Papers

We theoretically investigate the spatial response of chiral Andreev edge states at a quantum Hall-superconductor interface probed by scanning gate microscopy. Using the Bogoliubov-de Gennes formalism, we demonstrate that the localized potential of the scanning gate tip decouples the chiral Andreev edge states from the superconductor into independent electron and hole paths. In the $\nu=2$ regime, this local separation forces the quasiparticles to accumulate a relative magnetic phase in the normal region, resulting in Aharonov-Bohm-type conductance oscillations that are captured by a geometric analytical model. At higher filling factors (e.g., $\nu=4$), the tip progressively interacts with spatially distinct edge channels, dynamically altering the Andreev-induced mode mixing and modifying the transport resonances. Furthermore, we show that these highly localized interferometric signatures survive in the presence of disorder, even when conventional non-local transport oscillations are masked.

We prove zero density estimates for $L$-functions of cuspidal automorphic representations $\pi$ of $\mathrm{GL}_2(\mathbb{A}_{\mathbb{Q}})$. We show that $N_\pi(\sigma, T) \ll T^{\frac{5}{2}(1 - \sigma) + o(1)}$, where $N_\pi(\sigma, T)$ denotes the number of zeros $\rho = \beta + i\gamma$ of $L(s,\pi)$ with $\beta \geq \sigma$ and $|\gamma| \leq T$. The key input is an extension of the Guth$\unicode{x2013}$Maynard argument to Dirichlet polynomials whose coefficients satisfy a weaker $\ell^q$ bound.

We solve Nikolski's norm-controlled inversion problem for measure algebras of locally compact Abelian groups. For every $\delta>1/2$ there is a constant $C_M(\delta)$, independent of the group $G$, such that $\|\mu\|_{M(G)}\le1$ and $\inf_{\gamma\in\widehat G}|\widehat\mu(\gamma)|\ge\delta$ imply that $\mu$ is invertible in $M(G)$ and $\|\mu^{-1}\|_{M(G)}\le C_M(\delta)$. Together with Nikolski's negative results, this shows that $1/2$ is the sharp universal threshold. The main step is a uniform inversion theorem for Fourier algebras $A(K)$ of compact Abelian groups; its proof is based on a profile decomposition and a resolvent argument. We also obtain norm-controlled inversion for Toeplitz operators with non-vanishing Wiener symbols of winding number zero and give direct endpoint constructions showing failure of norm control at $\delta=1/2$ in $A(K)$ for every infinite compact Abelian group $K$, and in the unitization of $L^1(G)$ for every nondiscrete locally compact Abelian group $G$.

In this paper, we introduce the class of weakly localized operators on the weighted Bergman spaces induced by two-sided doubling weights. We first establish the characterization of the compactness of the operator belonging to the norm closure of weakly localized operators in terms of the boundary vanishing of the Berezin transform. Our approach here is different from the methods in the literature since the “translation" operators are not served us a direct applicable tool in our setting. We further characterize the compactness of Toeplitz operators with bounded symbols in the two-weight setting, a boundedness criterion of a two-weight Bergman projection is also presented.

Localizing an autonomous underwater vehicle without pre-deployed seabed transponders, or direct access to onboard vehicle sensors remains a core challenge. We present a receiver-passive 3-D localization and spatial mapping system utilizing a single floating surface buoy equipped with a hydrophone array and an inertial measurement unit (IMU). The central difficulty is that surface wave motion induces six-degree-of-freedom (6-DOF) perturbations that rotate the array between snapshots, degrading conventional subspace processing. We resolve this by introducing a fixed-point iterative MUltiple SIgnal Classification algorithm (FP-MUSIC) that uses IMU measurements to de-warp snapshot covariances prior to direction-of-arrival estimation. Furthermore, we employ a subspace-projected wideband matched filter to resolve beacon ranges and use power asymmetry for independent front-back identification. Evaluations across simulated sea states demonstrate that FP-MUSIC substantially reduces localization error relative to uncompensated methods and sustains robust 3-D tracking and vehicle orientation estimation under wave-induced motion. At moderate sea state, FP-MUSIC increases the 2-m beacon-separation accuracy from approximately 45% to 75%.

We establish a Donsker-Prokhorov invariance principle in some Besov spaces. Specifically, we show that polygonal line processes associated with partial sum processes of a triangular array of row-wise independent random variables converge in distribution to Brownian motion, extending earlier results in the literature.

