Papers

We prove an \(L\log L\)-endpoint Cwikel estimate on quantum tori $\mathbb T_\theta^d$, i.e., for every \(d\geq 1\) and \(x\in L\log L(\mathbb T_\theta^d)\), \begin{equation*} (1-\Delta)^{-\frac{d}{4}}\lambda(x)(1-\Delta)^{-\frac{d}{4}}\in\mathcal{L}_{1,\infty} \end{equation*} with a bound depending only on the dimension, where $\lambda(x)$ is the left multiplication by $x$. The main ingredient is an abstract weak Hilbert–Schmidt estimate for multiplication operators in finite von Neumann algebras whose \(L_2\)-spaces admit orthonormal bases of unitaries. When $\theta=0$, our argument also gives a simpler proof of the $L\log L$ Cwikel estimate on classical tori. For $d\geq2$, we use the $C^*$-algebraic principal symbol map to prove Weyl's laws for finite sums \[ \sum_{\ell=1}^N S_\ell(1-\Delta)^{-d/4}\lambda(x_\ell) (1-\Delta)^{-d/4}T_\ell, \] where $x_\ell\in L\log L(\mathbb T_\theta^d)$ and $S_\ell,T_\ell\in\Pi$, the $C^*$-algebra generated by $\lambda(C(\mathbb T_\theta^d))$ and Fourier multipliers with continuous symbols homogeneous of degree zero. We also prove a semiclassical Weyl's law for critical Schrödinger operators with $L\log L$ potentials.

Kemp and Ramgoolam proposed a basis of monomials for a commutative semisimple algebra with generators \(C_1,\ldots, C_L\), constructed from a degeneracy graph which records how the eigenspaces of \( C_1\) are successively split by \( C_2,\ldots, C_L\). They showed that the basis property is equivalent to the non-vanishing of the determinant of the matrix of values of these monomials on the projector basis, and conjectured a factorised formula for this determinant, as a product of powers of eigenvalue differences. We prove this formula for any number of generators. The proof uses graph analogues of Newton polynomials. Their evaluation matrix is not triangular in general, but expanding the rows of the children of each node of the penultimate layer in a Newton basis of their eigenvalues makes it block triangular, with diagonal blocks given by smaller degeneracy graphs in which the final layer is removed. The formula then follows by induction on the number of layers. We show that the exponents in the determinant are overlap volumes of higher-dimensional Young diagrams attached to the nodes of the graph: intervals, ordinary Young diagrams, plane partitions and their higher analogues. Finally, the block triangular structure gives an explicit recursive inversion of the evaluation matrix, expressing the projectors in the monomial basis.

The problem of maximizing vertex-degree-based topological indices over trees with a prescribed order and maximum degree $\Delta$ has traditionally relied on ad-hoc combinatorial arguments. In this paper, we establish a unified linear programming approach that systematically resolves these extremal problems. By formulating the optimization over degree-pair multiplicities, the search for an extremal tree reduces to minimizing a discrete dual penalty. Treating vertex degrees as continuous variables yields an analytical single-variable test condition that guarantees global optimality. We apply this framework across a diverse collection of invariants. We prove that the standard basic feasible solution is optimal for the Tangent Sombor, $SO_1$, $SO_2$, Albertson, and $\sigma$-irregularity indices for all $\Delta \ge 3$. However, the dual optimality conditions establish exact analytical thresholds at which the standard basis loses optimality, necessitating a transition to alternative basic feasible solutions. For the Arithmetic-Geometric ($AG$) index, the standard basis is optimal strictly for $\Delta \ge 9$; for smaller maximum degrees ($3 \le \Delta \le 8$), optimization shifts to an alternative edge basis, allowing us to explicitly construct and characterize all extremal trees across congruence classes, including chemical trees ($\Delta \in \{3, 4\}$). Conversely, for the Triangle Volume index ($SO_{11}$), the standard basis is optimal exclusively for $\Delta \le 8$, while for $\Delta \ge 9$ we analytically derive a novel, degree-dependent optimal basis to explicitly construct its extremal configurations.

We address a question of Canary–Taylor and Canary–Minsky–Taylor for sufficiently symmetric compact hyperbolic $3$-orbifolds with totally geodesic boundary. We give sufficient conditions on the symmetries of such an orbifold to show that, among the complete convex cocompact structures on the interior, the structure with Fuchsian ends is a critical point (respectively, a strict local minimum) for the critical exponent. We also provide an infinite family of non-rigid examples fulfilling these conditions.

