Papers

We give a simplified proof of the polynomial improvements for integer 3SUM and all-pairs shortest paths (APSP) established by Alman and Vassilevska Williams, with weaker exponents. The central ingredient is selected-entry matrix multiplication: computing a prescribed set of entries of a product of dense rectangular matrices. Starting from Schönhage's inner/outer-product identity, we construct a multiplication rule that shares intermediate products among many outputs. We apply this rule recursively and bound the work by counting the requested entries at each level. Applied to incidence matrices, the algorithm solves Offline Set Disjointness over a small universe. Known deterministic reductions then yield truly subquadratic integer 3SUM and truly subcubic APSP for polynomially bounded integer inputs.

Generative policies offer expressive action distributions for online reinforcement learning (RL), but many diffusion- and flow-based policies require costly iterative sampling. We propose DriftSpread, a one-step generative policy based on Drifting for online maximum-entropy (MaxEnt) RL. To incorporate entropy without a tractable policy density, we build on an energy-based view of Drifting and derive critic-guided Drift and entropy-induced Spread from the negative MaxEnt objective. Our method learns these updates sequentially in the same actor, using action regression for value improvement and distribution matching to capture the effect of entropy. The actor updates require neither direct samples from the critic-defined Boltzmann target nor policy-density or score estimation. Under exact fitting with a fixed critic, we prove convergence to the Boltzmann target in the small-step limit followed by the long-time limit. On MuJoCo, DriftSpread achieves the highest aggregate normalized return among all evaluated methods, while outperforming the diffusion- and flow-based baselines in both training and inference efficiency.

Feedback presentation shapes how LLMs use evaluator judgments during translation refinement. We introduce a machine translation protocol that keeps the evaluator score, main issue, and recommendation fixed while varying feedback form and supporting detail. Across seven models and six translation directions, low-load feedback outperforms high-load feedback in all 42 translator-direction settings under xCOMET. On the 29,400-record rewriting subset, mean xCOMET scores multiplied by 100 are 83.30 for low-load feedback and 80.67 for high-load feedback; the full-matrix original-translation reference mean is 83.46. MetricX-24 supports the low-load advantage on a stratified subset. A no-feedback second pass yields a small improvement over the original translations, and direct evaluator prompting reproduces the low-load advantage without a separate rewriting stage. These results show that compact feedback limits the quality loss from evaluator-guided refinement in the evaluated high-resource MT settings.

Domain generalised semantic segmentation (DGSS) has recently benefited from combining multiple foundation models, such as vision-language and vision foundation models, to leverage their diverse representations for generalisation. Existing multi-model approaches share a common design choice: cross-model fusion is restricted to depth-aligned layers, implicitly assuming that mutually beneficial information resides at the same depth across models. However, foundation models pretrained under different objectives may develop distinct representation hierarchies, making the optimality of depth-aligned fusion questionable. Moreover, we observe that the foundation models exhibit different depth-wise domain sensitivity. Motivated by these observations, we propose a Sparse All-Layer Connector (SALC) that enables each layer to access information from preceding layers of another model. As the number of accessible layers grows with depth, dense aggregation may hurt generalisation. SALC therefore learns to adaptively select a sparse subset of informative layers. We further introduce a candidate dropout regularisation that strengthens sparsity and encourages SALC to explore diverse layers, leading to more robust selection. Across four foundation model combinations and three DGSS evaluation settings, the proposed design demonstrates stronger generalisation than conventional depth-aligned fusion. The code is available at https://github.com/RuoyuGuo/SALC.

Centrality measures rank nodes even when the observed differences can be explained by ordinary structural variation. We therefore study a preceding question: whether a network contains a statistically identifiable central structure at all. We formulate this question as a network-level hypothesis testing issue using the adjacency spectral radius as the test statistic and estimate an empirical p-value from randomized networks generated under a specified null model. The framework is evaluated using G(N,M), Chung-Lu, and two implementations of the configuration model on simulated benchmarks and 55 real networks. Controlled experiments further vary the number and connection weights of designated central nodes while holding network size and density fixed. The results show a clear negative association between central-structure strength and the empirical p-value. They also demonstrate that rejection decisions depend substantially on the structural properties preserved by the null model: the detected network sets are not simply nested across models.

