Let $\mathcal H$ be a complex Hilbert space and $\mathcal F_s (\mathcal H)$ the real vector space of all self-adjoint finite rank bounded operators on $\mathcal H$. We generalize the famous Wigner's theorem by characterizing linear maps on $\mathcal F_s (\mathcal H)$ which preserve the set of all rank $k$ projections. In order to do this, we first characterize linear maps on the real vector space $\mathcal H_{0, 2k}$ of trace zero $(2k) \times (2k)$ hermitian matrices which preserve the subset of unitary matrices in $\mathcal H_{0, 2k}$. We also study linear maps from $\mathcal F_s (\mathcal H)$ to $\mathcal F_s (\mathcal K)$ sending projections of rank $k$ to finite rank projections. We prove some properties of such maps, e.g. that they send rank $k$ projections to projections of a fixed rank. We give the complete description of such maps in the case $\dim \mathcal H = 2$. We give several examples which show that in the general case the problem to describe all such maps seems to be complicated.

In this paper, we develop a quantum-optical treatment of acceleration radiation for atoms freely falling into a Schwarzschild black hole when the ambient field is a massive spin-1 (Proca) field. Building on the HBAR framework of Scully and collaborators, we analyze two detector realizations: a charged-monopole current coupling and a physical electric-dipole coupling, both within a cavity that isolates a single outgoing Schwarzschild mode prepared in the Boulware state. Using a near-horizon stationary-phase analysis, we show that the thermal detailed-balance factor governing excitation versus absorption is universal and depends only on the near-horizon Rindler coordinate transformation. At the same time, the absolute spectra acquire distinctive Proca signatures: a hard mass threshold, polarization-dependent prefactors, and axial/polar graybody transmissions. Promoting single-pass probabilities to escaping rates yields a master equation whose steady state is geometric and whose entropy flux obeys an horizon brightened acceleration radiation-style area-entropy relation identical in form to the scalar case, with all vector-field specifics entering through the radiative area change. Our results provide a controlled pathway to probe longitudinal versus transverse responses, mass thresholds, and the role of polarization-resolved graybody transmission in acceleration radiation. More precisely, we derive the universal near-horizon kernel and show how the Proca transmission data enter the escaping probabilities, rates, and entropy flux; a dedicated numerical computation of the axial/polar graybody profiles is left for future work. This sets the stage for extensions to rotating backgrounds, alternative exterior states, and detector-engineering strategies.

We perform the first simultaneous Bayesian inference of the temperature-dependent heavy-quark spatial diffusion coefficient $2πT\mathcal{D}_s$ and the scaled jet transport coefficient $\hat{q}/T^3$ in the quark-gluon plasma, utilizing $D$-meson nuclear modification factor $R_\text{AA}$ and elliptic flow $v_2$ data from Pb-Pb collisions at $\sqrt{s_\text{NN}} = 5.02\ \text{TeV}$. The analysis employs a unified improved Langevin transport model that incorporates both collisional and radiative energy loss, followed by coalescence plus fragmentation hadronization. The posterior distributions of the parameters of $\hat{q}/T^3$ and those of $2πT\mathcal{D}_s$ are well constrained, and compared with the results of theoretical models or other experimental data extraction, respectively. The $30-50\%$ centrality data provide significantly stronger constraints than the $0-10\%$ data. The extracted ratio $\hat{q}/κ$ between the quark jet transport and heavy-quark diffusion coefficients exhibits a non-monotonic temperature dependence, deviating from the value $2$ estimated from the definition, with a value interval spanning 0.25--0.8 corresponding to the mean values of the inferred parameters. This work establishes a data-driven quantitative relationship between these two fundamental transport properties in the same observables, offering crucial insight into their interplay in the strongly coupled medium.

We show that 3d gravity on manifolds that are topologically a Riemann surface times an interval $Σ_{g,n}\times I$ with end-of-the-world branes at the ends of the interval is described by a random matrix model, namely the Virasoro minimal string. Because these manifolds have $n$ annular asymptotic boundaries, the path integrals naturally correspond to spectral correlators of open strings upon inverse Laplace transforms. For $g=0$ and $n=2$, we carry out an explicit path integration and find precise agreement with the universal random matrix expression. For Riemann surfaces with negative Euler characteristic, we evaluate the path integral as a gravitational inner product between states prepared by two copies of Virasoro TQFT. Along the way, we clarify the effects of gauging the mapping class group and the connection to chiral 3d gravity.

