Papers

Quantum logic is usually presented as a non-classical departure from ordinary reasoning forced on us by quantum mechanics, with classical logic kept as the secure starting point. We argue for the opposite order of explanation in a finite and fully computable setting. The free orthomodular lattice on two generators has ninety-six elements, the direct product of a six-element non-distributive factor and a sixteen-element Boolean factor. Reading the first factor as a register of contexts and the second as Boolean content, we obtain a calculus whose elements are context–bit-vector pairs and whose operations act component by component. With this calculus we establish three results. First, we classify the six layers by commutativity, identifying the central kernel of context-neutral propositions together with a dual central layer in which all complementary contexts are present. Second, we show that orthocomplementation rearranges the layers exactly as the complementation of the small factor rearranges its elements, which makes the duality among the layers rigid rather than accidental. Third, we prove that the operation forgetting the context is a surjective homomorphism of orthocomplemented lattices whose quotient is the classical Boolean algebra, so that classical logic is a six-to-one, information-losing image of the contextual calculus.

A $q$-kernel of a digraph $D$ is an independent set $Q\subseteq D$ such that every vertex of $D$ is reachable from $Q$ by a directed path of length at most $q$, which is a natural generalization of kernels and quasikernels. In this paper, we establish three results on generalized quasikernels. Firstly, we prove that any $n$-vertex source-free bipartite oriented graph with no directed 4-cycles has a quasikernel of size at most $17n/35$. Secondly, we show that every digraph with no $(r-1)$-source set contains $r$ pairwise disjoint $(3r-2)$-kernels, where $r\ge 2$. At last, we consider unicyclic digraph with a directed cycle of length $2\ell$ and bipartition $U\cup V$, and we prove that for every odd integer $q\ge 3$, there exist two $q$-kernels $Q_U\subseteq U$ and $Q_V\subseteq V$ such that \[ |Q_U|+|Q_V| \le 2\cdot \frac{\lceil \ell/(q+1)\rceil}{\ell} |V(D)|. \] These results confirm two conjectures and give an affirmative answer to a question posed by Spiro in European Journal of Combinatorics 133 (2026), 104307.

This paper proposes a novel continuous aperture array (CAPA)-assisted integrated communication and navigation (ICAN) framework for low Earth orbit (LEO) satellite constellations. Within this framework, an electromagnetic-based collaborative transmission model is developed, in which multiple satellites equipped with CAPAs simultaneously radiate downlink data streams and navigation reference signals over shared spectrum. Building upon this, the achievable communication rate and the navigation Cramer-Rao bound (CRB) are derived, which explicitly characterize the intrinsic coupling between the dual-function beamformers and system performance. To improve the positioning accuracy with communication quality of service guarantee, a joint beamforming optimization problem is formulated to minimize the average CRB subject to transmit power budgets and minimum rate constraints. To tackle the inherent infinite-dimensionality of the CAPA beamformer design, an ICAN channel subspace is introduced to equivalently transform the formulation into a tractable finite-dimensional problem, which is then efficiently solved via an iterative convex optimization algorithm. Finally, numerical results demonstrate that the proposed CAPA-assisted beamforming design algorithm significantly outperforms conventional discrete phased array architectures and other benchmark schemes, yielding notable improvements in ICAN performance.

We study the $F$-threshold of differential power filtrations of monomial ideals. We give the existence of this number and show that it is upper bounded by the maximum height of the minimal primes of the underlying monomial ideal. As an application of our results, we show that the dimension of the polynomial ring and the $F$-threshold of the differential power filtration of the monomial maximal ideal are the same.

We study scattering and inverse scattering generated by sources and penetrable media with corner singularities. We introduce the notion of split corners, a local model that unifies geometric singularities and coefficient discontinuities arising in source and medium scattering. Within this framework, we establish scattering and inverse scattering results that extend the classical theory to split corners, allowing the source or medium contrast to approach different limiting values in distinct sectors meeting at the corner tip. For source scattering, we prove that every admissible two-split corner necessarily radiates in a general bounded inhomogeneous background, thereby generalizing the classical corner radiation principle to configurations involving both geometric corners and jump discontinuities. For penetrable media, we establish analogous results together with explicit compatibility conditions for incident waves of arbitrary vanishing order. As consequences, we obtain several uniqueness results in inverse source and inverse medium scattering, including recovery of polygonal convex hulls from a single far-field measurement.

