Papers

Clinical decision-support outputs can lack an au- ditable link between patient observations, encoded knowledge, conclusions, and recommendations. We present the CKG Clinical Explanation Engine, a downstream layer for a frozen, training-free clinical knowledge-graph reasoner that converts patient inference states and disease knowledge into typed facts, explicit rule-application traces, provenance-linked conclusions, and policy-licensed recommendations. The design separates measurement availability, representation completeness, and disease-specific activation; consequently, observed zero-activation evience is not treated as missing and partial representation is distinct from unobserved evidence. Optional language generation is restricted to symbolically licensed content. Across five usable workbooks (6,720 patients; 20,160 patient-disease traces; 1,021,440 feature-evidence rows), IG-range validity and knowledge provenance were 100%, numerical cross-sheet fidelity was 100% (120,960/120,960), and exported logical/report trace completeness was 100% (20,160/20,160). Availability representation consistency was 99.7028% (1,018,404/1,021,440); all 3,036 disagreements were confined to three systematic feature-cohort patterns. The corpus contained 86,783 observed zero-activation and 139,949 observed partially represented instances. A separate seeded 25-patient end-to-end audit completed without execution failure and passed all pre-specified trace, licensing, provenance, and state-consistency checks. These results establish structural and implementation auditability, not clinical correctness or utility.

An important question in learning in games is whether players can learn to achieve efficient outcomes, those that maximize the sum of their utilities, without communication or coordination. Completely uncoupled learning algorithms, which use only each player's own past actions and utilities, can guarantee this in broad classes of games, but they typically require every player to follow the same algorithm. In this work, we ask whether one can design welfare-maximizing learning algorithms that are robust to a deviant player. To study this question, we take a well-studied welfare-maximizing learning algorithm as a testbed and consider what happens when every player except one follows it. First, we consider a deviant player who either (i) always plays the same action or (ii) selects their action uniformly at random. We show that such a player can significantly alter the long-run behavior of this algorithm, leading to states that are far from welfare-maximizing, and that no completely uncoupled learning algorithm is robust to such a player in every game. Next, we consider a strategic deviant player who acts in their own interest and show that this player can adopt a different learning algorithm to shift the long-run outcome in their favor. Moreover, we show that under any completely uncoupled welfare-maximizing learning algorithm, there is a game in which some player benefits from switching to a different completely uncoupled learning algorithm. These results show that completely uncoupled welfare-maximizing learning is fragile, even to a single deviant player, suggesting that robustness may require communication or coordination.

We study the worst-case efficiency of the random-offerer mechanism in bilateral trade relative to the first-best gains from trade. We construct a family of independent buyer value and seller cost distributions whose ratio converges to $0.436943488\ldots$, improving the previous upper bound of $0.460242308\ldots$. Our construction combines recursively nested seller distributions with downward transport of buyer values. The recursive structure allows contributions from different scales to accumulate, leading to stronger hard instances. We analyze this construction through a recurrence and obtain an analytic characterization of the limiting constant.

A nonlinear observer is designed for a one-dimensional quasilinear diffusion equation with Neumann boundary conditions, using as an output the state at one end of the spatial domain. The observer consists of a copy of the system, with a nonlinear (in the output) correction term at the boundary. A Lyapunov-based analysis is performed to ensure global exponential stability of the zero equilibrium for the estimation error dynamics in the $L^1$ norm. The result holds for a general nonlinear diffusivity, under the assumption that it is lower-bounded by a positive constant, i.e., that the diffusion equation is uniformly parabolic.

We present Empirical Variational Autoencoder, a general generative framework for continuous-valued (i.e., non-vector-quantized) sequences. EVA is based on the evidence lower bound of the Variational Autoencoder (VAE) but learns autoregressive latent priors empirically from training data, which can be implemented only by an additional single linear layer on top of VAEs. By replacing the conventional standard-Gaussian constraint with the self-predicted priors, EVA significantly alleviates the latent distribution gap between prior and posterior which is typically observed in conventional VAEs, and leads to high-fidelity ancestral sampling for sequential data generation. Extensive experiments on image and sound synthesis demonstrate that EVA achieves competitive generation quality with autoregressive diffusion baselines despite its much faster inference time.