The leading corrections to General Relativity can be parametrized by higher-derivative interactions in a low-energy effective field theory, in a way that is general and agnostic to the precise UV completion of gravity. Using numerical methods, we compute the leading-order corrections to the Kerr metric across the entire range of sub-extremal values of spin and analyse their impact on physical quantities. We find that rapidly rotating black holes are most affected by the higher-derivative corrections, making them especially sensitive probes of new physics. A dataset of solutions and the code used to produce them are publicly available.

In this paper, we determine the $r$-colour size Ramsey number of the path $P_k$, up to constants. In particular, for every fixed $r \geq 2$ and $k \geq 100\log r$, we have \[ \widehat{R}_r(P_k)=Θ((r^2 \log r) \, k).\] Perhaps surprisingly, we do this by improving the lower bound on $\widehat{R}_r(P_k)$.

Let $A_q$ be the alternating central extension of the q-Onsager algebra, a comodule algebra over the quantum loop algebra of $sl_2$. We classify one-dimensional representations of $A_q$, and show that spin-j K-operators constructed in arXiv:2301.00781 act as K-matrices previously obtained in the literature. Using these K-operators and K-matrices, we construct universal spin-j transfer matrices generating commutative subalgebras in $A_q$. Within a technical conjecture, we derive their fusion hierarchy, the so-called universal TT-relations. On spin-chain representations of $A_q$, we show how the universal transfer matrices evaluate to spin-chain transfer matrices, and as a result we get explicit TT-relations for all values of spins for auxiliary and quantum spaces, any inhomogeneities, and general integrable boundary conditions. In particular, we derive previously conjectured TT-relations. Using the TT-relations, we show that n-th local conserved quantities of the spin-j chains of length N are polynomials of total degree 4Njn in two non-local operators of the q-Onsager algebra. As a result, we give an algorithm of explicit calculation of all local conserved quantities in terms of spin operators. Furthermore, using the universal TT-relations we derive exchange relations between spin-j Hamiltonians and the two non-local operators showing non-trivial symmetries for special boundary conditions, that they commute with all Hamiltonian densities. As another application of our universal TT-relations we propose universal T-system, Y-system and universal TQ-relations for $A_q$, and as a result, universal TQ for the q-Onsager algebra. For diagonal boundary conditions, we also obtain universal TT- and TQ-relations for a degenerate version of $A_q$ known as centrally extended augmented q-Onsager algebra. We finally discuss implications of our results for generalized Gibbs ensemble construction.

We construct parametrized isometric tensor network states -- referred to as skeletons -- that allow us to explore phases of abelian topological order and can be efficiently implemented on quantum processors. We obtain stable finite correlation length deformations of string-net fixed points, which are constructed both by conserving virtual symmetries of the tensor and by imposing local isometry constraints. They connect distinct topological phases via a shared critical point, thereby providing analytically tractable examples of phase transitions beyond anyon condensation. By mapping such classes of 2D tensor networks to 1D stochastic automata with local update rules, we show that expectation values of generalized Pauli strings of arbitrary weight can be efficiently computed using classical methods. Therefore these skeletons not only serve as an organizing principle for abelian topological order but also provide a non-trivial testbed for quantum processors.

High-redshift radio galaxies can provide insights into the structure formation and galaxy evolution at earlier cosmological epochs. TGSSJ1530+1049 was selected as a candidate high-redshift radio galaxy. Subsequent observations with the James Webb Space Telescope (JWST) presented in a companion paper (Saxena et al., 2026) have shown that it is located at a redshift z=4.0. The JWST data furthermore showed that the radio source is part of one of the densest structures of galaxies and ionized gas known at these redshifts. The complex system qualitatively resembles a massive (cluster) galaxy forming early through a rapid succession of mergers. TGSSJ1530+1049 is an unresolved source down to ~0.6" scale in multiple radio surveys. To reveal its high-resolution radio structure and allow for a detailed comparison with JWST observations, we studied its morphology at various angular scales with different radio interferometric instruments. We observed TGSSJ1530+1049 at milliarcsecond (mas) scale angular resolution with the European VLBI Network (EVN), and at ~100-mas scale resolution with the enhanced Multi-Element Remotely Linked Interferometer Network (e-MERLIN). We recovered a complex north--south oriented structure with steep-spectrum radio-emitting features, which are associated with lobes and hot spots of a jetted active galactic nucleus. The centre of the radio galaxy proved to be too faint at cm wavelengths to be unambiguously detected in our observations. Nevertheless, the linear size (~5.5 kpc) and the radio power place it among the so-called medium-sized symmetric objects, a smaller and/or confined version of larger radio galaxies. Comparison between the radio morphology and that of the ionized gas observed by JWST shows that the two are closely aligned. However, the optical emission line gas extends out to ~25 kpc, which is well beyond the detected radio structures. (Abridged)