We study the problem of recovering the correspondence between a collection of $n$ points in $\mathbb{R}^d$ and a noisy, permuted version of those points. In the high-dimensional regime $d=\omega(\log n)$, under a Gaussian model with noise variance $\sigma^2=d/(b\log n)$, prior work identifies $b=2$ as the threshold for almost exact recovery. We prove that this threshold is all-or-nothing: for every fixed $b<2$, no estimator recovers a positive fraction of the matching, and even estimating the matched point cloud in Euclidean distance is asymptotically no better than ignoring the correspondence. On the other hand, we consider a multi-view generalization of the problem where $K$ noisy, independently permuted copies of the same latent point cloud are observed. Here we show that a simple polynomial-time procedure recovers all relative matchings up to $o(n)$ errors whenever $b>K/(K-1)$. Thus multiple views can break the impossibility barrier $b=2$ for the original matching problem: in particular, for $3/2 < b < 2$, the two-view model has no nontrivial recovery, but a third view makes all latent correspondences efficiently recoverable.

We revisit the pathwise global well-posedness issue of the modulated Korteweg-de Vries equation (KdV) on the circle. In the previous work (2024), by combining the $I$-method and the sewing lemma, the second and fourth authors with C. Chouk, G. Li, and J. Li proved its global well-posedness in negative Sobolev spaces. This result was, however, restricted to the scaling subcritical regime $s > - \frac 32$ due to the use of the classical KdV scaling. In this paper, by noting that the modulated KdV enjoys additional one degree of freedom in its scaling symmetry thanks to the modulation term, we apply a non-KdV scaling to the unknown and prove that, given any $s \in \mathbb R$, the modulated KdV on the circle with a sufficiently irregular modulation is globally well-posed in $H^s(\mathbb T)$, thus going beyond the barrier of the scaling critical regularity $s = - \frac 32$.

We study the behavior of asymptotic cohomological functions on proper schemes over a field. We prove a formula for how asymptotic cohomological functions behave under perturbation by an object in the bounded derived category. Using the same method, we also prove that the volume of a line bundle on any proper scheme over a field always exists as a limit.

This letter develops a radio access network (RAN) framework for mixed discrete-continuous optimization problems that arise in user-centric cell=free massive multiple-antenna networks. The novel framework exploits the structural decomposition between discrete clustering decisions and continuous resource allocation variables by modeling the space of feasible serving states as a graph with Hamming-topology neighborhoods. A serving-state graph abstraction is introduced to enable topology-aware search-and-evaluate optimization procedures and a graph-based search-and-evaluate (GBSE) algorithm is devised along with their complexity analysis. Energy efficiency maximization at the RAN level is presented as an application of considered alongside the proposed framework and GBSE algorithm. Numerical results show that minimal Hamming neighborhoods offer an attractive trade-off between scalability and exploration capability in grap-based optimization and GBSE outperforms existing techniques.

A proper sub-box of $A=\{0,1,2\}^d$ is a product $S_1\times\dots\times S_d$ with each $\varnothing\neq S_i\subsetneq\{0,1,2\}$. A double cover is a finite multiset of proper sub-boxes covering every point of $A$ exactly twice; write $f(d)$ for the minimum size of a double cover. Leader, Milicevic and Tan asked whether $f(d)\ge 2^d$ for all $d$ (Question 4.1 of the PatternBoost paper of Charton-Ellenberg-Wagner-Williamson), analogous to the Alon-Bohman-Holzman-Kleitman partition bound $2^d$. No better than the trivial volume bound was previously known, for any $d\ge 2$. We prove the first nontrivial lower bounds. A modular refinement of the parity argument gives $f(d)\ge 2^{d+1}/(d+1)$; a slicing argument gives $f(4)\ge 19$, $f(5)\ge 33$, both above $2^d$, resolving the question for $d=4,5$ – the first cases beyond the trivially known $d\le 3$. A finer "line rigidity" argument yields $f(6)\ge 60$, breaking the profile-statistic barrier (capped at $57$, shown here). This is formally verified in Lean 4: $f(6)\ge 60$ is machine-checked on the three standard Mathlib axioms alone. On the upper-bound side, a dimension-lifting construction $f(r+3)\le 6\cdot 2^r+3f(r)$ gives $f(6)\le 81$ (improving the known $82$) and $f(d)\le(\tfrac65+o(1))2^d$ asymptotically; a refinement improves the constant to $\tfrac87$. This makes partial progress on PatternBoost's problem of reducing their constant $1.28$, and refutes the closed-form guess $f(d)=5\cdot 2^{d-2}+1$ from $d=7$ on. Together, $60\le f(6)\le 81$. Finally we isolate the construction-side obstruction – an "S+c=2^j+1" phenomenon, every skeleton sitting exactly one box past the partition bound – and show it is of a piece with the Leader-Milicevic-Tan question itself.

