In this paper, we focus on the study of immanantal polynomials for linear combination matrices composed of the degree matrix and adjacency matrix of a graph. First, applying the concept of vertex orientation for general graphs, we provide a combinatorial interpretation of the coefficients of the immanantal polynomials for the linear combination matrices of graphs, and we also characterize the bounds of these coefficients. These bounds implicitly encompass the existing results of Chan and Lam on trees and bipartite graphs. Furthermore, we give a solution to the open problem posed by Merris. Second, we characterize the first six coefficients of the hook immanantal polynomial. And the necessary and sufficient condition under which the linear combination matrices of two regular graphs have the same hook immanantal polynomial is proved. Third, we generalize the Frobenius--König theorem and the Laplace expansion theorem to immanants. Using these two theorems, we show that the star degree of a graph is always a lower bound for the multiplicity of a certain root of the immanantal polynomial of its linear combination matrix. Finally, we derive formulas for the first six coefficients of the hook immanantal polynomial for several important graph matrices.

As deep neural network (DNN) models are growing exponentially in size, their deployment on resource-constrained edge platforms is becoming increasingly challenging. In-memory-computing (IMC) with non-volatile memories (NVMs) has emerged as a potential solution by virtue of its higher energy efficiency compared to standard DNN hardware platforms. Amongst various NVMs, STT-MRAM is highly promising owing to its high endurance and other benefits. However, their IMC implementation is challenging because of their inherently low distinguishability. This issue is exacerbated due to array non-idealities and process-variations, leading to poor IMC robustness and severe inference accuracy degradation. To address this problem, we propose STRIDe - STT-MRAM-based IMC leveraging cross-coupling action to boost the bitcell-level high-to-low current ratio to up to 8000. We propose two flavors of STRIDe designs, both offering robust IMC for inputs and weights in {-1, 1}(XNOR-IMC) and {0, 1}(AND-IMC) regime. Our evaluations for STRIDe arrays show up to 3.86x and 1.77x sense margin (SM) improvement for XNOR-IMC and AND-IMC, respectively, and up to 27.6% read disturb margin (RDM) improvement over standard MRAM-IMC designs. The enhanced robustness of STRIDe translates to near-software inference accuracies (considering crossbar non-idealities and process variations) for ResNet18 BNN and 4-bit DNN trained on CIFAR10 dataset. We observe accuracy improvements of up to 70% (for BNN) and up to 35%(for 4-bit DNN) over standard MRAM designs, albeit with some energy-area-latency penalty.

The Comment criticizes the bifurcation analysis performed in the original paper on a Vlasov equation. This criticism can be traced back to a discrepancy in the definition of the paramagnetic phase. Apart from this discrepancy, there is no conflict between the Comment and the original paper.

Integrated sensing and communication (ISAC) has been receiving a notable interest as an energy- and spectrum-efficient enabler for simultaneous communication and sensing. Notably, reconfigurable intelligent surfaces (RIS) is among the key technologies enabling robust communication and sensing, particularly in environments without a line-of-sight (LoS). Recently, a new type of RIS, called beyond-diagonal RIS (BD-RIS), has drawn attention, offering additional degrees of freedom in controlling the propagation medium. In this paper, a novel secure BD-RIS-aided ISAC scheme is proposed and evaluated. The scheme is applicable to a multi-user multi-target ISAC network, where a dual-functional radar-communication (DFRC) base station (BS) simultaneously serves multiple downlink users and senses various targets that aim to eavesdrop on the legitimate signal transmitted to the users. The presence of a BD-RIS enables circumventing the absence of the LoS link and ensures secure transmission and sensing. To this end, an optimization problem is formulated aiming at maximizing a weighted sum of per-target reflected powers, subject to secrecy and transmit power constraints. Thus, by virtue of an alternating optimization (AO)- and Riemannian conjugate gradient-based approach, local optima for the BD-RIS scattering matrix, transmit signal beamforming matrices, and artificial noise covariance matrix are obtained. Numerical results highlight (i) the notable sensing gains of the BD-RIS-aided design with respect to its diagonal RIS (D-RIS)-based baseline and (ii) the improved secrecy-sensing trade-off, whereby the BD-RIS can ensure an increasing system secrecy without degrading the per-target reflected power.

