In this chapter we provide a pedagogical introduction to the main theoretical aspects related to topology and $θ$-dependence in Quantum Chromo-Dynamics (QCD), and to their phenomenological relevance in the Standard Model ($η^\prime$ physics, neutron electric dipole moment) and beyond (strong CP problem and the axion solution). We then provide an overview of the main analytic predictions for $θ$-dependence obtained using several different approaches (chiral effective theories, large-$N$ arguments, semiclassical methods) and their regimes of validity, as well as a selection of the most recent numerical results about QCD topology obtained via Monte Carlo simulations of the lattice-discretized theory.

A test of uniformity on [0,1] is developed for the setting of a single observation recorded with sufficient precision. Although consistency against general alternatives is not attainable with only one draw in the classical large-sample sense, a multiscale harmonic digit expansion provides a framework for structured inference. By aggregating trigonometric components across digit scales at Hadamard-gap frequencies, a quadratic test statistic is constructed whose null distribution converges to a chi-square law via a lacunary central limit theorem. Under departures from uniformity, the statistic is driven by Fourier components induced by digit-scale transformations of the observation, with detectability depending on their coherent accumulation as precision increases. The resulting procedure detects multiscale harmonic structure that remains invisible to classical digit-frequency methods.

The blazar 3C 279 is well known for its rapid and large-amplitude variability. On 20 December 2013, the source exhibited an orphan γ-ray flare characterized by a flux-doubling timescale of a few hours, a very hard spectrum, a time-asymmetric light curve with a slow decay, and no significant optical variability. We propose a new interpretation of this event based on a two-zone scenario in which a stationary emission region produces the quiescent emission, while a second zone accelerates within the broad-line region(BLR). We compute the time-dependent radiative output of both zones with the EMBLEM code, including synchrotron, synchrotron self-Compton, and external inverse-Compton processes, as well as bulk acceleration, adiabatic expansion, and a Fokker-Planck treatment of the electron distribution. This is the first attempt to precisely model the asymmetric γ-ray flux evolution during this flare. A model with a stationary region outside the dusty torus and an accelerating plasma blob reproduces the main features of the event: a short and intense γ-ray flare with a hard spectrum and no optical counterpart. The flare results from the variation of the external photon field in the blob frame as the blob crosses the BLR and reaches its terminal Lorentz factor not far from the inner radius of the BLR. Bulk acceleration and the propagation of a plasma blob within the jet provide a natural mechanism for producing high-energy flares and asymmetric light curves without requiring an ad hoc time-dependent particle injection. The model predictsa delayed EUV/X-ray enhancement once the blob exits the BLR. No very-high-energy data are available for this event, but if γ-rays were emitted in this band, a delay would be expected with respect to the Fermi-LAT flare.

We compare four polynomial-resource measurement strategies, (I) $Z$-basis-only, (II) nearest-neighbor $ZZ$ (NN), (III) multi-basis ($Z$, $X$, $Y$), and (IV) classical shadows, for classifying three quantum circuit families: IQP, Clifford, and Clifford$+T$. We find $Z$-only measurements outperform multi-basis and classical shadows across all qubit counts and all four classifiers evaluated, and the $O(\nqubits)$-feature NN strategy matches $Z$-only to within $0.02$ in Random Forest accuracy. The best result is a Random Forest accuracy of $0.91$ at 4--5 qubits under $Z$-only ($0.89$ for NN, $0.85$ for multi-basis, $0.67$ for shadows). All four strategies collapse to near-chance accuracy ($\approx 0.33$) above approximately 12 qubits under the quadratic shot budget $\shots = 16\nqubits^2$. These findings indicate that the discriminative signal between these circuit families is concentrated in local, nearest-neighbor $Z$-basis correlations, consistent with the diagonal gate structure of IQP circuits, and that additional Pauli correlator types or long-range correlations carry no compensating discriminative power for this task. We provide a formal theoretical framework showing that circuits with high diagonal fraction in a given basis concentrate their correlator structure in that basis, and that any deviation from the dominant basis incurs a provably higher estimator variance. These results establish that a quadratic shot budget is insufficient for reliable classification above approximately 12 qubits, but do not rule out the existence of a subquadratic or otherwise more efficient polynomial-resource strategy; whether any polynomial measurement protocol can classify these families at large qubit counts remains an open question.

