This paper analyzes a stochastic Allen--Cahn equation for the dynamics of biomolecular damage and repair. The system is driven by two distinct noise processes: a multiplicative cylindrical Wiener process, modeling continuous background stochastic fluctuations, and a jump-type noise, modeling the abrupt, localized damage induced by external shocks. The drift of the equation is singular and covers the typical logarithmic Flory-Huggins potential required in phase-separation dynamics. We prove well-posedness of the model in a strong probabilistic sense, and analyze its long-time behavior in terms of existence and uniqueness of invariant measures, ergodicity, and mixing properties. Eventually, we present an Euler--Maruyama scheme to simulate the model and illustrate how it captures fundamental biological phenomena, such as damage clustering, stress-induced topology perturbations, and damage dynamics.

The TV-Stokes model is a two-step variational method for image denoising that combines the estimation of a divergence-free tangent field with total variation regularization in the first step and then uses that to reconstruct the image in the second step. Although effective in practice, its mathematical structure and potential for parallelization have remained unexplored. In this work, we establish a rigorous functional-analytic foundation for the TV-Stokes model. We formulate both steps in appropriate infinite-dimensional function spaces, derive their dual formulations, and analyze the compatibility and mathematical consistency of the coupled system. In particular, we identify analytical inconsistencies in the original formulation and demonstrate how an alternative model resolves them. We also examine the orthogonal projection onto the divergence-free subspace, proving its existence in a continuous setting and establishing consistency with its discrete counterpart. Building on this theoretical framework, we develop the first domain decomposition method for TV-Stokes by applying overlapping Schwarz-type iterations to the duals of both steps. Although the divergence-free constraint gives rise to a global projection operator in the continuous model, we show that it becomes locally computable in the discrete setting. This insight enables a fully parallelizable algorithm suitable for large-scale image processing in memory-constrained environments. Numerical experiments demonstrate the correctness of the domain decomposition approach and its usability in parallel image reconstruction.

A general model is formulated for elasto-plastic materials undergoing linear kinematic hardening to describe microstructure evolution associated with phase transformations. Using infinitesimal strain theory, the model is based on variational principles for inelastic materials. In our work we combine the so-called dissipation distance, which describes an immediate phase transition in time via an underlying probability matrix. In addition, the volume fractions of the newly emerging phases are represented by Young measures to obtain a time continuous microstructure evolution. The model is verified employing a two-dimensional benchmark test implemented by the Finite Element Method (FEM).

Coin selection refers to the problem of choosing a set of tokens to fund a transaction in token-based payment systems such as, e.g., cryptocurrencies or central bank digital currencies (CBDCs). In this paper, we propose the Boltzmann Draw that is a probabilistic algorithm inspired by the principles of statistical physics. The algorithm relies on drawing tokens according to the Boltzmann distribution, serving as an extension and improvement of the Random Draw method. Numerical results demonstrate the effectiveness of our method in bounding the number of selected input tokens as well as reducing dust generation and limiting the token pool size in the wallet. Moreover, the probabilistic algorithm can be implemented efficiently, improves performance and respects privacy requirements - properties of significant relevance for current token-based technologies. We compare the Boltzmann draw to both the standard Random Draw and the Greedy algorithm. We argue that the former is superior to the latter in the sense of the above objectives. Our findings are relevant for token-based technologies, and are also of interest for CBDCs, which as a legal tender possibly needs to handle large transaction volumes at a high frequency.

We study the problem of differentially private (DP) computation of coreset for the $k$-means objective. For a given input set of points, a coreset is another set of points such that the $k$-means objective for any candidate solution is preserved up to a multiplicative $(1 \pm α)$ factor (and some additive factor). We prove the first computational lower bounds for this problem. Specifically, assuming the existence of one-way functions, we show that no polynomial-time $(ε, 1/n^{ω(1)})$-DP algorithm can compute a coreset for $k$-means in the $\ell_\infty$-metric for some constant $α> 0$ (and some constant additive factor), even for $k=3$. For $k$-means in the Euclidean metric, we show a similar result but only for $α= Θ\left(1/d^2\right)$, where $d$ is the dimension.