We prove finite-time gradient blow-up for the unforced incompressible porous media equation on the two-dimensional torus, removing the forcing used in the IPM blow-up constructions of Alpöge, Buckmaster, and Coiculescu \cite{lit:ABC} and Córdoba and Martínez-Zoroa \cite{lit:CMIPM}. The key is an identity that expresses the logarithmic growth of a layer's gradient as the negative pressure curvature transverse to its transported phase. A nearly vertical parent layer rotates and amplifies a new phase, with a gain exponential in the reciprocal parent tilt. An asymmetric periodic profile gives a uniform upper bound for the pressure Hessian, preventing rapid decay of the gradient magnitude. Each layer is seeded at time zero; its phase means are selected by a terminal displacement problem, and a finite order expansion is corrected to an exact unforced solution. The scale induction then shows blow-up of two derivative components at the origin and convergence to a terminal density in every $C^\eta$, $\eta<1$.

We prove that if the unit sphere of a separable infinite-dimensional real Banach space embeds uniformly into a Hilbert space, then the Banach space has Kasparov–Yu's Property (H).

Low-Rank Adaptation (LoRA) has become a standard approach for parameter-efficient fine-tuning, yet a fundamental practical question remains unresolved: how should the adapter rank be chosen? An overly small rank may lead to a poorly conditioned optimization landscape, whereas an unnecessarily large rank sacrifices the efficiency that motivates LoRA in the first place. Existing theoretical analyses provide only limited guidance on this trade-off, and their guarantees are typically established under restrictive theoretical settings. We address this gap by developing a substantially sharper landscape theory for LoRA, building on modern results from nonconvex low-rank matrix sensing. Our central insight is that the appropriate adapter rank should depend on the quality of the data-induced optimization geometry, rather than on the model alone. To formalize this connection, we introduce LoRA-RIP, a data-dependent restricted-isometry metric that characterizes the conditioning of the cross-entropy (CE) objective along LoRA-relevant low-rank directions. We prove that sufficient rank over-parameterization, with the required rank explicitly determined by the LoRA-RIP constant, eliminates spurious local minima, thereby extending existing RIP-based guarantees beyond the classical 1/3 regime. This characterization further enables principled data selection under a fixed rank budget. Experiments across language and vision tasks support these theoretical predictions, showing that rank and data quality are two coupled resources that should be jointly considered for more efficient and reliable LoRA fine-tuning.

I present a short approach to discretizing continuous unitary evolution. It gives a correspondence in both directions between continuous unitary evolutions and products of unitaries in finite dimensions, and I use it to reduce discrete adiabatic theorems to continuous ones. At fine steps, holding truncated samples of a regulated Hamiltonian on a suitable finite partition gives an approximating product. Conversely, rounding the clock identifies a product of $T$ unit-step samples with a continuous evolution, up to two half-step phases. I bound its operator-norm distance from the original evolution by $O(1/T)$ through cancellation within each time step. This assumes bounded first and second derivatives, bounded spectral width, and eigenvalue differences bounded away from nonzero multiples of $2\pi$, uniformly in $T$. For a unitary walk with these derivative bounds, a fixed common free spectral arc supplies a Hermitian logarithm path satisfying the comparison hypotheses. Continuous simulation guarantees then transfer to the phase-corrected product, and continuous adiabatic bounds to the product itself, with an explicit additive $O(1/T)$ error and each adiabatic theorem's hypotheses retained. For positive definite quantum linear systems of condition number $\kappa$, this comparison and continuous adiabatic theory give a rescaled qubitized walk with $O(\kappa/\delta)$ steps for leakage $\delta$, matching the scaling previously obtained through a discrete adiabatic theorem. These products need no discrete adiabatic theorem.