Pathology vision-language foundation models vary in performance across diseases and tasks, with no single model consistently performing best. The high cost of expert pathology annotation can also limit the labeled data available for task-specific adaptation. Combining complementary pretrained representations is a potential approach to these limitations, yet learning an effective fusion from few labeled examples remains challenging. We introduce Few-shot Fusion Foundation Models of Computational Pathology (FFM-CP), which is a framework that combines multiple pathology vision-language models in the few-shot learning setting. The framework first aligns heterogeneous representations using a closed-form Orthogonal Procrustes transformation estimated from corresponding support images. This alignment preserves within-model feature geometry without training an additional alignment network. Within the aligned space, a unified graph enables information exchange across backbones by jointly refining support-image features and visual and textual class prototypes. These refined representations support complementary text-prototype and case-retrieval branches that capture semantic class knowledge and within-class visual variation, respectively. Each branch learns to combine predictions from all ordered backbone pairs, allowing queries encoded by one model to draw on evidence represented by another. We evaluate three backbone combinations on six histopathology datasets at 4, 8, and 16 shots per class. FFM-CP achieves higher mean macro-F1 than the strongest individually adapted member of each fused set in 50 of 54 comparisons. These findings suggest that combining complementary pretrained representations can improve histopathological classification when annotations are limited.

Artificial intelligence workloads increasingly demand computing architectures combining high throughput, energy efficiency, and physical scalability. Here we present a signed incoherent optical matrix multiplier on thin-film lithium niobate that natively supports signed inputs and weights. We demonstrate closed-loop in situ backpropagation on a 16 X 16 photonic core, including forward computation, nonlinear operations, error propagation, and gradient computation. Measured physical outputs directly participate in optimization, thereby incorporating actual device responses and nonidealities into training. The core achieves 8-bit multiplication and 10-bit accumulation precision across 256 channels, maintains 10-bit accuracy in tiled 256 X 256 matrix computation, and further executes Transformer linear operations in a BERT-mini workload. This work presents a fundamental, scalable building block that addresses key limitations in practical optical computing, thereby opening avenues for large-scale, high-efficiency photonic neuromorphic systems.

We introduce an explicit deterministic collection of $N$ spherical points, $\cP_N\subset\mathbb S^2$, that we call the deterministic Diamond points. For every $0<\alpha<2$ we prove that there exists a constant $C_\alpha>0$ such that \[ 0\leq \frac{2^{\alpha+1}}{\alpha+2}N^2-\sum_{x,y\in\cP_N}|x-y|^\alpha \le C_\alpha N^{1-\alpha/2}. \] This result shows that $\cP_N\subset\mathbb S^2$ has a Riesz energy deficit of optimal order in this range of the parameter. In particular, for $\alpha=1$, Stolarsky's invariance principle implies that $\cP_N\subset\mathbb S^2$ has $L^2$ spherical cap discrepancy of asymptotically optimal order \[ D_{L^2}^C(\mathcal P_N)\asymp N^{-3/4}, \] making it, to our knowledge, the first explicit configuration proved to have this property. More generally, our result implies that the point set $\cP_N$ has Sobolev $H^{s}(\Sph)$ worst-case error of optimal order for all $1<s<2$.

We study a Kirchhoff-type nonlinear integro-differential equation in two spatial dimensions whose coefficients are allowed to depend on time, and we construct conservative discretizations in time for the associated initial–boundary value problem. We consider two symmetric three-layer schemes of Crank–Nicolson type, a locally linear one and a genuinely nonlinear one, and we show that each of them preserves a discrete analogue of the total mechanical energy of the homogeneous problem with constant coefficients. For the nonlinear scheme we establish uniform apriori bounds on the discrete solution and on its discrete velocity by working directly with the discrete energies, without invoking a nonlinear discrete Grönwall inequality, the constants still grow exponentially in the final time, and we prove local second-order convergence in time, both for the solution and for the central-difference approximation of its first time derivative. The nonlinear system arising at each time level is solved by a fixed-point iteration: given iterates at the two preceding levels that satisfy the apriori bounds, that equation has exactly one solution and the iteration converges to it at a geometric rate once the time step is small enough. Since the apriori bounds are index-local, alternating them with that one-step solver constructs the trajectory stepwise, for time-dependent coefficients as well, on the local interval on which those bounds hold. Numerical experiments, carried out in a setting in which the spatial discretization contributes no error, exhibit the conservation of the discrete invariants, confirm the second order in time and verify the geometric convergence of the fixed-point iteration.

