Papers

Given a $\mathbb{Q}$-Gorenstein normal projective Calabi-Yau variety $Y$ with ordinary double points as singularities and a small crepant resolution $\pi\colon X\to Y$ with $X$ in the Fujiki class $\mathcal{C}$, we establish uniform estimates away from the exceptional locus where the given metrics may degenerate. When $Y$ is a threefold, we prove that these metrics also converge in a Gromov-Hausdorff sense to the metric completion of $Y$ on the regular locus. In particular, our results extend the work of Song on a conjecture of Candelas and de la Ossa.

The cyclic steepest descent method computes the exact steepest descent stepsize at the start of each cycle and reuses it for a fixed number $m$ of iterations. For strictly convex quadratics in $n$ variables, it has been suggested that the method converges $R$-superlinearly whenever $m\ge(n+1)/2$. We show that this cannot hold for every initial point, already at the threshold cycle length. For $n=m=2$ we exhibit an initial point from which the method follows the classical zigzag trajectory of steepest descent: the stepsize never changes and the error is halved at every iteration. The same mechanism yields linearly convergent trajectories for every Hessian with at least two distinct eigenvalues and every cycle length. For $n=3$ and $m=2$ we construct a second, genuinely three-dimensional counterexample: all components of the gradient remain nonzero, consecutive cycles use two different stepsizes, and every four iterations reduce the gradient exactly by the factor $1/49$. Both trajectories are given in closed form. Consequently, any $R$-superlinear convergence result for this method at the threshold must exclude a set of initial points, as recent almost-everywhere results do.

We prove that every event invariant under swaps has probability zero or one for either the free or the wired spanning forest on any connected, locally finite, infinite network. Applications include the constancy of the number of components and of the total number of excessive ends. For the free forest, these results resolve longstanding questions of Benjamini, Lyons, Peres, and Schramm [Ann. Probab. (2001), Question 15.7] on constancy of the component count and of Lyons, Morris, and Schramm [Electron. J. Probab. (2008), Question 7.7] on the zero-one law for one-endedness. In the latter case, we obtain the stronger conclusion that the total number of excessive ends is deterministic.

Let $A$ be a string algebra. We identify its Gorenstein-projective support $\tau$-tilting pairs with a distinguished class of maximal good collections and give a finite compatibility graph for them. Under the G-condition, we construct an explicit labelled tiled-surface model for $\mathrm{End}_A(T)$ for every nonzero basic Gorenstein-projective module $T$. Under the same condition, we show that $A$ is representation-finite if and only if $\End_A(T)$ is representation-finite for every basic Gorenstein-projective module $T$.

We examine the relationship between trees and narrow systems. In particular, we prove that if a tree of height $\kappa$ admits a narrow subsystem, it must contain a well-pruned subtree of height $\kappa$. We extend these results to two-cardinal trees.

We present a continuous Galerkin, Deferred Correction (cG-DeC) method for the equations of Lagrangian hydrodynamics. The proposed scheme combines a high-order continuous finite element discretization with an explicit Deferred Correction time integrator, yielding a formulation that avoids the inversion of a global sparse mass matrix at each update. To stabilize the method in the presence of strong shocks we introduce a modification of the hyperviscosity model that is compatible with the cG-DeC framework and likewise does not require a global solve to construct the finite element approximation to the polyharmonic operator. We further present an alternative reformulation of the DeC iteration which admits a simple recursive algorithmic structure and provides insight into previously observed convergence behavior of explicit DeC applied to hyperbolic partial differential equations. Conservation of momentum and total energy of the resulting fully discrete scheme is analyzed. A set of numerical experiments illustrates the accuracy and robustness of the cG-DeC scheme. Comparisons with a continuous Galerkin Runge-Kutta (cG-RK) method are provided.

Let $\delta^*_{k,r}$ be the fractional $K_r^{(k)}$-decomposition threshold in minimum codegree. For fixed $k\ge2$, $r\ge k+1$ and $\varepsilon>0$, we prove that every $n$-vertex $k$-uniform hypergraph $G$ with minimum codegree at least $(\delta^*_{k,r}+\varepsilon)n$ admits, with high probability, a fractional $K_r^{(k)}$-decomposition after retaining each edge independently with probability $p \ge C(\log n/n^{r-k})^{1/(\binom{r}{k}-1)}$. The bound on $p$ is sharp up to a constant factor, confirming the conjectured threshold order in the graph case. The proof develops a probabilistic matrix-scaling approach.