Recent years have seen growing application potential for Lattice-skin Plate Structures in advanced manufacturing fields such as aerospace and automotive engineering. For multiscale performance evaluation of such structures, conventional homogenization methods for lattice-filled volume structures are often used for equivalent analysis. However, in finite-thickness Lattice-skin Plate Structures, periodic boundary conditions imposed along the three orthogonal directions of the representative cell cannot adequately capture the boundary effect of the free surfaces in the thickness direction, which introduces bias into the prediction of effective properties. To reduce this bias, this study develops and open-sources a homogenization method for Lattice-skin Plate Structures, forming an open-source computational framework for this class of structures. Representative numerical examples show that the framework can stably extract effective plate/shell stiffness matrices and can be extended to predict multiphase material properties and analyze steady-state heat conduction. The tool provides an open and reusable analysis foundation for the high-fidelity design of multifunctional lightweight structures.

The ultrastrong coupling regime of cavity photons and quantum materials has emerged as a pathway to modify materials properties, however definitive signatures of ultrastrong coupling remain elusive. Focusing on the quantum photon statistics of light transmitted through a cavity-embedded superconductor, we show that a two-photon Higgs polariton at strong coupling realizes a photonic nonlinearity at the single terahertz photon level. We find that as light-matter coupling increases, the photon statistics show pronounced changes due to the formation of a hybrid photon-matter dark-cavity state with finite photon occupancy, producing testable signatures of ultrastrong coupling. We derive a non-Markovian input output relation and study the cavity-embedded superconductor 2H-NbSe2 as it approaches ultrastrong light-matter coupling. Our results reveal a diagnostic for ultrastrong coupling in the two-photon coincidence statistics that is absent in total counts.

DP-coloring (also called correspondence coloring) is a generalization of list coloring introduced by Dvořák and Postle in 2015. The DP-chromatic number of a graph $G$, $χ_{_{DP}}(G)$, is the analogue of the chromatic number of $G$ in the DP context and is bounded above by the degeneracy of $G$ plus one. Over the last two years a plethora of authors have introduced variations on the notion of degeneracy and used these new ideas to give improved bounds on the DP-chromatic number of certain families of graphs. Fractional DP-coloring is a generalization of fractional list coloring introduced by Bernshteyn, Kostochka, and Zhu in 2019. In this paper we introduce two analogues of the degeneracy of a graph to the fractional context, each of which bound its fractional DP-chromatic number from above. We use these analogues to bound the fractional DP-chromatic number of a variety of graphs including unicyclic graphs, some complete bipartite graphs, and sparse graphs.

We investigate the interaction of an antikaon ($K^-$) with the $^{6}$Li nucleus, described as an $α+ d$ cluster system. The study aims to explore the formation of the $Λ(1405)$ resonance through the $K^-d$ subsystem in the presence of a spectator $α$ particle. In the absence of dedicated experimental data for this reaction, particular attention is given to providing quantitative predictions for the manifestation of the $Λ(1405)$ structure in low-energy $\bar{K}N$ dynamics within a light nuclear environment. Employing different models of the $\bar{K}N-πΣ$ interaction, we calculate the $πΣn$ invariant-mass spectra and the $α$-particle missing-mass spectra, thereby identifying robust features of the $Λ(1405)$ signal and offering guidance for future experimental investigations.

Fine-tuning and naturalness, the sensitivity of low-energy observables to small changes in the fundamental parameters of a theory, are cornerstones of physics beyond the Standard Model. We propose a new measure of fine-tuning based on information theory. To each point in parameter space we associate a probability distribution over observables. Divergence measures encode the sensitivity of observables to model parameters and determine a Riemannian metric on parameter space. By Chentsov's theorem, the physically motivated metric is the Fisher information metric, up to scaling. We propose a rescaled fine-tuning matrix $\mathcal{F}_{ij}$ derived from the Fisher information matrix, whose non-zero eigenvalues serve as our measure of fine-tuning. When the number of observables exceeds the number of parameters, $\mathcal{F}_{ij}$ admits a natural geometric interpretation as the pullback of the Euclidean metric from observable space to the submanifold of admissible predictions, with large eigenvalues corresponding to highly stretched directions and indicative of fine-tuning. Our measure reproduces the familiar Barbieri--Giudice criterion as a special case, while generalising it to multiple correlated parameters. We illustrate its behaviour on dimensional transmutation, the Wilson--Fisher fixed point, a simple model of the hierarchy problem, and the electron Yukawa coupling, finding agreement with physical intuition in each case.