This paper studies discrete-time two-person nonzero-sum linear quadratic stochastic games with random coefficients. Using convex variational analysis, we derive necessary and sufficient conditions for the existence of open-loop Nash equilibria. When weighting matrices are indefinite, the classical first-order conditions are no longer sufficient for optimality; we introduce a global nonnegativity condition to restore sufficiency, which becomes a cornerstone of the subsequent analysis. To characterize the equilibria explicitly, we develop fully coupled forward-backward stochastic difference equations and a system of non-symmetric stochastic Riccati equations (FBS$Δ$Es) with constraints. that decouple the stochastic Hamiltonian system. A key technical contribution is the provision of sufficient conditions -- positive semidefiniteness of the Riccati matrices operators and structural non-degeneracy -- that guarantee the invertibility of a related operator, ensuring the well-posedness of the closed-loop feedback representation of the open-loop Nash equilibrium strategies. A distinctive feature of this work is the presence of fully random coefficients, which leads to fully nonlinear higher-order backward stochastic difference equations in the Riccati framework, in contrast to the algebraic Riccati equations in the deterministic setting.

Hyperbolic hypergeometric integrals are defined as Barnes-type integrals of products of hyperbolic gamma functions. Their reduction to ordinary hypergeometric functions is well known. We study in detail their degeneration to complex hypergeometric functions. Namely, using uniform bounds on the integrands, we prove that the univariate hyperbolic beta integral and the conical function degenerate to two-dimensional integrals over the complex plane.

We consider graph property testing in $p$-degenerate graphs under the random neighbor oracle model (Czumaj and Sohler, FOCS 2019). In this framework, a tester explores a graph by sampling uniform neighbors of vertices, and a property is testable with one-sided error if its query complexity is independent of the graph size. It is known that one-sided error testable properties for minor-closed families are exactly those that can be defined by forbidden subgraphs of bounded size. However, the much broader class of $p$-degenerate graphs allows for high-degree ``hubs" that can structurally hide forbidden subgraphs from local exploration. In this work, we provide a complete structural characterization of all properties testable with one-sided error in $p$-degenerate graphs. We show that testability is fundamentally determined by the connectivity of the forbidden structures: a property is testable if and only if its violations cannot be fragmented across disjoint high-degree neighborhoods. Our results define the exact structural boundary for testability under these constraints, accounting for both the connectivity of individual forbidden subgraphs and the collective behavior of the properties they define.

The $q$-Garnier system was first proposed by Sakai and its other directions of discrete time evolutions were given by Nagao and Yamada. Recently, it was shown that all of those directions of discrete time evolutions are derived from a birational representation of an extended affine Weyl group which arises from the cluster algebraic construction established by Masuda, Okubo and Tsuda. In this article, we investigate a degeneration structure of the $q$-Garnier system of fourth order by using confluences in quivers.

This thesis develops a decision-theoretic framework for extracting thermodynamic work from temporal correlations in quantum systems. We model a classical agent -- lacking quantum memory -- performing adaptive work extraction through continuous inference and decision-making under uncertainty. By introducing $ρ^*$-ideal protocols, we demonstrate that exploiting memory effects allows adaptive strategies to surpass non-adaptive bounds. We formalize this via the Time-Ordered Free Energy (TOFE), a novel upper bound for causal, adaptive operations that reveals a thermodynamic gap linked to adaptive ordered discord. Additionally, we tackle work extraction from unknown sources using reinforcement learning. By adapting multi-armed bandit algorithms, we show an agent can simultaneously learn an unknown i.i.d. quantum state and extract work, achieving polylogarithmic cumulative dissipation that significantly outperforms standard tomography. Overall, this work lays the foundation for predictive and learning-based quantum thermodynamics.