Replay and rollback attacks threaten cloud application integrity by reintroducing authentic yet stale data through an untrusted storage interface to compromise application decision-making. Prior security frameworks mitigate these attacks by enforcing forward-only state transitions (state continuity) with hardware-backed mechanisms, but they categorically treat all rollback as malicious and thus preclude legitimate rollbacks used for operational recovery from corruption or misconfiguration. We present Rebound, a general-purpose security framework that preserves rollback protection while enabling policy-authorized legitimate rollbacks of application binaries, configuration, and data. Key to Rebound is a reference monitor that mediates state transitions, enforces authorization policy, guarantees atomicity of state updates and rollbacks, and emits a tamper-evident log that provides transparency to applications and auditors. We analyze Rebound's security properties and show through an application case study -- with software deployment workflows in GitLab CI -- that it enables robust control over binary, configuration, and raw data versioning with low end-to-end overhead.

Finite-size general security proofs for quantum key distribution based on Rényi entropies have recently been introduced. These approaches are more flexible and provide tighter bounds on the secret key rate than traditional formulations based on the von Neumann entropy. However, deploying them requires minimizing the conditional Rényi entropy, a difficult optimization problem that has hitherto been tackled using ad-hoc techniques based on the Frank-Wolfe algorithm, which are unstable and can only handle particular cases. In this work, we introduce a method based on non-symmetric conic optimization for solving this problem. Our technique is fast, reliable, and completely general. We illustrate its performance on several protocols, whose results represent an improvement over the state of the art.

The ground state of the classical antiferromagnetic XX model in a magnetic field is calculated for spins mounted on the vertices of the icosahedron. The magnetization is characterized by two discontinuities as a function of the external field. For a wide field range above the first discontinuity the ground state is degenerate, with two spins related by spatial inversion aligned with the field and the rest forming two magnetization units in the form of pentagons. It is shown that the degeneracy originates from the coupling of the two pentagons, which introduces the triangle, associated with ground-state degeneracy, as an interaction unit in the icosahedron. The magnetization discontinuities are shown to evolve first from the coupling of isolated triangles and then from the coupling of the two spins related by spatial inversion.

Quantifying the energetic cost of implementing quantum operations is essential for assessing the efficiency and scalability of quantum sensing and information-processing technologies. Here, we introduce a framework for analysing the interplay between complexity and energy cost of quantum processes. In particular, we apply our framework to a sequential quantum phase estimation protocol, where a phase of physical significance is encoded in a quantum channel. The channel is applied to a probe state repeatedly until the probe is measured and the outcome leads to an estimate on the phase. We establish a trade-off relation between the total energy cost of the protocol and the number of times the channel is applied (complexity), while reaching a desired estimation precision. A sweet spot is located where the two quantities are co-optimised. The principles of our analysis can be adapted to benchmark the energetic requirements in other quantum protocols and devices.

Due to recent blackout and system split incidents in power grids worldwide, as well as increased system complexity in view of the energy transition, there has been increasing interest in re-evaluating existing Power System Restoration (PSR) plans. In restoration scenarios, due to low island inertia, it is necessary to ensure not only the static, but also the dynamic stability of the system. In this paper, we pose and solve a formulation of the optimal PSR problem including frequency dynamics. We validate the switching constraints for global optimality within a static version of the formulation using a brute-force tree search method. We apply the dynamic problem formulation to the IEEE 9-Bus model, and show that the optimal switching sequence using the static formulation would violate dynamic constraints, illustrating the importance of dynamic considerations in PSR planning.

Let $G$ be a group acting on a field $L$, and suppose that $L /L^G$ is a finite extension. We show that the category of semilinear representations of $G$ over $L$ can be described in terms of the category of linear representations of $H$, the kernel of the map $G \rightarrow \mathrm{Aut}(L)$. When $G$ is finite and $L$ has characteristic 0 this provides a character theory for semilinear representations of $G$ over $L$, which recovers ordinary character theory when the action of $G$ on $L$ is trivial.