Preparing high-fidelity Greenberger-Horne-Zeilinger (GHZ) states on noisy quantum hardware remains challenging due to cumulative gate errors and decoherence. We introduce Group-Majority-Voting (Group-MV), a dynamic-circuit protocol that partitions arbitrary coupling graphs, prepares local GHZ states in parallel, and fuses them via majority-voted mid-circuit measurements. The majority vote over redundant boundary links mitigates measurement errors that would otherwise propagate through classical feedforward. We evaluate Group-MV on simulated Heavy-hex and Grid topologies for 30 through 60 qubits under a realistic noise regime. Group-MV generalizes to arbitrary GHZ sizes on arbitrary coupling topologies, achieving 2.4x higher fidelity than the Line Dynamic method while tracking the unitary baseline within 3%.

Large-scale integration of distributed energy resources has led to a rapid increase in the number of controllable devices and a significant change in system dynamics. This has necessitating the shift towards more distributed and scalable control strategies to manage the increasing system complexity. In this work, we address the problem of partitioning a low-inertia power network into dynamically coherent subsystems to facilitate the utilization of distributed control schemes. We show that an embedding of the power network using the spectrum of the linearized synchronization dynamics matrix results in a natural decomposition of the network. We establish the connection between our approach and the broader framework of spectral clustering using the Laplacian matrix of the admittance network. The proposed method is demonstrated on the IEEE 30-bus test system. We consider the robustness of the clusters by analyzing the sensitivity of the small eigenvalues and their corresponding eigenspaces to perturbations caused by variation in the steady-state operating points of the network.

We provide a comparison test for meromorphic extensions, i.e., if two series are ``close enough" then the existence of a meromorphic extension of one to the entire complex plane ensures a similar extension for the other. We use this result to generate new examples of Dirichlet series admitting meromorphic extensions. Moreover, we demonstrate that our requirements are optimal by constructing a collection of counterexamples where the series are ``close but not enough": one series admits a meromorphic extension while the other possesses a natural boundary.

Precise gravitational measurements are vital for geophysics and inertial navigation, but compact gravimeters with high measurement bandwidth remain difficult to realize. We propose and analyze a chip-scale superconducting gravimeter in which a flux-tunable transmon qubit is coupled to a high quality factor ($Q_m$) nanomechanical beam. The beam is embedded in a SQUID loop placed in parallel with the qubit's flux-tunable SQUID; gravity induced beam displacement therefore modulates the qubit frequency through the SQUID flux and is mapped onto the qubit's geometric phase. A stroboscopic readout at mechanical revival times suppresses qubit mechanics dephasing, yielding a projected sensitivity of $10^2$--$10^3\,\mathrm{nGal}/\sqrt{\mathrm{Hz}}$ with sub-millisecond interrogation times. Electrical \emph{in situ} tunability and microwave-based calibration make this architecture a practical route toward compact, high-bandwidth on-chip gravimetry.

The efficiency of recombination is of crucial importance for the existence of ultracold plasmas (UCP), particularly, the ones formed in the magneto-optical traps. Unfortunately, the equilibrium thermodynamic treatment of the ionization-recombination processes is inappropriate for the evolving UCP clouds, while the straightforward kinetic simulation encounters the problem of huge difference in the spatial and temporal scales for free and bound motion of the electrons. As a result, only the "virtual" electron-ion pairs are usually reproduced in such modeling, and it is necessary to employ some heuristic criteria to identify them with the recombined atoms. It is the aim of this paper to present the first successful ab initio simulation of the non-equilibrium recombination in the evolving UCP plasmas. We employed a special algorithm, which is based on using the "scalable" reference frame, co-moving with the expanding substance. Then, the recombination events are identified by a series of sharp equidistant peaks in the kinetic and/or potential energies, which are caused by the captured electrons passing near the pericenters of their orbits; and this is confirmed by a detailed inspection of their trajectories. Thereby, we were able to trace the real - rather than "virtual" - electron-ion pairs, and the total efficiency of their formation was found to be about 20%, which is in agreement with the laboratory measurements.

