We construct data-driven solutions to the Hubble tension, in light of recent data from the Atacama Cosmology Telescope (ACT DR6) and the Dark Energy Spectroscopic Instrument (DESI DR2). We search for the minimal modification to the recombination history through a time-varying electron mass $m_e(z)$ that increases the best-fit $H_0$ inferred from CMB data toward the SH0ES value, without worsening the fit to the data. Using Planck and ACT data including lensing, we find a perturbative modification to $m_e(z)$ that fully resolves the Hubble tension, with the solution sharing the same qualitative oscillatory structure as in previous work using Planck data alone, demonstrating its robustness to the inclusion of more precise and independent CMB data. As a byproduct, the solution also eases the $S_8$ tension. Once DESI DR2 BAO data are added, however, perturbative modifications to $m_e(z)$ cannot fully resolve the Hubble tension. This reflects the same fundamental limitation: raising $H_0$ by modifying recombination generically lowers $Ω_m$, being inconsistent with late-time cosmological observations.

Plasmonic nanocavities enable the synchronization of spatially distant emissive dipoles through strong near-field coupling in sub-nm gaps. We report formation of a room-temperature synchronized dipole state in locally-ordered plasmonic nanogap 2D arrays under non-resonant continuous-wave pumping. Unlike lasers, photonic Bose-Einstein condensates, or exciton-polariton condensates, this system exhibits spatial coherence across the dipoles, while rapid radiative and non-radiative emission suppresses temporal photon coherence. A change of behaviour is observed with increasing pumping, marked by the spatial spread of g(1) coherence, but without spectral narrowing or directional emission. This driven-dissipative system exhibits fast temporal coherence decay and complex spatial correlations, offering a new platform for studying synchronization at room temperature. Combining ultralow mode volumes, high Purcell enhancement, and scalable ambient operation, it opens pathways for novel photonic and quantum technologies.

We derive equations of motion for scalar particles self-consistently interacting with a scalar field,including the radiation produced by the particles' acceleration. Our approach differs in three key aspects from current methods: (1) we assume a small but finite discretization length scale $h$, which allows us to treat the particle as a small but finite object, (2) we choose the state vector for the system before deriving equations of motion, and (3) we formulate the equations explicitly in the lab frame and not in a manifestly covariant manner. This approach, which is self-consistent, happens to greatly simplify the resulting equations and their derivation, and is directly suitable for numerical calculations. The result is an effective source method which generalizes to electrodynamics or general relativity in a straightforward manner (although we do not consider this here). We then provide two possible discretizations of these equations, based on finite volumes and spectral methods, and show results of one-dimensional calculations. These calculations show excellent agreement with analytic solutions.

We study the phenomenology of early QCD confinement during inflation, driven by a direct coupling between the inflaton and Standard Model gluons. This coupling dynamically raises the QCD confinement scale, making the axion sufficiently heavy to suppress isocurvature perturbations during the CMB epoch. As inflation proceeds, the confinement scale decreases and the axion becomes light, allowing de Sitter fluctuations during the late stages of inflation or post-inflationary thermal fluctuations to generate the observed dark matter abundance. In addition, QCD-induced corrections to the inflationary potential can shift the scalar spectral index towards smaller values, providing a further observational handle. We embed this mechanism in an $α$-attractor model of inflation and explore the resulting parameter space. We show that, in the minimal scenario with reheating into gluons, successful dark matter production requires deconfinement to occur shortly after the CMB window. Extensions involving reheating through heavy right-handed neutrinos generally require large Yukawa couplings, which induce sizable loop corrections that spoil inflationary dynamics. We show that this tension can be resolved in the presence of supersymmetry and derive constraints on the SUSY breaking scale that allow the mechanism to remain viable within plateau models of inflation. Treating the reheat temperature as a free parameter further enlarges the viable parameter space.

We study charge-dependent modifications of identical-pion and identical-kaon femtoscopic correlation functions from two sources: the residual Coulomb field of the charged source and isospin-related hadronic dynamics. The residual Coulomb effect is modeled with a modified Retiere--Lisa blast-wave source, where the same emitted particles are propagated with positive and negative charge signs through an effective residual field. The residual Coulomb field produces a small but systematic positive-to-negative splitting of the correlation functions, strongest at low $k_T$ and sensitive to the effective charge, spatial width, and expansion velocity of the residual source. It also modifies the height and shape of the correlation function, complicating interpretation of the fitted radii. UrQMD 3.4 cascade calculations for Au+Au collisions at $\sqrt{s_{NN}}=7.7~\mathrm{GeV}$ show that charge-dependent splittings can also appear even without a residual Coulomb field. For pions, this difference is mainly driven by the initial isospin composition, while for kaons it is strongly affected by different $K^+$ and $K^-$ production mechanisms and subsequent hadronic evolution. These results show that residual Coulomb and isospin-related effects can compete, and neither can be interpreted reliably without constraining the other within the same model framework.