We study a few-body system composed of self-propelling camphor surfers confined within a circular boundary. These millimeter-sized particles move in a regime where inertia and long-ranged interactions play a significant role, leading to surprisingly complex and subtle collective dynamics. These dynamics include self-organized bursts and glassy behavior at intermediate densities--phenomena not apparent from ensemble-averaged steady-state measures. By analyzing quantities like the overlap order parameter, we observe that the system exhibits dynamical slowing down as particle density increases. This slowdown is also reflected in the bursting activity, where both the amplitude and frequency of bursts decrease with increasing particle density. A minimal inertial active-particle model reproduces these dynamical steady states, revealing the importance of a new intermediate length scale--larger than the particle size. This intermediate scale is critical for the formation of structures resembling caging and plays a key role in the glass-like transition. Our results describe a macroscopic analog of an active glass with the additional phenomena of density-dependent bursting.

Neuromorphic computing demands synaptic elements that can store and update weights with high precision while being read non-destructively. Conventional ferroelectric synapses store weights in remnant polarization states and might require destructive electrical readout, limiting endurance and reliability. We demonstrate a ferroelectric MEMS (FeMEMS) based synapse in which analog weights are stored in the piezoelectric coefficient $d_{31,eff}$ of a released Hf$_{0.5}$Zr$_{0.5}$O$_2$ (HZO) MEMS unimorph. Partial switching of ferroelectric domains modulates $d_{31,eff}$, and a low-amplitude mechanical drive reads out the weight without read-disturb in the device yielding more than 7-bit of programming levels. The mechanical switching distribution function follows a Lorentzian distribution as a logarithmic function of partial poling voltage ($V_p$) consistent with nucleation-limited switching (NLS), and the median threshold extracted from electromechanical data obeys a Merz-type field-time law with a dimensionless exponent $α= 3.62$. These relationships establish a quantitative link between mechanical weights and electrical switching kinetics. This mechanically read synapse avoids depolarization and charge-injection effects, provides bipolar weights (well suited for excitatory and inhibitory synapses), directly reveals partial domain populations, and offers a robust, energy-efficient route toward high-bit neuromorphic hardware.

We derive new families of quantum null energy inequalities (QNEIs), i.e. bounds on integrated null energy, in quantum field theories in two and higher dimensions. These are universal, state-independent lower bounds on semi-local integrals of $\langle T_{vv} \rangle$, the energy-momentum flux in a null direction, and the first of this kind for interacting theories in higher dimensions. Our ingredients include the quantum null energy condition (QNEC), strong subadditivity of von Neumann entropies, defect operator expansions, and the vacuum modular Hamiltonians of null intervals and strips. These results are fundamental constraints on null energy in quantum field theories.

MINFLUX microscopy allows for localization of fluorophores with nanometer precision using targeted scanning with an illumination profile with a minimum. However, current scanning patterns and the overall procedure are based on heuristics, and may therefore be suboptimal. Here we present a rigorous Bayesian that offers maximal resolutions from either minimal detected photons or minimal exposures. We estimate using simulated localization runs that this approach should reduce the number of photons required for 1 nm resolution by a factor of about four.

Given a measure equivalence coupling between two finitely generated groups, Delabie, Koivisto, Le Maître and Tessera have found explicit upper bounds on how integrable the associated cocycles can be. We extend these results to the broader framework of unimodular compactly generated locally compact groups. We also generalize a result by the first-named author, showing that the integrability threshold described in these statements cannot be achieved.

An interaction between ferromagnetic and superconducting orders, to be realized in a two dimensional ferromagnetic superconductor, is proposed obeying necessary symmetry principles. This interaction allows us to formulate a duality, similar to the Boson-vortex duality in 2+1 dimensional superfluid. In the dual theory the Skyrmion and the vortex excitations interact with each other via an emergent gauge field. The static interaction potential is attractive for a Skyrmion and a vortex with opposite topological charges. This interaction can lead to formation of bound pairs of the mentioned topological excitations. Furthermore, we argue that such pairing implies that a Magnus force acting on the vortex induces a transverse, Hall-like drift motion of the Skyrmion, which we term the vortex-drag induced Skyrmion Hall effect. Possible experimental manifestations of this effect are also discussed.

We introduce a higher-derivative action for dark matter whose energy-momentum tensor describes an imperfect fluid with nonzero pressure, energy flux, and anisotropic stress. In the limit where the higher-derivative couplings are switched off, the energy-momentum tensor reduces to pressureless dust. A systematic derivation follows from extending the singular conformal transformation used in the mimetic dark matter scenario to include higher-derivative terms while the resulting action is general and does not rely on the mimetic framework. On a homogeneous cosmological background, the dynamics coincides with that of pressureless dust, while in the presence of inhomogeneities the higher-derivative terms generate nonzero acceleration and vorticity, making it possible to avoid the formation of caustic singularities even if the strong energy condition satisfies. In particular, within the mimetic realization these terms can resolve the usual caustic pathology of mimetic dark matter.