We identify a symmetry that enforces every symmetric model to have a Fermi surface. These symmetry-enforced Fermi surfaces are realizations of a powerful form of symmetry-enforced gaplessness. The symmetry we construct exists in quantum lattice fermion models on a $d$-dimensional Bravais lattice, and is generated by the on-site U(1) fermion number symmetry and non-on-site Majorana translation symmetry. The resulting symmetry group is a noncompact Lie group closely related to the Onsager algebra. For a symmetry-enforced Fermi surface $\cal{F}$, we show that this UV symmetry group always includes the subgroup of the ersatz Fermi liquid L$_{\cal{F}}$U(1) symmetry group formed by even functions ${f(\mathbf{k})\in\mathrm{U}(1)}$ with ${\mathbf{k}\in \cal{F}}$. Furthermore, we comment on the topology of these symmetry-enforced Fermi surfaces, proving they generically exhibit at least two noncontractible components (i.e., open orbits).

We prove that under the heat semigroup $(P_τ)$ on the Boolean hypercube, any nonnegative function exhibits a uniform tail bound that is better than Markov's inequality. Specifically, for any $τ> 0$, $n \geq 1$, $η> e^3$, and $f: \{-1,1\}^n \to \mathbb{R}_+$ with $\int f dμ> 0$, we have \begin{align*} \mathbb{P}_{X \sim μ}\left( P_τf(X) > η\int f dμ\right) \leq c_τ\frac{ (\log \log η)^{\frac32} }{η\sqrt{\log η}}, \end{align*} where $μ$ is the uniform measure on the Boolean hypercube $\{-1,1\}^n$ and $c_τ$ is a constant that depends only on $τ$. This result resolves Talagrand's convolution conjecture up to a dimension-free $(\log \log η)^{\frac32}$ factor. Our proof uses the reverse heat process on the Boolean hypercube, a coupling construction with carefully engineered perturbations of jump rates and a time-smoothed anti-concentration estimate.

We study the quantum relaxation dynamics for a lattice version of the one-dimensional (1D) $N$-flavor Gross-Neveu (GN) model after a Hamiltonian parameter quench. Allowing for a system-reservoir coupling $γ$, we numerically describe the system dynamics through a time-dependent self-consistent Lindblad master equation. For a closed ($γ=0$) finite-size system subjected to an interaction parameter quench, the order parameter dynamics exhibits oscillations and revivals. In the thermodynamic limit, our results imply that the order parameter reaches its post-quench stationary value in accordance with the eigenstate thermalization hypothesis (ETH). However, time-dependent finite-momentum correlation matrix elements equilibrate only if $γ>0$. Our findings are consistent with the system being described by a pertinent Generalized Gibbs Ensemble (GGE) and, accordingly, highlight subtle yet important aspects of the post-quench relaxation dynamics of quantum many-body systems.

Large Language Models (LLMs) have shown potential for solving mathematical tasks. We show that LLMs can be utilized to generate proofs by induction for hardware verification and thereby replace some of the manual work done by Formal Verification engineers and deliver industrial value. We present a neurosymbolic approach that includes two prompting frameworks to generate candidate invariants, which are checked using a formal, symbolic tool. Our results indicate that with sufficient reprompting, LLMs are able to generate inductive arguments for mid-size open-source RTL designs. For 84% of our problem set, at least one of the prompt setups succeeded in producing a provably correct inductive argument.

Noncentrosymmetric superconductors (NCSs) with Rashba spin-orbit coupling (SOC) and in-plane magnetic fields have emerged as natural platforms for realizing both the bulk superconducting diode effect (SDE) and the Josephson diode effect (JDE) - phenomena characterized by unequal critical currents in opposite directions due to the simultaneous breaking of time-reversal and inversion symmetries. Using the quasiclassical Eilenberger formalism, we systematically investigate both the bulk SDE and the JDE in a clean NCS with Rashba SOC and in-plane magnetic fields. For the bulk system, we find that the diode efficiency can nominally approach its maximal value at the critical endpoint of the first-order Lifshitz transition between weak and strong helical phases featuring finite-momentum Cooper pairs, the latter marked by the emergence of Bogolyubov Fermi surfaces (BFSs). In a Josephson junction, we show that finite-momentum pairing in the superconducting leads is the dominant mechanism behind the JDE in short junctions, whereas in long junctions it is primarily governed by the Zeeman field in the normal region. In the long-junction regime, the diode efficiency additionally oscillates between positive and negative values as a function of magnetic field at low fields, providing a route toward a highly tunable Josephson diode. At higher fields, the onset of BFSs in the strong helical phase leads to a sharp suppression of both the JDE and the Josephson current when the current direction is aligned with momenta along the BFS, resulting in strong anisotropy. We propose that this anisotropy in the Josephson current offers an alternative method for detecting BFSs, applicable to systems with or without a JDE.