We present a new method for solving the relativistic Vlasov--Maxwell system of equations, applicable to a wide range of extreme high-energy-density astrophysical and laboratory environments. The method directly discretizes the kinetic equation on a high-dimensional phase-space grid using a discontinuous Galerkin finite element approach, yielding a high-order, conservative numerical scheme that is free from the Poisson noise inherent to traditional Monte-Carlo methods. A novel and flexible velocity-space mapping technique enables the efficient treatment of the wide range of energy scales characteristic of relativistic plasmas, including QED pair-production discharges, instabilities in strongly magnetized plasmas surrounding neutron stars, and relativistic magnetic reconnection. Our noise-free approach is capable of providing unique insight into plasma dynamics, enabling detailed analysis of electromagnetic emission and fine-scale phase-space structure.

Assuming $\mathfrak p=\mathfrak c$, we show that for every Eberlein compact space $L$ of weight $\mathfrak c$ there exists a short exact sequence $0\to c_0\to X\to C(L)\to 0$, where the Banach space $X$ is not isomorphic to a $C(K)$-space.

We present an algorithm turning any term of a linear quantum $λ$-calculus into a quantum circuit. The essential ingredient behind the proposed algorithm is Girard's geometry of interaction, which, differently from its well-known uses from the literature, is here leveraged to perform as much of the classical computation as possible, at the same time producing a circuit that, when evaluated, performs all the quantum operations in the underlying $λ$-term. We identify higher-order control flow as the primary obstacle towards efficient solutions to the problem at hand. Notably, geometry of interaction proves sufficiently flexible to enable efficient compilation in many cases, while still supporting a total compilation procedure. Finally, we characterize through a type system those $λ$-terms for which compilation can be performed efficiently.

Augmented Reality (AR) can simulate various visual perceptions, such as how individuals with colorblindness see the world. However, these simulations require developers to predefine each visual effect, limiting flexibility. We present ShadAR, an AR application enabling real-time transformation of visual perception through shader generation using large language models (LLMs). ShadAR allows users to express their visual intent via natural language, which is interpreted by an LLM to generate corresponding shader code. This shader is then compiled real-time to modify the AR headset viewport. We present our LLM-driven shader generation pipeline and demonstrate its ability to transform visual perception for inclusiveness and creativity.

In recent years, Homotopy Type Theory (HoTT) has had great success both as a foundation of mathematics and as internal language to reason about $\infty$-groupoids (a.k.a. spaces). However, in many areas of mathematics and computer science, it is often the case that it is categories, not groupoids, which are the more important structures to consider. For this reason, multiple directed type theories have been proposed, i.e., theories whose semantics are based on categories. In this paper, we present a new such type theory, Twisted Type Theory (TTT). It features a novel ``twisting'' operation on types: given a type that depends both contravariantly and covariantly on some variables, its twist is a new type that depends only covariantly on the same variables. To provide the semantics of this operation, we introduce the notion of dependent 2-sided fibrations (D2SFibs), which generalize Street's notion of 2-sided fibrations. We develop the basic theory of D2SFibs, as well as characterize them through a straightening-unstraightening theorem. With these results in hand, we introduce a new elimination rule for Hom-types. We argue that our syntax and semantics satisfy key features that allow reasoning in a HoTT-like style, which allows us to mimic the proof techniques of that setting. We end the paper by exemplifying this, and use TTT to reason about categories, giving a syntactic proof of Yoneda's lemma.

Lithium plays a unique role in astrophysics, as it is a powerful diagnostic for the physics and evolution of low-mass stars, Galactic archaeology, and cosmology. We review the Li observations in stars at different phases of their evolution, the strengths and the limitations of the current theoretical stellar models to explain the Li abundance data, our understanding of the Li sources and of the evolution of Li through- out the Galactic history. Key takeaways from the current state of the research in the field are: 1) Stellar evolution models accounting for fundamental transport processes of chemical species and angular momentum hold the promise of providing a common stellar Li depletion explanation to the Li abundance patterns observed in all Galactic stellar populations, including the dip and the plateau(s). 2) Novae are most probably the main source of Li in the Galaxy, on observational (but not yet theoretically established) grounds. 3) Radial migration of stars in the Galactic disk holds the key to understand many aspects of the Li evolution in the Milky Way.