Spatio-temporal graph models typically encode each snapshot with a GNN and then connect the resulting representations through a temporal module. This space-then-time design is effective, yet it does not explicitly represent how a pattern moves across the graph. We show empirically that, for a propagating process, the same present field can lead to different futures when its recent rate of change differs, motivating an explicit representation of motion in the predictive state. We introduce WaveGSSM, a second-order graph state-space model that maintains two coupled latent states at each node, one for the current pattern and one for its temporal rate of change. A graph-wave transition updates the motion state through graph interactions and uses it to advance the pattern state, coupling spatial propagation and temporal evolution within a single rollout. We evaluate WaveGSSM on four temporal-graph benchmarks and global weather forecasting. It consistently achieves the best mean performance across the temporal-graph benchmarks and reduces the geopotential RMSE by 20.2% on average for 1- to 5-day weather forecasts relative to a backbone-matched snapshot model, while better preserving large-scale atmospheric patterns.

In this paper, we present a preference-adaptive receding-horizon control framework for autonomous driving that accounts for passenger preferences and motion-sickness susceptibility. We formulate a finite-horizon optimal control problem with adaptive weights for speed, acceleration comfort, and motion sickness, while maintaining a fixed weight for collision risk. We predict motion sickness online using an individualized model on the Motion Illness Symptoms Classifi- cation (MISC) scale and update the preference weights offline from emotion-derived pairwise comparisons using Bayesian inference. A deterministic safety supervisor checks the planned trajectory and modifies the control command when necessary. We evaluate the framework using three simulated passenger profiles under three motion-sickness susceptibility levels. The learned weights yield distinct closed-loop behaviors, and the mean evaluation emotion score improves in seven of nine scenarios. No collisions occur in the full-method experiments, while the safety supervisor intervenes in 1.621% of the learning frames.

We study the problem of finding the closest singular hypersurface to a given nonsingular one from algebro-geometric point of view. The set of singular hypersurfaces $\Xi$ of a given degree is a projective variety dual to the Veronese variety. However, as was shown by Raffalli [Raf14], the Bombieri-Weyl (also known as apolar) distance from a general real hypersurface to $\Xi$ could be minimized at a singular point of $\Xi$. We study the singular locus of $\Xi$ and its irreducible component, consisting of hypersurfaces with cuspidal singularities. This variety is projectively dual to the tangential variety of the Veronese variety. We compute its Euclidean Distance Degree using topological tools and deduce a formula for the number of critical points of the distance function to $\Xi$ from a general hypersurface. We also initiate the study of the analogous problem for higher osculating varieties to the Veronese variety. Furthermore, we establish a new extremal property of Chebyshev polynomials (conjectured in [Raf14]) by characterizing real-rooted binary forms that maximize the distance to $\Xi$.

In this paper we prove the existence, uniqueness and regularity of classical solutions \break of a semilinear parabolic equation with a mixed local and nonlocal diffusion operator \break $\mathcal{L} = -(-\Delta)^s + \Delta$ and Dirichlet boundary conditions. Here, $(-\Delta)^s$ is the integral fractional laplacian and $\Delta$ is the classic local laplacian. We then study the interior regularity of said solutions and conclude that they are Hölder continuous in both space and time, and they are $C^{2,\alpha}_{loc}$ in space for all positive times.

Consider the linear regression of a response $Y$ on explanatory variables $X_1,\dots,X_p$ with a positive definite joint correlation matrix. Let $\mathbf{R}$ be the correlation matrix of the explanatory variables and $\mathbf{r}$ the vector of their correlations with $Y$. The ellipsoid $E=\{\mathbf{x} : \mathbf{x}^{\top}\mathbf{R}^{-1}\mathbf{x}=1\}$ is inscribed in the cube $[-1,1]^p$ and touches the face $x_j=1$ at $T_j$, the $j$th column of $\mathbf{R}$. The point $P=\mathbf{r}$ lies inside $E$, and three kinds of regression quantities are read off this diagram by elementary geometric constructions. The multiple correlation is $|OP|/|OP'|$, where $O$ is the origin and $P'$ is the point where the ray from $O$ through $P$ meets $E$ (for $P\ne O$). The standardized regression coefficients are the coordinates of $P$ in the basis $T_1,\dots,T_p$. The partial correlation between $X_i$ and $Y$ given the other explanatory variables is the position of $P$ on the chord of $E$ through $P$ parallel to the $x_i$-axis, on a linear scale from $-1$ to $1$ increasing with $x_i$; furthermore, the parallel chord through the origin has half-length $1/\sqrt{\mathrm{VIF}_i}$. Examples illustrate sign reversal and standardized coefficients outside the interval $[-1,1]$.

