We consider the identity of the abelian sandpile group of finite approximation graphs of the Sierpinski gasket, and we show that the second-order term in the scaling limit converges to the path distance to the nearest corner on the Sierpinski gasket. The proof relies on a decomposition of the identity of the sandpile group into the sum of a constant function and the Laplacian of the graph distance on the approximating graphs.

Optical Fabry-Perot cavities are crucial tools for metrology experiments, where they achieve extreme length stability, and for some atomic physics experiments, where tunability to atomic transitions enables atom-light interactions. However, achieving both frequency stability and tunability in a single cavity has remained a challenge, forcing metrology experiments exploiting atom-cavity interactions to rely on external active feedback systems to stabilize the length of the cavity. Here, we describe a piezoelectrically-tunable cavity with a cancellation of the coefficient of thermal expansion at around $5^\circ\mathrm{C}$, achieving fractional frequency instabilities at the $4\times 10^{-13}$ level for 1~s integration time. This advance eliminates the need for external stabilization in many atom-cavity experiments, making this design ideal for applications such as ultra-stable superradiant lasers and other cavity quantum electrodynamics experiments.

One-dimensional integrable fermions can be classified into massless and massive regimes, and the $R$-operator for the latter can be constructed from that of the former. Here, I define integrable massless fermions by the simultaneous satisfaction of the Yang-Baxter equation (YBE) and Shastry's decorated YBE (DYBE) by the $R$-matrix. This notion is strictly more general than Maassarani's `free-fermion algebra', yet more restrictive than the notion of free fermions in exactly solvable quantum models or in integrable two-dimensional classical vertex models dual to quantum spin chains. Within this framework, there emerge two archetypal mechanisms for opening a spectral gap and generating massive fermions: (i) breaking time-reversal symmetry by coupling to external field, and (ii) introducing time-reversal symmetric interactions. These paradigms are realized, respectively, in the XY chain in a longitudinal field and in the Hubbard model, both of which possess non-relativistic, bivariate $R$-matrices. Integrability conditions on local Hamiltonians for both massless and massive fermions are identified, and schematic procedures for uniquely determining their $R$-matrices are proposed.

The primary entropic measures for quantum states are additive under the tensor product. In the analysis of quantum information processing tasks, the minimum entropy of a set of states, e.g., the minimum output entropy of a channel, often plays a crucial role. A fundamental question in quantum information and cryptography is whether the minimum output entropy remains additive under the tensor product of channels. Here, we establish a general additivity statement for the optimized sandwiched Rényi entropy of quantum channels. For that, we generalize the results of [Devetak, Junge, King, Ruskai, CMP 2006] to multi-index Schatten norms. As an application, we strengthen the additivity statement of [Van Himbeeck and Brown, 2025] thus allowing the analysis of time-adaptive quantum cryptographic protocols. In addition, we establish chain rules for Rényi conditional entropies that are similar to the ones used for the generalized entropy accumulation theorem of [Metger, Fawzi, Sutter, Renner, CMP 2024].

LLM agents are highly vulnerable to Indirect Prompt Injection (IPI), where adversaries embed malicious directives in untrusted tool outputs to hijack execution. Most existing defenses treat IPI as an input-level semantic discrimination problem, which often fails to generalize to unseen payloads. We propose a new paradigm, action-level causal attribution, which secures agents by asking why a particular tool call is produced. The central goal is to distinguish tool calls supported by the user's intent from those causally driven by untrusted observations. We instantiate this paradigm with AttriGuard, a runtime defense based on parallel counterfactual tests. For each proposed tool call, AttriGuard verifies its necessity by re-executing the agent under a control-attenuated view of external observations. Technically, AttriGuard combines teacher-forced shadow replay to prevent attribution confounding, hierarchical control attenuation to suppress diverse control channels while preserving task-relevant information, and a fuzzy survival criterion that is robust to LLM stochasticity. Across four LLMs and two agent benchmarks, AttriGuard achieves 0% ASR under static attacks with negligible utility loss and moderate overhead. Importantly, it remains resilient under adaptive optimization-based attacks in settings where leading defenses degrade significantly.

We address the classical (degenerate or non-degenerate) center problem posed by Poincaré in the 19th century for monodromic singularities of analytic families of planar vector fields $\mathcal{X}$. We prove that every analytic center admits a Laurent inverse integrating factor $V$ in weighted polar coordinates. Moreover, we show that when $\mathcal{X}$ has no local curves of zero angular speed, the Poincaré map is analytic. Based on this result, we derive a theoretical procedure to determine parameter constraints within the family that characterize centers without curves of zero angular speed. Applications to nontrivial families that have resisted other methods are also provided.