The discovery of Chair44 (Tsiokos, 2026) settled the three-dimensional einstein problem with a strongly aperiodic polyhedral monotile in $\mathbb{R}^3$. This note extends the underlying mechanism—the rep-$2^N$ chair $C_N = [0,2]^N \setminus (1,2]^N$ with corner/socket markings—to $\mathbb{R}^N$. Besides expository material (the rep-$2^N$ dissection and a conditional strong-aperiodicity theorem under lattice registration and hierarchical enforcement), the note makes a new computational contribution. We introduce a frame-marking formalism in which the marking of a tile is its full orientation frame and the matching rule is the contact language generated by the substitution itself; this makes the search for matching rules finite in every dimension. We give a finite certificate (coarsening closure, tightness, and a two-shell enclosure analysis) whose validity implies that every lattice-registered tiling by the marked tile is uniquely hierarchical, hence strongly aperiodic. For $N=3$ the certificate passes: it yields explicit facet matching rules on the 24 panels of $C_3$ (135 admissible facet-contact triples) and reproduces, from first principles and independently of published constructions, the Chair44 statistics 2388 $\to$ 44 admissible contacts (30 occurring), 33 one-shell clusters, 15 extendable, each forcing a unique supertile. Among the 2187 homochiral frame assignments of the 3D substitution with a translated central child, the certified one is unique up to conjugation. For $N=4$ the same pipeline is run on several structured families of frame assignments (canonical, $D_4$-, $Z_2\times Z_2$- and $Z_4$-symmetric, and a lift of the 3D solution); none is coarsening-closed, and we report the failure data. A self-similar marking of $C_4$ thus remains an explicitly finite, open computational problem, which we state precisely. Code: https://github.com/dimkadimon/Monotile-RN

A magic square of order $n$ is a square array of $n^2$ distinct integers such that the integers of every row, every column and the two diagonals have the same common sum. If a magic square remains a magic square even when each integer is replaced by its square, it is called bimagic. If, however, all the rows and all the columns of the aforementioned square of squares add up to a common sum, but the two diagonals do not, the original magic square is called pseudo-bimagic. Except for one family of $6 \times 6$ pseudo-bimagic squares, parametrized families of bimagic squares have not been published. In this paper we construct parametrized families of $6 \times 6$ pseudo-bimagic squares and $8 \times 8$ bimagic squares. We use these families to construct, for any positive integer $m$, multi-parameter families of pseudo-bimagic squares of order $6m$ and bimagic squares of order $8m$. We also give numerical examples of $6 \times 6$ and $12 \times 12$ pseudo-bimagic squares and an $8 \times 8$ bimagic square.

We study the Dirichlet problems with unbounded boundary data. Our main theorem proves interior continuity from a local biholomorphic deformation satisfying a one-sided compatibility condition near the infinite locus and barriers on the associated boundary cluster set. The deformed domains need not be nested, and the proof uses neither a Runge hypothesis nor a pseudoconvex ambient neighbourhood. We also prove nonmonotone Perron–Bremermann convergence outside an intrinsic neighbourhood pluripolar hull, a disk-fibre hull representation, stability under simultaneous nonmonotone variations of the domain and obstacle, and uniqueness in a fixed obstacle class.

Large language models can translate business descriptions into optimization models, but executable code may misrepresent constraints or objectives. A solver can then return an optimal solution to the wrong problem. Even when the solution satisfies the intended operating rules, a better plan may exist. For organizations that repeatedly use optimization modeling, an LLM-based framework should produce accurate formulations at low cost and, ideally, run locally. We study how to verify improvements and allocate attempts across LLMs that differ in price and capability. We develop OSCAR (Optimization modeling by Simulator, Coder, And Reviewer), which uses an offline Simulator certified against labeled decision examples to compare candidates and continues searching beyond feasibility. We model the search for the next certified improvement as sequential decisions under unobserved difficulty: which LLMs to call and when to stop. In a simplified known-prior setting, we give conditions under which cost-ordered escalation is optimal. For general menus, we derive a prior-free competitive guarantee. On five benchmark problems, OSCAR achieves 95% to 100% accuracy at the reported settings using two small open-weight LLMs, each deployable locally on a single GPU. Their single-attempt accuracies average 29% and 48%. In five runs per problem, Codex and Claude Code incur average token costs 3.1 and 5.8 times OSCAR's, respectively. OSCAR supports open-weight models locally or in the cloud, depending on budget and confidentiality requirements. Firms should maintain labeled decision examples of feasible and infeasible decisions to clarify plain-language operating rules. OSCAR follows these labels when an LLM's interpretation conflicts with them. As LLM capabilities and prices change, OSCAR's simple operating rules and adjustable settings help firms adapt their model choices and benefit from these advances.

