Let $G$ be a finite group acting on a finite set $X$. This group action splits $X$ into disjoint orbits. The Burnside process is a Markov chain on $X$ which has a uniform stationary distribution when the chain is projected to orbits. We initiate the study of the Burnside process on Catalan structures. We consider two special cases: the first where the state space is the set of parking functions of length $n$ and $G = S_n$ is the symmetric group on $[n]$, such that $G$ acts by permuting coordinates, and the second where the state space is the set of labeled Dyck paths of length $2n$ and $G = S_n$ acts by permuting labels. The resulting Burnside processes give novel algorithms for sampling, respectively, an increasing parking function and a Dyck path approximately uniformly at random. Our main result shows that both processes are rapidly mixing, with mixing times upper bounded by $O(n \log n)$. As an application, we show how our Burnside process can be used to sample triangulations of an $(n+2)$-gon approximately uniformly at random.

This is the first part of a series of papers proving the nonlinear stability of a one-parameter family of continuously self-similar $C^{1,α}$ naked singularity solutions, with $0<α\ll1$, to the spherically symmetric Einstein-scalar field equations. The stability holds for initial perturbations lying in a small open neighborhood of the data generating these naked singularity solutions, measured in a localized Hölder topology. These continuously self-similar naked singularity spacetimes were previously constructed by Christodoulou [D. Christodoulou, Examples of naked singularity formation in the gravitational collapse of a scalar field, Ann. of Math. 140 (1994), 607--653], who also proved their instability to black hole formation under sufficiently rough perturbations [D. Christodoulou, The instability of naked singularities in the gravitational collapse of a scalar field, Ann. of Math. 149 (1999), 183--217], thereby verifying weak cosmic censorship within a rough functional framework. In complete contrast, in this paper, we obtain the first nonlinear stability of these naked singularity spacetimes under general perturbations of the same regularity as the background. We rely on the linearized stability result established in the companion paper [J. Singh and W. Zheng, Nonlinear stability of continuously self-similar naked singularities for the Einstein--scalar field equations II: linearized stability]. Our result underscores the decisive role of the functional framework in formulating the Weak Cosmic Censorship conjecture.

Dependence among marginally constrained observations can break a finite-sample barrier. To formalize this phenomenon, we introduce the \emph{minimum list entropy coupling} $H(P\|Q_1,\dots,Q_m)$, the minimum conditional entropy $H(X|Y_1,\dots,Y_m)$ over all joint distributions with prescribed discrete marginals $X\sim P$ and $Y_i\sim Q_i$. Unlike classical formulations based on independent observations, our model allows $Y_1,\dots,Y_m$ to be arbitrarily dependent while keeping each marginal fixed. This enlarged coupling space reveals a sharp dichotomy: independent observations reduce residual uncertainty exponentially, whereas dependent observations can eliminate it exactly after finitely many samples. We characterize this zero-entropy regime through necessary and sufficient conditions and give concrete structural criteria under which it occurs. In particular, under mild support assumptions, zero entropy is achieved with $O(\log(1/P_{\min}))$ observations, where $P_{\min}$ is the minimum nonzero mass of $P$. We also develop a greedy algorithm with monotone approximation guarantees for computing $H(P\|Q_1,\dots,Q_m)$. Finally, we show that the same framework formalizes finite-sample limits in distribution-matching representation learning and randomness extraction, where zero entropy corresponds to exact recovery and exact extraction.

In this article we prove a derived version of the Marsden-Weinstein-Meyer symplectic reduction theorem. We model the symplectic quotient as a dg-groupoid. We then construct the reduced symplectic form inside the Bott-Shulman complex of the groupoid. Finally, we show that the reduced form satisfies a derived analogue of the non-degeneracy condition.

The study of optimal preemption policies for status update systems has been a recurring topic in the age of information (AoI) literature, where threshold-based structures have been shown to be optimal under a generate-at-will update generation model under certain assumptions. In this work, we study the effectiveness of threshold-based policies for a system with random update arrivals. In this regard, we introduce an analytical framework for evaluating the AoI of multi-threshold preemption policies and present interesting characteristics of the structure of the optimal preemption policy. We show the effectiveness of these threshold-based policies over the traditional probabilistic preemption policies and single-threshold policies, where we observe that significant gains in terms of AoI can be obtained by utilizing both the age of the packet and the age of the system when designing these preemption policies.