The hierarchy and proximity are key dimensions of urban relational processes, but their interplay in shaping intercity interactions and the underlying structures of city systems remain unclear. We develop a novel geometric model of city systems embedding intercity mobility into a latent hyperbolic geometry, which unravels the measures of hierarchy and proximity accounting for their interplay. It is successfully validated against 12 different nationwide intercity mobility datasets. We find a bottom-up emergence of city hierarchies, along which the variations of city-hinterland relations are non-stationary in terms of their nesting and range properties. Such non-stationarity originates from trade-offs between city hierarchy and hinterland range in determining the formation of city-hinterland structures. Hierarchy- and proximity-dominated urban processes can be elucidated from examining dynamics of the trade-offs. The revealed urban relational processes of city systems are at the core of the emerging science of cities and crucial for spatial planning and regional policymaking.

Characterizing quantum states is essential for validating quantum devices, yet conventional quantum state tomography becomes prohibitively expensive as system size grows. Direct tomography offers a distinct route by enabling selective access to individual complex density-matrix elements, with a particular advantage for sparse target states and some verification tasks. Here we introduce a direct quantum state tomography scheme combining strong-measurement estimation with a fan-out coupling architecture. It enables mutually commuting interactions between system qubits and a single meter qubit, thereby achieving constant circuit depth, independent of system size. Notably, the involutory fan-out coupling reduces to the identity under repetition, enabling straightforward noise scaling for quantum error mitigation. We experimentally validate the scheme on a superconducting quantum processor via the IBM Quantum Platform, demonstrating four-qubit state reconstruction and single-circuit GHZ-state fidelity estimation up to 20 qubits with error mitigation. Consistent results with standard tomography and improved efficiency establish our scheme as a promising approach to reconstructing full quantum states and scalable verification tasks.

We formulate and prove a Riemann--Hilbert correspondence between two categories: wild difference modules and wild Stokes-filtered $\mathscr{A}_{\rm{per}}$-modules. This correspondence is motivated by the Riemann--Hilbert correspondence for germs of meromorphic connections in one variable due to Deligne--Malgrange. It also generalizes the Riemann--Hilbert correspondence for mild difference modules.

The Bott index has become an indispensable tool to probe the topology of quantum matter, particularly in systems lacking translational symmetry. Constructed from a plaquette operator, it retains the phase information while discarding the amplitude. Here we introduce and develop the Bott metric, which captures this complementary amplitude information and provides a measure of the underlying quantum metric of the system. We show that, in the thermodynamic limit, the Bott metric converges to the trace of the integrated quantum metric. Our framework provides a new route to reveal the quantum metric structure in non-periodic systems, which we illustrate using representative examples ranging from disordered to amorphous models. More broadly, our definition of the Bott metric unifies the notion of topological invariants and quantum metric under the same overarching plaquette operator construction.

Structural defects, particularly grain boundaries, play a crucial role in governing charge transport and the optoelectronic properties of metal halide perovskites, thereby limiting the performance of devices. Solar cells incorporating templated FA0.9Cs0.1PbI3-xClx show significant improvements in grain orientation and steady-state power conversion efficiency; however, the underlying mechanisms remain unclear. In this study, we address this gap by employing a suite of tailored low-dose electron microscopy techniques to investigate the templated FA0.9Cs0.1PbI3-xClx film, revealing that it exhibits a preferred crystallographic orientation along the <001> zone axis, with arbitrary grain rotations about that axis, indicative of a Volmer-Weber growth mechanism. We determine the atomic structure of the resulting high-angle and low-angle grain boundaries. We also reveal the presence of edge dislocations and their associated strain fields, demonstrating the compressive strain on one side of the dislocation core and tensile strain on the opposite side. Furthermore, we find dislocations associated with stacking faults. These atomic-level insights uncover which grain boundaries and intra-grain defects are likely to act as recombination centres or modify band gaps, crucial for understanding which defects influence the performance of perovskite solar cell devices.