We demonstrate high-efficiency telecom frequency conversion of heralded atomic biphoton wavepackets using a diamond-type atomic ensemble. By placing a 2.5 MHz heralded-photon spectrum within the high-efficiency region of the converter response, we achieve a conversion efficiency of 79.4(2.6)% while maintaining strong time-resolved correlations and well-defined temporal wavepackets. For a broader 17.4 MHz input bandwidth, the conversion efficiency is reduced to about 55%, whereas the temporal waveform remains largely preserved. This behavior reflects the nearly flat central response of the converter, which mainly causes spectral-edge loss rather than temporal-mode distortion. These results identify spectral matching as an effective route to efficient and low-distortion telecom conversion of narrowband quantum light from atomic systems.

This article surveys a selection of key conceptual and interpretational developments in quantum mechanics, tracing the theory from its foundational postulates to contemporary discussions of measurement, nonlocality, and the emergence of classicality. Beginning with the structure of Hilbert space and the postulates governing state evolution and measurement, the epistemic stance of the Copenhagen interpretation and its modern reformulations are examined. The Einstein-Podolsky-Rosen argument, Bell's theorem, and Hardy's paradox are then discussed as probes of locality and realism, alongside the deterministic but explicitly nonlocal de Broglie-Bohm theory. The measurement problem and the implications of contextuality are analyzed in relation to objective collapse models, which introduce new physical dynamics to account for definite outcomes. Finally, the role of decoherence in the suppression of interference and the emergence of classical behavior is explored, together with the interpretational frameworks of many-worlds and consistent histories. This material aims to provide a coherent introductory overview of how several of the most prominent interpretations address the central concern of what quantum mechanics tells us about the nature of physical reality.

In safety-critical autonomous systems, data freshness presents a fundamental design challenge. While the Logical Execution Time (LET) paradigm ensures compositional determinism, it often does so at the cost of injected latency, possibly degrading the age of data on high-frequency control loops. Furthermore, heterogeneous, multi-rate, task dependencies is typically guaranteed inefficiently through oversampling. This paper proposes a Task-based scheduling framework extended with data freshness constraints. Unlike traditional models, scheduling decisions are driven by the lifespan of data. We introduce a formal methodology to decompose Data Dependency Graphs into dominant paths by tracing the strictest data freshness constraints backward from the actuators. Based on this decomposition, we propose an offset search algorithm that synchronizes multi-rate, multi-dependencies, task chains. This approach enforces end-to-end data freshness without the artificial latency of LET buffering, a trade-off between data freshness and execution determinism. We formally prove that this offset-based alignment preserves the 100% schedulability capacity of Global EDF while addressing data freshness guarantees.

A quantum mechanical setting consisting of a frozen qubit composed with a fully thermalized chaotic system of N states is proposed, with potential relevance to quantum control. Observing the states of the composed system selectively retaining the states leads to the observation of novel localization in the subsystem. At a tuning parameter of 1.0, implying no selection, the system exhibits Wigner-Dyson level spacing statistics, indicative of quantum chaos. As the tuning parameter is reduced and selection occurs at a cutoff, the nearest-neighbor level spacing distribution develops heavier tails, a signature of suppressed spectral mixing and the emergence of non-thermal dynamics. In these regimes, the eigendensity develops a pronounced "cat-ears" structure, reflecting the formation of spatially localized bimodal eigenstates. These topological features persist without transitioning to Poisson statistics, indicating a transition from quantum chaos to a non-thermal, novel many-body localized (MBL) regime-referred to as Wigner Cat Phases. The proposed mixed random matrix ensemble offers a practical probe for sustaining this novel quantum localization setting. Results from our rigorous spectral statistics analysis show how "cat-ears" form in spectral densities based on the degree of selection or disorder and indicate that gap ratio statistics must be used with caution in detecting the full integrable limit due to the possibility of heavy-tailed Wigner-Dyson distributions.

We investigate a class of first-order scalar field theories minimally coupled to a Carrollian connection that are defined intrinsically on the Carrollian plane, i.e., the theories are not defined via limits of Lorentzian theories. The theories built are invariant under the extended Carrollian transformations which include supertranslations. The symmetry allows for a large class of Lagrangians, independence of spacetime coordinates is all that is required. However, invariance under supertranslations (which include boosts as linear supertranslations) forces the energy density to be static and the momentum density to vanish -- this precludes on-shell propagation of fields. Thus, to have propagating theories, one must move beyond single field theories that are minimally coupled to the geometry.