Steadily shearing a non-Brownian suspension forms a memory of direction, while shearing back and forth forms a memory of amplitude. Each memory is evident in the systems response to further shear, exemplifying its strong history-dependence. By combining the steady and oscillatory experiments, we show these memories are distinct but intersecting aspects of the same non-equilibrium physics: they can coexist, yet a specific amplitude suppresses directional memory and makes the system symmetric. Combined with prior results from disordered solids, our work presents a simple motif for limited memory capacity in non-equilibrium matter.

Controlled commands -- computations whose execution depends on a separate input -- play a central role in reversible Boolean circuits and quantum circuits. However, existing formalisms typically treat control only implicitly, entangled with other aspects of computation. From a semantic perspective, control is most naturally expressed in semisimple rig categories, which -- unlike standard circuit models such as props -- support both parallel and conditional composition. We present a construction that freely adjoins an explicit syntactic notion of control to a circuit theory specified as a suitable prop, subject to eight universally quantified equations. Our main result is that these equations are sound and complete for the intended semantics of control: the resulting theory satisfies a universal property, identifying it exactly as the circuit subtheory of the free semisimple rig completion. The proof combines coherence for rig categories with a new method based on induction over Gray codes. We illustrate the usefulness of the framework by showing that it simplifies several existing sound and complete axiomatisations of quantum circuits, isolating a small and conceptually clean set of generators and equations. In addition, the same equations yield a sound and complete axiomatisation of the multiply controlled Toffoli gate set, that is universal for reversible Boolean circuits.

Altermagnets have zero net magnetization yet feature spin-split bands. Here, we investigate how slow lattice vibrations (phonons) influence both the intrinsic and externally induced spin polarizations in two-dimensional $d$-wave altermagnets. For the induced spin polarization, we employ a Rashba continuum model with electron-phonon coupling (EPC) treated at the static Holstein level and analyze the spin Edelstein effect using the Kubo linear-response formalism to probe EPC-induced contributions. We find that, under a specific symmetry-lowering pattern such as a piezomagnetically active strain that explicitly breaks the inherent $C_4 \mathcal{T}$ symmetry, moderate-to-strong EPC progressively suppresses the induced polarization via both intraband and interband channels, with a threshold coupling marking the onset of complete spin Edelstein depolarization. The depolarization arises from a phonon-induced energy renormalization that leads to a complete collapse of the Fermi surface. While depolarization can occur even in the Rashba non-altermagnetic phase, it remains isotropic. The presence of altermagnetism makes it anisotropic and breaks the conventional antisymmetry between spin susceptibilities that occurs with pure spin-orbit coupling, rendering the effect highly relevant for spintronic applications. We further investigate how the phonon coupling to the altermagnetic order, Rashba spin-orbit strength, and carrier doping collectively tune the depolarization. Our findings demonstrate that static phononic effects offer a powerful means for on-demand control of spin polarization, enabling reversible switching between spin-polarized and depolarized states--a key functionality for advancing spin logic architectures and optimizing next-generation spintronic devices.

Correlated structures are intimately connected to intriguing phenomena exhibited by a variety of disordered systems such as soft colloidal gels, bio-polymer networks and colloidal suspensions near a shear jamming transition. The universal critical behavior of these systems near the onset of rigidity is often described by traditional approaches as the coherent potential approximation - a versatile version of effective-medium theory that nevertheless have hitherto lacked key ingredients to describe disorder spatial correlations. Here we propose a multi-purpose generalization of the coherent potential approximation to describe the mechanical behavior of elastic networks with spatially-correlated disorder. We apply our theory to a simple rigidity-percolation model for colloidal gels and study the effects of correlations in both the critical point and the overall scaling behavior. We find that although the presence of spatial correlations (mimicking attractive interactions of gels) shifts the critical packing fraction to lower values, suggesting sub-isostatic behavior, the critical coordination number of the associated network remains isostatic. More importantly, we discuss how our theory can be employed to describe a large variety of systems with spatially-correlated disorder.