Distributed Constraint Satisfaction Problems (DCSPs) model distributed decision-making in multi-agent systems, where agents assign values to local variables while satisfying shared constraints. However, many DCSP solvers still rely on broad, repeated peer-to-peer communication and are largely reactive, with limited support for strategic coordination or learning. This increases communication overhead, can lead to stagnation, and reduces the success of hard instances. Hence, we propose Dynamic Conflict Clustering-Incomplete (DCC-Incomplete), a repair-based solver that combines conflict-centred decomposition, structured coordination, strategic repair, and lightweight learning for stagnation recovery. DCC-Incomplete groups the unsolved part of the problem into conflict clusters and restricts communication via a coordination layer to agents within the same conflict region. It also introduces memory-guided restart, enabling agents and clusters to pause, revise, and adapt repairs based on neighbouring conflict states. A thorough evaluation of DCC-Incomplete on distributed graph-colouring benchmarks with planted satisfying assignments shows that its memory-guided restart improves the success rate from 37.50% to 61.11%, reduces median runtime from 0.565s to 0.367s, and cuts median communication from 12.493 to 6.503 messages per constraint. These results suggest that dynamic conflict clustering, combined with lightweight strategic and learning-guided repair, improves both communication efficiency and robustness in incomplete DCSP solving.

A subharmonic mixer topology is introduced that uses quarter-wave stubs to create virtual RF/LO grounds and provide impedance matching in close proximity to the antiparallel Schottky barrier diode pair, thereby reducing the electrical length and enabling broadband operation. The RF/LO matching topology is demonstrated at 600 GHz and 1200 GHz, with good agreement between measured and simulated performance. The 600-GHz mixer demonstrates approximately 20% RF bandwidth with a DSB receiver noise temperature below 1300 K across most of the measured band, while the 1200-GHz implementation achieves DSB receiver noise temperatures below 3000 K over a broad frequency range. Both mixers are characterized at room temperature with approximately 2 mW of LO power. The 600-GHz mixer is used in the Submillimeter Wave Instrument (SWI) onboard ESA's Jupiter Icy Moons Explorer. Overall, these results demonstrate state-of-the-art room-temperature performance for broadband receiver front ends targeting atmospheric research and remote-sensing applications.

We study the communication cost of exact sampling under Campbell's exponential coding criterion. Given a target distribution $P$ and a proposal distribution $Q$, the sender and receiver share i.i.d. $Q$-distributed samples, and the sender communicates an index so that the selected sample has distribution exactly $P$. We introduce a new information measure Rényi channel simulation divergence $R_\alpha(P\Vert Q)$, a Rényi generalization of the channel simulation divergence. We then show that it lower bounds the Campbell cost of every exact sampler and is achievable, via a new layered sampling scheme, within an additive constant depending only on the Campbell parameter. This yields a tighter one-shot characterization than previous bounds based on Rényi divergences of different orders. For i.i.d. product distributions, under suitable non-degeneracy and moment conditions, we further show that the gap beyond the leading Rényi-divergence term is $\frac{1}{2}\log n+O(1)$.

Stationary scattering admits a dynamical formulation in which the scattering data are extracted from the evolution operator for an effective non-stationary and non-unitary quantum system. This provides a mathematical method of solving scattering problems. We consider the possibility of realizing such a quantum system and relating the scattering information to outcomes of quantum measurements performed on it. To this end, we examine stationary scattering problems in one dimension where the scattering data consist of reflection and transmission amplitudes and the corresponding effective quantum system is a non-unitary two-level system. The states of this system identify points in a Bloch sphere $S^2$ and their evolution yields trajectories in $S^2$. Quantum measurements turn out to be capable of providing information about the reflection amplitudes. We identify configurations displaying reflectionlessness with situations where one or both of the poles of $S^2$ undergo cyclic evolutions. Spectral singularities and coherent perfect absorption correspond to dynamical spin flips which map one of the poles of $S^2$ to the other.