During the core-collapse of a rotating massive star, the standing accretion shock instability (SASI) favours the development of non-axisymmetric motions which can imprint specific frequency signatures on the neutrino and gravitational wave signals.This study establishes analytical approximations for the eigenfrequencies of the dominant SASI modes. It also explains the physical mechanism responsible for the further destabilization of prograde SASI modes by differential rotation. A perturbative analysis is used to calculate the eigenfrequencies of a stalled accretion shock in spherical geometry, taking into account the rotation of the collapsing stellar core. The formulation of the perturbative equations as a self-forced oscillator is extended to include differential rotation and interpret the results physically. The oscillation frequency of the dominant mode weakly depends on the detailed formulation of neutrino emission if the shock radius exceeds ~1.5 times the radius r_nabla of maximum deceleration. Analytical expressions are obtained for the one and two-armed spiral modes with a 10% accuracy in this regime. The effect of differential rotation is explained by the role of phase mixing between the advective forcing and the acoustic structure. The radial wavelength of vorticity perturbations associated with the prograde mode is increased by differential rotation, leading to a better phase match with the large radial scale of the acoustic structure. Even when rotation is too modest to involve a corotation radius, its adverse effect on phase mixing can be significant at small radius due to the steep inward increase of the rotation frequency ~1/r^2 in the region of stationary accretion. In the regime of stronger rotation involving a corotation radius, the stationary phase approximation sheds light on the dominant advective-acoustic coupling, located between the corotation zone and the shock.

We investigate the interacting non-Hermitian Su-Schrieffer-Heeger (SSH) model, focusing on the interplay between topology and charge ordering. Using a real-space topological marker, charge correlations, and the complex many-body spectrum, we map out the phase diagram under periodic and open boundary conditions. We show that the topological marker remains a robust diagnostic of non-Hermitian topological phases in the presence of interactions and consistently signals their breakdown at the onset of a charge density wave (CDW). We further demonstrate that non-Hermiticity enhances interaction effects: While moderate changes occur under periodic boundary conditions, open boundary conditions lead to a pronounced amplification of staggered charge correlations near exceptional points. This enhancement arises from the accumulation of low-energy states near exceptional points, which promotes electronic instabilities and strengthens CDW tendencies.

Non-reciprocal supercurrents enable non-dissipative rectification, holding great promise for superconducting electronics. Conventionally, this non-reciprocity, termed the superconducting diode effect, requires the simultaneous breaking of time-reversal and parity symmetries. Here, we propose a superconducting triode effect in an asymmetric quantum-dot Josephson junction coupled to an additional metallic top gate, which breaks the parity symmetry while explicitly preserving time-reversal symmetry. We demonstrate that the supercurrent across this junction exhibits a strong non-reciprocal effect that can be continuously manipulated via the top gate to achieve an ideal unidirectional supercurrent, thus manifesting a superconducting triode effect. Furthermore, under radio-frequency radiation, this junction exhibits highly asymmetric Shapiro steps, realizing fully quantized supercurrent rectification. Our work not only provides an alternative physical mechanism for the superconducting diode effect observed in Josephson junctions with explicit time-reversal symmetry, but also introduces a new tuning knob to manipulate supercurrent non-reciprocity.