Informal and privatized transit services, such as minibuses and shared auto-rickshaws, are integral to daily travel in large urban metropolises, providing affordable commutes where formal public transport is inadequate and other options are unaffordable. A defining feature of these systems is their decentralized organization, with drivers providing service in response to rider demand and earning opportunities. While this structure helps fill critical mobility gaps, it can also generate inefficient service patterns when profit-driven driver route choices do not align with system-wide mobility goals. We develop an analytically tractable game-theoretic framework to study incentives underlying informal and privatized transit systems with a fixed menu of routes, quantify efficiency losses from decentralized driver route choice, and design incentive mechanisms to mitigate these inefficiencies. Here, profit-maximizing informal operators (drivers) decide where to provide service and cost-minimizing commuters (riders) decide whether to use these services. Within this framework, we establish tight price of anarchy bounds showing that decentralized, profit-maximizing driver behavior can lead to bounded yet substantial losses in cumulative driver profit and rider demand served and that these losses can be mitigated through targeted interventions: budget-balanced cross-subsidization, which uses route-specific tolls/subsidies to shape driver payoffs, and fare optimization, which changes rider demand and driver margins through centrally regulated route-level fares. Finally, numerical experiments based on a real-world informal transit system in Nalasopara, India, reinforce these findings.

Two-dimensional Fermi liquids at low temperatures have been theoretically established to exhibit an odd-even effect in the collective quasiparticle relaxation rates where even-parity deformations of the Fermi surface decay at a much faster rate than odd-parity ones. A predicted consequence of this effect is a new tomographic transport regime that mixes hydrodynamic and collisionless transport. In the presence of a magnetic field, however, the tomographic regime is expected to evolve towards conventional transport regimes as soon as the cyclotron radius becomes smaller than the dominant odd-parity mean-free path. In this work, we examine this transition from the point of view of collective modes, using a numerically exact solution of the linearized Boltzmann equation within a generalized relaxation time approximation for the odd-parity and even-parity modes. In the absence of a magnetic field, the transverse conductivity exhibits two diffusive tomographic collective modes, and we find that at a critical magnetic field one of these two tomographic modes disappears. Which tomographic mode persists depends on the Landau parameters, and becomes increasingly dominated by hydrodynamic modes at high fields. We corroborate our analysis using a variational approach for the Fermi surface deformation that captures the angular structure of the deformation and the critical magnetic field strength. The collective modes discussed here can in principle be observed by examining the damping of longitudinal and transverse current responses in finite magnetic fields.

Lagrangian measurements of tracer particle dispersion in stratified turbulence are presented from a large-scale experiment achieving both high buoyancy Reynolds numbers and low Froude numbers -- a regime characteristic of oceanic conditions. Stratification has a pronounced effect on the vertical particle dispersion, which is observed to be constrained to distances on the order of the buoyancy scale $w_{\mathrm{std}}/N$, where $w_{\mathrm{std}}$ is the standard deviation of the vertical velocity and $N$ is the Brunt-Väisälä frequency. As expected in strongly nonlinear, stratified turbulence, the frequency spectrum of the Lagrangian velocity becomes isotropic at frequencies higher than $N$. The spectral decay follows a $1/f^3$ scaling, which contrasts with the $1/f^2$ behavior typical of homogeneous isotropic turbulence. At time scales corresponding to internal waves, the statistics of velocity increments remain Gaussian, consistent with the weakly nonlinear regime of wave turbulence. At smaller scales, however, the flow exhibits strongly non-Gaussian statistics, indicative of fully nonlinear turbulent dynamics driven by wave breaking.

Carrollian manifolds offer an intrinsic geometric framework for the physics in the ultra-relativistic limit. The recently introduced Carrollian Lie algebroids are generalised to the setting of $ρ$-commutative geometry, (also known as almost commutative geometry), where the underlying algebras commute up to a numerical factor. Via $ρ$-Lie-Rinehart pairs, it is shown that the foundational tenets of Carrollian geometry have analogous statements in the almost commutative world. We explicitly build two toy examples: we equip the extended quantum plane and the noncommutative $2$-torus with Carrollian structures. This opens up the rigorous study of noncommutative Carrollian geometry via almost commutative geometry.