Plasma-based accelerators offer high accelerating gradients and scalability through staging or long plasma sources, which makes them good candidates for future accelerator and collider concepts. Proton-driven accelerators in particular have the potential to bring particles to high energy in a single stage. In the quasilinear regime - where the plasma wake is only partially evacuated - a witness bunch of electrons drives a cavitated wake, which acts to preserve the emittance of the portion of the witness inside this self-blowout. In the case of a misalignment between the driver and witness, this behaviour can persist, but its effectiveness is reduced. In this paper, we study transverse witness dynamics in this regime, and develop analytical models to describe the witness motion, and develop a metric to estimate emittance preservation based on a single parameter which estimates the density of the witness after phase mixing. Particle in cell simulations using the AWAKE Run 2c baseline parameters show excellent agreement with the predictive models developed. This work allows alignment constraints to be set both for the AWAKE experiment and other wakefield acceleration schemes operating in the quasilinear regime.

Saturn's moon Titan exhibits remarkable parallels to the Earth in many geophysical and geological processes not found elsewhere in the solar system at the present day. These include a nitrogen atmosphere with a condensible gas - methane - replacing the Earth's water, leading to an active meteorology with rainfall and surface manifestations including rivers, lakes and seas, and the dissolution of karstic terrain. Other phenomena such as craters, dunes, and tectonic features are found elsewhere - e.g. on Mars and Venus - but their continuing alteration by pluvial, fluvial and lacustrine processes can be studied only on Earth and Titan. Meanwhile Titan also hosts an interior liquid water ocean with similarities to the Earth as well as to ocean worlds such as Europa and Enceladus. Our focus in this review paper is twofold: to describe the geophysical and geological parallels between Earth and Titan, and to evaluate the yet-underexploited possibilities for field analog research to gain new knowledge about these processes. To date, Titan's much colder temperature and different atmospheric and crustal materials have led to a skepticism that useful analogs can be found on Earth. Our conclusion, however, is that a much larger range of useful analog field work is possible and this work will substantially enhance our knowledge of both worlds. Such investigation will supplement the existing sparse data for Titan returned by space missions, will greatly enhance our understanding of such datasets, and will help to provide science impetus and goals for future missions.

Free-electron interactions with light and matter have long served as a cornerstone for exploring the quantum and ultrafast dynamics of material excitation. In recent years, this paradigm has evolved from a classical description of radiation and acceleration toward a fully quantum framework, transforming our understanding of light-matter interactions at the single-electron level. These advances have opened new opportunities in high-resolution imaging, ultrafast spectroscopy, interferometry, and the coherent shaping of electron wavepackets. This review surveys stimulated interactions between slow electrons and light, encompassing free-space and near-field mediated mechanisms. We discuss how free-space optical fields coherently modulate electron momentum and energy, and how near-field coupling in nanophotonic and plasmonic structures enables strong, phase-matched, efficient momentum exchange with the electron wavepacket. We further describe electron recoil, which is significant in the slow-electron regime, and temporal and spatial wavepacket shaping that enhances coupling efficiency and extends access to quantum-coherent regimes. Building on these foundations, we outline emerging frameworks including hybrid optical-electrostatic modulation, ponderomotive laser-based aberration correction, and optical electron interferometry. By unifying these developments, stimulated electron-light interactions provide a versatile route to precise beam control, quantum-state engineering, and tailored light-matter coupling, with implications for ultrafast spectroscopy, nanoscale metrology, attosecond pulse generation, electron-photon entanglement, and the creation of nonclassical states of light.

This paper investigates the motion of a single photon in a two-dimensional plane under closed and open boundary conditions. We employ two methods to construct the Hilbert space: Method A, based on the standard second-quantization formalism, and Method B, based on a non-standard approach. By eliminating redundant quantum states, we obtain a reduced Hilbert space with significantly lower dimensionality, thereby improving the efficiency of numerical simulations. In a closed system, the two methods are equivalent, and their unitary evolution results are identical. The probability distribution diffuses outward from the center and exhibits a significant rebound after reaching the boundary. In an open system, Method B, by incorporating more dissipation channels, provides a more accurate description of the photon escape process at the boundary. The probability curves obtained from the two methods completely overlap before reaching the boundary. After the boundary is reached, a slight difference appears, but this difference does not amplify with evolution and tends to converge in the later stage. Method B yields a slightly higher dissipative-state probability, indicating that the photon escapes faster. Visualization of the two-dimensional probability distribution shows that the three scenarios (closed system, open system with Method A, and open system with Method B) exhibit identical probability distributions before reaching the boundary. After the boundary is reached, the open systems exhibit significant probability loss, which increases rapidly with evolution. The probability distribution patterns of the two open systems are highly similar, exhibiting synchronized evolution.