We present a comprehensive analysis of the Orion star-forming complex (OSFC), combining structural, kinematic, and dynamical information to constrain the present-day state and future evolution of its stellar substructures. Using \textit{Gaia} DR3 astrometry and complementary radial velocities from high-resolution spectroscopic surveys, we derived three-dimensional velocity distributions and structural parameters for 13 young clusters. For the stellar component, we estimated a correction of the present-day mass function for observational incompleteness and calculated the virial state, $α_{\rm vir}$, finding that all clusters are supervirial. Direct $N$-body simulations initialized from the present-day global parameters and evolved for 300~Myr in a Galactic potential suggest a separation of the OSFC clusters into two regimes: seven clusters with $α_{\rm vir}\lesssim 7$ evolve in a Galactic-potential-regulated regime that retains a bound core for $\gtrsim 170$ Myr as long-lived open clusters, whereas six clusters with $α_{\rm vir}\gtrsim 7$ enter an internal-dynamics--dominated regime, dissolving before 120 Myr and rapidly populating the Galactic stellar field. For both regimes, a control test indicates negligible cluster--cluster interactions under current OSFC conditions. Finally, long-lived clusters show low-amplitude modulations in the bound fraction correlated with the Galactic vertical motion, consistent with disk-crossing tidal heating and the temporary recapture of marginal members. These results highlight the OSFC as a natural laboratory where heterogeneous initial conditions give rise to persistent open clusters and dispersing groups.

Quantum algorithms for basic numerical linear algebraic tasks have proven essential for translating diverse problems to a unified quantum computational context. Many of these tasks -- e.g., applying a polynomial function to the spectrum of a matrix embedded in a unitary process (a so-called block encoding), or taking linear combinations of block encodings -- are well-addressed by techniques like quantum singular value transformation (QSVT) or linear combination of unitaries (LCU). However, there exist useful matrix transforms whose realization by existing quantum algorithms is unclear or inefficient. In this work we construct improved quantum algorithms for some of these transforms, the simplest of which is a polynomial function applied element-wise. We show the space required to compute quantum element-wise transforms can be reduced exponentially in the degree of the applied function compared to prior work, and raise and rectify errors in previous constructions. We present our algorithms alongside applications to machine learning, simulation, and signal processing.

High-rate quantum low-density parity-check (qLDPC) codes are a leading candidate for fault-tolerant quantum computing. They feature higher encoding rates than planar alternatives such as the surface code, but their implementation often entails significant hardware hurdles like the need for long-range couplers. We leverage the flexibility of a trapped-ion quantum computer to demonstrate nine quantum error-correcting codes with starkly different qubit connectivity requirements on a single device without any hardware reconfiguration. These experiments span three families of quantum error-correcting codes: qLDPC codes, topological codes, and concatenated codes. With a qLDPC code encoding 4 logical qubits into 18 physical qubits, we achieve a logical error rate up to $9\times$ better than a previous demonstration of a similar code on superconducting solid-state qubits. Moreover, our implementation exhibits breakeven performance, with some instances achieving qubit lifetimes comparable to or slightly exceeding that of our trapped-ion qubits. We use a novel implementation of the optical-metastable-ground (OMG) architecture for addressable mid-circuit measurement and reset, which enables us to perform these experiments without any ion transport or dedicated coolant ions, requirements that typically consume a large fraction of the runtime or ion count of trapped-ion quantum computers.

In celebration of the 2025 UN International Year of Quantum Science and Technology, this Resource Letter surveys the rapidly-growing field of scholarship in quantum information science and engineering (QISE) education. It is primarily written as a guide for educators wishing to get started teaching QISE using research-based teaching methods, as well as for discipline-based education research (DBER) practitioners looking to get started in this field. Topics covered include scoping the field of QISE education, research into student reasoning in QISE, research-based and research-inspired curricular materials from the high school to graduate level, research-based assessments, simulation and gamification tools, and tools for incorporating discussion of the societal and ethical implications of quantum technologies into the classroom.

Mirror-assisted optical interference can improve the fidelity of Young's double-slit interference. Here we discuss an electron analogue: particle-hole interference in superconductors produced by a single impurity near a line defect, terrace edge, or phase boundary. Quasiparticle waves scattered directly from the impurity interfere with waves reflected by the boundary, generating a ``ghost'' interference pattern that combines conventional $2k_F$ Friedel oscillations with additional hyperbolic fringes. Compared to the recently studied two-impurity Young's interference, this effect appears already at first order in the impurity potential and is therefore parametrically stronger. The resulting spatial modulation extends beyond $λ_F/2$ and is directly sensitive to the quasiparticle structure of the paired state, including possible Fermi-surface anisotropy of the superconducting order parameter. These findings point to boundary-assisted impurity interference as a robust local probe of superconducting electronic order, with clear signatures accessible to STM/STS measurements.