Let $q=2^e$ with $e\ge2$ even, put $d=(q^2+q+1)/3$, and let $\sigma(X)=X+X^d+X^{dq}$ be the permutation of $\mathbb F_{q^2}$ introduced by Ding, Qu, Wang, Yuan, and Yuan. For $\alpha\in\mathbb F_q^*$, define the Boolean function \[ f_\alpha(x)=\operatorname{Tr}_{q^2}\bigl(\alpha(\sigma^{-1}(x))^3\bigr), \qquad x\in\mathbb F_{q^2}. \] In this paper, we determine the complete Walsh distribution of $f_\alpha$ in the remaining cubic case $\alpha\in(\mathbb F_q^*)^3$. More precisely, these functions are not bent but are $2$-plateaued: their Walsh values are precisely $0$ and $\pm 2q$, with exact multiplicities. The main new tool is a completion method for the outside Walsh coefficients: the punctured Fourier transform arising from the outside reduction is filled on the missing line, a modification invisible to outside frequencies, and the completed function is then identified with a Boolean component of a Kasami APN monomial. The APN property supplies a fourth-moment identity which, together with the known subfield spectrum and a Hasse divisibility congruence, forces the pointwise cubic spectrum.

A graph is homogeneously traceable if each vertex is an endpoint of a Hamiltonian path. Chartrand, Gould, and Kapoor (1979) proved irregular homogeneously traceable nonhamiltonian graphs exist for every order $n\ge 9$. Hu and Zhan (DAM, 2022) considered the $3$-regular and $4$-regular cases and asked which order $n$ can be realized by a $k$-regular homogeneously traceable nonhamiltonian graph. Recently, Liu and Qiao (DAM, 2026) showed that $n=p(k-1)+q\ge 6(k-1)+q$ can be realized if $k\ge 5$ is odd and $q\in\{0,2,4,6\}$, or $k\ge 6$ is even and $q\in\{0,1,...,6\}$. In this paper, we show that for any $k\ge 6$ and $n\ge 6(k-2)$, there exists a $k$-regular homogeneously traceable nonhamiltonian graph of order $n$.

Motivated by a theorem of Silverman, we consider the following problem. Let $A$ be an abelian variety over a global field $K$. Given a non-torsion point $P \in A(K)$, for a sufficiently large positive integer $n$, whether there exists a place $v$ of $K$ such that the order of the reduction of $P$ modulo $v$ is $n$? In this article, we first show that this holds for an elliptic curve over a global function field of positive characteristic $p>3$ and for sufficiently large positive integers $n$ coprime to $p$. In the second part of the paper, we consider its relative version over $\mathbb C$. More precisely, let $\pi: \mathcal{A} \rightarrow S$ be an abelian scheme over some variety $S$ over $\mathbb C$, and let $P$ be a non-torsion section of $\pi$. If the Betti map associated to $P$ is generically submersive, then for every sufficiently large $n$, there is a point $s$ in $S(\mathbb C)$ such that $P(s)$ is a point of order $n$ in the corresponding fiber.

In large-scale computing systems, jobs often demand heterogeneous server allocations: large jobs that occupy a substantial fraction of the servers are of high importance and are thus latency-sensitive, while small jobs fill in the remaining capacity to maintain throughput. To model this dynamic, we introduce the half-batch multiserver-job (MSJ) framework, a queueing model in which large jobs arrive according to a Poisson process and require all servers simultaneously, while small jobs, each needing only one server, are always available. We prove that, in the half-batch MSJ model, the Pareto frontier for large-job mean response time and small-job throughput admits a simple and exact characterization. It is generated by a family of convoy policies, under which the system serves small jobs until $k$ large jobs have arrived and then switches to serving large jobs, together with convex combinations of neighboring convoy policies. Our result is fully general and non-asymptotic, holding for every stable arrival rate $\lambda$, every number of servers $n$, and every large-job size distribution $S$.