Given a polyhedron (planar, $3$-connected graph) $G$, we investigate its common neighbourhood graph con($G$). For cubic ($3$-regular) polyhedra, we show that the planarity of con($G$) depends on the number of odd faces of $G$, and on their adjacency. We then prove that for all other polyhedra, con($G$) is non-planar. We introduce a novel concept for polyhedra (and more generally, for plane graphs) $G$, namely the `facial common neighbourhood graph' facecon($G$). Its definition takes into account pairs of vertices with a common neighbour on the same face of $G$. It is a spanning subgraph of con($G$), that coincides with con($G$) for cubic polyhedra. It also generalises the reverse construction of the radial graph. As part of our investigation, we also prove a technical result of independent interest: if a maximal planar graph (triangulation of the sphere) has exactly two vertices of odd degree, then they are not adjacent. We also answer several questions in extremal graph theory. Fixing the number of vertices, we characterise the polyhedra $G$ such that con($G$) is planar and the number of edges in con($G$) is minimal/maximal. We address the same problem for facecon($G$), and prove that if it is maximal planar, then $G$ has no face of length greater than $6$. We notably characterise and explicitly construct all polyhedra $G$ of maximal face length $4$ such that facecon($G$) is maximal planar.

Gravitational models with non-minimal couplings involving functions of the matter Lagrangian and curvature have become popular in recent decades. By coupling the matter Lagrangian directly to the gravitational Lagrangian, one hopes to construct theories that can explain dark energy or dark matter without introducing additional sources. When this matter Lagrangian describes a perfect fluid, some technicalities are involved in its variational formulation. We present a careful derivation of the gravitational field equations together with the complete set of fluid equations. The fluid's equations allow us to define thermodynamic quantities such as temperature, chemical potential and number density and thus allow us to understand the effects of the non-minimal couplings on these quantities. We demonstrate the non-equivalence of the Lagrangian formulations of Schutz and Brown for these types of models and provide a detailed interpretation of our results.

Very recently, Green and Sawhney obtained a quasipolynomial bound in the Furstenberg--Sárközy theorem for square differences by proving an ''arithmetic level-$d$'' inequality, thereby yielding a greatly improved density increment scheme. We adapt their method to general intersective polynomials $h\in\mathbb{Z}[x]$ and obtain an analogous quasipolynomial upper bound for the largest subset of $\{1,2,\dots,X\}$ whose difference set contains no nonzero element of the form $h(n)$ with $n\in \mathbb{Z}$. This is the best quantitative upper bound presently known for sets lacking intersective polynomial differences. In contrast to the square case, extending the method to general intersective polynomials requires performing a density increment iteration in which the underlying polynomial changes at each step; a key contribution of this paper is to show that the arithmetic level-$d$ inequality remains effective uniformly across all auxiliary polynomials arising in the iteration. We also develop smoothly weighted versions of the exponential sum estimates of Rice.

Traditional scientific modeling typically begins with fixed, instance-wise effective equations and then carries out equation-specific analysis and computation, a procedure that becomes exceptionally challenging in complex applications such as multiscale systems. We propose an alternative paradigm by learning mesoscopic dynamics within a mathematically constrained hypothesis class. Building upon a generalized Onsager principle, we introduce a unified framework encompassing both dissipative and conservative mesoscopic dynamics. We establish uniform and a priori theoretical guarantees, including global well-posedness, asymptotic stability, unique factorization identifiability, and discrete energy dissipation, applicable to all spatio-temporal evolution equations within this hypothesis class prior to all learning stages. Data from each problem instance is then used to guide the identification of members within our hypothesis class, giving rise to accurate, robust and interpretable dynamical models. We empirically validate this framework on both data from continuum PDE models as a check, and on data arising from microscopic chain models for which exact meso-scale models are unknown. The proposed approach not only acts as an effective dynamics learner, but also offers vital interpretable diagnostics of the underlying physics.

The wolf note is an acoustic instability that occurs in large bowed string instruments when a strong body resonance interacts with the vibrating string, producing amplitude modulation and loss of tonal control. Various wolf suppressors - tuned mass dampers attached to the string or to the instrument body - are used in practice to mitigate this effect. In this paper, we propose a mathematical model describing the coupled dynamics of a string and a two-dimensional body equipped with one or two wolf suppressors. Both string and body include elastic (second-order) and stiffness (fourth-order) contributions and can be excited either by plucking or bowing. Three performance indicators are introduced: The first one perceives the wolf-tone appearance, the second one quantifies the attenuation of the notes possibly caused by the wolf suppressor, and the third one measures the acoustic fidelity (in terms of spectrum) with respect to the original instrument. The proposed numerical tests give insights about optimal tuning and placement of one or two suppressors, achieving effective wolf-note suppression while preserving as much as possible the global tonal balance.