The study of quartic couplings of electroweak gauge bosons not only provides a test of the Standard Model (SM) predictions, but also can look for signals of new physics beyond the SM. We present a comprehensive study of anomalous quartic gauge couplings in same-sign $W^\pm W^\pm$ production via vector boson scattering at the LHC. The analysis is carried out within the framework of the SM Effective Field Theory, exploiting polarization and spin-correlation effects encoded in angular asymmetries in addition to conventional kinematic observables. We demonstrate that spin-correlation asymmetries provide sensitivity to anomalous $WWWW$ interactions that is comparable to that obtained from the transverse mass distribution of the $WW$ system. By identifying a minimal set of the most sensitive asymmetries, we show that the dominant constraints on the Wilson coefficients can be captured with a reduced number of observables. A combined analysis of angular asymmetries and kinematic information leads to improved limits compared to either approach alone. The impact of unitarity considerations is also examined by imposing invariant-mass cut-offs on the $WW$ system, allowing us to determine unitarity-safe regions for the anomalous couplings.

We present a new static solution describing a charged black string immersed in a Kiselev-type quintessence fluid and a cloud of strings. The metric and field equations are solved for a general quintessence state parameter, with explicit results provided for the physically relevant case $w_q = -2/3$. We analyze the event-horizon structure and the Kretschmann scalar, verify energy-condition constraints, and derive thermodynamic properties including the Hawking temperature and heat capacity to identify stability regimes. Finally, we investigate the photon cylinder for null geodesics. The solution generalizes known charged black-string spacetimes by simultaneously including quintessence and a string-cloud parameter.

Context. Neutral helium (He I) produces several spectral lines that are widely used for solar diagnostics. The role of collisions between He I atoms and neutral hydrogen (H I) in the modeling of solar He I lines remains insufficiently quantified. Accurate determination of collisional rates affecting atomic polarization is needed for solar spectropolarimetry. Aims. Our aim is to provide a set of multi-level and multi-term collisional depolarization, polarization-transfer, and population-transfer rates, due to isotropic collisions with neutral hydrogen, for He I levels and terms involved in the main solar He I diagnostic lines. Methods. The calculations are performed within the frozen-core approximation, in which the inner 1s electron is treated as a core with Lc = 0, Sc = 1/2, and Jc = 1/2, while the outer electron is treated as the active valence electron. Results. We compute both multi-level rates, describing depolarization and polarization transfer between fine-structure J-levels, and multi-term rates, which additionally account for coherences between different J-levels belonging to the same term. Conclusions. Our results provide the collisional input needed for the statistical equilibrium equations (SEE) of the polarization of the main He I solar lines, including the 10830 Å, D3 5876 Å, and related triplet transitions, and allow a quantitative reassessment of the role of neutral-hydrogen collisions in He I spectropolarimetry.

We propose a new concept for quantum thermal logic gates -- analogous to classical electronic logic gates -- that exploit the heat current in a coupled quantum-dot system tunnel-coupled to metallic thermal reservoirs for logic operations in quantum circuits. We obtained a remarkable one-to-one correspondence with the structure of classical electronic logic gate circuits. An experimental setup is presented that demonstrates a realizable nano-electronic quantum circuit architecture for implementing such quantum thermal logic operations.

Strontium ruthenate (Sr$_2$RuO$_4$, SRO) has been the subject of intense research as a model quasi-two-dimensional metal with strong electronic correlations and potential exotic multi-component superconductor. Yet the nature of the superconducting state and its emergence remain debated, despite highly detailed knowledge of the normal-state electronic properties. Here we use diffuse neutron and x-ray scattering to uncover self-organized structural inhomogeneity on the picosecond timescale in SRO. We show that these structural correlations do not originate from extrinsic disorder but rather involve correlated displacements of oxygen atoms in the quintessential RuO$_2$ planes. Moreover, the observed displacement pattern is consistent with distortions due to orbital order that we obtain in first-principles calculations, which suggests that orbital effects could play a pivotal role in the physics of SRO. The appearance of such dynamic inhomogeneity may be relevant for a wide range of prominent oxides with similar lamellar structures, such as the cuprates and nickelates.

We present a novel approach to eigenvector universality for generalized Wigner matrices. Our main consequences are asymptotic normality of joint eigenvector projections everywhere in the spectrum as well as a quantitative lower bound on the largest entry of an eigenvector. In the case of smooth entries, we are able to obtain joint normality of an explicit growing number of eigenvector projections, and we are also able to obtain an explicit rate of convergence in Kolmogorov distance. This is based on a new analysis of the Dyson vector flow which does not rely on the eigenvector moment flow.

