The rapid expansion of the Internet of Things (IoT) has intensified cybersecurity challenges, particularly in mitigating Distributed Denial-of-Service (DDoS) attacks at the network edge. Traditional Intrusion Detection Systems (IDSs) face significant limitations, including poor adaptability to evolving and zero-day attacks, reliance on static signatures and labeled datasets, and inefficiency on resource-constrained edge gateways. Moreover, most existing DRL-based IDS studies overlook sustainability factors such as energy efficiency and carbon impact. To address these challenges, this paper proposes two novel Deep Reinforcement Learning (DRL)-based IDS: DeepEdgeIDS, a label-free Autoencoder-DRL hybrid, and AutoDRL-IDS, a supervised LSTM--DRL model. Both DRL-based IDS are validated through theoretical analysis and experimental evaluation on edge gateways. Results demonstrate that AutoDRL-IDS achieves 94% detection accuracy using labeled data, while DeepEdgeIDS attains 98% offline evaluation accuracy through label-free anomaly detection and online mitigation feedback. This study introduces a carbon-aware, multi-objective reward formulation that supports supervised reward optimization for AutoDRL-IDS and label-free online reward learning for DeepEdgeIDS, enabling sustainable real-time IDS operation in dynamic IoT networks.

We investigate the high-energy, small-angle limit of two-body gravitational scattering. Using power counting arguments and dispersion relations in an effective field theory for the Regge regime, we derive the general loop expansion that determines how the leading Regge logarithms and their complex structure arise as a power series in $t/s$. Focusing on the tower of multi-H diagrams that govern the leading logarithmic behavior, we compute the leading double logarithm at four loops (5PM) using both effective field theory methods and the multi-Regge expansion, finding complete agreement. Finally, using the aforementioned dispersion relations, we extract the single logarithmic contribution to the imaginary part of the eikonal phase at 5PM in the Regge limit.

Strong evidence for an isotropic, Gaussian gravitational wave background (GWB) has been found by multiple pulsar timing arrays (PTAs). The GWB is expected to be sourced by a finite population of supermassive black hole binaries (SMBHBs) emitting in the PTA sensitivity band, and astrophysical inference of PTA data sets suggests a GWB signal that is at the higher end of GWB spectral amplitude estimates. However, current inference analyses make simplifying assumptions, such as modeling the GWB as Gaussian, assuming that all SMBHBs only emit at frequencies that are integer multiples of the total observing time, and ignoring the interference between the signals of different SMBHBs. In this paper, we build analytical and numerical models of an astrophysical GWB from circular, inspiralling binaries inclined relative to the line-of-sight of the observer, without the above approximations, and compare the statistical properties of its induced PTA signal to those of a signal produced by a Gaussian GWB. We show that finite population and windowing effects introduce non-Gaussianities in the PTA signal, which are currently unmodeled in PTA analyses.

We propose a refinement of the Kontsevich-Soibelman operator for a class of ``special'' 4d $\mathcal{N}=2$ superconformal field theories characterized by the following conditions: (1) their Coulomb branch admits a source/sink chamber, i.e., a chamber in which the BPS quiver consists of only source and sink nodes, (2) The nodes with valency greater than 2 of the BPS quiver in a source/sink chamber are either all sources or all sinks. We present strong evidence that the trace of this refined operator is related to the Macdonald index of the theory. In particular, we conjecture closed form expressions for the Macdonald indices of the $(A_1,\mathfrak{g})$ Argyres-Douglas theories for any simply-laced Lie algebra $\mathfrak{g}$.

It is usually believed that symmetry breaking in photonic systems leads to weaker optical confinement, such as in the case of metasurfaces when bound states in the continuum are replaced by quasi-bound states with lower quality factors (Q factors). Here we show that symmetry breaking can instead enhance light trapping by strengthening in-plane nonlocal coupling pathways. We consider finite-size arrays of optical resonators supporting Mie resonances (a Mie-tronics platform) and employ diffraction and multiple-scattering analyses. We demonstrate that diffractive bands and Mie-tronics supermodes originate from the same underlying Mie resonances but differ fundamentally in their physical nature. Finite arrays exhibit Q-factor enhancement driven by redistributed radiation channels, and reversing the trends predicted by infinite-lattice theories. We reveal that controlled symmetry breaking opens new electromagnetic coupling channels, enabling polarization conversion in nonlocal metasurfaces. These novel findings establish a unified wave-physics platform linking both scattering and diffraction theories. Also, they outline the design principles for multi-functional metasurfaces that exploit nonlocality for advanced light manipulation, computation, and emission control.