In this paper, we investigate a bistatic multiple-input multiple-output (MIMO) integrated sensing and communication (ISAC) system over block-fading channels, focusing on the scenario where the sensing and communication receivers (Rxs) are co-located. Under the assumption of unknown channel state information (CSI) at the Rx, two schemes are considered: pilot sensing (PS) and data-aided sensing (DAS). The communication rate-sensing distortion functions for both schemes are characterized. For the DAS scheme, a closed-form asymptotic expression for the sensing distortion is derived by using random matrix theory (RMT). The asymptotic performance analysis explicitly quantifies the significant gains of the DAS scheme, revealing a strict $3$ dB effective SNR improvement in the low-SNR regime and a strictly faster performance scaling rate in the high-SNR limit compared to the PS scheme.

Enabling continued data-center growth under increasing grid stress motivates closer coordination between flexible computing demand and co-located battery energy storage systems (BESS) to improve site operations and provide grid services. This paper develops a robust co-optimization framework for day-ahead operation of data centers with co-located BESS under utility-imposed interconnection limits on peak load and ramping. The model jointly considers deadline-constrained computing workloads, managed through workload scheduling and dynamic voltage and frequency scaling (DVFS), together with degradation-aware BESS dispatch to enable cost optimization and participation in ancillary-service markets. Case studies based on real-world market and workload data show that the proposed framework yields feasible day-ahead schedules across a range of operating conditions, with substantially larger benefits when interconnection constraints become binding. Under baseline conditions, BESS value is derived from both ancillary-service participation and improved workload and energy management. Under stressed peak-load and ramping limits, however, the daily value of BESS increases by a factor of two or more, driven primarily \revise{by BESS actions to reduce the potential incompletion in the schedulable workload while complying with interconnection constraints}. Under tight peak-load caps, workload composition also matters where a higher share of non-schedulable jobs can increase operating cost by more than 25\% relative to more flexible workload mixes. \revise{Additionally, DVFS studies further show that processor-level control is a material flexibility lever under tight load limits.} These results demonstrate that coordinated compute-storage flexibility can materially expand the operational headroom and grid value of data centers, especially under increasingly scarce grid capacity.

We study the \textit{Kaluza--Klein} descent of non-invertible higher-form symmetry defects from eleven-dimensional Supergravity to Type IIA Supergravity. Starting from the eleven-dimensional construction of non-invertible Supergravity defects, we show that the full defect, including its auxiliary topological sector, admits a pushforward along the M-theory circle in the zero-mode Supergravity regime. The seven-dimensional \textit{Chern--Simons}-like auxiliary theory descends to a six-dimensional \(BF\)-type sector. We also show that the compactification of the eleven-dimensional \textit{Bianchi} sector splits into an invertible \(H_{[3]}\)-sector and a twisted non-invertible \(\widetilde F_{[4]}\)-sector, controlled by \(d\widetilde F_{[4]}+H_{[3]}\wedge F_{[2]}=0\). The resulting Type IIA defect algebra contains both an invertible Picard subgroup and non-invertible \(BF\)-dressed defects, with charged probes identified through the standard M-theory/Type IIA brane dictionary.

In this paper, the conditions of monogenicity are weakened for functions with values in a three-dimensional commutative algebra over the field of complex numbers. By monogenicity we mean continuity and the existence of the Gâteaux derivative.

We consider 3D micropolar flows with possible vanishing spin viscosity and investigate the decay of the energy for large times. We compute first the exact $L^2$-asymptotic profile, as $t\to+\infty$, for solutions to the linear 3D micropolar equations, up to the second order. For the nonlinear micropolar system, we first establish the existence of restricted Leray solutions. This new notion of solutions is required because it is not known whether the weak finite energy solutions verify a strong energy inequality. Next, we study the large-time behavior of restricted Leray solutions, and prove that they behave asymptotically in $L^2$ like their linear counterpart, up to the critical algebraic decay rate $O(t^{-5/2})$ for the energy. Applying a remarkable linear enstrophy identity, we show that the microrotation field exhibits faster decay in $L^2$ than the velocity field, allowing us to impose our hypothesis on the velocity field only and not on the angular velocity.