Continuous Vision (CV) systems underpin real-time applications such as autonomous driving and augmented reality, where latency, throughput, and energy are tightly constrained on mobile platforms. Modern CV SoC pipelines, however, still serialize image capture and processing, leading to high end-to-end latency. Prior work reduces this latency by predicting future frames and running pixel-domain backend inference speculatively, but incorrect predictions force re-execution on real frames, increasing energy and complexity. We present MVP, a motion-predictive speculative vision pipeline that operates entirely in the motion domain. Instead of forecasting full images, MVP predicts future motion vectors and uses them to extrapolate perception results from previously processed frames before the next frame arrives. A lightweight hardware extension in the Image Signal Processor (ISP) reuses existing motion-estimation logic to predict motion with minimal area and energy cost. MVP introduces a scheduling model that treats motion extrapolation as the default path, while full backend inference runs periodically in the background for drift correction off the critical path. It also supports optional frontend scaling, allowing the system to reduce sensor sampling under low or predictable motion to save energy. We evaluate MVP on object detection, demonstrating up to 66.8% reduction in tail latency and 46% energy savings, at a small accuracy cost.

We develop a multiscale entropy framework based on the wavelet conditional renormalization group that decomposes the total Shannon entropy into contributions from different spatial scales. Generative modeling enables accurate estimation of these entropies from configurations or snapshots, even in nonequilibrium systems whose underlying probability distributions are unknown. Applications to the equilibrium $\varphi^4$ model and nonequilibrium active Model B+ show that multiscale entropies reveal criticality and spatial organization obscured in the total entropy. These entropies serve as scale-dependent order parameters for characterizing phase transitions and pattern formation.

Finite-shot optimization can make noisy variational quantum eigensolver energies look artificially accurate. We studied six hardware-tractable molecular active spaces on two IBM Heron r3 processors. Seven optimizer-selected values fell below the exact active-space eigenvalue by up to 17.36 mHa. The cell-balanced fraction of the source-observed best-final advantage not retained on later fixed-parameter measurement was 0.989 (approximate 95% session-level interval [0.906, 1.072]); alternative aggregations span 0.892-0.989. Thus little of the selected advantage remained on average. Measurements occurred 5.6-38.4 days later, so parameter-dependent drift remains a competing explanation. Best-observed hardware VQE values require prompt independent confirmation before supporting accuracy claims.

The fixed-angle conjecture for the quantum approximate optimisation algorithm (QAOA) is shown to be false for depth-2 QAOA on 9-regular graphs, and proven to be correct for depth-1 on any regular graph, as well as for any depth on all 2-regular graphs.

A camera and an IMU are the minimal sensor setup for metric localization and dense mapping, yet classical visual–inertial filters must wait for parallax before they start and then retain only sparse landmarks. Feed-forward geometry models, in contrast, predict dense structure from a few images but provide neither metric scale nor gravity. We present DAVIO, which uses a single multi-view depth model, Depth Anything~3, for both start-up and mapping. At start-up, a five-image window and preintegrated IMU measurements form a feature-free linear system. Its robust, conditioning-checked solution bootstraps a VIO filter through buffered replay. During tracking, the filter's metric poses condition the depth model. Residual scale is corrected only along viewing rays, which preserves the metric camera baselines, and a gravity-preserving submap graph with drift-gated revisits refines the map. On EuRoC, DAVIO starts markedly earlier, reduces the localization error, and maps more accurately than SOTA feed-forward mappers given identical poses. On building-scale ORI sequences, DAVIO is on bar or better than SOTA mappers on the same odometry, and degrades far less when GT poses are replaced by real odometry. We release the code of DAVIO, a real-time dense metric SLAM system, to the community.

Using the idea of Venkataramana's construction of infinite families of arithmetic orthogonal hypergeometric groups, we construct some of the very first examples of infinite families of arithmetic symplectic hypergeometric groups that do NOT satisfy the arithmeticity criterion of Singh and Venkataramana. For example, we show that the hypergeometric groups associated to the pairs of polynomials $(x-1)^4P_m(x^{9})$ and $(x^4+x^3+2x^2+x+1)Q_m(x^{9})$, where $P_m$ and $Q_m$ are integral polynomials of degree $2m$ such that the corresponding pair determines a Zariski dense symplectic hypergeometric group, are arithmetic in $\operatorname{Sp}(18m+4)$ for any integer $m\in\mathbb{N}$.