Hive is a boardless two-player strategy game in which the pieces form the playing surface. We prove that deciding whether White can force a win in single-Queen $n$-Hive, the generalization with exactly one Queen Bee per player, is EXPTIME-complete, even when all pieces are already placed. Our hardness proof reduces from unbounded two-player Constraint Logic: Grasshoppers encode edge orientations, while the One Hive rule enforces the vertex constraints. We obtain the EXPTIME upper bound by considering positions up to translation. The result answers the single-Queen question raised by Andel and Rin. Keeping the same inventory and movement rules, it also gives EXPTIME-completeness for multi-Queen variants. To the best of our knowledge, this is the first use of unbounded two-player Constraint Logic to prove EXPTIME-completeness for a natural problem.

For any $n\geq 1$, we construct a continuum of non-conjugate actions by homeomorphisms of $\operatorname{SL}_{n+1}(\mathbb{Q})$ on the $n$-sphere $S^{n}$ that do not extend to a continuous action of $\operatorname{SL}_{n+1}(\mathbb{R})$. We do this by adapting a blow-up construction of Bowden and Mann. We motivate studying these actions and note down some related observations.

A quantum-dot array prepared in a timed Dicke state exhibits directional radiation asymmetry, which has often been interpreted as evidence of reciprocity breaking through quantum-state preparation. We show that this interpretation is incorrect: reciprocity concerns transmit-receive response, not necessarily pattern symmetry. Moreover, the first-order coherence of any single-excitation state is exactly equivalent to the power pattern of a classical phased array driven by conjugate weights, making the observed asymmetry classically reproducible. The genuine quantum signature instead appears in second-order correlations, where a single-excitation state violates the classical lower bound on photon coincidences. We therefore propose a quantum-antenna certificate combining nonzero directional asymmetry with photon antibunching at the beam peak. We further derive a tight asymmetry bound, establish the correct mirror operation for planar arrays, correct the collective decay rate with dipole-dipole coupling, and derive a saturation law for direction-finding precision. All theoretical results are validated through independent numerical implementations.

Fourier neural operators predict power-system transient trajectories with high accuracy. We extend this line of work by examining the choice of spatial spectral basis, studying both the graph Fourier transform derived from the grid's own coupling network and learnable bases trained from that starting point and from a Fourier control. Evaluated in one controlled pipeline across three protocols spanning unseen fault locations and shifted operating regimes, the learnable bases improve on a parameter- and budget-matched DFT variant by 7-14% in distribution, with per-case significance on both in-distribution protocols and no loss on the third; the gain comes from adaptation rather than from the coordinates alone. We further characterize the regime dependence of this gain, identifying its cause, locus, and continuity, and show that a resolution requires no new model: averaging the learned and frozen variants gives the best shift-robust predictor, while their disagreement provides a label-free alarm that flags every shifted case with no false alarms.

High-order direct arbitrary Lagrangian–Eulerian discontinuous Galerkin (ALE-DG) methods can suffer severe order reduction under nonsmooth mesh motion despite satisfying the geometric conservation law (GCL). To address this limitation, we propose a new ALE-DG framework applicable to moving triangular and quadrilateral meshes. This framework successfully overcomes the uniform Lipschitz constraint inherent in existing direct ALE-DG methods, retaining high-order accuracy even when the mesh-velocity gradients are unbounded. This capability is driven by transport polynomial exactness (TPE($k$)), which is a structural property stronger than the GCL. TPE($k$) ensures the exact transport of polynomials of degree at most $k$ under arbitrary mesh motion. To incorporate TPE($k$) into a direct ALE-DG method, we introduce a coefficient-evolution (CE) formulation. Reconciling strict local conservation, an indispensable requirement for accurate shock capturing, with the preservation of the TPE(\(k\)) property constitutes the principal technical challenge. We resolve this by evolving the coefficients using the third-order strong-stability-preserving Runge–Kutta method, paired with a conservative correction executed via Simpson's rule integration. Crucially, this correction strictly vanishes for exactly transported quadratic polynomials, thereby ensuring that our third-order scheme rigorously satisfies the TPE($2$) property. Numerical experiments demonstrate that the proposed scheme retains third-order accuracy when the discrete mesh-velocity Lipschitz estimate grows as $\mathcal O(h^{-1})$, whereas a representative existing direct ALE-DG scheme exhibits order reduction under the same mesh motion. Tests with discontinuous solutions further demonstrate robust and high-resolution performance under random and Lagrangian mesh motions.