The shadow of a black hole serves as a pristine window into the strong-gravity regime, with cuspy feature emerging as a smoking-gun signature of physics beyond the Kerr paradigm. In this paper, we extend the work of [arXiv:2601.15612 [gr-qc]] and study the detailed properties of the cuspy shadow by using the parametric expressions of the shadow boundary. From a topological perspective, we provide a rigorous topological classification of these shadows, categorizing them into distinct ``rectangular" and ``8-shape" topologies. Crucially, we establish a formal gravity/thermodynamics correspondence by mapping the cuspy shadow to the swallowtail behavior observed in thermodynamic free energy. We demonstrate that the self-intersection of the shadow boundary, marking a geometric phase transition, can be precisely determined through three independent but equivalently thermodynamic-like approaches. Furthermore, we analytically derive the critical exponents governing the emergence of these cusps, revealing that they are consistent with the mean-field universality class. Our results suggest that the observational features of black hole shadows are deeply rooted in the underlying gravitational thermodynamics, offering a novel framework to probe the fundamental nature of spacetime.

Quantum simulation and computing traditionally has been based on two main paradigms, namely, digital and analog. In the digital paradigm, usually single and two-qubit gates (where qubit is an acronym for quantum bit) are employed as building blocks for scalable, universal quantum computing, although errors add up fast and error correction will be ultimately needed for scaling up. In the analog paradigm, large analog blocks are normally employed for a unitary dynamics that carries out the computation, enabling quantum operations on many qubits with reduced errors, but with the drawback of a limited choice of evolutions and lack of universality. In the past decade, a new paradigm has emerged, showing interesting possibilities for quantum simulation and computing in the near and mid term. This is the paradigm of digital-analog quantum technologies, which proposes to combine the best of both paradigms: large analog blocks, provided by native interactions of the employed quantum platform, enabling scalability, combined with digital gates, allowing for more versatility and, ultimately, universality. In this Perspective, I give an overview of the evolution of the field along the past decade, and an outlook for its future possibilities.

Quantum clock synchronization (QCS) aims to establish a shared temporal reference between distant nodes by exploiting uniquely quantum phenomena such as entanglement, single-photon interference, and quantum correlations. In contrast to classical synchronization and time-transfer techniques, which are limited by signal propagation delays, atmospheric disturbances, and oscillator drift, QCS protocols offer the potential to surpass classical precision bounds and enhance resilience against adversarial manipulations. As precise and secure time synchronization underpins distributed quantum networks, navigation systems, and emerging quantum Internet infrastructures, understanding QCS principles, capabilities, and implementation challenges has become increasingly important. This survey provides a unified and critical overview of the rapidly growing QCS research landscape, highlighting fundamentals, protocol types, enabling resources, performance constraints, security considerations, and practical implementations of QCS. We first introduce the theoretical underpinnings of QCS, including entanglement-assisted time transfer, Hong-Ou-Mandel interference-based synchronization, and quantum slow-clock transport. We then categorize the main QCS protocols, ranging from ticking-qubit and entanglement-based schemes to time-of-arrival correlation methods, conveyor-belt synchronization, and quantum-enhanced two-way time transfer. This organization clarifies the relationships between protocol families and their achievable precision advantages over classical methods. Key quantum resources such as spontaneous parametric down-conversion-based entangled photon pairs, Greenberger-Horne-Zeilinger and W multipartite states, squeezed and frequency-entangled light, quantum frequency combs, and quantum memories are reviewed in the context of scalability and robustness.