The groundbreaking metric age of information (AoI) has been introduced to measure information freshness in communication networks. As transformational as it is, AoI metric falls short in some applications, such as remote monitoring, since it is a semantic-agnostic metric which does not consider the dynamics of the random process. There is a need to quantify the performance of a remote estimator via a metric that combines freshness and semantic aspects. To this end, in this paper, we introduce a novel metric coined age of staleness (AoS) that measures when the last time that the current estimation was correct. First, we analyze a simple scenario where an $n$-ary symmetric Markov source is observed by a monitor via a constant sampling rate, obtain a closed-form expression for the AoS, and show that it is a monotonically decreasing function of the sampling rate. Next, we consider multiple distinct Markov sources, and formulate an optimization problem, where the remote monitor allocates the total sampling rate to tracking the sources. Although the optimization problem is non-convex, its structure is suitable for obtaining a near-optimal solution using the polyblock algorithm, which leverages the monotonicity of the objective function. While the new AoS metric could be applicable in many scenarios, we believe it is particularly well-suited for a digital twin network (DTN) where multiple physical objects (POs) are monitored with a total sampling rate constraint to maintain a digital representation of them, namely, their digital twin (DT).

Achieving a just and sustainable transition requires the pursuit of multiple social and environmental targets. Two primary barriers impede this process: (1) targets are often in conflict with each other, and (2) policies aimed at these targets are commonly planned in isolation, neglecting complex interdependencies in the system. To address these challenges, we propose a general modeling framework that evaluates the holistic impact of policies and decision-making on sustainability targets while capturing system interdependencies in a policy-target network. Inspired by Kauffman's NK fitness landscape, our framework takes the form of a multi-objective optimization model that employs a dynamic evolutionary algorithm in conjunction with network analysis. Our algorithm accounts for tradeoffs between conflicting targets by dynamically reallocating resources to the most impactful and efficient policies. One key finding indicates that increasing resources generally enhances performance, but marginal gains stagnate at a point of diminishing returns. Sensitivity analysis reveals that the system is primarily driven by three factors: budget constraint, network density (interconnectivity), and policy efficacy. This study serves as a foundational step towards developing a decision-support tool that assists policymakers in achieving optimal outcomes for problems with a large number of dynamically interacting targets.

We investigate the role of the four viscosity parameters, in fluids where the particles possess a microstructure (micropolar flows) and are allowed to rotate in a two-dimensional setting. We first establish the existence of global finite energy solutions, satisfying the classical energy equality, for arbitrary initial data in $L^2$, in the case of a spin viscosity $γ\ge0$, and we construct the asymptotic profiles of the solution as $t\to+\infty$. We deduce the remarkable fact that the large time behavior only depends on the kinematic viscosity $μ$, and not on the other parameters $χ$ (vortex-viscosity), $γ$ (spin viscosity) and $κ$ (gyroviscosity) of the model. Our primary tool is a new enstrophy-like identity of independent interest, involving the difference between the fluid vorticity and the micro-angular velocity. Another consequence of our analysis is the identification of scenarios where the presence of micro-rotational effects significantly enhances dissipation, thereby slowing down the fluid motion at large times.

We make rigorous the physics prediction that lattice Yang-Mills theories with gauge groups which have trivial centers do not satisfy Wilson's criterion for quark confinement. Specifically we prove that $\mathrm{SO}(3)$ lattice Yang-Mills theory does not satisfy Wilson's criterion in a strong coupling regime.

We establish the joint $*$-convergence of a random circulant matrix and a specific deterministic diagonal matrix. We also show that the empirical spectral distributions of skew-circulant and left skew-circulant random matrices converge weakly a.s.~to complex Gaussian and symmetrized Rayleigh distributions, respectively. The $*$-convergence of symmetric Toeplitz and Hankel random matrices is well known. So is the weak convergence of their random spectrum. However, not much is known about the limits. We exploit the connections of circulant, reverse circulant, and left skew-circulants with the Hankel and Toeplitz matrices, to show the $*$-convergence of the random symmetric Toeplitz matrix to the sum of two non-commutative self-adjoint variables, each having a real Gaussian distribution. A similar result holds for the non-symmetric Toeplitz matrix, but the variables are not self-adjoint and have a complex Gaussian distribution. The random Hankel matrix is shown to converge in $*$-distribution to a sum of two self-adjoint variables, each of which has a symmetrized Rayleigh distribution. As a consequence of these results, we also obtain a different proof of the convergence of the empirical spectral distribution of symmetric Toeplitz and Hankel matrices, and a slightly different way of expressing the moments of the limit spectral distribution.

