it can be used to allow for the experimental observation of a peculiar peak in the boson momentum distribution at finite momentum.

The small flow boundary property (SFBP), introduced by Burguet for fixed-point free topological flows, is a non-trivial generalization of the small boundary property (SBP). We characterize when a time-discretization of such a flow satisfies the SBP and deduce that an SFBP flow admitting an aperiodic time-discretization has vanishing mean dimension. Furthermore, we introduce a new quantity, \textit{flow-generated entropy}, for a factor between a flow and a time-discretization, quantifying the dynamical complexity inherited from the flow itself. This is used in order to establish that any time-discretization of a flow with SFBP admits factors of arbitrarily small flow-generated entropy separating any fixed pair of distinct points. The argument relies on a construction of a dynamical version of the Cantor staircase function. Finally, the appendix includes proofs of fundamental properties of the marker property which have not yet appeared in the literature.

In this article, we explore to what extent the geometry of gradient Ricci solitons can be carried over to the geometry of non-gradient Ricci solitons. We use energy function $E$ of the soliton as a tool to study this. We use Omori-Yau maximum principle to investigate the bounds on the energy function. One of the main results obtained is that a non-steady Ricci soliton with symmetric covariant derivative is gradient. We explore non-gradient Ricci solitons of constant scalar curvature. We prove a weighted $L^1$-Liouville type theorem with respect to the energy function under mild assumptions on the scalar curvature. Finally, we show that measure $e^{-E} d \mathrm{Vol}_{g}$ for complete shrinking Ricci soliton is finite, partially generalizing a result by Aaron Naber. Subsequently, this implies that nongradient shrinking Ricci solitons also have finite fundamental groups similar to their gradient counterparts.

It is well known that finite groups with exactly two character degrees have an abelian derived subgroup and, consequently, are solvable. Let $G$ be a finite group and $N$ a normal subgroup of $G$. In this paper, we prove that normal subgroups possessing exactly two degrees of minimal $G$-invariant characters are solvable. Furthermore, it is shown that if these degrees are $\{1, f\}$ for some integer $f$, then either $f$ is a prime power or the commutator subgroup $[N,G]$ is abelian. Whether $[N, G]$ is abelian when $f$ is a prime power remains an open problem. Specifically, we prove that this holds when $f=p$.

We present a simple and robust framework to assess the information content of the power spectrum of a marked density field within the Effective Field Theory of Large Scale Structure, providing a physically motivated regime of validity of perturbation theory for such fields. We optimize the choice of mark to maximize the constraining power on cosmological parameters, focusing on the resulting reduction of parameter uncertainties. After marginalizing over the small-scale counterterms, we find improvements in the parameter error bars by a factor of approximately $1.30$-$2.3$, $1.3$-$8.3$, and $1.2$-$2.5$ for $\Omega_m$, $\sigma_8$, and $h$, respectively, depending on redshift and smoothing scale. The optimized mark exhibits a simple functional form, for which we provide a physical interpretation in terms of its response to the underlying density field. These results demonstrate the potential of marked statistics as a tool to extract additional cosmological information beyond standard two-point analyses and open a pathway for further exploration of nonlinear transformations of the density field.

At the intersection of rising wealth inequality and intensifying environmental pressures, we investigate a reverse causal relationship that has received comparatively little attention: wealth inequality may not only be a consequence of environmental crises, but also act as a structural obstacle to the ecological transition itself. We develop a stylized agent-based model in which heterogeneous agents, whose initial wealth follows a Pareto distribution, allocate their income between either a Brown or a Green sector through a utility function. The function is designed to capture the trade-off between short-term returns and exposure to long-term systemic risks. A central ingredient is that wealthier agents perceive themselves as less vulnerable to environmental shocks, thereby reducing the amount of resources available for the transition. We show that, beyond inequality thresholds compatible with those observed in most developed countries, the economy remains locked in a Brown regime, even when a substantial share of agents is sensitive to externalities. We then assess a set of stylized fiscal policies (basic income, carbon taxation, Green incentives, and a combined scheme) and find that their effectiveness depends strongly on the inequality regime and on the regressivity embedded in the fiscal mechanism, revealing multidimensional trade-offs between transition speed, cumulative environmental destruction, growth, and fiscal pressure.