A fundamental problem in fault-tolerant quantum computation is the tradeoff between universality and dimensionality, exemplified by the the Bravyi-König bound for $n$-dimensional topological stabilizer codes. In this work, we extend topological Pauli stabilizer codes to a broad class of $n$-dimensional Clifford hierarchy stabilizer codes. These codes correspond to the $(n+1)$D Dijkgraaf-Witten gauge theories with non-Abelian topological order. We construct transversal non-Clifford gates through automorphism symmetries represented by cup products. In 2D, we obtain the first transversal non-Clifford logical gates including T and CS for Clifford stabilizer codes, using the automorphism of the twisted $\mathbb{Z}_2^3$ gauge theory (equivalent to $\mathbb{D}_4$ topological order). We also combine it with the just-in-time decoder to fault-tolerantly prepare the logical T magic state in $O(d)$ rounds via code switching. In 3D, we construct a transversal logical $\sqrt{\text{T}}$ gate in a non-Clifford stabilizer code at the third level of the Clifford hierarchy, located on a tetrahedron corresponding to a twisted $\mathbb{Z}_2^4$ gauge theory. Our constructions surpass the Bravyi-König bound by achieving the logical gates in the $(n+1)$-th level of Clifford hierarchy in $n$ spatial dimension.

We explore the growth of structure in multi-species wave (and particle) dark matter. We derive the evolution of the power spectrum of total density contrasts for an arbitrary number of component species, density fractions, and initial field power spectra. We also derive cross-spectra for density correlations across or within individual species. Our framework includes cold and warm wave dark matter, which can give rise to significant intrinsic Poisson-like density fluctuations along with scale-dependent evolution connected to the free-streaming and Jeans scales. Such dark matter components could be globally or locally misaligned scalar fields as well as multi-component fields with spin $>0$. The framework also includes cold and warm particle dark matter in the appropriate limits.

While numerous models exist which explain certain aspects of irradiation-induced nanopatterning on semiconductors, a comprehensive theoretical explanation has remained elusive. However, it is increasingly apparent that such a model will require understanding the dual influence of the collision cascade initiated by ion implantation: first, as a source of material transport by sputtering and atomic displacements occurring over short time scales, and, second, as a source of defects permitting viscous flow within the thin, amorphous layer that results from sustained irradiation over longer time scales. To better understand the latter, we develop several asymptotic approximations for coupling the local ion flux experienced by the amorphous layer to the layer's evolving free interface. From these and the physical hypothesis of irradiation-induced anisotropic plastic flow, or ``ion-hammering", we derive a generalized Kuramoto-Sivashinsky-type equation for the evolving free surface. With physically plausible parameters, the present model achieves good quantitative and qualitative agreement with several aspects of experimental observations of nano-pattern formation during irradiation of silicon by argon, krypton and xenon, and with projectile energies from 500eV to 2keV. Disagreements between model and experiment are discussed, as are implications for future directions.

Ultra-cool T- (T$_{\mathrm{eff}} \approx$ 500 - 1200 K) and Y-dwarfs (T$_{\mathrm{eff}}$ $\lessapprox 500$ K) have historically been found only a few hundred parsecs from the Sun. The sensitivity and wavelength coverage of the NIRCam instrument on board the James Webb Space Telescope offer a unique method for finding low-temperature brown dwarfs in deep extragalactic datasets out to multiple kiloparsecs. Here we report on the selection of a sample of 41 brown dwarf and brown dwarf candidates across the JWST Advanced Deep Extragalactic Survey (JADES) in the GOODS-S and GOODS-N regions. We introduce a new open-source Bayesian tool, the Near-Infrared Fitting for T and Y-dwarfs (\texttt{NIFTY}), to derive effective temperatures, metallicities, and distances from JWST photometry. We find that 31 candidates have fits consistent with T-dwarf temperatures out to 5 - 6 kpc, and 10 candidates have fits consistent with Y-dwarf temperatures out to 1 - 2 kpc. The majority of the sources are best fit with sub-solar metallicity models, consistent with them being subdwarfs in the Milky Way thick disk and halo. We report proper motions for nine brown dwarf candidates (three are newly presented), and calculate the number density of T- and Y-dwarfs as a function of temperature and distance above the Milky Way midplane. We further discuss how Y-dwarfs can serve as contaminants in the search for ultra-high-redshift galaxies. Together, these results demonstrate the power of deep JWST extragalactic imaging to probe the coldest substellar populations far beyond the solar neighborhood, providing new constraints on the Milky Way's structure and brown dwarf demographics.