We show that space-time modulation of electromagnetic potentials enables Klein tunneling far below the static threshold. The derived kinematics reveal oblique transitions that can connect opposite-energy continua without requiring their overlap, yielding a velocity-tunable Klein gap where transmission vanishes within a finite velocity window and reemerges beyond. The associated reduction in energy thresholds -- by up to four orders of magnitude -- suggests the potential for experimental realization using flying-focus fronts and relativistic electron beams.

We present a complete one-loop analysis of charged scalar boson pair production in photon-photon collisions, $γγ\to H^\pm H^\mp$, within the framework of the Inert Doublet Model (IDM). The calculation is carried out in the on-shell renormalization scheme and incorporates both weak corrections and QED effects, including soft and hard photon radiation. Virtual loop contributions and real emission processes are computed using the Feynman diagrammatic method, ensuring the cancellation of ultraviolet and infrared divergences. To properly account for the Coulomb singularity that arises in the QED sector near threshold, we introduce the resummed cross section based on the Sommerfeld factor. The IDM parameter space is explored under theoretical consistency conditions, collider limits, and dark matter constraints, and three representative scenarios are studied in detail. We find that the magnitude of the quantum corrections is strongly controlled by the absolute value of the trilinear scalar coupling $λ_{h^0 H^+ H^-}$, which correlates with the charged scalar mass. When all constraints are applied, the weak corrections are typically in the range of $-12\%$ to $-7\%$ at $\sqrt{s}=250$~GeV, and between $-15\%$ and $6\%$ at $\sqrt{s}=500$~GeV. At higher energies, such as $\sqrt{s}=1$~TeV, the corrections can become very large, ranging from about $-20\%$ up to $+60\%$. Our findings highlight the significant role of higher-order effects in photon-photon collisions and establish $γγ\to H^\pm H^\mp$ as a promising process to investigate the charged scalar sector of the IDM at future high-energy photon colliders. Several benchmark points are proposed to facilitate future experimental searches.

Galaxy observations suggest that mergers of supermassive black hole binaries (SMBHBs) are rare events, with rates of order one per decade across the observable Universe. We present a framework to search for merging SMBHBs in pulsar timing array (PTA) data using a physically complete waveform model including inspiral, merger, ringdown, and gravitational-wave memory. This enables a unified treatment of continuous emission and the non-oscillatory memory signal. Using simulated PTA datasets, we demonstrate parameter estimation for representative systems with chirp masses of $10^8$ and $10^{10}~M_\odot$ at distances of $3$ Mpc to $100$ Mpc respectively. For sufficiently strong signals, we recover binaries with log Bayes factors >10 and constrain chirp mass and luminosity distance, subject to their characteristic degeneracy. Sky localization uncertainties of a few degrees could potentially enable electromagnetic follow-up and multi-messenger observations of SMBHB mergers. We further demonstrate that commonly used memory burst approximations lead to biased strain amplitudes and inferred source parameters when compared to the full SMBHB waveform, even when optimally tuned. These results establish a pathway for searching for SMBHB mergers with PTAs using complete waveform models.

We extend a recently defined measure of symmetry breaking, the entanglement asymmetry, to higher-form symmetries. In particular, we focus on Abelian topological order in two dimensions, which spontaneously breaks a 1-form symmetry. Using the toric code as a primary example, we compute the entanglement asymmetry and compare it to the topological entanglement entropy. We find that while the two quantities are not strictly equivalent, both are sub-leading corrections to the area law and can serve as order parameters for the topological phase. We generalize our results to non-chiral Abelian topological order and express the maximal entanglement asymmetry in terms of the quantum dimension. Finally, we discuss how the scaling of entanglement asymmetry correctly detects topological order in the deformed toric code, where 1-form symmetry breaking persists even in a trivial phase.