Pattern formation in reaction-diffusion systems is traditionally analyzed under isothermal assumptions, overlooking the dynamical role of temperature in systems where reactions generate and dissipate heat. Here, we investigate non-isothermal reaction-diffusion dynamics by coupling activator-inhibitor kinetics to a dynamically evolving temperature field that modulates reaction rates through Arrhenius-type dependencies. This coupling introduces an additional feedback mechanism that influences stability and pattern selection. Through analytical and numerical analysis of the Cholrine dioxide-Iodine-Malonic acid (CDIMA) and Schnakenberg models, we demonstrate that thermal feedback modifies dispersion relations by enhancing instability growth rates and shifting pattern selection toward shorter wavelengths. Beyond these intrinsic effects, we identify a boundary-mediated mechanism in which thermal constraints qualitatively alter global dynamics. In particular, fixed-temperature boundaries induce nonstationary behavior in the CDIMA system, whereas the Schnakenberg model exhibits robust stationary patterns. These results establish thermal-kinetic coupling as a general mechanism for controlling pattern formation and highlight the role of boundary-mediated heat exchange as a tunable parameter for spatiotemporal organization.

The dissipation in the relaxation of an ensemble of conservative stochastic systems towards the steady state is quantified by the free energy difference. Functional inequalities within the framework of [F. Otto and C. Villani, J. Funct. Anal. 173, 361 (2000)] are here revisited which connect the free energy dynamics and optimal transport, offering a geometric perspective on the instantaneous speed of relaxation in the presence of potential barriers. These are illustrated with numerical relaxation experiments on Landau-Ginzburg potentials.

Simulating high-enthalpy fractured geothermal reservoirs is challenging due to the complex coupled processes of non-isothermal, multiphase, multicomponent flow, strongly nonlinear thermodynamics, and the dominant role of fractures. These complexities are amplified by mineral scaling, such as halite precipitation, which can impair reservoir permeability and well productivity. To address this, we present a new compositional flow model based on a persistent set of primary variables (pressure, enthalpy, and overall salt mass fraction). The formulation naturally handles phase transitions without manual switching, enhancing numerical stability. The model integrates a discrete fracture-matrix approach and employs an efficient, robust correlation-based phase-behaviour linearisation of saltwater thermodynamics, replacing expensive on-the-fly phase separation calculations. It incorporates the Kozeny-Carman relation to dynamically model porosity and permeability reduction from halite precipitation. Implemented in the open-source PorePy framework, the model is verified through a 1D salt dissolution benchmark against the established closed-source simulator CSMP++, showing strong agreement across geothermal conditions involving transitions between single- and multi-phase regions. Application to a 2D halite-saturated fractured reservoir with injection and production demonstrates the model's capability to predict halite precipitation patterns and their impact on permeability damage and energy recovery. Numerical results further show the model's value in predicting operational challenges such as wellbore blockage and the role of fracture connectivity. The model thus provides an open-source numerical tool for analysing complex heat and mass transport with mineral scaling in high-enthalpy fractured geothermal systems.

The laboratory test process of HbA1c measurement is a time-consuming and invasive method. The HbA1c parameter is the most important feature to predict the level of diabetes. Although invasive methods are irritating in the case of frequent measurements. Moreover, HbA1c measurement is only possible at the diagnostic centre, followed by medical protocol. Hence, it is still challenging to measure the HbA1c frequently at remote locations, where diagnostic centres are not easily available. Therefore, an intelligent and new non-invasive HbA1c measurement system, iGLU 5.0, is proposed for instant diagnosis of the HbA1c value without prior measurement setup. The proposed measurement device is based on optical spectroscopy for the collection of glucose values in different formats. The glucose values have been collected in fasting, postprandial, and random formats. The glucose value has also been collected using the oral glucose tolerance test (OGTT), along with the average blood pressure value, correspondingly. These four formats of glucose values, along with blood pressure, were used to predict the estimated average glucose (eAG) using an optimized prediction model. Further, the predicted average glucose is converted into an HbA1c value using a standard formula. The eAG prediction models have been trained and validated using 2000 samples of healthy, prediabetic and diabetic people to analyze the optimized prediction model. 94% and 96% accuracy have been examined during training and cross-validation of optimized DNN model, respectively. A 0.3 mean absolute difference has been identified from predicted HbA1c values using the proposed DNN model. The novel non-invasive HbA1c prediction system is useful for instant diagnosis without irritation for smart healthcare.