We present a model-independent test of anisotropy in the cosmic distance duality relation (CDDR), $D_L=(1+z)^2 D_A$, using the Pantheon+ type Ia supernova sample and baryon acoustic oscillation (BAO) data. The angular diameter distance is reconstructed via Gaussian Processes, enabling an estimate of $η(z)=D_L/[D_A(1+z)^2]$ without assuming a background cosmology. We also allow for a possible isotropic evolution, parameterized as $η(z)=1+η_1 z$, and find a redshift-dependent deviation whose significance depends on the assumed supernova calibration. Anisotropy is modeled through a dipole modulation and constrained using a full covariance-based likelihood. To assess statistical significance, we construct null realizations that preserve both the redshift distribution and the survey selection function. We find that the observed dipole amplitude is consistent with isotropic expectations and lies below the levels induced by statistical fluctuations and survey geometry. We obtain a robust 95\% upper bound $A_{95}=0.025$, stable across different supernova calibration choices. We find no evidence for intrinsic anisotropy in the CDDR. Our results highlight the importance of accounting for survey selection effects in anisotropy searches and provide a viable framework for testing directional deviations in cosmological relations.

Neural-network quantum states have recently emerged as a powerful method for solving quantum many-body problems, with notable successes in lattice systems. Here, we extend this approach to strongly interacting few-body problems in continuous space, and demonstrate its capability by computing the Efimov states and associated few-body bound states. Using a fully connected feedforward neural network with Jacobi coordinates as inputs, combined with a projection method, we compute the ground and first excited states for three- to six-body systems of identical bosons at unitarity, as well as a mass-imbalanced fermionic system consisting of two identical fermions and a third particle. The obtained energies of the ground and first excited states agree well with previously reported results. Furthermore, the proposed approach also reproduces key features of Efimov states, including the discrete scale invariance, the characteristic geometric structure of the wave function, and the critical-mass behavior in mass-imbalanced fermionic systems. Our method can be readily applied to a broad class of strongly correlated few-body problems in continuous space.

In this work, we investigate the collective electrostatic effects of one-dimensional (1D) Janus MoSTe nanotubes and their impacts on the band alignment of nanotube heterostructures. Using first-principles calculations based on Density Functional Theory, we find that the Janus nanotube generates a large and uniform electrostatic potential of over 1.3 V within the nanotube pores, which is accumulative for double wall nanotubes. We develop an analytical model to provide a quantitative understanding of the electrostatic potential and its dependence on the quadrupole moment and nanotube radius. For double wall MoSTe nanotube, we find a substantial band edge shift of about 1.0 eV for the inner tube originated from the electrostatic effects, leading to a type-II band alignment. These results demonstrate that the electrostatic effects of 1D nanotubes can be used to tune the electronic properties and band alignment of 1D nanotube heterostructures for optoelectronic and catalytic applications.

We study chimera baryons, fermion bound states composed of two (hyper)quarks transforming in the fundamental and one in the antisymmetric representation of a non-Abelian gauge group. While in QCD they coincide with ordinary baryons, in composite Higgs models (CHMs) with top partial compositeness, spin-1/2 chimera baryons serve as partners of the top quark and are responsible for its large mass. We perform non-perturbative lattice calculations of the low-lying spectrum of the chimera baryons, in a specific realization of CHMs based on a Sp(4) gauge theory. In the quenched approximation, we present the numerical results in the continuum and massless limits. Then, for dynamical fermions, we measure the spectrum and matrix elements by employing a newly developed spectral density analysis for several choices of the lattice parameters.

Affordances, originating in psychology, describe how an object's design influences the physical and cognitive actions users may take. Past work applied affordance theory to visualization to explain how design decisions can impact the cognitive actions of visualization readers. In this work, we demonstrate that affordances can complement effectiveness rankings by further explaining the root causes behind visualizations' task performance. To do so, we conduct a case study on static normal probability density function plots, identifying their current affordances. Next, we identify the optimal affordances for a common probability-comparison task and develop a novel affordance-driven visualization, the Croissant Chart, to support them. We empirically validate the design's effectiveness through a preregistered study (n = 808), demonstrating how affordances can inform predictable changes in task performance. Our findings underscore the potential for affordance-based approaches to enhance visualization effectiveness and inform future design decisions.