We prove the continuity of the pluricomplex Green functions introduced by F. Lárusson and R. Sigurdsson on a class of bounded (B)-regular domains that includes all (B)-regular convexifiable domains.

One way to make AI systems safe is to shape what the system is: its objective and dispositions. We take a complementary route: treat the agents' characteristics as partly unknown and ask what structure of interaction ensures that bad collective outcomes are not equilibria. Mean field games suit this when many interchangeable agents are coupled through an aggregate. We introduce a program for using them in AI safety and carry one example through end to end: the July 2026 incident in which about 1,200 agents in an OpenAI evaluation coordinated on an improvised message board and 684 attacked a third party's infrastructure. We model the decision to attack as a mean field game of optimal stopping whose gain is a product: belief that provenance will be audited, times reachability of the record, minus the perceived hazard. The central result is an exact threshold on the belief. No agent attacks unless the population's confidence that provenance is checked exceeds $\pi^{**} = \eta/(\eta + \psi + \varepsilon a \overline{M})$, where $\eta$ is the perceived hazard, $\psi$ and $\varepsilon a \overline{M}$ measure how far one attacker and the collective can alter the record, and $\overline{M}$ is the peak population. Below it, no attack is the unique equilibrium for all agent parameters. The threshold survives every enrichment we consider. We then use the per-agent record to discipline the model. Its features, a stable minority attacking for thirty hours and then a pivot in which most of the board joined within a day, motivate each refinement. The account that emerges is heterogeneous belief meeting a sequence of public discoveries, each lowering the belief at which attacking paid. A few coordinating agents made those discoveries, so the model describes the several hundred who responded, not the few who produced them; a major-player version is left to future work.

Understanding how macroscopic wealth distributions emerge from microscopic transaction rules among agents is a central problem in econophysics. While many traditional models restrict agents' wealth to non-negative values, debt is an essential feature of realistic economic systems. In this paper, we introduce a stochastic transaction mechanism into the agent-based model with a central-bank-mediated collective debt limit, as studied in [14]. In our model, each agent probabilistically determines whether to transfer a coin, depending on their current asset or debt level. From a probabilistic perspective, the resulting system can be formulated as a zero-range process with $\mathbb{Z}$-valued site occupation numbers, where negative values represent debt. We prove that, under an appropriate scaling, the wealth distribution in the stationary state converges to either a two-sided Gamma distribution or a Gamma distribution, depending on the asymptotic behavior of the function governing coin transfers. Our results rigorously identify these previously unconsidered limiting distributions and provide a mathematical framework for understanding how stochastic monetary exchanges give rise to macroscopic wealth distributions.

In this paper, we study the presence of perfect powers in orbits of polynomials, or of semigroups generated by a finite family of polynomials, defined over a number field. In particular, we correct and generalize a result of Ahmadi et al. (2012), showing that multiplicative shifts of perfect powers in orbits can be detected by a finite algorithm. We also give a finiteness result on elements in orbits satisfying certain multiplicative relations.

We prove that a local minimal hypersurface $M^3$ in the unit sphere $S^4$ with constant scalar curvature and three distinct principal curvatures at every point must be locally congruent to the minimal Cartan isoparametric hypersurface in $S^4$, which provides a positive answer to the Bryant conjecture \cite{ChangLocal}. The key ingredient is to prove that, for each fixed \(S_0>3\), the set $\mathscr{G}_{S_0}$ of fourth-order jets of normalized local graph solutions satisfying \(|A|^2\equiv S_0\) is compact, where \(A\) denotes the shape operator.