Polymers are indispensable materials with applications ranging from electronics to medicine owing to their versatility, which can be tailored by adjusting their chemical composition and architecture. The design space for these compounds is vast and governed by factors such as monomer classes, copolymer configurations (e.g., linear, branched, random, and alternating), chain size, stoichiometry, and material properties (e.g., density, refractive index, solubility, and Poisson's ratio). Exploring this space requires efficient computational methodologies for polymer science. To address this challenge, we introduce PolyGraphPy, an open-source Python framework that integrates atomistic simulations with machine learning for accurate property prediction and property-guided polymer design. The framework automates Density Functional Tight Binding calculations to efficiently construct structured datasets for monomers, homopolymers, and alternating copolymers. For property prediction, PolyGraphPy employs Bayesian Graph Neural Networks (GNNs) with stochastic graph representations to predict target properties, such as static polarizability, while providing robust uncertainty quantification. Furthermore, the platform incorporates two complementary generative models for the de novo design of targeted molecules: a SELFIES-based Generative Pretrained Transformer (GPT) and a Genetic Algorithm (GA) based on BRICS graph fragmentation. Demonstrated on a dataset of acrylates, PolyGraphPy provides a highly customizable end-to-end pipeline that reduces computational costs and accelerates data-driven polymer informatics.

We present a comparative study of four supervised machine learning regression algorithms -- Support Vector Regression (SVR), Gradient Boosting Regression (GBR), Gaussian Process Regression (GPR), and Multilayer Perceptron (MLP) -- for predicting the proton structure function $F_2^p(x, Q^2)$ using high-precision BCDMS experimental data. Unlike conventional methods that solve the DGLAP evolution equations, our data-driven framework directly captures the complex nonlinear dynamics of partonic structure. To ensure statistical robustness, we employ $k$-fold cross-validation and perform thorough hyperparameter optimization. Our results show that the MLP and GPR models achieve superior predictive accuracy. In particular, MLP exhibits the highest sensitivity to nonlinear gradients, while SVR proves most stable against experimental uncertainties. The close convergence of training and validation metrics confirms that the models learn the underlying QCD physics without overfitting to statistical fluctuations. This work highlights the potential of ML-based regression as a complementary tool for structure function analysis and kinematic extrapolation in high-energy physics.

We formulate stationary-density-preserving nonreversible perturbations of Fokker--Planck dynamics as gauge fields that deform relaxation spectra while leaving the invariant state fixed. When detailed balance holds, a similarity transformation maps the reversible Fokker--Planck operator to a Witten-Laplacian-type supersymmetric Hamiltonian; nonreversible gauges then appear as non-Hermitian perturbations that preserve the zero mode but modify the excited spectrum. This operator viewpoint gives a common language for relaxation gaps, circulating probability currents, hypocoercive acceleration, and finite control costs. We represent admissible gauge currents by antisymmetric tensor fields and identify the detailed-balance-violating Ohzeki--Ichiki force as a constant symplectic example whose infinite-strength limit is Hamiltonian dynamics. The continuous-time spectral gap alone does not select a finite gauge strength, so we introduce a finite-time regularized objective and an actor--critic procedure for learning the gauge. An exactly solvable anisotropic Gaussian Ornstein--Uhlenbeck benchmark separates the spectral transition from the finite-time optimum and shows that the learned gauge recovers the Lyapunov-equation optimum. A double-well benchmark then illustrates the same constrained selection in a nonconvex metastable landscape. Stochastic gradient methods enter this framework as physically relevant Fokker--Planck systems: mini-batch noise acts as an effective diffusion tensor, and adaptive methods such as Adam correspond to metric choices with possible nonequilibrium currents.