This paper introduces a Markov chain-based approach for the analysis and optimization of spare-management policies in large-scale satellite constellations. Focusing on the direct strategy, we model spare replenishment as a periodic-review reorder-point/order-quantity policy, where spares are deployed directly to constellation planes. The stochastic behavior of satellite failures and launch vehicle lead times is captured through Markov representations of both failure and replenishment dynamics. Based on this efficient and accurate framework, we construct and solve an optimization problem aimed at minimizing operational costs. The effectiveness of the proposed method is demonstrated through a case study using a real-world mega-constellation.

We study the possibility that the Right-handed neutrino is a five-dimensional state propagating along a micron size extra dimension, as required in the dark dimension proposal. We work out the signatures of R-neutrino production in KATRIN experiment and compare them with those of a sterile neutrino which manifests by a kink in the electron energy spectrum of the beta-decay at a value corresponding to the sterile neutrino mass. We explore the allowed parameter space of the compactification scale and the R-neutrino bulk mass versus the Yukawa coupling, and show that a large part of it is within KATRIN's sensitivity. When the bulk mass is much smaller than the compactification scale, several kinks could be observed corresponding to the positions of the R-neutrino Kaluza-Klein excitations, while for large bulk mass there will be effectively one kink at the position of the bulk mass.

The increasing heterogeneity of student populations poses significant challenges for teachers, particularly in mathematics education, where cognitive, motivational, and emotional differences strongly influence learning outcomes. While AI-driven personalization tools have emerged, most remain performance-focused, offering limited support for teachers and neglecting broader pedagogical needs. This paper presents the FACET framework, a teacher-facing, large language model (LLM)-based multi-agent system designed to generate individualized classroom materials that integrate both cognitive and motivational dimensions of learner profiles. The framework comprises three specialized agents: (1) learner agents that simulate diverse profiles incorporating topic proficiency and intrinsic motivation, (2) a teacher agent that adapts instructional content according to didactical principles, and (3) an evaluator agent that provides automated quality assurance. We tested the system using authentic grade 8 mathematics curriculum content and evaluated its feasibility through a) automated agent-based assessment of output quality and b) exploratory feedback from K-12 in-service teachers. Results from ten internal evaluations highlighted high stability and alignment between generated materials and learner profiles, and teacher feedback particularly highlighted structure and suitability of tasks. The findings demonstrate the potential of multi-agent LLM architectures to provide scalable, context-aware personalization in heterogeneous classroom settings, and outline directions for extending the framework to richer learner profiles and real-world classroom trials.

Cuprate superconductors show various collective charge correlations that are intimately connected with their electronic properties. In particular, charge order in the form of an incommensurate charge density wave (CDW) order with an in-plane wavevector $δ_{\text{CDW}} \approx $ 0.23--0.35~r.l.u. appears to be universally present. In addition to CDW, dynamic charge density fluctuations (CDF) are also present with wavevectors comparable to $δ_{\text{CDW}}$. CDFs are present up to $\sim300\;$K and have relatively short correlation lengths of $ξ\sim 20$\;Å. Here we use Cu-$L_3$ and O-$K$ resonant inelastic X-ray scattering (RIXS) to study the doping dependence of CDW and CDFs in La$_{2-x}$Sr$_x$CuO$_4$. We fit our data with (quasi)elastic peaks resulting from the CDW and up to four inelastic modes associated with oxygen phonons that can be strongly coupled to the CDFs. Our analysis allows us to separate the charge correlations into three components: the CDW with wavevector $δ_{4a-\text{CDW}} \approx 0.24$ and two CDF components with $δ_{4a-\text{CDF}} \approx 0.24$ and $δ_{3a-\text{CDF}} \approx 0.30$. We find that for $T \approx T_c$ the CDW coexists with the CDFs for dopings near $x=p \sim 1/8$. The $4a$-CDW disappears beyond $x=0.16$ and the $4a$-CDF beyond $x=0.19$, leaving only a weak $3a$-CDF at the highest doping studied, $x=0.22$. Our data suggest that low-energy charge fluctuations exist up to doping $x=0.19=p^{\star}$, where the pseudogap disappears, however, we find no evidence that they are associated with a quantum critical point.