We study the variance of the free energy in the Gaussian Sherrington-Kirkpatrick model at zero external field. For every system size and inverse temperature $\beta>0$, we prove $2N^2\operatorname{Var} F_N^{\mathrm{off}}(\beta) \le \log\chi+(\beta^2-1)(\chi-1), \chi=N\mathbb{E}\langle R_{1,2}^2\rangle_\beta$, where $R_{1,2}$ is the replica overlap and $F_N^{\mathrm{off}}=N^{-1}\log Z_N^{\mathrm{off}}$ omits diagonal disorder. The proof uses an exact identity for the edges incident to one spin, Schur complements, and Gaussian disorder interpolation. At criticality the bound reduces to $2N^2\operatorname{Var} F_N^{\mathrm{off}}(1)\le\log\chi$. An explicit nonnegative integral measures its gap. A second decomposition relates this gap to a mixed cubic overlap functional. Existing moment and cavity estimates imply that its required integral is bounded. Critical susceptibility asymptotics give the sharp logarithmic upper coefficient and show that the gap converges exactly when the variance has a constant term. We extend an entropy bound throughout bounded critical temperature windows and establish overlap moment stability on a smaller interval.

Interconnected financial systems are vulnerable to cascading failures arising from cross-holdings and nonlinear contagion, making the analysis and mitigation of systemic risk a challenging computational problem. In this work, we develop a unified optimization framework for financial network analysis based on Ising models and Quadratic Unconstrained Binary Optimization (QUBO). Starting from the Elliott Golub Jackson financial network model, we extend equilibrium valuation to incorporate threshold-induced failures, formulate the Maximum Cascade Failure Problem, and derive an equivalent QUBO representation. We then formulate the Optimal Bailout Allocation Problem as a controlled Ising model and transform the resulting bi-level optimization into a single joint QUBO that simultaneously determines equilibrium failures and optimal interventions under budget constraints. To characterize the influence of individual institutions, we introduce bailout susceptibility as a response-based measure of systemic importance and develop a susceptibility-driven greedy intervention strategy. Numerical simulations demonstrate equilibrium valuation, worst-case cascade identification, optimal bailout allocation, and susceptibility analysis on financial networks of varying sizes. Beyond optimization, the Ising representation provides a general statistical-mechanical framework for analyzing financial contagion, enabling the application of response theory, Monte Carlo methods, and other techniques developed for interacting spin systems. The proposed framework establishes a unified approach for systemic risk analysis that is compatible with classical annealing, quantum-inspired optimization, and emerging quantum annealing technologies

In this paper we establish a Banach-type fixed point theorem for mappings that contract a general nonnegative functional rather than a metric. The principal assumption is that the functional dominates the underlying metric through a simple lower bound, together with a compatibility condition at limits. Our result provides a common framework that recovers recent fixed point theorems in perturbed metric spaces and for polynomial contractions as particular cases, while showing that neither symmetry nor the triangle inequality is essential for the contraction argument. Consequently, the contractive quantity may be interpreted as a general discrepancy or divergence measure rather than a metric. The results highlight that the essential ingredient behind Banach's contraction principle is not the metric structure itself, but the ability of the contracted functional to control the geometry of the underlying space. This perspective provides a simple and flexible framework for obtaining fixed point results for broad classes of non-metrical contractive quantities.

Let $D$ be a noncommutative division ring and let $R = M_{m}(D)$ with $m > 1$. We characterize additive mappings $f,$ $g:R\rightarrow R$ satisfying the identity $f(X) = X^{n}g(X^{-1})$ for every invertible element $X$ in $R$, where $n$ is a nonnegative integer. We show that the solutions are precisely given by $f = g$ and $f(X) = Xf(I)$ for all $X$ in $R$. Moreover, if $n \neq2$, both $f$ and $g$ are identically zero.

OpenAI's recent disproof of the Erdős unit distance conjecture marked a milestone for AI in mathematics. It also inspired another breakthrough: a human disproof of the Erdős–Szemerédi sum-product conjecture over $\mathbb R$. In this paper, we present a simple agent built on GPT-5.5 Pro. Using a problem-agnostic, three-stage prompting pipeline – proof-plan proposal, proof construction, and review – the agent autonomously generated correct proofs that the sum-product conjecture is false over $\mathbb R$ in 7 of 8 independent trials; in the remaining trial, it identified an unresolved gap in its argument. The seven proofs are diverse: some are close to existing unit-based constructions, while others avoid units by using $L^p$-type regions of algebraic integers. The system used an average of 132.4k reasoning tokens per trial. We release the code, intermediate outputs, and generated proofs, providing a reproducible, data-contamination-free case study in autonomous proof generation.