We establish finite-dimensional von Neumann type inequalities for commuting matrix tuples in noncommutative operator balls. Let $\Omega=\mathbb{D}_Q$ be the NC operator ball of a linear pencil $Q$ with linearly independent coefficients. If $\Omega(1)$ admits a transitive subgroup of automorphisms with NC lifts and $\Omega(1)$ has the bounded $H^\infty$-Gleason property at the origin, then $\|f(T)\|\leq C_{\Omega,n}\|f\|_{H^\infty(\Omega(1))}$ for every commuting tuple $T\in\Omega(n)$, with $C_{\Omega,n}=(2\rho_{\Omega}K_{\Omega})^{n-1}$; here $K_{\Omega}$ is the Gleason constant and $\rho_{\Omega}$ is the sum of the suprema of the coordinate functions on $\Omega(1)$. The scalar coordinate bounds persist at every matrix level. The hypotheses hold for products of row and column balls, where $\rho_{\Omega}=d$, yielding an unconditional estimate $(2dK_{\Omega})^{n-1}$. For a general type I Cartan domain $R_{p,q}$, we obtain only a conditional result: the same bound holds provided the scalar domain has the bounded $H^\infty$-Gleason property at the origin, which we do not establish when $\min(p,q)\geq 2$.

Partially optimal assignments (also called persistencies) allow combinatorial optimization problems to be reduced by fixing a subset of variables to values consistent with at least one optimal solution. For the quadratic unconstrained binary optimization (QUBO) problem, such assignments can be computed efficiently via autarkies of quadratic posiforms. However, the largest autarky domain and thus the number of fixed variables can depend on the specific posiform representation. We show that every quadratic posiform with the greatest possible constant term (the roof-dual bound) yields the largest autarky domain obtainable from any quadratic posiform. By computing such a roof-dual-optimal quadratic posiform and a corresponding autarky with largest domain in polynomial time, one can therefore attain the maximum partial optimality obtainable by quadratic posiform autarkies in polynomial time. This justifies the computation of roof-dual-optimal quadratic posiforms and corresponding autarkies in popular polynomial-time algorithms and establishes that finding additional partial optimality in QUBO requires going beyond autarkies of quadratic posiforms.

We study nonconvex-strongly-concave minimax problem $\min_{\mathbf{x}\in\mathbb{R}^{d_x}}\max_{\mathbf{y}\in\mathbb{R}^{d_y}}f(\mathbf{x},\mathbf{y})$ in which the objective has $\rho$-Lipschitz Hessian and condition number $\kappa$. We propose envelope cubic regularized Newton method (ECRN) which applies a refined inexact cubic regularized Newton method to the envelope of the primal function $\Phi(\cdot) : = \max_{\mathbf{y}\in\mathbb{R}^{d_y}}f(\cdot,\mathbf{y})$. We show that ECRN finds approximate second-order stationary points of $\Phi(\cdot)$ with $\widetilde{\mathcal{O}}(\sqrt\rho\,\kappa^{3/4}\varepsilon^{-3/2})$ second-order oracles, improving the condition number dependence $\mathcal{O}(\kappa^{2})$ achieved by the best known second-order methods. Our improvement comes from the key observation that the Hessian Lipschitz parameter of the envelope is $\mathcal{O}(\rho\kappa^{3/2})$, which is sharper than the order of $\mathcal{O}(\rho\kappa^{3})$ for the function $\Phi(\cdot)$. We also conduct a lower bound on second-order methods to show that our complexity is optimal up to a logarithmic factor.

Let $S_m$ be the Schur multiplier with symbol $m$. We characterize the boundedness of $S_m:\mathcal S^1\to\mathcal S^{1,\infty}$ by approximation of weighted columns in a common finite-dimensional subspace. If $W(m)$ is the weak $(1,1)$ bound and $\fD(m)$ is the column approximation constant, then $\tfrac12W(m)\le\fD(m)\le2W(m)$. These estimates hold uniformly under amplification. Our second main result is the one-dimensional Hörmander–Mikhlin–Schur weak $(1,1)$ estimate. This settles the one-dimensional case of the weak $(1,1)$ problem stated in Parcet's ICM 2026 survey. The proof uses two adjacent dyadic grids and block decompositions indexed by finite binary trees. We apply Cuculescu's projections to the associated block-diagonal filtration. The resulting tree estimate is independent of the depth and grows as $2^{r/2}$ with the descendant complexity $r$.

Let $G=C_2\times C_2$ and $A=\{0,1\}$. We characterize the submonoid of $\text{CA}(G;A)$ generated by lazy and invertible cellular automata. A cellular automaton belongs to this submonoid if and only if, whenever it maps the two free $G$-orbits onto the same free orbit, it maps both constant configurations and some two-point orbit to the same constant. The proof gives a factorization using elementary orbit collapses and three two-cell lazy maps. The criterion yields the cardinality $59\,712$ and identifies the maps missed by a weaker constant-separation invariant. Adjoining one idempotent generates the full monoid, and the generated submonoid is contained in exactly one maximal proper submonoid, of cardinality $65\,024$. The submonoid generated by lazy maps alone has $4\,007$ elements, and for both generating sets the maximum shortest-word length is six. The criterion and the cardinalities $59\,712$ and $4\,007$ are formally verified in Lean.