Astrophysical black holes are usually surrounded by an accretion disk. At least parts of these accretion disks consist of a plasma in which light rays with different energies are dispersed. However, we usually do not know the exact configurations of these plasmas. In this paper we will now use the example of a Schwarzschild black hole embedded in an inhomogeneous pressureless and nonmagnetised plasma to investigate how the structural changes of the photon rings can help us to determine the properties of a plasma surrounding a black hole using multifrequency observations. For this purpose we will use a simple analytic model which describes a plasma whose electron density increases towards the equatorial plane when we approach the event horizon. For the chosen model we will first derive and then analytically solve the equations of motion. Then we will place an observer in the domain of outer communication and introduce an orthonormal tetrad to relate the constants of motion to latitude-longitude coordinates on the observer's celestial sphere. In the next step we will use the analytic solutions to investigate the geometric structures of the direct image as well as the photon rings of first and second order on the observer's celestial sphere. We will write down a lens equation and calculate the redshift and the travel time. We will compare the obtained results to results for photon rings in vacuum and in the presence of a homogeneous plasma. Finally, we will discuss which of these quantities can be used to extract information about the properties of the plasma.

Sanctioning blockchain addresses has become a common regulatory response to malicious activities. However, enforcement on permissionless blockchains remains challenging due to complex transaction flows and sophisticated fund-obfuscation techniques. Using cryptocurrency mixing tool Tornado Cash as a case study, we quantitatively assess the effectiveness of U.S. Office of Foreign Assets Control (OFAC) sanctions over a 957-day period, covering 6.79 million Ethereum blocks and 1.07 billion transactions. Our analysis reveals that while OFAC sanctions reduced overall Tornado Cash deposit volume by 71.03% to approximately 2 billion USD, attackers still relied on Tornado Cash in 78.33% of Ethereum-related security incidents, underscoring persistent evasion strategies. In this paper, we identify three significant, structural limitations in current sanction enforcement practices: (i) fragmented censorship in blockchain consensus and application layer; (ii) the complexity of obfuscation virtual asset services exploited by users; and (iii) the susceptibility of naive binary sanction classifications to dusting attacks. Our analysis and findings contribute to ongoing discussions around regulatory effectiveness in Decentralized Finance by providing empirical evidence, clarifying enforcement challenges, and informing future compliance strategies in response to sanctions and blockchain-based security risks.

This paper presents a PID tuning method based on step response curve fitting (PID-SRCF) that utilizes L2-norm minimization for precise reference tracking and explicit transient response shaping. The algorithm optimizes controller parameters by minimizing the root-mean-square error between desired and actual step responses. The proposed approach determines optimal PID parameters by matching any closed-loop response to a desired system step response. Practically a first-order plus time delay model or a second-order system with defined settling time and overshoot requirements are preferred. The method has open-source implementation using constrained nonlinear optimization in MATLAB. Comparative evaluations demonstrate that PID-SRCF can replace known analytical methods like Ziegler Nichols, Lambda Tuning, Pole Placement, Dominant Pole and MATLAB proprietary PID tuning applications.

We predict an anomalous thermally activated superradiance in molecular aggregates within polaritonic environments. In contrast to free space, the collective emission is enhanced when either the exciton-phonon coupling or the temperature increases. This counterintuitive phenomenon is captured by a microscopic theory that combines macroscopic quantum electrodynamics with a modified polaron quantum master equation approach, revealing a nontrivial interplay among excitons, phonons, and polaritons.

Plasma lensing provides compact focusing of electron beams, since they offer strong focusing fields (kT/m) in both planes simultaneously. This becomes particularly important for highly diverging beams with a large energy spread such as those typically originating from plasma accelerators. The lens presented here is a nonlinear active plasma lens, with a controlled focusing-strength variation purposely introduced in one transverse direction. This lens is a key element of a larger transport lattice, core of the ERC project SPARTA, which aims to provide a solution for achromatic transport between plasma-accelerator stages. We report on preliminary experimental results from the CLEAR facility at CERN, which aims to probe the magnetic field structure of the lens using an electron beam, in search of the desired nonlinearity, together with 2D plasma simulation results.

The large gradients of plasma-wakefield accelerators promise to shorten accelerators and reduce their financial and environmental costs. For such accelerators, a key challenge is the transport of beams with high divergence and energy spread. Achromatic optics is a potential solution that would allow staging of plasma accelerators without beam-quality degradation. For this, a nonlinear plasma lens is being developed within the SPARTA project. As a first application of this lens, we aim to implement an achromatic spectrometer for electron bunches produced by a laser-wakefield accelerator. This will greatly improve the resolution across the typically one to tens of percent energy spread bunches and therefore help diagnosis and optimization of the plasma interaction. We report on progress in designing such an experiment.