This paper investigates the growth of the sequences $(\|T^n(I-T)^k\|)_{n\ge 1}$ for bounded linear operators on Banach spaces under various resolvent conditions. Focusing on $\alpha$-Ritt operators that are also strongly Kreiss bounded, we show that the growth estimates established by Nevanlinna for Kreiss bounded operators can be significantly refined within the Hilbert space setting, nearly attaining the optimal rates obtained by Seifert for the more restrictive power-bounded case. We further introduce the class of $(\alpha, \beta)$-RK operators as a generalization of both Ritt and Kreiss-type conditions. For these operators, we derive comprehensive growth estimates which, for certain ranges of $\alpha$ and $\beta$, yield improvements over existing bounds in the literature. Particular attention is given to $\beta$-Kreiss operators, for which we provide a characterization via Cesàro-type means. We show that, in contrast to the well-known case $\beta = 1$, the power growth estimate $\|T^n\| = O(n^{\beta})$ is sharp whenever $\beta > 1$. The optimality of our estimates is discussed in several cases, relying on constructions and techniques developed by Nevanlinna, Spijker, and Borovykh. We conclude by providing a characterization of Ritt operators that appears to be absent from the literature.

We show that every Polish extended metric measure space arises as an inverse limit of metric measure spaces up to isomorphism. We then prove that synthetic Ricci curvature lower bounds and several functional inequalities, including the log-Sobolev, Talagrand, Poincaré, and dimension-free Harnack inequalities are stable under inverse limit. We discuss applications to infinite-dimensional spaces, including abstract Wiener spaces and their quotient spaces.

We investigate a class of quintessence models in which the dark-energy scalar field interacts with sterile neutrinos responsible for neutrino mass generation through the seesaw mechanism. Radiative corrections involving sterile neutrinos induce Higgs-quintessence mixing and provide an effective portal between quintessence and Standard Model particles. We calculate the corresponding one-loop mixing amplitude and show that its structure depends on the relation between the characteristic momentum transfer and the sterile-neutrino mass. In the low-momentum regime the mixing is effectively kinetic, while at high energies it acquires a mass-mixing form. A notable result is that the overall suppression of the induced mixing is governed by the physical neutrino mass scale, leading to a predictive relation between neutrino properties and Higgs-quintessence interactions. The resulting loop-induced effects are found to be suppressed by factors comparable to those controlling the tree-level quintessence-neutrino interaction. The induced mixing generates effective couplings of quintessence to Standard Model fermions and gauge bosons, leading to modified Higgs, W and Z boson processes and opening decay channels for the quintessence field, which may prove to be cosmologically important. While the corresponding effects remain well below current experimental sensitivities in the minimal heavy-seesaw scenario, the framework establishes a direct connection between dark-energy dynamics, neutrino mass generation and Higgs-sector phenomenology and provides a basis for studying scenarios with lighter sterile neutrino states where observable effects may be enhanced.

We completely characterize when three equal-order subgroups of a finite abelian group share a common transversal. Consequently, we determine when three full-rank lattices of equal volume with a discrete sum in $\mathbb{R}^d$ admit a common fundamental domain, answering, for $n=3$, an open question from 1997.