We prove measurable analogues of Whitney's classical theorems on weak isomorphisms of finite graphs. In the setting of locally finite graphings, we introduce a notion of weak isomorphism as an edge-measure-preserving Borel bijection that preserves cycles and hyperfinite subgraphs, modulo null sets. We first show a rigidity theorem, proving that for weakly 3-connected infinitely-ended graphings, every weak isomorphism is induced by an isomorphism of graphings. To our knowledge, this gives the first general sufficient condition in measurable combinatorics for the existence of an isomorphism between two given graphings. Next, we give a full measurable version of Whitney's theorem, showing that every weak isomorphism between graphings can be implemented by countably many measurable Whitney operations, which we introduce in this setting. The proofs require new measurable-combinatorial tools, including a careful analysis of infinitely-ended subforests. This work further develops the limit theory of matroids recently initiated by Lovász.

We investigate the problem of covert quickest change detection in a Bayesian and infinite-horizon setting. A legitimate entity seeks to detect a change in the state of a discrete memoryless channel as quickly as possible by actively probing it. Simultaneously, the entity must ensure its probing remains covert from an adversary monitoring the channel for active sensing. We introduce the expected covertness budget (ECB) as an analytically tractable covertness metric that bounds from above the relative entropy between the observation sequences induced by active and passive sensing. Under constraints on both the probability of false alarm (PFA) and the ECB, we establish a second-order asymptotic converse bound on the average detection delay as the PFA constraint approaches zero, for any positive ECB constraint, explicitly quantifying the maximum square-root-order covert sensing gain possible. Furthermore, we propose an achievability scheme utilizing a constant-sensing-probability Shiryaev-type policy and show that it matches the second-order asymptotic converse. We illustrate our result with a numerical example.

We study finite systems of vectors whose frame operator matrices are unitarily equivalent, via explicit and computationally efficient unitary transformations, to block-diagonal matrices. We call such systems block-equivalent. We show that a Gabor system $\mathcal{G}=\mathcal{G}(g,L\times K)\subset \mathbb C^N$ is block-equivalent when either the modulation set $L$ or the translation set $K$ is a subgroup of $\mathbb Z_N$. We also characterize situations in which the frame operator matrix becomes diagonal. Finally, we show that geometric conditions on subsets of $\mathbb Z_N$ force certain diagonals of the frame operator matrix of $\mathcal{G}$ to vanish, yielding additional sparsity and block structures.

The future behavioural fate of a forced nonlinear system can depend sensitively on the forcing profile as well as natural fluctuations within the system. This is especially the case for rate-induced tipping, where the forcing pushes the system to a basin boundary of a future behaviour and small changes in the forcing can lead to drastically different eventual behaviours. This sensitivity may be present only for a limited period of time, for example when the forcing is most rapidly changing. Moreover, critical slowing down based methods fail to be informative in such cases. We investigate a geometric early warning to evaluate when a system is in such a sensitive state. This involves computing the R-tipping indicator, namely the signed distance to an approximate R-tipping threshold. The latter is a dynamic state that embeds knowledge of the system and future behaviour of the forcing. We contrast this with early warnings of bifurcation-induced tipping, where tipping is associated with passing a threshold on slow variation of forcing. As an example, we consider methods of early prediction of the future state for a 3-box model of the Atlantic Meridional Overturning Circulation (AMOC) with specified rapid forcing. We show that the skill of the geometric early warning compares favourably with simple thresholds.

For a fixed flow-based generative model under a small inference budget, sample quality can depend strongly on where the sampler spends its few function evaluations. Flow matching and Schrödinger bridges define probability paths, yet their inference grids are usually heuristic or inherited from one-endpoint diffusion. We derive a conditional-marginal entropy-rate objective for bridge-aware discretization, separating endpoint-conditioned bridge geometry from marginal flow evolution, and use it to build a training-free entropic inference-time scheduler from first principles. For Gaussian Brownian bridges this rate is closed-form and U-shaped, motivating boundary-heavy nonuniform grids. On trained two-dimensional bridge/flow models, the estimated profile recovers the predicted shape and improves 10-step ODE-Heun MMD over linear by 18.1%, with a paired 22.7% SDE-Heun improvement in the same low-NFE sweep. On EDM/CIFAR-10, the entropic time-discretization gives the best tested five-step FID (186.3 \pm 4.0 versus 200.5 \pm 2.9 for linear and 238.0 \pm 5.3 for cosine). On AlphaFlow protein generation, entropic conditional-marginal (cond-marg) scheduling shows advantage in low-NFE regimes on both CAMEO22 and ATLAS benchmarks. These results support entropy-rate scheduling as a practical low-budget allocation signal for high-dimensional bridge and flow samplers.