A goodness-of-fit test tells whether a logistic risk model fits, but the analyst also needs to know what is wrong: an overall risk that is too high is repaired by updating the intercept, risks that are too extreme by recalibration, a curved relation between score and risk by a flexible recalibration, and misfit within groups of patients only by revising the model. We split any misfit into these four parts, which are orthogonal and match the calibration hierarchy. Each part gets a group of score tests that sees that part only, and the failing parts are named by closed testing. In external validation, in large samples and for small departures, a part is wrongly named with probability at most the chosen level, and the highest part named points to the lightest update that removes the misfit. Checking a model on its development data never sees an error in the overall risk or the calibration slope, and with binary or skewed covariates it can mistake misfit within patient groups for a curved link. External validation sees all four parts, for any prediction model. In simulations it kept its error rate when the model was right and for small departures, and usually pointed to the right update. On intensive-care data its verdicts for one model differed in-sample, on similar patients and after transport.

It follows from the compactness of the connectedness locus that the locus of PCF polynomials in the moduli space of polynomials of degree d (at least 2) over the complex number is a countable union of zero-dimensional subvarieties. Over fields of characteristic p the analogous claim is certainly false, so it is natural to ask how large the components of the PCF locus can be. We show that the components of the PCF locus in the moduli space of polynomials of degree d over an algebraically closed field of characteristic p>0 have dimension at most d/p and that this bound is sharp, in the strong sense that for every p and d there are infinitely many distinct PCF components whose dimension is the greatest integer no greater than this bound (except in the case $p=d=2$).

This article aims to design a distributed economic dispatch (ED) scheme based on marginal cost (MC) consensus for a battery energy storage system (BESS) network considering the utility grid switching. Due to the low control accuracy and slow convergence speed of previous schemes, some important changes have been made in this article. Firstly, the feedback coupling term is introduced into the closed-loop error to further improve the stability and convergence rate of the controller. Then, a proportional integral (PI) controller with a reset mechanism is implemented in the designed scheme, which improves the speed and control accuracy of MC consensus. The performance of a control system with a reset mechanism is well guaranteed by analyzing the eigenvalues of the corresponding matrix. Finally, some tests focus on UG switching, time phased pricing, load switching, a BESS/agent plug and play, and large-scale system applications to illustrate the effectiveness and progressiveness of the design scheme.

Let $\mathbf{T}=(T_1,\ldots,T_n)$ be a commuting tuple of bounded linear operators on a separable Hilbert space $\mathcal{K}$, and let $\{a_\alpha\}_{\alpha\in\mathbb{Z}_+^n}$ be strictly positive weights. Under mild assumptions on the weights, we characterize the weighted frame vectors and weighted Parseval frame vectors, that is, those vectors $\xi$ for which $\{a_\alpha \mathbf{T}^\alpha\xi\}_{\alpha\in\mathbb{Z}_+^n}$ is a frame or a Parseval frame. We reduce the problem to characterizing surjective module maps from the reproducing analytic Hilbert module associated with the weights onto the module $\mathcal{K}$, equipped with the module structure defined by $\mathbf{T}$. It is shown that weighted Parseval frame vectors differ by a unimodular scalar whenever they exist. If the kernel defined by the weights has the complete Nevanlinna–Pick property, then any two associated weighted frame vectors are similar whenever they exist. We also focus on the case of the Hardy kernel on the polydisc, that is, the case $a_\alpha=1$ for all $\alpha\in\mathbb{Z}^n_+$. This is related to the famous Douglas–Foias problem: whether two similar quotient modules of the Hardy spaces $H^2(\mathbb{D}^n)~(n>1)$ are equal. Our result shows that the answer is negative. This leads to the conclusion that frame vectors for representations of $\mathbb{Z}_+^n$ $(n>1)$ may fail to be similar. It means that, for the case $a_\alpha=1$, there exists a tuple $\mathbf{T}=(T_1,\dots,T_n)$ with $n>1$ on $\mathcal{K}$ and two distinct frame vectors for $\mathbf{T}$ which are not similar.