The entanglement of curves within a 3-periodic box provides a model for complicated space-filling entangled structures occurring in biological, chemical and physical systems. Quantifying the complexity of the entanglement within these models enhances the characterisation of these structures. In this paper, we introduce a new measure of entanglement complexity through the untangling number, reminiscent of the unknotting number in knot theory. The untangling number quantifies the minimum distance between a given 3-periodic structure and its least tangled version, called ground state, through a sequence of operations in a diagrammatic representation of the structure. For entanglements that consist of only infinite open curves, we show that the generic ground states are crystallographic rod packings, well known in structural chemistry.

This Element offers a practical guide to estimating conditional marginal effects-how treatment effects vary with a moderating variable-using modern statistical methods. Commonly used approaches, such as linear interaction models, often suffer from unclarified estimands, limited overlap, and restrictive functional forms. This guide begins by clearly defining the estimand and presenting the main identification results. It then reviews and improves upon existing solutions, such as the semiparametric kernel estimator, and introduces robust estimation strategies, including augmented inverse propensity score weighting with Lasso selection (AIPW-Lasso) and double machine learning (DML) with modern algorithms. Each method is evaluated through simulations and empirical examples, with practical recommendations tailored to sample size and research context. All tools are implemented in the accompanying \texttt{interflex} package for \texttt{R}.

A program to search for radio emission from dwarf-novae-type cataclysmic variables was conducted with the South African MeerKAT radio telescope. The dwarf novae RU Pegasi, V426 Ophiuchi and IP Pegasi were detected during outburst at L-band (1284 MHz central frequency). Previously, only one cataclysmic variable was radio-detected at a frequency this low. We now bring the number to four. With these three newly found radio-emitters, the population of dwarf novae confirmed to be radio-emitting at any frequency reaches 10 systems. We found that the radio luminosity is correlated with the optical luminosity. For V426 Ophiuchi and RU Pegasi we found a radio decline contemporary with the outburst's optical decline. The peak radio luminosity of dwarf novae in outburst is very similar to that of novalike Cataclysmic Variables and no correlation with orbital period is seen.

Let $p$ be a prime, $E$ be a supersingular elliptic curve defined over $\bar{\mathbb{F}}_p$, and $\mathscr{O}$ be its (geometric) endomorphism ring. Earlier results of Chevyrev-Galbraith and Goren-Love have shown that the successive minima of the Gross lattice of $\mathscr{O}$ characterize the isomorphism class of $\mathscr{O}$. In this paper, we extend this work and show that the value of the third successive minimum $D_3$ of the Gross lattice gives necessary and sufficient conditions for the curve to have its $j$-invariant in the field $\mathbb{F}_p$ or in the set $\mathbb{F}_{p^2} \setminus \mathbb{F}_p$, as well as finer information about the endomorphism ring of $E$ when its $j$-invariant belongs to $\mathbb{F}_p$ and $p \equiv 3 \pmod{4}$. We end our article with an investigation of the geometry of Gross lattices of supersingular elliptic curves.

We consider a process $X^\ve$ that solves a stochastic Volterra equation with an unknown parameter $θ^\star$ in the drift function. The Volterra kernel is singular, and includes as an example, $K\_0(u)=c u^{α-1/2} \id{u>0}$ with $α\in (0,1/2)$. It is assumed that the diffusion coefficient is proportional to $\ve \to 0$. From an observation of the path $(X^\ve\_s)\_{s\in[0,T]}$, we construct a Trajectory Fitting Estimator, which is shown to be consistent and asymptotically normal. We also specify identifiability conditions insuring the $L^p$ convergence of the estimator.

Accurate covariance forecasting is central to portfolio allocation, risk management, and asset pricing, yet many existing methods struggle at medium-term horizons, where shifting market regimes and slower dynamics predominate. We propose a deep learning framework that combines three-dimensional convolutional neural networks, bidirectional long short-term memory layers, and multi-head attention to capture complex spatio-temporal dependencies. Using daily data on 14 exchange-traded funds from 2017 through 2023, we find that our model reduces Euclidean and Frobenius distance metrics by up to 20\% relative to classical benchmarks (e.g., shrinkage and GARCH approaches) and remains robust across distinct market regimes. Our portfolio experiments demonstrate significant economic value through lower volatility and moderate turnover. These findings highlight the potential of advanced deep learning architectures to improve medium-term covariance forecasts, offering practical benefits for institutional investors and risk managers.