Many types of bounded data defined on the unit interval arise naturally as ratios of the form $X/(X + Y)$. In the existing literature, the main statistical models proposed for this type of bounded data typically based on the assumption that the random variables $X$ and $Y$ are independent. However, this assumption is often unrealistic in practical applications, where $X$ and $Y$ tend to be correlated due to shared underlying mechanisms or common sources of variability. In this paper, we overcome such limitations and propose a model in which the marginal distributions of the two components are linked by a copula, leading to a more flexible and realistic representation of unit-interval data. In particular, in the proposed model, $X$ and $Y$ are dependent gamma random variables whose joint distribution is specified via Morgenstern's bivariate distribution}, allowing for positive and negative correlations between the components. The mathematical properties and practical applications are rigorously investigated. The resulting distribution exhibits a wide range of shapes, accommodating different degrees of skewness and, for some parameter configurations, more complex density structures. A Monte Carlo simulation study is carried out that shows the good performance of the maximum likelihood estimator in several scenarios of parameter choices. The potential and limitations of efficient likelihood-based computations are also discussed. We evaluate the effectiveness of the new model and its estimates in modeling real-world datasets related to economics.

A search is presented for massive long-lived particles in events featuring at least one displaced vertex and at least one displaced muon, using proton-proton collision data collected by the ATLAS detector at the Large Hadron Collider from 2022 to 2024 at a centre-of-mass energy of 13.6 TeV. The data sample corresponds to an integrated luminosity of 164 fb$^{-1}$. The analysis targets scenarios in which long-lived particles decay inside the ATLAS inner detector, resulting in a topology of at least one massive, displaced vertex (DV) with multiple associated tracks, and at least one muon with a large transverse impact parameter relative to the primary interaction point. The muon is not required to be associated with the DV. Two signal regions are defined by the transverse distance of the reconstructed DV from the interaction point. Background contributions are estimated by using fully data-driven techniques. No significant excess above the expected background is observed. Upper limits at 95% confidence level are set on the visible cross-section and on the production cross-sections of several benchmark models of $R$-parity-violating supersymmetry.

Let $R$ be a noetherian algebra over a Cohen--Macaulay ring $S$ admitting a canonical module $ω$, and assume that $R$ is maximal Cohen--Macaulay over $S$. We prove that the category of finitely generated Gorenstein projective $R$-modules coincides with the left $\mathrm Ext$-orthogonal class of the thick subcategory generated by $R$ and ${\mathrm Hom}_S(R,ω)$. As an application, finitely generated Gorenstein projective $R$-modules form the left half of a hereditary cotorsion pair. In the case of Cohen--Macaulay local rings, this yields an affirmative answer to a question of R. Takahashi. We further characterize when $R$ is left weakly Gorenstein. Finally, we prove that a Cohen--Macaulay local ring is Gorenstein if and only if the right $\mathrm Ext$-orthogonal class of finitely generated Gorenstein projective modules coincides with the category of finitely generated modules of finite projective dimension.

We study quantum and classical systems associated with the quantum corner symmetry group $\mathrm{QCS}=\widetilde{\mathrm{SL}}(2,\mathbb{R})\ltimes \mathrm{H}_3,$ which arises in the context of quantum gravity. We relate quantum observables -- specified by representation-theoretic data -- to their classical counterparts using generalized Perelomov coherent states and the framework of Berezin quantization. This procedure links abstract representation-theoretic input to geometric classical observables, such as area. We conclude by applying the formalism to static, spherically symmetric spacetimes admitting a horizon.

Identifying the parameters of nonlinear state-space models from input-output data typically requires solving a highly non-convex optimization problem, which is prone to slow convergence and suboptimal local solutions. This work improves the reliability and efficiency of the estimation process by decomposing the overall optimization problem into a sequence of tractable subproblems. Starting from a linear baseline model, nonlinear residual dynamics are first estimated using a guided residual search (GRS) and subsequently refined through multiple-shooting optimization. Experiments on two benchmarks show competitive performance with state-of-the-art black-box methods and improved convergence over naive initialization.