We investigate the formation of bound states between dark-antidark solitary waves in two-component dipolar Bose-Einstein condensates. The excitation spectrum contains density and spin branches, and a rotonic feature of the spin branch enables long-range soliton interactions, giving rise to multiple bound states for a single pair, each with a distinct separation. We show that these bound states originate from periodic modulations of the inter-soliton potential, while individual solitons are surrounded by spatial spin-density oscillations. Both features provide direct signatures of the spin roton. Collisions between unbound solitons probe this potential, with dipolar interactions enforcing universal bouncing at low velocities, independent of soliton sign, whereas nondipolar solitons may either transmit or bounce. This distinct behavior offers a realistic path to confirming spin rotons experimentally.

Achieving tritium self-sufficiency is a critical challenge for future fusion power plants. The BABY 1L experiment, part of the LIBRA project at MIT, aims to benchmark tritium breeding and release in molten salt breeder systems under deuterium-tritium (DT) neutron irradiation. Building on the initial \SI{100}{mL} campaign, BABY 1L introduces a tenfold increase in breeder volume, improved thermal and gas handling systems, and enhanced neutron diagnostics, including a proton recoil telescope. We report on results from four irradiation experiments using sealed-tube DT neutron generators, with tritium collected by water bubblers measured via liquid scintillation counting. Experimentally determined Tritium Breeding Ratios (TBRs) were compared to OpenMC neutronics simulations, showing very good agreement. The measured TBR values demonstrate a six-fold improvement over the \SI{100}{mL} experiments, largely attributed to the increased solid angle and improved measurement fidelity. We also investigate tritium release dynamics and identify diffusion-limited transport as the dominant regime in the salt volume in the temperature range 630-750 \si{\celsius}. Additionally, we observe that the introduction of hydrogen in the helium carrier gas significantly accelerates tritium release, consistent with an isotopic exchange mechanism. All analysis is conducted through the open-source \texttt{libra-toolbox} \cite{libra-toolbox}, which streamlines simulation, data processing, and validation across experimental campaigns. These results provide critical insights into the design and operation of future liquid breeder systems and demonstrate the maturity of the BABY platform as a testbed for tritium breeding studies.

Urban traffic anomalies, such as collisions and disruptions, threaten the safety, efficiency, and sustainability of transportation systems. In this paper, we present a simulation-based framework for modeling, detecting, and predicting such anomalies in urban networks. Using the Simulation of Urban MObility (SUMO) platform, we generate reproducible rear-end and intersection crash scenarios with matched baselines, enabling controlled experimentation and comparative evaluation. We record vehicle-level travel time, speed, and emissions for both edge- and network-level analysis. Building on this dataset, we develop a hybrid forecasting architecture that combines bidirectional long short-term memory networks with a diffusion convolutional recurrent neural network to capture temporal dynamics and spatial dependencies. Our simulation studies on the Broadway corridor in New York City demonstrate the framework's ability to reproduce consistent incident conditions, quantify their effects, and provide accurate multi-horizon traffic forecasts. Our results highlight the value of combining controlled anomaly generation with deep predictive models to support reproducible evaluation and sustainable traffic management.

Mass-modeling methods are used to infer the gravitational field of stellar systems, from globular clusters to giant elliptical galaxies. While many methods exist, most require assumptions about the form of the underlying distribution function or data binning that leads to loss of information. With only line-of-sight (LOS) data, many methods suffer from the well-known mass-anisotropy degeneracy. To overcome these limitations, we develop a new, publicly available mass-modeling method, GravSphere2. This combines individual stellar velocities from LOS and proper motion (PM) measurements to solve the Jeans equations up to fourth order, without any data binning. Using flexible functional forms for the anisotropy profiles at second and fourth order, we show how including additional constraints from a new observable - fourth-order PMs - fully closes the system of equations, breaking the mass-anisotropy degeneracy at all orders. We test our method on mock data for dwarf galaxies, showing how GravSphere2 improves on previous methods. GravSphere2 recovers the mass density, stellar velocity anisotropy, and logarithmic slope of the mass density profile within its quoted 95% confidence intervals across almost all mocks over a wide radial range (0.1 < R/Rhalf < 10). We find GravSphere2 outperforms simple mass estimators, suggesting that it is worth using even when only a few LOS velocities are available. With 1,000 tracers without PMs, GravSphere2 recovers the logarithmic density slope at Rhalf with 12% (25%) statistical errors for cuspy (cored) mock data, enabling a distinction between the two. Including PMs, this improves to 8% (12%). With just 100 tracers and no PMs, we recover slopes with ~ 30% (20%) errors. GravSphere2 will be a valuable new tool to hunt for black holes and dark matter in spherical stellar systems, from globular clusters and dwarf galaxies to giant ellipticals and galaxy clusters.