Ultralight axions (ULAs) are well-motivated dark matter particle candidates that arise in many extensions of the Standard Model of particle physics. ULAs with mass $m_\mathrm{a} \lesssim 10^{-27}$ eV have been strongly constrained by cosmic microwave background (CMB) observations in temperature and polarization. We fit recent measurements of gravitational lensing of the CMB from \textit{Planck}, the Atacama Cosmology Telescope (ACT) and the South Pole Telescope (SPT-3G) using a state-of-the-art simulation-calibrated nonlinear clustering model for ULAs. We derive the strongest constraints on ULAs in the mass range $10^{-26}\;\mathrm{eV}\leq m_\mathrm{a}\leq 10^{-24.5}\;\mathrm{eV}$. ULAs of this mass have been shown to alleviate tensions between inferences of the matter clustering if they compose a few percent of the total dark matter content of the Universe. We conclude that ULAs with a mass of $10^{-26}$ eV make up less than 1.5\% of the dark matter and $10^{-25}$ eV axions make less than 9\% (both at 95\% confidence level). We identify a slight preference for non-zero axion density at $10^{-24.5}$ eV at $2.1σ$. We find that the preference for ULAs is largely driven by a few data points and that further investigation of non-linear ULA physics is needed to confirm or rule out this signal.

While core-collapse supernovae have been widely discussed as potential neutrino sources, definitive observational evidence has remained elusive. In this work, we report evidence of high-energy neutrino emission in the direction of supernova SN 2021foa, which is one of the closest and brightest interacting supernovae observed to date. Using the second data release of muon track data from the IceCube Neutrino Observatory, we conducted a time-dependent analysis and identified a neutrino clustering that temporally coincides with the optical peak of SN 2021foa, occurring approximately 16 to 22 days after the discovery date, with a maximum test statistic of $\sim 28.2$. Monte Carlo simulations indicate that the probability of observing such a neutrino excess by chance is $\sim6.7 \times 10^{-5}$, corresponding to a significance of $\sim4.0\,σ$. The spatial and temporal correlation strongly suggests that the neutrinos originate from the supernova. SN 2021foa is a unique "flip-flop" supernova; its spectra repeatedly transitioned between hydrogen-rich (Type IIn) and helium-rich (Type Ibn) phases within 50 days post-peak, reflecting a violent and complex mass-loss history of its progenitor. The inferred neutrino energy exceeds the optical radiative energy and ejecta kinetic energy of the supernova by orders of magnitude, suggesting that the neutrino emission is likely powered by a delayed central engine driving a jet that is choked within the dense circumstellar medium.

This study presents a novel framework for estimating the central sea-level pressure ($P_{\rm c}$) of tropical cyclones (TCs) using infrared radiometers. We leverage the long-overlooked combination of high spatial resolution and sounding capability of the Moderate Resolution Imaging Spectroradiometer (MODIS) to measure warm-core anomalies in TC eyes. We develop the MODIS Thermal Infrared Sounding (MOTIS) framework, which performs instrument-specific preprocessing and estimates $P_{\rm c}$ using multiple linear regression. MOTIS yields $r^2 = 0.945$ and RMSE = 4.3 hPa for high-intensity TCs ($\overline{P}_{\rm c} = 937$ hPa) with clear eyes observed, outperforming all existing methods for intense TCs. We construct a dataset of 3288 (1082 clear-eye) MOTIS estimates from 2002 to 2025 and demonstrate its potential to improve the quality of Best Track $P_{\rm c}$, roughly halving uncertainties in the absence of pressure observations. Although MODIS is nearing the end of its mission, the MOTIS framework could be extended to next-generation geostationary sounders to provide accurate real-time $P_{\rm c}$ estimation for high-intensity TCs.

We propose a resolution of the firewall paradox in JT gravity by incorporating topology change into canonical quantization under relational time evolution. The gravitational Hamiltonian acts on the black hole interior through a pair of pants interaction, mapping between a single interior sector and a connected two interior sector. To describe dynamics in the interior while keeping track of the exterior, we pass to an extended phase space description obtained by splitting the bulk Hilbert space across the event horizon. The split introduces boost edge modes at the horizon, which the Hawking modes become gravitationally dressed to. Covariance of the resulting crossed product algebra provides a precise gravitational realization of the firewall: a one sided boost of the interior edge mode relative to the exterior holding the matter fixed, or equivalently, a relative phase between the interior and exterior Hawking partners holding the edge modes fixed. Although each topology changing transition is exponentially suppressed, evolution over a Page time causes the connected two interior branch to dominate. One of these is the naive semiclassical interior, which we show carries a nontrivial one sided boost upon gluing the interior back to the exterior, and hence a firewall. The other interior is shown to be a zero mode of the one sided boost generator. Gluing the interior back to the exterior quotients by the gravitational constraints, which annihilates the firewall branch. On the surviving branch, we show the horizon vacuum measurement and the early radiation purity measurement become the same Dirac observable. Equivalently, we show that Page time dynamics induces a large diffeomorphism on the connected branch under which the operator algebra of the interior Hawking partner and that of the decoded early radiation are identified.