Young star-forming clumps (SFCs) emit strongly in the ultraviolet (UV), making UV imaging ideal for detecting them. The Ultraviolet Imaging Telescope (UVIT) onboard AstroSat, with 1.5arcsec resolution, has enabled the characterization of recently formed (up to 300 Myr) SFCs on tens of parsec scales in nearby galaxies. The spatial distribution of SFCs with different ages can provide insights into the hierarchy of star formation. This study presents a semi-novel approach to characterize SFCs in two nearby spiral galaxies, NGC 5457 and NGC 1313, by combining UVIT FUV data with g-band data from the Dark Energy Camera Legacy Survey (DECaLS). We tested and optimized our method on NGC 5457 and showed that after proper background subtraction, the FUV-g color of SFCs can serve as an equally reliable age indicator as the widely-used FUV-NUV color. Next, we parametrized the star formation hierarchy in NGC 5457 using the two-point correlation function (TPCF) and found good agreement between the hierarchy parameters derived using FUV-NUV and FUV-g based ages. Using our FUV-g based SFC ages, we also constrained the global hierarchy parameter of NGC 1313 for the first time. The development of our FUV-g based method is motivated by the fact that the NUV channel of the UVIT is not operational, and there is a wealth of archival UVIT FUV-only observations of nearby galaxies. This work demonstrates the potential of our method in constraining the SFC ages and investigating hierarchical star formation in nearby galaxies using FUV and optical observations.

Large-spin two-dimensional magnets are generally expected to develop conventional long-range order once the dominant exchange scale becomes appreciable. The layered spin-$3/2$ maple-leaf compound Na$_2$Mn$_3$O$_7$ defies this expectation: despite sizable antiferromagnetic interactions and no evident disorder, it exhibits no magnetic ordering and displays two well-separated thermodynamic crossover scales. We show that this behavior originates from a crystallography-driven molecularization of the magnetic degrees of freedom. The low-symmetry structure partitions the Mn sublattice into inequivalent exchange pathways, generating a pronounced hierarchy that nearly isolates antiferromagnetic hexagons. Magnetic correlations therefore develop in two stages: first within individual hexagons at a scale set by the dominant exchange, and only at much lower temperatures do frustrated inter-hexagon couplings attempt to establish coherence across the lattice. While isolated hexagons reproduce the two-step thermodynamic structure, the experimentally relevant temperature scales emerge only once the hexagons are embedded in the frustrated two-dimensional network. The resulting quantum ground state is magnetically disordered, characterized by strong intra-hexagon correlations and rapidly decaying inter-hexagon correlations. These results identify crystallographic inequivalence as a materials-level mechanism for stabilizing molecularized and quantum-disordered states even in large-spin two-dimensional magnets.

We propose a scheme to correctly incorporate the contribution of the chiral anomaly in the D3/D7 model to calculate chiral transport phenomena. To ensure the D7-brane wraps S^5 appropriately and the Wess-Zumino term is switched on, we allow the D7-brane to rotate in the compactified extra directions and perform the analysis accordingly. To demonstrate that this calculation procedure works well, we specifically compute the magnetoresistance in the D3/D7 model. We find that a finite axial chemical potential is realized and the negative magnetoresistance is enhanced by the anomaly contribution.

We investigate the phase structure of the single-flavor Gross--Neveu model with Wilson fermions using the Grassmann corner transfer matrix renormalization group (CTMRG). The path integral is formulated as a two-dimensional Grassmann tensor network and approximately contracted by the Grassmann CTMRG algorithm. We investigate the phase diagram by varying the fermion mass and the four-fermion coupling, using the pseudoscalar condensate as an order parameter for the $\mathbb{Z}_{2}$ parity symmetry breaking phase. The universality classes of the phase boundaries are identified through the central charge $c$ obtained via scaling analysis of the entanglement entropy. Furthermore, we extract the quantity related to the entanglement spectrum from the converged CTMRG environments, allowing us to distinguish the topological insulator phase and the trivial phase. The resulting phase structure suggests that the Aoki phase is separated from the other phases by critical lines characterized by $c=1/2$, while the critical lines with $c=1$ separate the topological insulating and trivial phases. Our numerical results also indicate that the Aoki phase does not persist in the strong-coupling regime for the single-flavor theory.

We theoretically investigate the motion of a domain wall and a vortex in type-II superconductors driven by inhomogeneities of temperature or spin accumulation. The model consists of the time-dependent Ginzburg-Landau equation and the thermal or spin diffusion equation, whose transport coefficients, such as the thermal and spin conductivities and the spin relaxation time, depend on the order parameter and interpolate between their values in the superconducting and normal states. Numerical and analytical calculations indicate that the domain wall moves toward the higher-temperature region or the region with larger spin accumulation, where the order parameter is suppressed. We also derive analytical expressions for the vortex velocity and confirm the predicted direction of vortex motion by numerical simulations. The dynamics of these topological defects can be understood as processes that reduce the loss of condensation energy. We also analyze the driving force, viscous force, thermal force, and force due to the spin accumulation gradient on the basis of momentum balance relations.