Extending the scalar sector of the Standard Model is a well-motivated approach to exploring physics beyond the Standard Model. In this work, we investigate the phenomenology of the Three Higgs Doublet Model at a future muon collider. The scalar spectrum of the 3HDM comprises three CP-even Higgs bosons, two CP-odd Higgs bosons, and a pair of charged Higgs states. Focusing on Higgs pair production via muon-antimuon annihilation, we study the production and decay of neutral scalar states through the process $μ^+μ^- \to ϕ_i ϕ_j$, assuming a mass hierarchy in which the SM-like CP-even Higgs is the lightest state. We analyze several benchmark scenarios leading to $b\bar{b}b\bar{b}$ and $b\bar{b}t\bar{t}$ final states, and perform a cut-and-count analysis at a center-of-mass energy of $\sqrt{s}=3$ TeV. Our results demonstrate that a future muon collider provides a sensitive and promising environment to probe extended Higgs sectors, with neutral scalar states in the mass range of $200-400$ GeV being discoverable with $5σ$ significance for integrated luminosities of $\mathcal{O}(1-4 \ \mathrm{ab}^{-1})$.

Light meson spectra are studied in a Dyson-Schwinger/Bethe-Salpeter approach to QCD. By varying the interaction strength of three sets of models for the quark-antiquark interaction, the transition from the chiral symmetric to the chirally broken regime in the vacuum is studied. The simplest type of these models leads to degenerate meson spectra for a large domain of the strength parameter. The more sophisticated and thus more realistic models show significantly smaller parameter domains for which degenerate meson spectra are obtained. The underlying mechanism for obtaining and then lifting degeneracies is traced back to the location of the quark propagators' poles, in particular, whether they are beyond or within the domain of integration in the Bethe-Salpeter equation. In view of this mechanism the potential relation of the obtained degeneracies to the dynamical emergence of symmetries is discussed, adding thereby another point of view on the conjectured chiral spin symmetry of QCD in the temperature domain right above the crossover.

Differential privacy (DP) implementations are notoriously prone to errors, with subtle bugs frequently invalidating theoretical guarantees. Existing verification methods are often impractical: formal tools are too restrictive, while black-box statistical auditing is intractable for complex pipelines and fails to pinpoint the source of the bug. This paper introduces Re:cord-play, a gray-box auditing paradigm that inspects the internal state of DP algorithms. By running an instrumented algorithm on neighboring datasets with identical randomness, Re:cord-play directly checks for data-dependent control flow and provides concrete falsification of sensitivity violations by comparing declared sensitivity against the empirically measured distance between internal inputs. We generalize this to Re:cord-play-sample, a full statistical audit that isolates and tests each component, including untrusted ones. We show that our novel testing approach is both effective and necessary by auditing 12 open-source libraries, including SmartNoise SDK, Opacus, and Diffprivlib, and uncovering 13 privacy violations that impact their theoretical guarantees. We release our framework as an open-source Python package, thereby making it easy for DP developers to integrate effective, computationally inexpensive, and seamless privacy testing as part of their software development lifecycle.

We determine all restrictions on the dimension of the fixed locus of a diagonalizable group acting on a smooth projective variety that arise from the Chern numbers of the ambient variety. We reduce the problem to finding lower bounds for actions of p-groups, which we achieve by analyzing the equivariant cobordism ring with the help of the concentration theorem. To do so, we construct enough explicit examples of actions that realize the expected lower bound. We then prove that this family is maximal in the equivariant cobordism ring, in an appropriate sense.

Machine learning training places immense demands on cluster networks, motivating specialized architectures and co-design with parallelization strategies. Recent designs incorporating optical circuit switches (OCSes) are promising, offering improved cost, power efficiency, and long-term bandwidth scaling than packet switches. However, most existing approaches rely on costly high-radix OCSes and/or combine them with packet switches to achieve competitive performance at scale. Unfortunately, high-radix OCSes are both expensive and slow to reconfigure, limiting both scalability and performance. We propose Arrays of Cheap Optical Switches (ACOS), which bring application co-design directly to the structure of the reconfigurable fabric. Using low-radix OCSes as building blocks, ACOS supports the forms of reconfiguration needed in training clusters including topology selection, workload adaptation, and failure resilience. The cost of ACOS scales with supported topologies and adaptations rather than with port count, breaking past the scalability barriers of current specialized ML networks. We show through simulation that ACOS-based deployments match the performance of fully provisioned packet-switched networks when training state-of-the-art LLMs at scale, while delivering significant cost savings using existing off-the-shelf OCSes, with strong bandwidth scaling and higher cost savings in the future.