In this note we consider linear functionals on an unital commutative R-algebra. We give an integral representation of a nonnegative functional on an Archimedean cone where we do not assume that this cone is a semiring or a quadratic module. We also give a solution of the moment problem on a product of intervals and determine conditions for a functional to be a moment functional on a compact of characters.

Grain-boundary (GB) local stress is central to the initiation and evolution of long-term creep damage in polycrystalline superalloys. Owing to the high-dimensional nonlinear relationships between the GB stress response and multiple crystallographic, microstructural, and micromechanical characteristics, it remains challenging to identify the key characteristics governing GB stress and to elucidate their mechanisms of influence. Dislocation-climb-affected crystal-plasticity finite-element simulations of minimal grain clusters are combined with an integrated causation-guided machine-learning framework, in which mechanics-informed descriptors are analyzed by causation entropy to identify governing mechanisms and then distilled into a reduced-order regression form for interpretable prediction of GB normal stress. Among 18 physically motivated characteristics, the GB inclination angle, the slip transmission, the climb-related Schmid-type indicator, and the elastic-modulus mismatch are found to be dominant, revealing the coupled roles of interfacial geometry, crystallographic compatibility, creep stress relaxation, and micromechanical contrast. The identified characteristics hierarchy and functional representation remain effective under multiaxial loading and can be extended to tricrystal systems through physically interpretable nonlocal augmentation when a purely local description becomes insufficient, demonstrating strong physical consistency and robust generalizability across physical conditions. The extracted candidate functions also improve surrogate-model performance across multiple machine-learning model classes, providing supporting evidence for the physical relevance and efficiency of the identified representation. The proposed methods demonstrate strong potential for the development of interpretable machine-learning models and for the study of microscale nonlocal damage.

Fractional boundary value problems are often used to model complex systems and processes characterized by memory effects and anomalous diffusion. In this paper, we consider fractional boundary value problems involving the Riesz-Caputo operator, which is particularly suited for modeling physical phenomena exhibiting symmetric diffusive effects. We provide an integral representation of the solution to prove existence and uniqueness of the fractional differential problem. We introduce a B-spline collocation method to approximate the solution of the problem and provide a convergence analysis, with both theoretical insights and numerical experiments.

We introduce the median uniformity $\mathcal U_{\mathrm m}$, an intrinsic precompact convex uniform structure on a median algebra. It is Hausdorff under natural assumptions, for instance for finite-rank median algebras. In the Hausdorff case, its uniform completion yields the Minimal Median Compactification (MMC). The induced topology $τ_{\mathrm m}$ provides a natural higher-rank analogue of the interval topology on linearly ordered sets and of the shadow topology on rank-one median algebras. When all intervals in the median algebra $X$ are finite, the MMC is the unique proper median compactification of $(X,τ_{\mathrm m})$; in particular, it coincides with the Roller compactification. We apply this uniform framework to continuous actions of a topological group $G$ by median automorphisms. We show that the MMC is a median $G$-compactification. In the finite-rank case, the resulting compact $G$-system is Rosenthal representable and hence dynamically tame.

This is the second part of a series of papers proving the nonlinear stability of a one-parameter family of continuous self-similar $C^{1,α}$ naked singularity solutions, with $0<α\ll1$, to the spherically symmetric Einstein-scalar field equations. These solutions were constructed by Christodoulou and are known to be unstable under sufficiently rough perturbations due to the blue-shift instability mechanism. In complete contrast to the previous instability results, we establish the linearized stability for those naked singularity spacetimes under perturbations of the same regularity as the background, revealing the central role of regularity in determining the strength of the blue-shift instability mechanism, and showing that it is not triggered at the regularity level of the background spacetime. The linear analysis carried out in this paper provides the foundation for the nonlinear stability result established in the companion paper [W. Zheng, Nonlinear stability of the continuous self-similar naked singularities for the Einstein-scalar field equations I: main results]. Together with that companion paper, this yields the nonlinear stability of these continuously self-similar naked singularities.