In this paper, we prove that every connected sesqui-regular graph with parameters $(n,k,c)$, where $c\geq3$, and with smallest eigenvalue in $[-3,-2)$ is $1$-integrable if its valency is sufficiently large. The bound on $c$ is optimal: for every integer $t\geq2$, the Cartesian product of the Shrikhande graph and $K_t$ is a connected sesqui-regular graph with parameters $(16t,t+5,2)$ and smallest eigenvalue $-3$, but it is not $1$-integrable.

In this paper, we prove that on a smooth complex projective curve of positive genus, the Deligne–Hitchin twistor spaces for the structure groups $\mathrm{GL}(n,\mathbb{C})\ (n\geq1)$ and $\mathrm{SL}(n,\mathbb{C})\ (n\geq2)$ admit no algebraization by a complex scheme locally of finite type. When the genus is at least two, the same holds for their stable loci and hence for the corresponding Hitchin twistor spaces. Nevertheless, the Deligne–Hitchin twistor spaces admit $\mathbb{R}_{\mathrm{alg}}$-definable complex analytic structures when the rank or the genus is one.

Power swings in large Data Centers (DTCs) running Artificial Intelligence (AI) workloads can excite poorly damped modes in power systems. The resulting forced oscillations can lead to flicker, equipment disconnection, or blackouts. This paper investigates the risks of such load fluctuations for different system strengths and damping conditions. Using an analytical approach based on transfer functions, we identify critical DTC locations in the power grid at which load fluctuations could induce the largest forced oscillations and further characterize the harmonic spectrum of the resulting system response. Depending on the frequency and magnitude of the DTC load fluctuations, forced oscillations can become unbounded. The underlying instabilities are classified into saddle-node bifurcations of the forced periodic response and impasse-surface encounters, using Floquet multipliers and the minimum singular value of the algebraic Jacobian. Furthermore, the impact of different duty cycles and harmonic components beyond the fundamental oscillation frequency in square-wave load profiles is analyzed. Finally, the interaction of two oscillating DTCs is investigated for different locations and forcing frequencies, considering both synchronized and unsynchronized operation. The findings can help system operators to define new regulations on the maximum load fluctuations permitted for DTC facilities at specific grid locations, without negatively affecting the stability and operation of the system.

We evaluate the Hellinger-distance coherence of the two-flavor neutrino state subject to Lindblad damping of the mass-eigenstate interference term, and obtain a closed expression in terms of the flavor-basis density-matrix elements. In the two-flavor vacuum treatment with negligible wave-packet separation, the undamped state is pure, and the smallest eigenvalue of the damped state grows linearly with $\epsilon=1-\kappa$, where $\kappa=e^{-\Gamma L}$. An expansion about the pure state then yields a nonanalytic $\sqrt{\epsilon}$ correction to the Hellinger coherence, with coefficient $K=2\sqrt{2}\,u\sin2\theta/\sqrt{1-2u}$, whereas the flavor-transition probability, the $l_1$ coherence, the concurrence, the entanglement of formation and the local quantum Fisher information all respond linearly in $\epsilon$. Using representative Daya Bay, KamLAND and MINOS working points together with a current $90\%$ C.L. bound on energy-independent damping, we find fractional changes in the Hellinger coherence of $0.18\%$, $9.40\%$ and $21.51\%$, against $0.0009\%$, $0.43\%$ and $1.81\%$ for the concurrence. The comparison concerns the response of the quantifiers to the damping parameter and carries no implication about experimental resolution. The same mechanism operates, with a different coefficient, for a correlated dephasing channel acting on the flavor coherence, where the non-Markovian regime produces damped revivals.