Graph-based approaches to approximate nearest neighbor search (ANNS) enable fast, high-recall retrieval on billion-scale vector datasets. Among them, the Sparse Neighborhood Graph (SNG) is widely used due to its strong search performance. However, the lack of theoretical understanding of SNG leads to expensive tuning of the truncation parameter that controls graph sparsification. In this work, we present OPT-SNG, a principled framework for analyzing and optimizing SNG construction. We introduce a martingale-based model of the pruning process that characterizes the stochastic evolution of candidate sets during graph construction. Using this framework, we prove that SNG has a maximum out-degree of \(O(n^{2/3+ε})\), where \(ε>0\) is an arbitrarily small constant, and an expected search path length of \(O(\log n)\). Building on these insights, we derive a closed-form rule for selecting the optimal truncation parameter \(R\), thereby eliminating the need for costly parameter sweeping. Extensive experiments on real-world datasets demonstrate that OPT-SNG achieves an average \(5.9\times\) speedup in index construction time, with peak improvements reaching \(15.4\times\), while consistently maintaining or improving search performance.

The surprising excess of missing-energy events far beyond all standard-model expectations in the weak decays of the charged ground-state $B^+$ meson into some charged strange meson, rather recently observed by the Belle-II experiment, may (easily) be explained by the decay of the $B$ meson into the strange meson and a pair of dark-matter fermion and antifermion, mediated by an (intermediate) scalar or vector boson. Thorough inspections of both the total and the differential widths of these decays provide, among others, a simple means for the (straightforward) discrimination of such mediator boson's scalar or vector nature.

We study a spectral analog of the Turán problem for simplicial complexes. Specifically, we consider the extremal problem of maximizing the signless Laplacian spectral radius among simplicial complexes without holes. We determine the structure of the simplicial complex attaining the maximum spectral radius, extending classical extremal results for graphs without cycles to the setting of higher-dimensional simplicial complexes. More generally, we establish an upper bound on the signless Laplacian spectral radius of simplicial complexes with prescribed Betti numbers. As an application, using the connection between the signless Laplacian spectral radius and the face numbers of a simplicial complex, we derive bounds on Turán numbers for both hypergraphs and simplicial complexes. Our technique involves the canonical Alexander dual of perfect matchings and coloring of simplicial complexes.

We develop the theory of antisymmetric Raman response, defined as the difference between the Raman signals of two scattering geometries related by an exchange of mutually perpendicular incoming and the outgoing photon polarizations. Such responses, finite in orthorhombic or lower symmetry systems, are related to cross-susceptibilities of two Raman operators and are characterized by the absence of intraband terms. This is in contrast to standard Raman responses which measure auto-susceptibilities where both intra- and interband processes contribute. We illustrate the theory with examples from the charge density wave rare-earth tritellurides and the excitonic insulator Ta$_2$NiSe$_5$. Our theory establishes antisymmetric Raman response as a unique tool to probe microscopic features such as interband energy scales and to detect reflection symmetry breaking.

In this work we apply a procedure based on the quantum imaginary time evolution method to solve the unit-disk maximum independent set problem. Numerical simulations were performed for instances of 6, 8 and 10-qubits graphs. We have found that the failure probability of the procedure is relatively small and rapidly decreases with the number of shots. In addition, a theoretical upper bound for the failure probability of the procedure was obtained.

We study a colored generalization of the famous simple-switch Markov chain for sampling the set of graphs with a fixed degree sequence. Here we consider the space of graphs with colored vertices, in which we fix the degree sequence and another statistic arising from the vertex coloring, and prove that the set can be connected with simple color-preserving switches or moves. These moves form a basis for defining an irreducible Markov chain necessary for testing statistical model fit to block-partitioned network data. Our methods further generalize well-known algebraic results from the 1990s: namely, that the corresponding moves can be used to construct a regular triangulation for a generalization of the second hypersimplex. On the other hand, in contrast to the monochromatic case, we show that for \emph{simple} graphs, the 1-norm of the moves necessary to connect the space increases with the number of colors.

We study locally flat, compact, oriented surfaces in $4$-manifolds whose exteriors have infinite cyclic fundamental group. We give algebraic topological criteria for two such surfaces, with the same genus $g$, to be related by an ambient homeomorphism, and further criteria that imply they are ambiently isotopic. Along the way, we prove that certain pairs of topological $4$-manifolds with infinite cyclic fundamental group, homeomorphic boundaries, and equivalent equivariant intersection forms, are homeomorphic.