Surgical world models must turn prospective robot actions into visually faithful, structurally consistent instrument motion. We introduce SurgCast, which predicts future instrument skeletons from current structure, robot state, and future actions, then uses the forecast to control video generation. Dual-path injection combines skeleton geometry with derived, instrument-specific motion descriptors through geometric residuals and conditioning modulation. Both paths complement native action conditioning and are retained in a four-step causal student. On SutureBot and SRTH-Porcine-Cholei, the teacher and student improve all seven reported quality and instrument-consistency metrics over their respective direct-action baselines; teacher FVD decreases by 37.6% and 67.3%. Ablations show that future actions improve skeleton prediction and that combining both injection paths performs best across all six metrics under predicted skeletons. These results support using predicted articulated structure and its motion semantics as a control interface for surgical video prediction. Project videos and examples are available at https://surgcast.netlify.app/.

This paper explores topological phase transitions in the momentum space of the $\kappa$-deformed $\phi^4$ lattice model within the mean-field approximation. We show that the double-well structure of the momentum space gives rise to degenerate momentum states at high energies and that this change is marked by a phase transition. We investigate this transition using both numerical and analytical methods, with the analytical approach based primarily on singularity theory. Our numerical analysis shows that the energy at which the topological phase transition occurs is related to the noncommutative parameter by the simple power law $E \sim 1/\alpha^{4}$. By mapping the constant-energy-level surface to the $A_{3}$ singularity, we obtain the same result analytically. We also present results unique to noncommutative systems, showing that the fraction of particles in degenerate momentum states has a nonzero commutative limit. As a nonperturbative effect, this result provides a testable prediction for the existence of noncommutative spacetime.

Let $p = 4q + 1$ be a prime integer, where $q$ is an odd prime. Let $L = \mathbb{Q}(\zeta_p)^+$ denote the maximal totally real subfield of the $p$-th cyclotomic field $\mathbb{Q}(\zeta_p)$. In this paper, we prove that if $U_L^+=U_L^2$ (i.e., every totally positive unit is a square in $L$), then for any non-trivial subfield $K \subseteq L$ and any non-trivial totally positive unit $\varepsilon \in U_K^+$, the absolute trace satisfies the sharp lower bound $\Tr_{K/\mathbb{Q}}(\varepsilon) \ge p - 4$. As an application, we show that a spherical fusion category $\C$ with global dimension $\dim(\mathcal{C})=p=4q+1$ is pointed, where $2$ is a primitive root modulo $q$, or $p=4q+1<2^{22}$ and $p\neq 29$.

Under a limited context budget, a long-term agent must recall not merely memories that are individually relevant, but evidence that is collectively sufficient for the current query. Conventional memory access retrieves, scores, and truncates memories, often spending multiple context slots on redundant facts while leaving complementary evidence behind. We cast the read phase of long-term memory as query-conditioned evidence composition: each memory is valued by what it adds beyond the evidence already selected. We instantiate this operation with MemPursuit, a training-free mechanism that uses the residual of a sparse convex query reconstruction as a representation-space proxy for what the current memory set is missing. By reusing the query and candidate embeddings already available from retrieval, MemPursuit progressively composes nested evidence contexts without invoking a language model or cross-encoder to score each query–memory pair. The effectiveness of our method is verified on 3 popular benchmarks in comparison to strong baselines. In fixed-budget LoCoMo experiments, MemPursuit improves end-to-end accuracy by 7.2 percentage points at Top-10 and 5.5 percentage points at Top-20, with consistent gains across multiple memory frameworks, while achieving a 32.9$\times$ speedup in median reranking latency over fully batched BGE-v2-m3. Our implementation is available at https://github.com/FengQi-HITSZ/MemPursuit.