The SYNTAX score is an established tool for assessing coronary artery disease and guiding revascularization treatment decisions. However, its manual estimation from coronary angiography videos by clinical experts is time-consuming and subject to inter-reader variability. While machine learning has shown promise in automating this process, prior work has primarily focused on lesion detection, characterization, or binary disease classification, leaving direct SYNTAX score prediction relatively unexplored. We propose SynSeq, a video-based method for direct SYNTAX score prediction. It combines targeted preprocessing with a tailored training strategy using a zero-inflation-aware loss and linear target scaling. Evaluated on the public CardioSyntax dataset, SynSeq significantly outperforms previous state-of-the-art methods, improving $R^2$ by 0.55, reducing prediction bias by 93.1% and achieving more consistent performance across annotations from three independent expert graders. In addition, SynSeq achieves a weighted $F_1$-score of 0.80 for revascularization treatment recommendations, slightly below inter-expert agreement. These results demonstrate the potential of SynSeq to provide consistent, automated SYNTAX score assessment and reliable decision support for coronary revascularization planning.

Perceiving objects in the environment is a fundamental capability of autonomous robots. In dark or low-light environments, external cameras often fail to reliably perceive object positions and geometry; when visual sensing is unavailable, completing target search, recognition, and grasping through touch alone becomes a key robot manipulation capability. This task must simultaneously address container-scale spatial exploration and object-scale fine-grained geometric perception, which is particularly challenging for low-degree-of-freedom parallel grippers. However, a unified framework remains lacking for connecting container-scale spatial exploration with object-scale fine-grained geometric perception and grasping. To address this challenge, we present GLoTouch, a global-to-local haptic perception and manipulation framework built on a parallel gripper. In the global stage, the gripper holds a passive long-reach probe, combining force measurements with known tool geometry to localize contacts and actively estimate candidate-object positions, coarse contours, and heights. In the local stage, the robot sets down the probe and uses the bilateral visuotactile sensors on the same gripper to directly acquire local haptic observations, which are matched against a given target 3-D model without object-specific training. We evaluate the framework in both simulation and real-robot experiments. Source code will be open-sourced.

Let $0 \xrightarrow{}I \xrightarrow{ } A \xrightarrow{ }A/I \xrightarrow{} 0$ be a short exact sequence of C*-algebras. In this paper, we prove that if $\dim({\rm Cu}(I))\leq n$ and $\dim({\rm Cu}(A/I))\leq m$, then $\dim({\rm Cu}(A))\leq n+m+1$, this result gives an affirmative answer to a problem concerning the Cu-semigroup of C*-algebras raised by Thiel and Vilalta. We further show that certain comparison and divisibility properties satisfied by $I$ and $A/I$ are inherited by the extension algebra $A$. As an application, we provide an alternative proof of equivalence that $A$ is pure if and only if $I$ and $A/I$, a result originally proven by Perera, Thiel, and Vilalta.

We prove that, for real Banach spaces of dimension greater than one, each of the identities $\|Id+T^2\|=1+\|T^2\|$ and $\|Id-T^2\|=1+\|T^2\|$, required for every rank-one operator $T$, characterizes the Daugavet property, thereby answering a question posed by Kadets, Martín, and Merí (2007). Consequently, every extremely non-complex Banach space of dimension greater than one has the Daugavet property, answering a question of Martín and Merí (2011). We also compare pointwise versions of the square Daugavet properties with Daugavet points, $\Delta$-points, and $\nabla$-points.

We establish a Hamiltonian-to-pancyclic analogue of Bondy's theorem for graph systems under an aggregate degree condition. Let $\G=(G_1,\ldots,G_n)$ be a graph system on a common $n$-vertex set $V$, and write $\delta(v)=\min_{i\in[n]}d_{G_i}(v)$. If $\G$ contains a rainbow Hamilton cycle and \[ \sum_{v\in V}\delta(v)\ge \left\lceil\frac{n^2}{2}\right\rceil-1, \] then $\G$ is rainbow pancyclic, unless $n$ is even and every member is the same balanced complete bipartite graph. For even $n$ the threshold is exact at the integer level. Unlike the usual transversal Dirac- or Ore-type hypotheses, our condition is not layerwise: the member attaining $\delta(v)$ may depend on $v$, and some vertices may have $\delta(v)<n/2$. Relative to a fixed rainbow Hamilton cycle, we count shortcuts whose colors are released by the Hamilton arcs they replace. A missing cycle length forces complementary shortcut supports to cross-intersect. A counting gap settles even shortening, while equality or near equality in odd shortening yields a distance-two exchange whose orbits force the balanced bipartite obstruction. At the lower integer threshold an exact defect identity shows that only one or two units of slack are available. \noindentKeywords: graph system; rainbow cycle; pancyclicity; Hamilton cycle; extremal graph theory.