Magic-state distillation is a major resource cost in fault-tolerant quantum computing. The cost of a magic-state factory depends strongly on its failure rate, which grows with the number of input magic states. Although symmetry-restricted methods have recently made distance two searches tractable, distance three and above have remained elusive at moderate input counts. We develop symmetry- and AI-assisted methods to search this regime. We present a unified binary-matrix formulation encompassing both triorthogonal-code and direct circuit searches. We show that distance at least three is equivalent to nonzero, pairwise distinct syndromes, separating the choice of syndromes from the search for compatible output gates. We restrict the syndrome search using group symmetry and language-model agents, followed by deterministic solving and independent verification. Our searches yield 699 factory classes, including 564 new ones. These include factories for pure-T states and factories with entangled outputs comprising combinations of T, CS, and CCZ magic states. The pure-T factories [[63, 11, 3]] and [[850, 128, 6]] achieve the lowest overhead exponents we know among protocols with at most 100 and 1000 inputs, respectively, with $\gamma = 1.589$ and $\gamma = 1.057$. Our [[1715, 287, 6]] factory, with $\gamma = 0.998$, is the smallest known pure-T factory with $\gamma < 1$. We also provide a context directory of search briefs and campaign notes with which readers can train their own agents and tailor the search to their requirements. With these results, we begin constructing an active, open-source repository of magic-state distillation protocols for the quantum community, supplemented by our methods and data, for the practical fault-tolerant quantum computing regime.

We present a generalization of the Mermin-Peres magic square game to $N$ players, in which the square is replaced by an $N$-dimensional hypercube. We show that the winning probability of the optimal classical strategy remains $8/9$ for all $N$, while for $N \ge 3$, a generalization of the quantum strategy for $N = 2$ achieves a winning probability between $1/3$ and $3/4$. This loss of quantum advantage for $N \ge 3$ is a consequence of monogamy.

We give a closed-form finite-dimensional construction of quantum channels with real algebraic entries exhibiting arbitrarily large violations of minimum-output Rényi entropy additivity uniformly above any prescribed positive Rényi order, using a different finite realization based on direct products of free groups and fixed-field high-girth graphs. More precisely, for every $p_0>0$ and $g>0$, we construct one channel and one Bell input for which the additivity gap is greater than $g$ simultaneously for all $p\in[p_0,\infty]$. With $r=\lceil2g\rceil$, the construction uses $15r$ output qubits and $O(r2^{30r}+r^2(1+p_0^{-1}))$ input qubits. Compared with the stated parameters of Shou and Gorshkov \cite{ShouGorshkov2026}, our construction uses fewer output qubits for every target gap and has a smaller asymptotic input-qubit bound as the gap grows with the positive cutoff fixed. At the von Neumann order, a separate specialization gives an explicit channel with gap greater than $1/1024$ using $37608898$ input qubits and $8$ output qubits.

Peer feedback promotes active learning, critical reflection, and skill development, but its effectiveness is often limited by the quality of feedback students provide. Recent advances in LLMs offer new opportunities to support peer feedback by generating more coherent and actionable feedback while preserving human oversight. This paper presents AICoFe, an AI-based collaborative feedback system designed to support teacher, peer, and self-assessment in higher education. AICoFe integrates rubric-based evaluations, GenAI-supported feedback, Learning Analytics dashboards, and video recordings to foster reflective learning. The system combines quantitative scores and qualitative observations to generate structured feedback focused on strengths, areas for improvement, and actionable recommendations, which teachers can review and curate. An evaluation with 65 undergraduate and master's students shows high satisfaction with the coherence and usefulness of the feedback, as well as the excellent usability, indicating that AICoFe effectively supports peer feedback in authentic educational settings.