This paper investigates how bistatic delay information improves the observability and localization performance of 3D moving targets in distributed Wi-Fi sensing. Motivated by emerging IEEE 802.11bf WLAN sensing, we consider one transmitter, multiple distributed receivers, and a single target with unknown position and velocity. We compare Doppler-only sensing with delay-Doppler sensing, in which delay-capable receivers provide additional bistatic path-length information for target localization. We derive the Fisher information matrix (FIM) and the position equivalent FIM (EFIM) for both sensing formulations. The resulting observability analysis shows that Doppler-only localization generically requires six receivers, whereas delay-Doppler localization can become locally identifiable with three receivers under non-degenerate geometry. We further study mixed receiver architectures in which only a subset of receivers provides delay estimates. Simulations with Wi-Fi-like bandwidths confirm the transition and quantify the roles of bandwidth, Doppler noise, receiver geometry, and delay-capable receiver availability.

Small language models (SLMs) used as on-device agents must balance accuracy and latency as their context grows. Mixed precision helps manage this tradeoff by assigning different precisions to weight tensors according to their size and sensitivity. However, these choices depend on the operating conditions, so a configuration selected for a short context may no longer provide the best tradeoff at a longer one. We calibrate a timing model from measurements at different precisions and context lengths, and combine it with tensor sensitivities to choose configurations under a latency or accuracy budget. We argue that fewer bits do not always mean faster execution, so our predictor estimates latency from calibrated execution costs rather than weight bytes alone. Under a latency budget, reading the larger key-value (KV) cache leaves less time for weight operations, leading the allocator to select lower precisions. This gives a schedule of tensor changes that can be prepared before execution. Our runtime follows the schedule while retaining the session's KV cache. We evaluate executed schedules on a laptop GPU, including a longer-context schedule with three weight formats and a tool-use benchmark measuring full response time and correctness. Under an accuracy budget, measured error from a quantized KV cache determines the budget left for weights. The allocator can raise or lower weight precision as this estimate changes. These experiments show where context-aware precision helps and where accuracy recovery remains limited.

Long-horizon robotic manipulation is inherently composed of semantically meaningful sub-tasks. While Movement Primitives (MPs) can model these components, their composition typically relies on manual engineering or sample-inefficient reinforcement learning. Existing supervised methods are limited by the lack of MP labels in standard datasets and often fail to generalize across discrete and rhythmic movements. We propose Neural Dynamic Composition (NDC), a supervised framework to compose Dynamic Movement Primitives (DMPs) using only raw trajectories for supervision, given a single example of each primitive. Our approach exploits parallel combinations of DMPs as a differentiable relaxation of categorical primitive selection, and utilizes a warp-invariant loss formulation to identify primitives despite spatial or temporal variations. A staged training curriculum is employed to learn the selection policy and DMP parameters, unifying discrete and rhythmic movements in one framework. Evaluated on 2D handwriting and 3D robotic tasks, our method outperforms baselines on most tasks and metrics. An ablation study demonstrates how each design choice contributes to the system's performance.

Knowledge graph (KG) foundation models (KGFMs) predict links on unseen graphs zero-shot. KGFMs have been pioneered by ULTRA which builds a relation graph from co-occurrences that match motif templates, and passes messages over it to obtain relation representations. More recent KGFMs such as MOTIF and TRIX increase the size of this relation graph, and FLOCK replaces message-passing with probabilistic random-walk ensembles. However, larger relation graphs mean more messages, and more walks mean more sequences to encode, so they improve performance at higher training and inference cost. This cost hinders their use in more complex reasoning tasks, such as complex query answering. To reverse this trend, we design MERIT, the first KGFM with a transformer relation encoder: every relation name attends to every other, and the motifs enter as a soft prior rather than as edges. Over 56 benchmarks, MERIT matches the zero-shot MRR of SoTA KGFMs while being 6x faster than TRIX and over 100x faster than FLOCK at inference. Fine-tuned for complex query answering like UltraQuery, MERITQuery improves over it at matched cost.

We prove that every holomorphic isometric embedding, with respect to the Bergman metrics, from a complex unit ball of maximal dimension into irreducible classical bounded symmetric domain of rank at least two that is not biholomorphic to a Type IV domain is equivalent to Mok's embedding. This completely resolves a problem of Mok.