Lens spaces are useful test examples in spectral geometry because their spin Dirac eigenspaces admit explicit congruence descriptions. We use these descriptions to study equivariant $η$ invariants for three-dimensional lens spaces with the round metric and the standard coordinate-torus action, retaining the spin-Fourier character of each eigenspace rather than only the ordinary scalar $η$ value. For the square family $L(\ell^2,\ell-1)$ and $L(\ell^2,2\ell-1)$, with $\ell\geq 5$ odd, we obtain a residual-circle equivariant $η$ separation: the ordinary $η$ values agree, and the first derivative of the residual $η$ germ vanishes by symmetry, but the second derivative is nonzero. For $L(25,4)$ versus $L(25,9)$, the normalized second derivative is $-6080$. Thus, the residual-circle equivariant $η$ germ detects a distinction invisible to the ordinary $η$ invariant. The calculation uses spin-Fourier residues directly; perturbative Hessian signs serve only as motivation and are not part of the invariant.

Let $\mathcal{A}$ be a (not necessarily rational) conformal net with a faithful action of a finite group $G$. Let $\text{Rep}^G(\mathcal{A})$ be the $G$-crossed balanced $\mathrm{W}^*$-tensor category of $G$-twisted representations of $\mathcal{A}$ as introduced in arXiv:2606.03623. We show that there is an equivalence of balanced $\mathrm{W}^*$-tensor categories $(\text{Rep}^G(\mathcal{A}))^G\cong \text{Rep}(\mathcal{A}^G)$ between the $G$-equivariantization of $\text{Rep}^G(\mathcal{A})$ and the category of representations of the fixed-points conformal net $\mathcal{A}^G$. This generalizes to the non-rational case the equivalence of braided tensor categories $(\text{Rep}^G(\mathcal{A}))^G\cong \text{Rep}(\mathcal{A}^G)$ for $\mathcal{A}$ rational appearing (in the language of localized endomorphisms) in arXiv:math/0403322, and it also includes the balances.

We construct a periodic weighted graph whose discrete Laplacian has a spectrum with precisely $n$ gaps. Moreover, we show that by an appropriate choice of the weights, the endpoints of these gaps, as well as the upper edge of the spectrum, attain the prescribed values. The underlying graph has a brush-like geometry: it consists of an infinite chain of vertices, each of which is connected to $n$ additional pendant vertices by extra edges. Semi-explicit formulae for the weight coefficients are provided: some of the coefficients are determined explicitly, while others are given as roots of an explicitly determined polynomial.

We propose a phenomenological mass estimator for an intermediate ultralight axion dark-matter component using thin stellar streams. The central observation is an analytic relation linking three length scales in a fuzzy-dark-matter stream progenitor: the tidal radius of a bound dark core, its gravitational Bohr radius, and the axion de Broglie wavelength evaluated at the stripped-star velocity scale. If the stream width is (w=C_w r_{\rm t}), with (C_w=O(1)), then [ \frac{r_{\rm t}}{r_{\rm B}} =========================== \frac{4π^2}{C_w^2} \left(\frac{w}{λ_{\rm dB}}\right)^2 . ] Thus (λ_{\rm dB}\sim w) automatically implies (r_{\rm t}/r_{\rm B}\sim25)--(40): the dark wave core survives well inside the tidal boundary while the extended stellar envelope is stripped into a thin stream. This wave--tide locking gives a no-simulation axion-mass estimator, [ m_a \simeq 2.69\times10^{-19},{\rm eV} \left(\frac{\sqrt{2}}{q_κ}\right) \left(\frac{R}{10,{\rm kpc}}\right) \left(\frac{38,{\rm pc}}{w}\right)^2 \left(\frac{w}{\ell_{\rm ripple}}\right) \left(\frac{220,{\rm km,s^{-1}}}{v_c}\right). ] In the simplest locked case (\ell_{\rm ripple}\sim w), a homogeneous first-pass set of narrow stream widths points to (m_a) of a few (10^{-19},{\rm eV}) and hidden-core masses of a few (10^3)--(10^4M_\odot). This is not a detection claim; it is a falsifiable phenomenological test for an ultralight axion component that can be sharpened by homogeneous stream catalogs and Schrödinger--Poisson simulations.