Statistical agencies frequently release frequency tables derived from microdata, but small frequency cells may lead to disclosure risks. We present \texttt{iLBA}, an open-source \textsf{R} package for confidential dissemination of aggregated frequency tables. The package implements the Information-Loss-Bounded Aggregation (iLBA) algorithm, which combines Small Cell Adjustment (SCA) at the finest level table with an aggregation procedure that introduces controlled ambiguity while bounding information loss. The software enables users to construct masked finest level tables, generate confidential aggregated tables for selected variables, and obtain masked frequencies for single-cell queries. By providing an accessible implementation of the iLBA method, the package facilitates reproducible and efficient disclosure control for tabular data derived from microdata.

We analyze 18 quadrillion models for the joint pricing of corporate bond and stock returns. Strikingly, we find that equity and nontradable factors alone suffice to explain corporate bond risk premia once their Treasury term structure risk is accounted for, rendering the extensive bond factor literature largely redundant for this purpose. While only a handful of factors, behavioral and nontradable, are likely robust sources of priced risk, the true latent stochastic discount factor is dense in the space of observable factors. Consequently, a Bayesian Model Averaging Stochastic Discount Factor explains risk premia better than all low-dimensional models, in- and out-of-sample, by optimally aggregating dozens of factors that serve as noisy proxies for common underlying risks, yielding an out-of-sample Sharpe ratio of 1.5 to 1.8. This SDF, as well as its conditional mean and volatility, are persistent, track the business cycle and times of heightened economic uncertainty, and predict future asset returns.

Moist heatwaves and convective storms frequently co-occur, posing compound risks. Although historically concentrated in the tropics, these moist weather extremes are projected to intensify substantially towards the midlatitudes, with regions downstream of major highland terrains, including northeastern Asia and eastern North America, emerging as hotspots of future change. Yet their physical drivers remain uncertain. Here we show that the intensification of concurrent moist heat and convection extremes in the midlatitudes is tightly constrained by changes in low-level atmospheric inversions. Specifically, we find that amplified warming over western highlands is transported downstream by prevailing westerlies, strengthening low-level thermal inversions and raising the attainable maxima of moist heat and convection. Targeted model experiments confirm the critical role of orographically elevated heating in driving these extremes. Our results reveal a mechanistic pathway for compound extremes and highlight low-level inversions as a key factor for emerging midlatitude risks of moist heat and severe weather under climate change.

Phase transitions constitute fundamental mechanisms underlying abrupt or qualitative changes in the collective dynamics of interacting units across a wide range of natural and engineered systems. In dynamical networks, such transitions lead to significant reorganization in the coordinated behavior of coupled elements. In adaptive dynamical networks, the connectivity evolves dynamically in response to the states of the nodes, resulting in a coevolution of structure and dynamics. In this work, we report two distinct forms of heterogeneous nucleation that give rise to single-step and multi-step phase transitions toward global synchronization in finite-size adaptive networks with connection delays. We demonstrate that the nature of the nucleation transition is governed by both the presence and magnitude of the delay, as well as the class of natural frequency distribution. Using a collective coordinate framework, we develop a mean-field description of cluster dynamics and derive an analytical upper bound condition for the existence of two-cluster states, which shows excellent agreement with numerical simulations. Furthermore, we extend the analysis to systems with distributed delays and obtain corresponding analytical conditions. Our results provide a theoretical framework for understanding synchronization transitions in adaptive networks with time-delayed interactions.

We propose a novel mechanism in which leptonic CP-violating phases are generated dynamically through the radiative breaking of non-invertible selection rules. In this framework, tree-level mass matrices, initially constrained by a CP-like symmetry within a non-invertible structure, acquire flavor-dependent phases once loop corrections are incorporated. Furthermore, these corrections can also generate mass terms, thereby addressing the mass hierarchy problem. As an illustrative example, we employ the Inverse Seesaw (ISS) model to demonstrate how the Majorana mass of the light sterile neutrino $N_L$ arises via this mechanism while simultaneously realizing CP violation. Although our analysis is carried out within the ISS framework, the mechanism has broader implications, potentially offering new perspectives on CP-related problems such as the strong CP problem, leptogenesis, and baryogenesis. This work thus establishes a foundation for exploring the dynamical breaking of non-invertible selection rules as a novel origin of CP violation in particle physics.