Iterative methods are extremely popular for solving linear ill-posed inverse problems. While deterministic convergence, or more precisely, semi-convergence of such methods is widely studied, the problem of quantifying uncertainty in the resulting reconstructions due to random noise in the observed data has received far less attention. In this work, we study uncertainty quantification for the Landweber iteration in linear inverse problems using the Radon transform as a motivating example. We interpret the Landweber reconstruction statistically by analyzing how uncertainty in the measured data is propagated through the reconstruction map. This perspective is closely related to generalized fiducial inference, where uncertainty about the parameter is induced by inverting the relation between the observed data and the unknown (fixed) quantity. We combine the resulting stochastic uncertainty with a theoretical bound on the regularization bias to construct confidence intervals for the reconstructed solution. The evaluation of the resulting uncertainty estimates require repeated operations with the forward operator that can become expensive in large-scale problems. To reduce this cost, we use randomized singular value decomposition to obtain a low-rank approximation of the normal operator associated with the Radon transform and incorporate this approximation in quantifying uncertainty. {{In particular, this requires that the additional randomization error introduced due to the use of such randomized techniques be included in our approach for uncertainty quantification.}} Numerical results show that the proposed method provides accurate reconstructions and reliable uncertainty estimates at substantially reduced computational cost.

In the fair dueling game, two players take turns shooting at each other, with the turn order defined by the following fairness criterion: the next shot is assigned to the player who is less likely to have won the duel so far. Cooper and Dutle showed that when both players have the same probability $p$ of a successful shot, the shooting sequence converges to the Thue–Morse sequence as $p \rightarrow 0$. We extend this model by forcing the sequence to begin with an arbitrary finite prefix before reverting to this fairness criterion. We prove that this initial disruption fundamentally alters the sequence's eventual behavior: in the $p \rightarrow 0$ limit, unless the initial prefix is itself a prefix of the Thue–Morse sequence or its complement, the resulting sequence is eventually periodic. Moreover, the least period is a power of 2 that is at most $4^{n+1}$, where $n$ is the length of the prefix, and the repeating block is itself a prefix of the Thue–Morse sequence.

We study the spectral properties of the harmonic modification of the quantum mechanical Kronig-Penney model motivated by the theory of the dressing chain. In particular, we show that in the special cases the corresponding ground states can be given explicitly and all spectral gaps have the same length with the ends expressed as zeros of the corresponding Weber-Hermite functions.

Brandt [Discrete Math. 183 (1998) 17–25] conjectured that the least signless Laplacian eigenvalue of every regular triangle-free graph of order $n$ is at most $4n/25$. Using flag algebras, Balogh, Clemen, Lidický, Norin and Volec [SIAM J. Discrete Math. 37 (2023) 1173–1179] established the stronger bound $15n/94$ for all triangle-free graphs. We investigate the effect of additionally excluding pentagons and prove that every $\{C_3,C_5\}$-free graph $G$ of order $n$ satisfies $\qmin(G)<0.0569n$, without any regularity assumption. The proof combines seven-vertex flag inequalities with local Rayleigh constraints that retain the least eigenvalue throughout the counting argument. An exact integer certificate establishes the required inequality.

The finite group isomorphism problem asks whether two finite groups of order $N$ are isomorphic. The first algorithm, attributed to Tarjan (see Miller, STOC '78), runs in time $N^{\log N + O(1)}$. Despite intensive study, the current best known algorithm has a running time of $N^{(1 / 4 + o(1))\log N}$ (Rosenbaum, '13). $p$-groups of class $2$ have been recognized as the major bottleneck for faster group isomorphism. Recent progress has led to $N^{o(\log N)}$-time algorithms for $p$-groups of class $2$ where $p$ is odd (Sun, STOC '23; Ivanyos–Mendoza–Qiao–Sun–Zhang, FOCS '24; Grochow–Qiao–Stange–Sun, STOC '25). However, the case of $p=2$, which represents the majority of $p$-groups of class 2 assuming a well-known conjecture in group enumeration, remained elusive, with essentially no progress until now. In this paper, we present an algorithm for testing the isomorphism of two 2-groups of Frattini class 2 of order $N$ in time $N^{O((\log N)^{1/2})}$. To our knowledge, this is the first $N^{o(\log N)}$-time isomorphism algorithm for a class of $2$-groups that constitutes logarithmically almost all $2$-groups, in the sense that $\lim_{N \to \infty} \frac{\log(\text{\# 2-groups of Frattini class 2 and order } \leq N)}{\log(\text{\# 2-groups of order} \leq N)} = 1$. As our main tool, we present the first non-trivial algorithms for the quadratic form space/tuple isometry problems over $\mathbb{F}_2$. These algorithms rely on combinations of combinatorial and algebraic ideas, including finite matrix group algorithms developed by Luks (FOCS '92). As far as we know, this is the first time that matrix group algorithms are used to make progress on the worst-case complexity of $p$-group isomorphism.