To meet the ever-increasing demands of the cybersecurity workforce, AI tutors have been proposed for personalized, scalable education. But, while AI tutors have shown promise in introductory programming courses, no work has evaluated their use in hands-on exploration and exploitation of systems (e.g., ``capture-the-flag'') commonly used to teach cybersecurity. Thus, despite growing interest and need, no work has evaluated how students use AI tutors or whether they benefit from their presence in real, large-scale cybersecurity courses. To answer this, we conducted a semester-long observational study on the use of an embedded AI tutor with 309 students in an upper-division introductory cybersecurity course. By analyzing 142,526 student queries sent to the AI tutor across 396 cybersecurity challenges spanning 9 core cybersecurity topics and an accompanying set of post-semester surveys, we find (1) what queries and conversational strategies students use with AI tutors, (2) how these strategies correlate with challenge completion, and (3) students' perceptions of AI tutors in cybersecurity education. In particular, we identify three broad AI tutor conversational styles among users: Short (bounded, few-turn exchanges), Reactive (repeatedly submitting code and errors), and Proactive (driving problem-solving through targeted inquiry). We also find that the use of these styles significantly predicts challenge completion, and that this effect increases as materials become more advanced. Furthermore, students valued the tutor's availability but reported that it became less useful for harder material. Based on this, we provide suggestions for security educators and developers on practical AI tutor use.

This article considers a mean field game model inspired by crowd motion models in which agents aim at reaching a given target set and wish to minimize a cost consisting of an individual running cost, an individual cost depending on the arrival time at the target set, and an interaction running cost, which takes the form of pairwise interactions with other agents through both positions and velocities. We subsume this game under a more general class of games on abstract Polish spaces with pairwise interactions, and prove that the latter games have a variational structure (in the sense that their equilibria can be characterized as critical points of some potential functional) and admit equilibria. We also discuss two a priori distinct notions of equilibria, providing a sufficient condition under which both notions coincide. The results for the games in abstract Polish spaces are applied to our mean field game model, and a numerical illustration concludes the paper.

The accuracy and precision of high-energy spallation models play a crucial role in the design and development of new applications and experiments, as well as in data analysis. We discuss the complementarity between parameter optimisation and model bias estimation approaches within a Bayesian framework. This is illustrated using the IntraNuclear Cascade model of Liège (INCL) together with the Ablation model (ABLA), for which these two approaches for model bias estimation have been applied independently in previous works.

Complex organic molecules are key markers of molecular diversity, and their formation conditions in protoplanetary disks remain an active area of research. These molecules have been detected on a variety of celestial bodies, including icy moons, and may play a crucial role in shaping the current composition of the Galilean moons. Experimental studies suggest that their formation could result from UV irradiation or thermal processing of NH3:CO2 ices. In this context, we investigate the formation of complex organic molecules in the protosolar nebula and their subsequent transport to the Jupiter system region. Lagrangian transport and irradiation simulations of 500 individual particles are performed using a two-dimensional disk evolution model. Based on experiments with UV irradiation and thermal processing of CO2:NH3 ice, this model allows us to estimate the estimate the potential for the formation of complex organic molecules through these processes. Almost none of the particles released at a local temperature of 20 K (corresponding to ~12 AU from the Sun) reach the location of the system of Jupiter. However, when released at a local temperature of 80 K (~7 AU), approximately 45% of the centimetric particles and 30% of the micrometric particles can form complex organic molecules via thermal processing, subsequently reaching the location of the system of Jupiter within 300 kyr. Assuming that the Galilean moons formed in a cold circumplanetary disk around Jupiter, the nitrogen-bearing species potentially present in their interiors could have originated from the formation of complex organic molecules in the protosolar nebula.

Feedback-driven quantum reservoir computing has so far been studied primarily in gate-based architectures, motivating alternative scalable, hardware-friendly physical platforms. Here we investigate a linear-optical quantum reservoir architecture for time-series processing based on multiphoton interference in a reconfigurable interferometer network equipped with threshold detectors and measurement-conditioned feedback. The reservoir state is constructed from coarse-grained coincidence features, and the feedback updates only a structured, budgeted subset of programmable phases, enabling recurrence without training internal weights. By sweeping the feedback strength, we identify three dynamical regimes and find that memory performance peaks near the stability boundary. We quantify temporal processing via linear memory capacity and validate nonlinear forecasting on benchmarks, namely Mackey-Glass series, NARMA$-n$ and non-integrable Ising dynamics. The proposed architecture is compatible with current photonic technology and lowers the experimental barrier to feedback-driven QRC for time-series analysis with competitive accuracy.