We derive an explicit formula for the sectional curvature of the space ${\cal M}(M)$ of finite measures on a Riemannian manifold M. The space ${\cal M}(M)$ is equipped with the Hellinger-Kantorovich metric $HK$. Even in the case M=R^n, the curvature is comprised of two parts: the `lifted part' is negative, and the `twisted part' is positive. It will be analyzed in detail for the multidimensional torus. Our general approach to sectional curvature in geodesic spaces also leads to new insights into the curvature of the space $P_2(M)$ of probability measures on M equipped with the Kantorovich-Wasserstein metric $W_2$.

Classical high-order backward differentiation formula (BDF) methods for the Landau-Lifshitz-Gilbert (LLG) equation often suffer from restrictive stability constraints, requiring small time steps and imposing stringent lower bounds on the damping parameter. These limitations become particularly severe for schemes of order higher than three. In this paper, we develop a class of high-order generalized BDF (GBDF) schemes for the LLG equation, including both semi-implicit and fully explicit treatments of the gyromagnetic term. The proposed schemes significantly improve stability properties and substantially relax the damping parameter constraints, but introduce essential difficulty in its analysis compared to the classical BDF schemes. We construct a novel multiplier which enables us to carry out a energy-based error analysis. This approach yields optimal-order error estimates under considerably weaker assumptions on the damping parameter than those required for classical BDF schemes. Numerical experiments are presented to confirm the theoretical results, and demonstrate that the proposed GBDF schemes achieve higher accuracy, enhanced stability, and much wider admissible damping regimes compared to classical high-order BDF methods.

The ML literature contains many distinct concepts falling under the heading of 'AI alignment'. After noting three concepts of AI alignment in the context of their corresponding research programs, we claim that realistic interventions may promote 'AI alignment' under one conception while being actively counterproductive from the perspective of others. We suggest that tensions between alignment ideals emerge due to differences in background threat-models, alongside differences in normative orientations. In light of our analysis, researchers aiming to further the goal of 'AI alignment' should do five things. First, they should not conflate distinctions of policy and distinctions of scientific scope; second, methodological disagreements should be acknowledged explicitly; third, researchers should distinguish between 'AI alignment' as a high-level ideal and specific 'alignment proxies' used in empirical research; fourth, they should use more granular concepts to identify both the source and nature of possible AI harms/benefits; fifth, they should explicitly acknowledge the diversity of 'alignment' concepts in both empirical work and in communication with non-technical audiences.

Configuration tuning is critical to database performance, yet automatic database tuning remains challenging due to high-dimensional knob spaces, substantial online tuning cost, unreliable textual hints derived from Large Language Models (LLMs) or community documents, and the difficulty of exploiting the remaining optimization room after initialization. Hence, we propose PLRTune, a staged database tuning system that leverages workload-specific domain knowledge to identify a reduced search space and further optimize within this promising region. First, we develop an importance pre-sampling and reranking strategy to identify the dominant knob subset in a workload-specific manner and derive a compact state representation. Second, we design an execution-guided hint refinement technique to improve the initialization quality of documentation-guided tuning. Finally, we propose a post-tuning refinement stage that leverages Twin Delayed Deep Deterministic Policy Gradient (TD3) to explore the dominant knob subset and further exploit the remaining optimization room. We evaluate PLRTune on MySQL and PostgreSQL across diverse benchmark workloads. Compared with state-of-the-art approaches, PLRTune achieves the best final result on all tested workloads, improving over the corresponding best-performing alternative by 9.50% on average. Moreover, PLRTune reaches the strongest baseline's best performance level 9.03 times faster on average across workloads, demonstrating its practical runtime efficiency without sacrificing final tuning quality.

We investigate the linear and nonlinear rheology of dense chiral fluids composed of self-spinning particles under external shear. Using particle-based simulations of a two-dimensional Lennard-Jones model with transverse interactions, we show that chirality acts as an intrinsic source of fluctuations and shear. In the solid regime, spinning fluidizes the system, weakening hexatic order. In the liquid regime, the viscosity is quantitatively described by a Green-Kubo relation upon replacing the temperature by a chirality-dependent effective temperature. Beyond linear response, flow curves collapse when expressed in terms of the ratio between imposed shear and spinning rates, revealing a thinning-by-spinning mechanism. At large forcing, this correspondence breaks down and a pronounced handedness asymmetry emerges: when transverse interactions oppose the imposed shear, stresses relax through the formation of string-like flow channels. Our results identify chirality as a generic mechanism for fluidization and provide a unified framework for understanding the rheology of dense chiral suspensions.