We study the inverse problem of determining a connected, closed Riemannian manifold of dimension at least two from wave observations. The sources are supported in an open set $S$, and the resulting waves are observed on an open receiver set $R$, which may be at a positive distance from $S$. We prove uniqueness up to isometry within the class of manifolds that satisfy a subcritical spectral lower bound: $\|\phi|_R\|_{L^2(R)} \ge ce^{-C\lambda^\theta}\|\phi\|_{L^2(M)}$ for every eigenfunction $\phi$, where $\lambda$ is the corresponding eigenvalue, $\theta\in[0,1/2)$, and the constants $c,C>0$ do not depend on $\phi$. If a bound of this type holds also on $S$, then uniqueness holds within the class of all connected, closed manifolds. By the spectral inequality, the bound always holds with $\theta=1/2$. Uniqueness nevertheless fails at this endpoint: in every dimension $n\ge 2$, we construct a pair of nonisometric connected, closed Riemannian $n$-manifolds that produce identical wave observations.

We study endpoint mapping properties of low-energy wave operators for Schrödinger operators $H=-\Delta+V$ on $\mathbb R^n$, $n\ge3$. In dimension four, we prove that a zero-energy resonance prevents $L^1$ boundedness, whether or not zero is also an eigenvalue, under $|V(x)|\lesssim\langle x\rangle^{-\beta}$ with $\beta>10$. For a zero-energy eigenvalue without a resonance, we obtain a complete low-energy $L^p$ classification in every dimension $n\ge3$ under $\beta>n+4$. In particular, $L^\infty$ boundedness is equivalent to the vanishing of the zeroth, first, and harmonic second moments of $V\psi$ for every zero-energy eigenfunction $\psi$. The proof identifies the finite-rank obstruction and shows that the remaining eigenvalue correction cannot cancel its critical asymptotic profiles. The same conclusions hold for the full wave operators when the corresponding high-energy bounds are available.

A 35 input/single output quantum phase neural network (QPNN) is proposed to recognize the English letters' A 'and' B 'composed 7*5 of pixels. Each node of the proposed QPNN has the input with 0 or 1. Through normalization and quantum phase encoding, each set of 35 with 0 or 1 digital inputs is encoded into a quantum superposition state described by 35 quantum phases. Then, a quantum rotation gate is used to introduce adjustable phase, and a controlled NOT gate is used to entangle the relative phase between the two inputs. The relationship between the input and output of the network is derived. This paper also derived the phase weight learning training algorithm with adaptive learning rate. The solutions with the recognition probability of 1 is designed, and the analytical expressions for all adjustable phase solutions with zero errors are not unique. The experimental performance is verified on the Qiskit platform. This paper uses phase drive to convert the probability calculation into an analytical polynomial function with adjustable rotation angles in the network, avoiding the bottleneck of generating all 2^35 complex exponential amplitudes and providing an effective way to solve the "exponential wall" problem, which provides a new implementation solution for the practical application of multi-input recognition problems.

We prove the transcendence of numbers that have $\beta$-expansions (in algebraic bases $\beta$) belonging to dendric shifts with pure discrete spectrum, under combinatorial conditions on their directive sequences of substitutions. This includes Arnoux-Rauzy sequences on arbitrary alphabets and Cassaigne-Selmer sequences. The proof uses Rauzy fractals and the subspace theorem.

Let \(H(A)\) be the Hattori space associated with \(A\subseteq\mathbb R\), and put \(B=\mathbb R\setminus A\). Kulesza announced at the 2018 Joint Mathematics Meetings that Lindelöfness of \(H(A)^{2n}\) forces Lindelöfness of \(H(A)^{2n+1}\) for every positive integer \(n\). The meeting abstract states this implication without a proof. This paper proves the case \(n=1\) in ZFC. The same one-sided geometry gives a pointwise half-space criterion for affine discreteness and, when \(H(A)^2\) is Lindelöf, a four-coordinate obstruction for uncountable closed discrete monotone traces. Known results on mixed Sorgenfrey–Euclidean products lead to hereditary-Lindelöf criteria for finite powers, a countable-power characterization, and a finite-power Lindelöf reduction. A transfinite construction gives \(A\subseteq\mathbb R\) for which \(H(A)^2\) and \(H(A)^3\) are Lindelöf, whereas \(H(A)^4\) contains a closed discrete affine line of cardinality \(\mathfrak c\).