We study a continuum Hatano--Nelson model with a saturating nonlinear nonreciprocity and analyze its stationary states via the associated phase-space flow. We uncover a global scenario controlled by a subcritical Hopf bifurcation and a saddle-node of limit cycles, which together generate a finite coexistence window. In this window, skin modes and extended states are both stable at a fixed energy $E$, separated by a nonlinear basin separatrix in phase space rather than a spectral (mobility-edge) mechanism in a linear system. An averaged amplitude equation yields closed-form predictions for the limit-cycle branches and the SNLC threshold. Building on the basin geometry, we introduce a basin-fraction order parameter that exhibits a first-order-like jump at SNLC. Intriguing physical phenomena in the coexistence window are also revealed, such as separatrix-induced long-lived spatial transients and hysteresis. Overall, our findings highlight that, beyond linear spectral concepts, global attractor-basin geometry provides a powerful and complementary lens for understanding stationary states in nonlinear non-Hermitian systems.

Preparing high-fidelity logical states is a central challenge in fault-tolerant quantum computing, yet existing approaches struggle to balance control complexity against resource overhead. Here, we present a complete framework for the fault-tolerant preparation of arbitrary logical states encoded in the four-legged cat code. This framework is engineered to suppress the dominant incoherent errors, including excitation decay and dephasing in both the bosonic mode and the ancilla via error detection. Numerical simulations with experimentally realistic parameters on a 3D superconducting cavity platform yield logical infidelities on the order of $10^{-4}$. A scaling analysis confirms that the logical error rate grows nearly quadratically with the physical error rate, confirming that all first-order errors are fully suppressed. Our protocol is compatible with current hardware and is scalable to multiple bosonic modes, providing a resource-efficient foundation for magic state preparation and higher-level concatenated quantum error correction.

In this paper, we show that the main algebraic assumption required to perform a fixed point argument for rough differential equations implies the algebraic assumption for the Bailleul flow approach. This assumption requires that the rough path associated with the equation is given by a Hopf algebra whose coproduct admits a cocycle and has a tree-like basis. We show that the Hopf algebra of multi-indices does not satisfy the cocycle condition. This is a rigorous result on the impossibility, observed in practice, of performing a fixed point argument for multi-indices rough paths and multi-indices in Regularity Structures.

Matrix-product operators (MPOs) appear throughout the study of integrable lattice models, notably as the transfer matrices. They can also be used as transformations to construct dualities between such models, both invertible (including unitary) and non-invertible (including discrete gauging). We analyse how the local Yang--Baxter integrable structures are modified under such dualities. We see that the $\check{R}$-matrix, that appears in the baxterization approach to integrability, transforms in a simple manner. We further show for a broad class of MPOs that the usual Yang--Baxter $R$-matrix satisfies a modified algebra, previously identified in the unitary case, that gives a local integrable structure underlying the commuting transfer matrices of the dual model. We illustrate these results with two case studies, analysing an invertible unitary MPO and a non-invertible MPO both applied to the canonical XXZ spin chain. The former is the cluster entangler, arising in the study of symmetry-protected topological phases, while the latter is the Kramers--Wannier duality. We show several results for MPOs with exact MPO inverses that are of independent interest.

Motivated by the $y$-ification of HOMFLY--PT homology by Gorsky and Hogancamp, and the $\mathfrak{sl}_2$-action of Gorsky, Hogancamp, and Mellit, we construct $y$-ifications of Khovanov homology and its equivariant versions within Bar-Natan's framework for tangles, and define an action of the element $e$ in $\mathfrak{sl}_2$ on these $y$-ifications. We then prove that our construction is compatible with the previous ones under Rasmussen's spectral sequence from HOMFLY--PT homology to Khovanov homology. Our construction is elementary and well suited to diagrammatic manipulations and algorithmic implementations. As a result, we verify directly that these additional structures distinguish pairs of knots with identical Khovanov homology and HOMFLY--PT homology, in particular the Conway knot and the Kinoshita--Terasaka knot.