We propose a nonparametric test for unobserved treatment effect heterogeneity in regression discontinuity designs. Under the null of no unobserved heterogeneity, a transformed outcome that imputes treated potential outcomes for untreated units must have a continuous conditional distribution at the cutoff. We convert this implication into an integrated conditional-moment restriction using characteristic functions, thereby allowing the conditional local average treatment effect to be an unrestricted function of covariates. We derive the asymptotic distribution of the test statistics via a $U$-process and establish the validity of a multiplier bootstrap procedure for calculating critical values. Monte Carlo experiments show well-controlled size and increasing power. Two empirical applications illustrate how the test distinguishes between heterogeneity explained by observables and that explained by unobserved factors.

Researchers in industry and academia use synthetic survey respondents powered by large language models as substitutes for human samples. These synthetic populations require validation against real-world data, so researchers often address them using ad hoc comparisons with human surveys. Inspired by the intention-behaviour gap in behavioural science, we argue that these validations test the wrong thing for most applied cases where decision makers commission synthetic research to anticipate consequential behaviour. To address this problem, we propose a validation framework with two requirements. First, every validity claim must state its level of correspondence with human data: does the sample predict what the represented people do, which of four diagnostics (location, dispersion, response process and structure) does the validation address, and does the validation compare against experimental effects? Second, researchers must report validity claims for subgroups, since these groups are often the most affected by consequential decisions and aggregate accuracy hides their misrepresentation. Our validation framework operationalises three justice dimensions (distributional, procedural, and recognition) as measurable quantities and treats within-persona counterfactual experiments as a design that itself requires validation. We then apply the framework to electric vehicle charging tariffs, before closing with a reporting checklist that researchers can use to make convincing validity claims.

We study the convergence rate of the heat-bath Glauber dynamics for the Discrete Gaussian Free Field on arbitrary connected finite graphs. We show that, when starting from super-harmonic initial conditions, the evolution enjoys a strong form of monotonicity. This allows us to get a sharp mixing profile as the size of the graphs diverges. More precisely, we show that mixing occurs at time $\frac{1}{2\lambda}\log(\mathcal{E})$ with window $\mathcal{O}(1/\lambda)$, where $\lambda$ is the spectral gap of the graph Laplacian and $\mathcal{E}$ is the energy of the super-harmonic initial condition. This result holds for arbitrary graphs that do not exhibit extreme connectivity properties (one way or the other). In particular, it holds for finite boxes of the grid $\mathbb{Z}^d$, in dimension $d\geq 3$.

Visual document retrieval (VDR) systems depend on page embeddings computed before deployment, which makes adaptation difficult when encoder parameters or corpus re-encoding are unavailable. Rerankers provide useful relevance signals, but conventional reranking applies them only to selected queries and candidate pages. We introduce Q-REACT, a query-side test-time adaptation method that converts limited reranker feedback into reusable retrieval improvements. Q-REACT learns a shared low-rank transformation that produces query-dependent residuals, combines adapted query scores with document-level context, and distills reranker preferences with a student distribution normalized over the complete task-specific page index. This design lets unscored pages compete through cached embeddings while keeping the encoders and page index fixed. Across eight ViDoRe V3 tasks and five open-weight and proprietary backbones, Q-REACT improves average retrieval over evaluated baselines at sparse and full-coverage budgets, transfers to held-out queries and tasks, and adds little inference overhead. The results show that finite reranker feedback can be amortized across a query collection without retraining or rebuilding the retriever.

We consider the central Hardy–Littlewood maximal operator in $\mathbb{R}^n$. For every integer $n\geq1$, we construct two bounded, nonnegative functions $f,g\in L^1(\mathbb{R}^n)$ that differ on a set of positive measure and satisfy $M_cf(x)=M_cg(x)$ at every point $x\in\mathbb{R}^n$. Thus the central maximal operator is not injective on the nonnegative cone of $L^1(\mathbb{R}^n)$. The functions can have compact support; when $n\geq2$, they can also be chosen radial, either with compact support or strictly positive throughout the space. The construction makes a small redistribution of mass invisible to the maximal operator by excluding averaging spheres that cross its support. We also give an explicit formula, in terms of the incomplete beta function, for the $n$-dimensional volume of the intersection of two balls with arbitrary radii and centers, and an elementary formula for the central maximal function of a ball in $\mathbb{R}^3$.