Let all pairwise distances of an arbitrary configuration of points in $\mathbb{R}^{\ell}$ be known exactly. We prove that every second-order stationary point of the complete-graph squared-stress objective over configurations in $\mathbb{R}^{k}$ has zero residual whenever $k > \ell$. The result holds for every number of points, without a genericity assumption, and allows repeated points and degenerate ground-truth configurations. It resolves the one-extra-dimension conjecture for complete, uniformly weighted data. The proof combines positivity of the residual stress, polar normalization of the candidate configuration, a row-wise rank-nullity argument, and propagation of zero curvature through the full Hessian. The theorem, including the passage from derivatives of the original quartic objective to its matrix formulation, has been formalized in Lean 4. The accompanying verification records include a kernel check with zero trust level and a transitive axiom audit.

LZ77 coders spend the same number of bits on the distance of every match, although short matches rarely pay off at long distances. With match-length-dependent windows, used in several practical codecs, a match of length $\ell$ may refer back at most $W(\ell)$ symbols: short matches get short distance fields, while long matches keep the full window $W$. This breaks the suffix closure that classical LZ77 analyses rely on: the tail of an admissible match need not be admissible. We bound the cost of this loss, and of matches missed by a practical hash search, by the phrase count of a single-window reference parse. Consequently, when the full window starts at length $O(\log W)$, greedy parsing keeps the convergence rates and the optimal finite-input rates of LZ77 under substring-complexity constraints, and when it starts at length $o(\log W)$, greedy parsing is universal for stationary ergodic sources. We then delimit what the windows gain. A calibrated schedule never increases the minimum code length and saves a polynomially small rate on some iid sources. The leading redundancy term is set by the length code instead: on uniform iid data, with logarithmic-cost lengths and a fixed full-window threshold, it is at least of order $(\log\log W)/\log W$ whatever windows the short matches use, and a short codeword for a growing longest match reduces it to $O((\log\log\log W)/\log W)$. We also quantify the gains of coding single-symbol runs by count and of Huffman coding the token fields, and obtain a consistent entropy-rate estimator by selecting the window.

We study a general class of fully non-linear elliptic Hessian equations on compact Riemannian manifolds. In particular, we replace the standard assumption that the equation is a symmetric function of the eigenvalues of the Hessian made by Caffarelli, Nirenberg and Spruck by a weaker one. After deriving a priori estimates for solutions under suitable assumptions on the equation, we solve it with a continuity method.

We show that the familiar Cherenkov burst from a finite superluminal charge track can be understood as a pair of interfering electromagnetic memory transitions. Endpoint accelerations of the track produce opposite memory steps in the transverse vector potential, such that the corresponding finite-frequency radiation is universally fixed, below the endpoint-acceleration-scale, by the memory amplitude and the time for which the temporary memory offset persists. As the Cherenkov direction is approached, the memory amplitudes grow while their separation in observer time shrinks, until the two vector-potential steps overlap and their divergent contributions cancel into a finite Cherenkov burst. This perspective opens the door to using the Askaryan effect as a setting in which individual macroscopic memory transitions may be resolved, providing a new experimental arena for memory-linked infrared physics.

Given a finite poset $\mathcal P$, its induced saturation number, $\text{sat}^*(n,\mathcal P)$ is the smallest size a family of subsets of $[n]$ can have such that it does not contain an induced copy of $\mathcal P$, but adding any other set to it creates one. The saturation numbers already show a sharp dichotomy – Freschi, Piga, Sharifzadeh and Treglown showed that for any given poset its saturation number is either bounded or at least $2\sqrt n$. The dominant conjecture is that in fact, the saturation numbers are either bounded, or exactly linear. In this paper we show that this dichotomy is true for all posets on at most 4 points. Most of these posets, most notably the butterfly and the diamond, had individually long resisted analysis, and the missing piece which this paper tackles is the poset comprised of a $\mathcal V$ and an isolated point, denoted by $\widehat{\mathcal V}$, for which we show linear saturation. The architecture of the proof uncovers the structure of an arbitrary $\widehat{\mathcal V}$- saturated family, namely an antichain with at most two chains above it, and an upside-down forest below it. Moreover, we anticipate that, at least partially, this structure is relevant to the wider class of posets that contain an isolated point.