We investigate finite time vacuum energy relaxation in conformally coupled scalar-tensor cosmology distinguishing the suppression accumulated along a physical trajectory from its asymptotic rate. For a canonical scalar with an exponential potential and constant conformal coupling, the autonomous system exhibits a stability exchange between scalar dominated and vacuum kinetic scaling solutions. In the power-law metric $f(R)=f_0R^n$ realization with $n>2$, the stable scalar dominated branch has a sustained relaxation rate below $0.579$ decades per Einstein frame e-fold. We additionally derive a finite time upper bound that accounts for depletion of the vacuum relative to the scalar potential. For the specified reference initial state $\Omega_{\Lambda,i}=0.99$ and $\Omega_{{\rm kin},i}=10^{-4}$, the bound excludes $60$ decades of suppression within $60$ e-folds throughout the $n>2$ family: $D_\Lambda(60)<47.7$. Numerical trajectories illustrate why this bound is conservative and why the transient contribution cannot be inferred from the final attractor alone. For the reference $f(R)\propto R^{2.14}$ solution, $D_\Lambda(60)\simeq7.53$ decades including a transient excess of about $3.27$ decades relative to uniform evolution at the asymptotic rate. The relaxed quantity is the dimensionless vacuum to gravitational scale hierarchy, its evolution also entails variation of the conformal frame Planck mass. The bounds are conditional on the homogeneous model and specified initial state, not a demonstration of complete cosmological viability.

Let $C_n^k$ denote the $k$th power of the cycle $C_n$. The equivalence covering number $\operatorname{eq}(G)$, studied by Alon, is the minimum number of equivalence subgraphs needed to cover all edges of $G$. King conjectured that $\operatorname{eq}(C_n^k)=\Omega(k)$ for sufficiently large $n$ relative to $k$. We study $\operatorname{eq}(C_n^k)$ and prove that, for every $n \geq 2k + 2$, we have $1 + \lceil\log_2(k+1)\rceil \leq \operatorname{eq}(C_n^k) \leq 1 + 2\lceil\log_2(k+1)\rceil$. We further show that, whenever $n\ge(k+1)\max\{41,5\lceil\log_2(k+1)\rceil\}$, \begin{equation*} \operatorname{eq}(C_n^k)\in\left\{1+\lceil\log_2(k+1)\rceil,\,2+\lceil\log_2(k+1)\rceil\right\}. \end{equation*} Moreover, for every integer $n\ge \max\{2k+2,k^2(k+2)\}$, the following exact values hold in the indicated cases: \begin{equation*} \operatorname{eq}(C_n^k)=\begin{cases} 1+\lceil\log_2(k+1)\rceil, & k+1\neq2^{\lceil\log_2(k+1)\rceil} \text{ or }(k+1)\mid n,\\[3pt] 2+\lceil\log_2(k+1)\rceil, & k+1=2^{\lceil\log_2(k+1)\rceil} \text{ and }2\nmid n. \end{cases} \end{equation*}

Sycophancy in large language models risks amplifying unsupported claims or beliefs across the knowledge ecosystem as these systems increasingly contribute to research synthesis, evaluation, and decision-making. A reliable model should revise its judgement when a user identifies a genuine flaw, but should resist pushback based on fallacies, emotion, or irrelevant appeals to authority. While sycophancy is usually measured on questions with a verifiable answer, the evaluative tasks in which it is most difficult to detect have no such ground truth. We introduce Big-Big-SycoBench, which measures sycophancy through argument discernment: the ability to revise an evaluation after valid criticism while resisting invalid pressure. The benchmark consists of 450 text artefacts with 5,400 arguments for raising or lowering their quality scores, each reviewed by one or more of twelve human annotators. An LLM scores each artefact twenty times to establish its baseline score and uncertainty. The model is then given a valid or invalid argument and asked to update its score. We only count a revision if it exceeds the model's own baseline variability. We use the resulting score delta to distinguish responsiveness to valid arguments from compliance with invalid ones. Across ten models, responsiveness is 88-95% for seven, 76-77% for Kimi K3 and GLM-5.3, and 32% for Opus 5.5. Compliance ranges from 0.2% for GPT-6 Astra to 95% for GPT-4o. Big-Big-SycoBench is intended to guide the development of models whose judgements can be corrected by sound arguments, but not displaced by unsupported pressure.