We study residue restrictions for a two-color partition series $P(q)=(q^4;q^4)_\infty S(q)$ arising from work of Andrews and Bachraoui on partitions with odd smallest part. Motivated by the explicit exponent structure in the Bailey-transform formulas for the associated generating function, we obtain elementary restrictions on the support of the coefficients. In particular, we show that the residue class $4\pmod8$ does not occur, and we prove a further refinement modulo $16$, yielding several vanishing classes for the coefficients of the series. Our argument is completely residue-theoretic and avoids the use of modular completions.

In recent years, the technique of convex integration has demonstrated that for some initial data (also referred to as 'wild initial data'), many PDE models of mathematical fluid mechanics allow for a multitude of admissible (weak) solutions. This paper is concerned with the question regarding how large the set of such wild initial data is for the isentropic Euler equations. We prove that wild initial data form a dense set in $L^r$ for any $r \in [1,\infty)$. In contrast to existing results in the literature, in this paper 'wild initial data' are data which give rise to infinitely many global-in-time weak solutions which are admissible in the sense that the local energy inequality holds. In other words, the set of initial data with infinitely many admissible weak solutions (independent of the choice of time interval) is dense. A novel part of the construction is that we use a measure-valued (dissipative) solution as the ansatz for the subsolution. This requires several new ideas, in order to ensure the required regularity of the subsolution and to obtain a lower bound for the density. Another crucial ingredient of the proof is that the (local) energy density and the energy flux are constructed as part of the convex integration scheme, in order to obtain solutions which adhere to the local energy inequality.

We demonstrate that the $D_{s0}^{*}(2317)$ can be described as a bound-state pole arising from the coupling between a discrete $c\bar{s}(1^3P_0)$ state and the $DK$ continuum state in the Lee-Friedrichs model. The elementariness and compositeness of the $D_{s0}^{*}(2317)$ are determined to be about $Z:X\approx 51.1 \%:48.9\%$, indicating a nearly equal admixture of compact quark-model state and hadronic molecular components. The decay width of $D_{s0}^{*}(2317)\rightarrow D_s^{+}\pi^0$ is evaluated within the quark rearrangement framework. The transition $c\bar{s}(1^3P_0)\rightarrow D_s\pi^0$ proceeds via the OZI-allowed $c\bar{s}(1^3P_0)-D_s\eta$ coupling followed by $\eta-\pi^0$ mixing, which serves as the primary source of isospin violation in this channel. The coupling between $DK$ and $D_s\pi^0$ is computed in a quark rearrangement model, where the isospin violation effect originates from the mass difference between the $D^0K^+$ and $D^+K^0$ thresholds. The parameters of the scheme shares the same ones with those of the underlying potential model and only the vacuum production strength is adjusted to reproduce the physical $D_{s0}^{*}(2317)$ mass. This calculation may shed more insight to the nature of exotic $D_{s0}^{*}(2317)$ state and its isospin-breaking decay properties.

We construct light-front transverse gravitational distributions associated with the scalar trace form factor and the non-scalar gluon energy-momentum tensor (EMT) combination selected by compact-quarkonium chromoelectric scattering. The construction is performed in the Drell--Yan frame, where $t=-{\Delta}_\perp^2$, and separates universal EMT form-factor shapes from the forward chromoelectric normalization. The resulting non-scalar distribution is normalized by the previously derived threshold strength $R_{\rm LF}^{\rm int}=N_{\rm nt}(0)/N_\theta(0)\simeq 0.15$, while its transverse profile is governed by the off-forward combination of $A_g(t)$, $D_g(t)$, and $\bar C_g(t)$. We show that scalar and non-scalar responses can have different transverse localization, and that near-threshold quarkonium production probes a chromoelectric EMT projection rather than an individual form-factor slope.