For the damped wave equation on the torus, when some geodesics never meet the positive set of the damping, energy decay rates are known to depend on derivative bounds and growth properties of the damping near the boundary of its support, as well as the geometry of the support of the damping. In this paper we obtain, sometimes sharp, energy decay rates for damping which satisfy more general non-polynomial derivative bounds and growth properties. We also show how these rates can depend on the geometry of the support of the damping. We prove general results and apply them to examples of damping growing exponentially slowly, polynomial-logarithmically, or logarithmically. The decay rates found for these examples interpolate between known sharp rates for purely polynomial damping. The proof relies on estimating the solution at very fine, non-poynomial, semiclassical scales to obtain resolvent estimates, which are then converted to energy decay rates via semigroup theory.

Alcohol Use Disorder (AUD) is a prevalent addictive disorder affecting an estimated 29.5 million Americans. It is characterized by impaired control over alcohol consumption despite negative consequences. The number of diagnostic criteria met by an individual typically determines the severity of AUD. Research into AUD focuses on understanding individual susceptibility differences and developing preventive strategies. Alcohol vapor inhalation has emerged as a promising method for pathophysiological investigations in animals, allowing researchers to control the dose and duration of alcohol exposure. This approach is crucial for studying the escalation of voluntary alcohol-drinking behavior. Current commercial systems for alcohol vapor generation have limitations, including combustion risks and the need to adjust multiple parameters. Other methods, like bubbling or blow-over evaporation, face challenges in maintaining equilibrium and avoiding aerosolization. To address these issues, a new type of ethanol vapor generating system is proposed that relies solely on temperature control, creating a vacuum into which ethanol evaporates under thermodynamic control. This approach eliminates the need to adjust multiple parameters and offers improved accuracy and precision in vapor dose delivery. We validated the system as anticipated, achieving stable ethanol vapor after a few priming cycles. Using a 1.2 L cylinder, we obtained approximately 3.6 L of saturated vapor/air mix in 1 minute. Gravimetric results showed that each cycle produced about 100 mg/L or ~10,000 ppm vapor-to-air mixture. The intended use of the ethanol vapor generator is to provide a concentrated ethanol vapor / air mixture to be further diluted before delivering to the animals.

Stochastic optimal control control problems with merely measurable coefficients are not well understood. In this manuscript, we consider fully non-linear stochastic optimal control problems in infinite horizon with measurable coefficients and (local) uniformly elliptic diffusion. Using the theory of $L^p$-viscosity solutions, we show existence of an $L^p$-viscosity solution $v\in W_{\rm loc}^{2,p}$ of the Hamilton-Jacobi-Bellman (HJB) equation, which, in turn, is also a strong solution (i.e. it satisfies the HJB equation pointwise a.e.). We are then led to prove verification theorems, providing necessary and sufficient conditions for optimality. These results allow us to construct optimal feedback controls and to characterize the value function as the unique $L^p$-viscosity solution of the HJB equation. To the best of our knowledge, these are the first results for fully non-linear stochastic optimal control problems with measurable coefficients. We use the theory developed to solve a stochastic optimal control problem arising in economics within the context of optimal advertising.

There are two major generalizations of the standard ordinal analysis: One is Girard's $Π^1_2$-proof theory in which dilators are assigned to theories instead of ordinals. The other is Pohlers' generalized ordinal analysis with Spector classes, where ordinals greater than $ω_1^{\mathsf{CK}}$ are assigned to theories. In this paper, we show that these two are systematically entangled, and $Σ^1_2$-proof theoretic analysis has a critical role in connecting these two.

Carleson and sparse collections of sets play a central role in dyadic harmonic analysis. We employ methods from optimization theory to study such collections. First, we present a strongly polynomial algorithm to compute the Carleson constant of a collection of sets, improving on the recent approximation algorithm of Rey. Our algorithm is based on submodular function minimization. Second, we provide an algorithm showing that any Carleson collection is sparse, achieving optimal dependence of the respective constants and thus providing a constructive proof of a result of Hänninen. Our key insight is a reformulation of the duality between the Carleson condition and sparseness in terms of the duality between the maximum flow and the minimum cut in a weighted directed graph.