We construct a new class of infinite-dimensional diffusions with values in a generalized Kingman simplex with finitely many marks. The model describes the temporal evolution of the relative frequencies of infinitely many types that are labeled by a finite number $H$ of marks, but unlabeled within each mark. We first establish a blockwise skew-product representation for a finite-type Wright-Fisher diffusion, extending the aggregation-renormalization self-similarity property of Dirichlet laws. The decomposition separates an $H$-dimensional Wright-Fisher diffusion governing the evolving random mark masses, from $H$ Wright-Fisher diffusions, each run on its own random clock, which describe the evolution of the relative frequencies within each mark. After ranking the within-mark frequencies in decreasing order, we identify the distributional limit as the number of types per mark tends to infinity and we derive an explicit form of its infinitesimal generator on a suitable domain. The limiting diffusion admits the multiple Poisson-Dirichlet distribution as a stationary distribution; it recovers the infinitely-many-neutral-alleles diffusion when all types share the same mark and yields a diffusion on the Thoma simplex when there are two marks.

A famous conjecture of Erdős asserts that for $k\ge 3$, the maximum number of edges in an $n$-vertex $k$-uniform hypergraph without $s+1$ pairwise disjoint edges is $\max\{\binom{n}{k}-\binom{n-s}{k},\binom{sk+k-1}{k}\}$. This problem has been central in extremal combinatorics, with substantial progress in the literature, including a complete solution for $k=3$ due to the first author. In this paper, we make progress towards the $4$-uniform case, proving the conjecture for $n\ge 5s$ and sufficiently large $n$, thereby taking a first step analogous to the $3$-uniform case. The main technical contribution is a stability result of independent interest. We further apply this stability to resolve two new instances of conjectures on the minimum $d$-degree threshold for matchings in $5$- and $6$-uniform hypergraphs, in a strengthened form.

Fluid phase equilibrium depends on the external constraints imposed on a system. In a closed system with fixed volume, depending on the average density, a vapor bubble may be stable, metastable, or unstable, with respect to the homogeneous liquid phase. In the case where the bubble is metastable, we study its lifetime, i.e. the average waiting time needed to observe bubble collapse, and the corresponding lifetime of the homogeneous liquid. For the smallest systems, we predict the possibility to observe phase flipping, when the fluid oscillates between states with and without bubble. We provide an example of phase flipping in a simulation of a Lennard-Jones fluid.

We compute the $\mathrm{MU}$-based syntomic cohomologies, mod $(p,v_1,\cdots,v_n)$, of all $\mathbb{E}_1$ $\mathrm{MU}$-algebra forms of the truncated Brown--Peterson spectrum $\mathrm{BP}\langle n\rangle$. As qualitative consequences, we resolve the Lichtenbaum--Quillen, telescope, and redshift questions for the algebraic K-theories of all $\mathbb{E}_{1}$ $\mathrm{MU}$-algebra forms of $\mathrm{BP} \langle n\rangle$. This extends work of the Hahn and Wilson. We also explicitly compute the algebraic K-theory of arbitrary $\mathbb{E}_{1}$ $\mathrm{MU}$-algebra forms of $\mathrm{BP}\langle 2\rangle$ at all primes $p\ge 5$ extending previous work of the author, Ausoni, Culver, Höning, and Rognes.A dditionally, we present a new computation of mod $(p, v_1, v_2, v_3)$ algebraic K-theory of arbitrary $\mathbb{E}_1$ $\mathrm{MU}$-algebra forms of $\mathrm{BP}\langle 3\rangle$ at all primes $p\ge 7$, the first explicit computation of algebraic K-theory of an $\mathbb{E}_1$-ring of height $3$.

We develop a framework, in the style of Adler, for interpreting the notion of "witnessing" that has appeared (usually as a variant of Kim's Lemma) in different areas of neostability theory as a binary relation between abstract independence relations. This involves extending the relativisations of Kim-independence and Conant-independence due to Mutchnik to arbitrary independence relations. After developing this framework, we show that several results from simplicity, $\text{NTP}_2$, $\text{NSOP}_1$, and beyond follow as instances of general theorems for abstract independence relations. In particular, we prove the equivalence between witnessing and symmetry and the implications from this notion to chain local character and the weak independence theorem, and recover some partial converses. Finally, we use this framework to prove a dichotomy between $\text{NSOP}_1$ and Kruckman and Ramsey's $\text{BTP}$ that applies to most known $\text{NSOP}_4$ examples in the literature.