SIREN (Ship Inviscid Resistance and Elevation eNgine) is a Rankine-source panel solver for the steady wave resistance of a ship, in the tradition of Dawson's method. The free-surface condition is linearised about a base flow, and the linearisation is iterated. In the present formulation the base flow is the solution of a Neumann problem on the actual boundary: sources on the entire hull, wet and dry, and on the free surface panels, with impermeability imposed on both, so that the free surface acts as a rigid lid at the position it currently occupies. This replaces the classical double-body flow, whose image about z=0 is meaningful only for a flat surface and a hull cut at z=0, and it removes the need for desingularised free-surface panels. In the non-linear iteration the free-surface panels are moved rigidly in the vertical to the relaxed wave elevation at their centroid, the hull panels are marked wet or dry against the moved surface, and the base flow is recomputed on the moved geometry; impermeability applies to the whole hull regardless of this marking, which is used only to select which panels enter the force integral, and under-relaxation is confined to the passes where the non-linear correction accumulates without being reset. The radiation condition is imposed by a one-row upstream shift of the collocation points, and the wave resistance is obtained by pressure integration and, independently, by transverse wave-cut analysis.

We systematically explore the relationship between edge-connectivity and Lin–Lu–Yau curvature for graphs. The intuition is that a connected graph with large Lin–Lu–Yau curvature should also have large edge-connectivity, and vice versa under suitable conditions. We prove that the edge-connectivity of a connected graph is bounded lower by the product of its minimum degree, its Lin–Lu–Yau curvature and the constant $\frac{3}{2}$. We also provide two lower bounds on the edge-connectivity of a graph that guarantee positive Lin–Lu–Yau curvature, and discuss the sharpness of these bounds.

Catastrophe bonds have become a central instrument for transferring catastrophic risk to capital markets, yet their payout structures are typically specified exogenously rather than optimized. In this paper, we treat the indemnity function as the decision variable and study the optimal design of CAT bonds from the sponsor's perspective. We establish the optimality of a piecewise linear indemnity function under both expected-disutility and spectral-risk-measure frameworks and thereby reduce an infinite-dimensional design problem to a one-dimensional optimization over the attachment point. The optimal design also eliminates the potential moral hazard associated with the discontinuity of threshold payout structures. We characterize the optimal attachment point under six risk criteria and obtain explicit or semi-explicit solutions. We further derive bounds for the optimal attachment point and develop comparative statics that provide practical design insights. Using U.S. severe storm losses from NOAA's Billion-Dollar Weather and Climate Disasters database for calibration, we show that the optimal CAT bond designs achieve substantial risk reductions across most risk criteria.

Paul Erdős conjectured that the only powers of two whose ternary expansions contain no digit 2 are 1, 4, and 256. Despite extensive computational verification, the conjecture remains open. We reformulate the problem using generalised base-4 repunits. Any admissible exponent is even, say \(n=2s\), and \[ 2^{2s}=4^s=3R_s(4)+1,\qquad R_s(4)=\frac{4^s-1}{3}. \] Thus \(2^{2s}\) omits the ternary digit 2 if and only if \(R_s(4)\) does. For each depth \(m\), the map \(\Phi_m:s\bmod 3^m\mapsto R_s(4)\bmod 3^m\) is a permutation of the residue classes modulo \(3^m\). Hence every prescribed \(m\)-digit ternary suffix has a unique canonical exponent seed modulo \(3^m\), and all exponents producing that suffix form an arithmetic progression. The lifting identity \[ R_{r+q3^m}(4)\equiv R_r(4)+q3^m\pmod{3^{m+1}} \] gives three lifts at each depth, exactly two of which preserve admissibility. The finite maps assemble into an isometric homeomorphism of the 3-adic integers conjugating \(x \mapsto x+1\) with \(y \mapsto 4y+1\). The unresolved global integer problem remains. No proof of the Erdős conjecture is claimed.