The Hubble tension is shaped not only by shifts between early- and late-time parameter estimates, but also by the stiffness of the constraints that define them. In this work, we analyze this geometric structure in the wCDM model by separating the discrepancy into two components: a parameter displacement and a directional Fisher curvature. Within the local Gaussian approximation, the quadratic tension along a given direction factorizes into the squared shift and the combined directional curvature contributed by the datasets. Applying this framework to Planck, DESI DR2, and SH0ES, we show that extending \LambdaCDM to wCDM primarily reshapes the Fisher geometry of the CMB constraint rather than opening a genuinely new route to concordance. Allowing the dark-energy equation-of-state parameter w to vary suppresses the leading Planck Fisher eigenvalue to only \sim 2.7 % of its \LambdaCDM value, while producing only a modest rotation of the dominant acoustic-scale eigenmode. The net effect is a strong softening of the effective acoustic rigidity. At the same time, high-precision late-time data, especially from DESI DR2, inject substantial curvature along the expansion-rate direction. This added stiffness acts as a geometric wall, closing off phantom-like escape routes and sharply limiting tension relief within the extended parameter space. Our results indicate that changes in the inferred H_0 tension under model extension are best understood as a reconfiguration of the constraint manifold rather than as evidence for new physical agreement. The shift-curvature decomposition thus offers a simple, fast, and physically transparent way to diagnose cosmological tensions.

The discovery of ambient-pressure superconductivity in compressively strained (La,Pr)$_3$Ni$_2$O$_7$ thin films has intensified efforts to identify the pairing mechanism. However, the symmetry of the superconducting order parameter and the character of the normal state remain unsettled. Here we combine bulk-sensitive terahertz (THz) time-domain spectroscopy with THz third-harmonic generation to present spectroscopic insights into these issues. Linear THz spectroscopy reveals a bulk superconducting response in the (La,Pr)$_3$Ni$_2$O$_7$ films, evidenced by the suppression of low-frequency spectral weight below the onset critical temperature, $T_\mathrm{c}^{\mathrm{onset}}$. A weak coherence peak near $T_\mathrm{c}^{\mathrm{onset}}$, together with substantial residual low-frequency conductivity as $T\to 0$, is consistent with disordered $s_{\pm}$-wave pairing. In the nonlinear regime, the third-harmonic signal rises sharply on cooling through $T_\mathrm{c}^{\mathrm{onset}}$, providing an independent signature of the transition. Strikingly, the nonlinear response persists above $T_\mathrm{c}^{\mathrm{onset}}$, pointing to either disorder-enhanced nonlinearity or a distinct correlated normal state. Motivated by angle-resolved photoemission spectroscopy on similarly grown films that identifies a comparable temperature scale, we associate the anomalous normal-state terahertz nonlinearity with a pseudogap. These results establish (La,Pr)$_3$Ni$_2$O$_7$ as a bulk superconductor with $s_{\pm}$-like pairing that coexists with, and may compete with, a distinct ordered state, providing a platform for exploring unconventional superconductivity beyond cuprates and pnictides.

We develop a novel primal heuristic for nonconvex Mixed-Integer Quadratically Constrained Quadratic Programs (MIQCQPs). The method is built around a convex approximation that is dynamically adjusted within a feasibility-pump-style alternating heuristic. Approximations are adjusted based on the structure of the MIQCQP instance. Additionally, parallelized local branching is incorporated to further refine detected solutions. This paper builds upon the second-place finalist submission in the 2025 Land-Doig MIP Computational Competition. Our results are validated with computational experiments on instances from QPLIB, finding feasible solutions for three previously unsolved cases and improving the best-known solutions for fifteen instances within five minutes of runtime.