We study a discrete version of the Sturm–Liouville eigenvalue problem on grids whose spacing may vary from point to point, using the discrete Prüfer transformation. We show that any eigenvalues of the problem are real and that there are finitely many of them. We then compare two numerical methods for computing the eigenvalues, regular shooting and Prüfer-based shooting, and find that the Prüfer method remains accurate on non-uniform grids where regular shooting loses accuracy or fails.

Let \(G\) be a finite simple graph. We study perfect matchings through two complementary viewpoints: reductions to bipartite permanent terms and the directed cycle covers counted by the ordinary adjacency permanent. The main identity is an explicit odd-cycle-indexed form of the classical cycle-cover expansion: it separates \(\perfmat(G)^2\), the even-cycle-cover contribution coming from superposing two perfect matchings, from the contribution of cycle covers containing odd cycles. This gives a nonnegative odd-cycle defect \(\delta_{\mathrm{odd}}(G)\). Combining the identity with the Bregman–Minc inequality yields a structural refinement of the Alon–Friedland degree-sequence bound. We prove product, positivity, and fractional-perfect-matching interpretations for the defect; show that for \(K_{2n}\) the defect asymptotically accounts for almost the entire Bregman–Minc target; and derive from this a derangement identity. We also study near equality in the Alon–Friedland bound: we classify all one-edge perturbations of the extremal graphs, record the uniform obstruction coming from \(K_4\), and formulate a sharp bounded-degree gap problem whose two natural candidate extremal graphs cross between maximum degrees \(9\) and \(10\).

We discuss the geometry of the two-dimensional sphere. Our perspective focuses on the set of conformal factors that rescale a given metric to a round metric. We find that there exists a unique symmetric and trace-free tensor on the sphere which complements the Ricci tensor to a divergence-free symmetric tensor. Using this tensor, we derive a linear second-order PDE whose solution space carries a natural Minkowski structure. We show that there are in fact infinitely many such spaces, all related by isometries which can be thought of as being parametrised by “4-vectors”. Finally, we show that every 2-sphere can be isometrically embedded in a Minkowski space in a canonical way.

A definition of harmonic variables for Laplace operators on Riemannian and pseudo-Riemannian spaces is introduced. Harmonic variables allow, in particular, the construction of particular (non-invariant group) solutions to a wide class of nonlinear differential equations. For Lie groups with a bi-invariant metric and a certain class of homogeneous spaces (e.g., symmetric spaces), an algorithm for constructing such variables is presented.

A negative Pell equation produces positive integers with $a_n^3+b_n^3+c_n^3=561N_n^3+(-1)^{n+1}$. We classify all positive solutions satisfying the natural linear constraints of the construction and prove an exact pairwise coprimality law, with exceptional factor $137$ in one class modulo $68$. The associated sextic $K=(1-13u+26u^2)^3+(6+182u^2)^3$ defines an elliptic K3 surface with six type-IV fibres. Its genus-four cubic cover has a geometrically simple Prym threefold; exact good-reduction arithmetic establishes simplicity over every finite extension of the residue field. We determine the full geometric Mordell-Weil group, the integral Neron-Severi and transcendental lattices, and the arithmetic Picard lattice over $\mathbb{Q}$, which is $U\oplus\langle-4\rangle^{\oplus4}$. Every two-cube representation by rational functions over $\mathbb{Q}$ is obtained from a multiple of one section, with numerator degrees $2m^2$ and denominator degree $2m^2-2$. Polynomial rigidity follows without a degree assumption. Along the specified Pell orbit all specialized points are nontorsion, their heights are $\frac{4}{3}|n|\log(8+\sqrt{65})+O(1)$, and infinitely many rational cubic-twist classes occur. We also identify the local zeta function at $19$ and the generic rank and rank-jump criterion for the corresponding two-parameter base-change family.

We provide an explicit analytical evaluation of Ramanujan-type series for $1/\pi$. Focusing on the singular moduli $k_{r}$ for $r \in \{5, 7, 13, 37\}$, we demonstrate that the underlying elliptic identities can be established through lower-degree modular transformations. Specifically, we resolve the case $r = 5$ via a combination of degree-2 and degree-3 modular transformations; the case $r = 7$ utilizing the modular transformation of degree 2; the case $r = 13$ via a combination of degree-2 and degree-7 modular transformations; and the case $r = 37$ via a combination of degree-2 and degree-19 modular transformations.