Photon loss makes Gaussian boson sampling easier to simulate classically. A standard approach replaces the lossy input by classical light, and it stays accurate while the mean number of surviving photons grows no faster than the square root of the number $N$ of squeezed inputs at fixed squeezing and accuracy. We prove that this scale is a limit of replacement by classical light. For explicit interferometers, no state with a nonnegative Glauber–Sudarshan $P$ representation, Gaussian or not and however correlated, keeps the error in the photon count distribution small beyond this scale. We then surpass this limit by approximating the detection instead of the input. The sampler evolves the state exactly between detections and, after each detection, replaces the conditional state by a Gaussian with the same covariance. For every interferometer its total variation error is $O(N\eta^3)$ at polynomial expected cost, where $\eta$ is the photon survival probability. The mean surviving photon number can therefore grow as $N^{2/3}$ at fixed accuracy. A separate pair construction reaches $N^{3/4}$ when every input mode is equally squeezed.

Rotational transitions of SiC$_6$ with $j=26$–$40$ have been detected in IRC+10216, but collisional rate coefficients for SiC$_6$ with He or H$_2$ were previously unavailable. We present rate coefficients for SiC$_6$ rotational excitation by He, calculated using a two-dimensional potential energy surface (PES) at the RCCSD(T)-F12 level under the rigid-rotor approximation. Close-coupling (CC) calculations were performed for low rotational levels, while the coupled-states (CS) method was used for higher levels. Convergence tests were conducted, and CC and CS scattering cross sections were compared. The CS rate coefficients cover rotational levels up to $j=40$ and temperatures of 3–60\,K. Although CC and CS rate coefficients differ by up to a factor of $\sim 2$ for some transitions, particularly those affected by weak coupling and resonances, they yield nearly identical excitation and brightness temperatures for the high-$j$ transitions examined, including $19\rightarrow18$. These results support the reliability of the CS method for non-LTE modelling of the high-$j$ transitions considered, for which CC calculations are computationally impractical. We use reduced-mass scaling of the SiC$_6$–He rate coefficients to estimate those for SiC$_6$–H$_2$ and model the observed transitions. A $\chi^2$ analysis of the observed line fluxes, excluding heavily blended features, yields a best-fitting of $n(\mathrm{H_2})=4.33\times10^5$\,cm$^{-3}$ and $N(\mathrm{SiC_6})=8.11\times10^{11}$\,cm$^{-2}$. The latter agrees excellently with the previous observational estimate of $(7.9\pm1.5)\times10^{11}$\,cm$^{-2}$, supporting the consistency of our non-LTE modelling with observational constraints.

Magnetic reconnection is a fundamental mechanism underlying the heating and a wide variety of eruptive phenomena in the solar atmosphere; however, characterizing its 3D aspects remains challenging. Our aim is to understand how 3D magnetic reconnection in unipolar regions drives localized heating and plasma ejections. To this end, we used a self-consistent 3D radiation-MHD simulation of a coronal bright point (CBP) performed with the Bifrost code. We focused on localized, intense heating events and upflows near a unipolar CBP footpoint associated with its parasitic polarity. 3D reconnection diagnostics were carried out using the parallel electric field and its integral along magnetic field lines. We then compared the resulting upflows with spectroscopic observations from Hinode/EIS. We find recurrent episodes of 3D magnetic reconnection without nulls at the footpoints of loops rooted near the CBP's parasitic polarity. A detailed analysis of one representative event places the reconnection region at heights z=[1.0,2.5] Mm, within a predominantly vertical magnetic field with localized concentrations of electric current. The dissipation of these currents efficiently heats the plasma to about 2 MK and produces strong gas-pressure gradients that drive upflows of a few tens of km/s. Blueshifts observed with Hinode/EIS near the parasitic polarity of an observed CBP are consistent with this scenario. The recurrence of these reconnection events are essential for heating CBP loops to million-degree temperatures. This form of 3D reconnection may operate more generally in the solar atmosphere, dissipating magnetic stress around strong flux concentrations and contributing to heating in apparently unipolar regions. The resulting pressure-driven upflows offer an indirect spectroscopic diagnostic of localized reconnection heating, which future observations with MUSE and Solar-C will be able to test.