We propose the One Scalar Theory (1ST), a minimalist framework where a single real singlet scalar field mediates the dynamical generation of the Higgs self-coupling and the top Yukawa coupling. Unlike generic portal models, the 1ST removes parametric freedom by locking production and decay modes to a single fundamental scale $\Lambda_0$, rendering the framework highly predictive with unique experimental signals. We demonstrate that the collider phenomenology is partitioned by the $2m_t$ kinematic threshold into di-Higgs and di-top resonance regimes. By recasting current ATLAS data, we set lower bounds on $\Lambda_0$ at the TeV scale and show that the High-Luminosity LHC will probe this scale up to $4$ TeV, providing a definitive test for the dynamical origin of the electroweak sector.

Sliver elements are an intrinsic difficulty of three-dimensional tetrahedral mesh generation and remain costly, and sometimes impractical, to eliminate completely. Although isolated degenerate elements do not necessarily prevent finite element convergence, connected clusters or sheets of slivers may impose artificial constraints on the discrete solution, leading to locking and severe loss of accuracy. In this work, we revisit the effect of slivers from the viewpoint of the finite element solution and propose a robust solver-side treatment based on the Tempered Finite Element Method (TFEM). The method limits the singular contribution of degenerate elements by introducing a lower bound on the Jacobian determinant, which can be interpreted as a vanishing added-volume correction. The resulting formulation prevents the effective element volume from falling below a threshold while preserving the relevant physical modes of the solution. We analyze the stiffness matrices of degenerate tetrahedra, identify the mechanisms responsible for locking in sliver bands, and assess the method on a range of representative physical problems, including incompressible flow, Cahn--Hilliard phase-field dynamics, transient wave propagation, and vibro-acoustic fluid--structure interaction. The numerical results show that TFEM consistently recovers accurate and physically meaningful solutions on meshes for which standard FEM exhibits locking or loss of convergence, providing a simple and broadly applicable alternative to exhaustive geometric sliver removal.

Helium nanodroplets provide a unique and versatile platform for investigating strong-field-driven nanoplasma dynamics. In this work, we present large-scale, GPU-accelerated particle-in-cell simulations using \textsc{PIConGPU} to study the interaction of pure helium nanodroplets containing up to $10^{6}$ atoms with intense near-infrared femtosecond laser pulses, and compare the results with single-shot velocity-map electron imaging and ion measurements. The simulations describe the plasma evolution from the first ionization events to collective electron motion, nanoplasma formation, and early expansion. We show that the calculated electron and ion observables reproduce the main features of the measured spectra in systems with similar cluster sizes and laser intensities. Our results demonstrate that \textsc{PIConGPU} captures the essential physics of nanoplasma formation previously addressed mainly with molecular-dynamics or TDDFT approaches, while remaining computationally efficient and applicable to much larger systems. This establishes \textsc{PIConGPU} as a powerful and scalable tool for connecting nanoplasma theory with experimentally accessible observables.

We have demonstrated that compact deformation sensors with heights and widths of about 100 micrometers can be fabricated by two-photon polymerization, using commercially available optical ferrules with embedded 125 micrometers-diameter optical fibers as the basic platform and the commercial photopolymers OrmoComp and FemtoBond as the fabrication materials.

This short paper has two goals. First, explaining a simple procedure (which is essentially folklore) that, sometimes, makes it possible to obtain a formula for the number of solutions to a system of multivariate polynomial inequalities over a finite field. Second, applying that procedure to prove a formula for the number of contiguous superregular $4 \times 4$ matrices over a finite field. The formula was previously conjectured by Appuswamy, Bazzani, Connelly, Ekaireb, Congero, and Zeger [Probability of super-regular matrices and MDS codes over finite fields, arXiv:2603.20983 ]. In addition, the same procedure is used to provide formulas for the number of contiguous superregular $3 \times 4$, $3 \times 5$, and $3 \times 6$ matrices over a finite field.