A jigsaw puzzle may be assembled in many different ways. Some assembly sequences remain connected throughout, while others temporarily build separate parts of the puzzle before joining them together. By representing the puzzle as a graph, these assembly sequences become vertex orderings that can be counted using recent results from graph theory. We apply this framework to enumerate three natural assembly strategies: connected assembly, assembly beginning from several disconnected pieces, and assembly in which new disconnected sections may be started during the process. The resulting counts reveal that, even for modest puzzle sizes, connectivity-preserving assembly sequences are greatly outnumbered by those that pass through disconnected intermediate stages.

We study finite field value distributions through the fixed-point statistics of monodromy groups. For a regular degree-\(n\) cover, omitted values are controlled by derangements. Thus natural symmetric monodromy gives support density \(1-D_n/n!\to 1-e^{-1}\), while the Cameron–Cohen bound gives the universal ceiling \(1-1/n\), attained by sharply \(2\)-transitive affine monodromy. We give an explicit polynomial realization of this optimal mechanism. For \(N=p^e\) and \(h\mid N-1\), set \[ \Lambda_{N,h}(U)=U\bigl(U^{(N-1)/h}-1\bigr)^h . \] Its geometric Galois closure is rational, \[ U=z^h, \qquad T=(z^N-z)^h, \] and its geometric monodromy is the affine group \((\F_N,+)\rtimes H_h\). For every extension \(\F_Q/\F_p\) we compute the complete fibre enumerator of \(\Lambda_{N,h}\) exactly, including the nonregular cases. In the full affine case \(h=N-1\), the polynomial \[ U(U-1)^{N-1} \] attains the Wan–Shiue–Chen upper bound for non-permutation polynomials over every finite field containing \(\F_N\); over arbitrary extensions we compute the exact defect from that bound.

In this paper, we assume that all the zeros of the Riemann zeta function are simple. Under this assumption we give an explicit formula for the function $\Phi(e^{-t})=\sum_{n=1}^{\infty}\mu(n)e^{-nt}$, as a function of the values of $\zeta(s)$ and $\zeta'(s)$ at the odd integers and as a function of the zeros of $\zeta(s)$. A structural feature distinguishes this formula from the classical explicit formula for the Mertens function: the poles of $\Gamma(s)$ collide with the trivial zeros of $\zeta(s)$, producing double poles whose residues contain a logarithmic term. Using this formula, we give a criterion for the Riemann hypothesis: the bound $O(x^{-1/2})$ on the transform implies the Riemann hypothesis unconditionally, while the converse direction requires additional hypotheses on the zeros. We also introduce special entire functions related to $\zeta(s)$ and show that they admit absolutely convergent closed forms as Möbius-weighted series of Bessel functions of rotated argument.

Recently, large language models (LLMs) have been successfully adopted in various fields, including wireless communications, robotics, and autonomous vehicles, owing to their outstanding adaptability and reasoning abilities. Despite their huge potential, the application of LLMs for mobility management is relatively scarce since it requires not only analyzing wireless measurements but also predicting dynamic user trajectories and making real-time handover decisions across densely deployed small base stations (SBSs). In this paper, we propose an environment-aware mobility management scheme based on large multimodal models (LMMs), which extend capabilities of LLMs to process multimodal sensing data. By leveraging LMMs, the proposed scheme extracts contextual information on the surrounding environments from RGB-D images to capture user equipment (UE) mobility patterns and identify signal reflections and blockages caused by static reflectors and dynamic obstacles. Using the extracted environmental information, the proposed scheme learns the intrinsic mapping from UE and SBS positions to channel capacity, referred to as channel capacity map (CCM), from which future channel capacities along UE trajectories are predicted. Based on the predicted channel capacities, we determine proactive handover decisions maximizing the cumulative channel capacities. Simulation results demonstrate that the proposed scheme achieves substantial channel capacity improvements over conventional deep learning (DL)-based approaches.

An integer $n \ge 1$ is cyclic if $\gcd(n,\varphi(n))=1$ (equivalently, if every group of order $n$ is cyclic), and Sophie Germain cyclic if both $n$ and $2n+1$ are cyclic. Let $C_\sigma(N)$ count the Sophie Germain cyclic integers in $[1,N]$. Cohen conjectured that $C_\sigma$ is subadditive, $C_\sigma(m+n) \le C_\sigma(m)+C_\sigma(n)$ for all $1 \le m \le n$ (his Conjecture 66), having checked $m,n \le 10^6$ without finding a counterexample. We give one: at $m=31$, $n=3928$, $C_\sigma(3959)=697 > 696 = C_\sigma(31)+C_\sigma(3928)$. The argument is short, and is verified by the Lean 4 kernel.