In this work, we present the Simpson-Visser black-bounce (SV-BB) spacetime with both electric and magnetic charges (the dyonic SV-BB) as an exact solution of general relativity coupled to Maxwell electrodynamics and an anisotropic fluid with vanishing radial pressure. We assume the anisotropic fluid represents an Einstein cluster, which consists of a dense system of particles moving in circular orbits that generate only tangential pressure and zero radial pressure. We show that the electromagnetic SV-BB metric, which provides a regular black hole spacetime, can be obtained as an exact solution to the Einstein field equations sourced by a combination of an Einstein cluster (which is responsible for providing the exotic matter required to keep a black-bounce throat open) and an electromagnetic field in the framework of Maxwell electrodynamics.

A model under pushback can correct itself, capitulate, or hold, and one flip rate counts a correction and a capitulation alike. Using SycoLens, a modular replay protocol, we test how user pressure and evaluation settings shape measured flip rates. Each measurement is one stateless replay of an item, a committed answer, and one scripted user line in a fixed form. Every effect is read against a matched control with the line deleted. Pushback wording, committed text, answer format, boundary distance, and ground truth become factors of one instrument; earlier instruments vary one to three of them. Across eleven frontier models from three providers and about 760,000 controlled replays, which models look sycophantic depends on how the user pushes back. Lines that assert the opposite verdict and lines that challenge the answer without asserting one rank the models almost unrelatedly. Flip effects grow several-fold near a model's boundary, yet items answered identically in every screening draw still carry about half of the most-affected totals. On arithmetic tasks where the truth is known, one model re-derives and corrects itself under pressure while another abandons correct answers without written work. On the model tested, a planted derivation lowers release of the answer it argues for, true or wrong, where a bare stated value does not; the wrong answer is corrected much more often than the true one is abandoned. Under a yes/no readout the rankings come closer, entangled with a pressure-induced shift toward "no". One score per model therefore compares different behaviours across models and benchmarks. We condense these dependencies into a reporting profile; the instrument, records, and analyses will be released upon publication.

We show that the two modular curves of level $49$ and genus $9$ left open by the work of Rouse, Sutherland and Zureick-Brown, and of Furio and Lombardo, have no non-CM rational points. Together with the theorem of Furio and Lombardo on $X_{ns}^{+}(49)$, this completes the classification of the $7$-adic images of Galois of non-CM elliptic curves over $Q$. The proof is a Kummer descent on the superelliptic equations $F(x,y) = k\,w^{7}$ of Furio and Lombardo, in the cases $7 \mid k$ that they left open. The covering curves are twists of the Fermat septic, they map to twists of the Klein quartic, and a $7$-adic computation shows that the twist attached to a solution is trivial. The rational points of the Klein quartic were determined by Hurwitz, who reduced the question to Fermat's Last Theorem for exponent $7$, proved by Lamé. Thus, the last open case of the $7$-adic part of Mazur's Program B rests on Fermat's Last Theorem for exponent $7$.

Neurons of the brain exhibit a rich repertoire of electrophysiology dynamics with the same repeated stimulus eliciting very different voltage responses from the same cell. One common approach in biophysically detailed models is to capture this variability through ensembles of deterministic parametrizations, at a cost of hundreds of thousands of CPU hours. Existing machine learning surrogates inherit the same limitation, where a stimulus is mapped to a single voltage response. We address this challenge by learning a conditional generative model for single-neuron electrophysiology, using flow matching with a velocity field conditioned on the input current. On biophysically detailed models of two human cortical interneuron types, the generated responses closely reproduce the electrophysiological feature distributions, spike-time structure, and excitability profiles, even matching the experimental recordings from the corresponding human cortical neurons. Near the firing threshold, firing and non-firing responses coexist at the same stimulus amplitude, and at high amplitudes, ensembles may split into low- and high-firing modes near depolarization block. We show that our model recovers both modes in each case, while a neural operator baseline suppresses spiking near threshold and blurs the gap between modes at depolarization block.