We study cache-aided linear computation in distributed systems where each source generates and stores its own data. An external receiver requests a linear combination of these data, and the sources use information cached before the request is known to reduce the communication needed to recover the requested result. In our model, each of the $N$ sources stores its own independent file over a finite field and has an additional cache of normalized size $M$. The receiver must recover any requested linear combination with zero error from the messages sent by the sources over separate noiseless links. We establish a lower bound on the worst-case communication rate over any finite field by analyzing several requests under the same cache placement. This bound determines the optimal rate when no additional cache is available. For binary files, we propose parity-aided placement (PAP), a coded-placement scheme that combines cached parities with uncoded symbols. For every $N\geq3$, we characterize its communication rate for each request and optimize its placement parameters to obtain a closed-form worst-case rate. The resulting achievable rate meets the lower bound for $0\leq M\leq1/(2N-1)$ and $M\geq(N-1)/N$, establishing the optimal memory–rate tradeoff in these two ranges.

In this paper we describe the quotients of several plactic-like monoids by the least congruences containing the relations $a^{\sigma(a)} = a$ for $\sigma(a) \geq 2$ for every generator $a$. The starting point of this paper is the recent paper of Reutenauer and the first author about the stylic monoid. This monoid is the quotient of the plactic monoid by the relations $a^2 = a$ for every generator $a$. The plactic-like monoids considered here are the plactic monoid itself, the Chinese monoid, the hypoplactic monoid, the sylvester monoid, the metasylvester and hyposylvester monoids. In each case we describe a set of normal forms and the idempotents. We obtain formulae for their cardinalities and we describe their isomorphism types.

Shute showed that the leading constant in the asymptotic formula for squareful values of the form $ax^2+by^2$ does not match with the conjecture established by Pieropan, Smeets, Tanimoto and Várilly-Alvarado. In this paper, $S$-squareful values of this form are considered for $S$ the set of primes that divide $a$ and $b$. We prove an asymptotic formula for these numbers and show that the leading constant of the asymptotic formula still does not match the conjecture. Furthermore, we consider the updated conjecture established by Chow, Loughran, Takloo-Bighash, and Tanimoto regarding the leading constant for Campana points in the given setting and show that this constant is consistent with our computed constant.

We study the Cauchy problem for the energy-critical Schrödinger map into the 2-sphere: \[ \partial_{t}u = u\times\Delta u, \quad (t,x)\in\mathbb{R}^{1+2}, \quad u(t,x)\in\mathbb{S}^2 \] for equivariant initial data of homotopy degree $k=1$. Given any integer $L\in\mathbb{N}_+$, we exhibit a set of initial data arbitrarily close to the ground state harmonic map in the energy norm such that the corresponding solution blows up in finite time with a quantized rate \[ \lambda(t)\sim \frac{\left(T-t\right)^L}{\left|\log(T-t)\right|^{\frac{2L}{2L-1}}} \quad\mbox{as}\ \ t\uparrow T. \] Our proof is inspired by the techniques developed for the energy-critical harmonic map heat flow in a similar context by Raphaël and Schweyer~[Anal. PDE 7 (2014), 1713–1805]. However, its adaptation to a Schrödinger-type equation requires the introduction of a new energy-Virial-Morawetz Lyapunov functional tailored to quantized slow blowup dynamics.

For each positive integer $N$, let $p_i$ be the $i$-th prime for $i\in\{1,\ldots,N\}$, and set $$ \Phi_N=\left\{\prod_{i=1}^{N}p_i^{a_i}:a_i\in\{0,1,\ldots,N-1\}\right\}. $$ Write $f(N)$ for the largest cardinality of a subset of $\Phi_N$ containing no two distinct legs of an integral right triangle. We prove $$ N^{N-1}(\log\log\log N)\ll f(N). $$

We study the finite basis problem for flat semirings of the form $S(W)$, where $W$ is a nonempty set of nonempty words, possibly infinite. We prove that $S(W)$ is finitely based if and only if the strict interior of every word in $W$ is linear; here the strict interior of a word is obtained by deleting its first and last letters. This completely resolves an open problem raised by Jackson et al.~(J. Algebra 611: 211–245, 2022).

For quandle rings, Bardakov–Passi–Singh and Elhamdadi–Nunez–Singh respectively formulated an analogue of Kaplansky's zero divisor conjecture and a weakened analogue of Kaplansky's idempotent conjecture involving the augmentation map. In this note, we verify these conjectures for medial quandles. The latter holds for a larger class of quandles; we characterize integral domains over which the statement holds for this class. We also characterize integral domains of characteristic 0 over which the statement holds for a given finite Alexander quandle.

We show that purely continuous microscopic diffusions can homogenize to a Markov process with genuine jumps. Consider a multiscale diffusion with time-dependent generator $\mathcal{L}_{0}+r_{t}^{\epsilon}\mathcal{L}_{1}$ driven by a regenerative random environment. When the integrated activity over a regeneration cycle has a regularly varying tail of index $\beta\in(0,1)$, the cumulative operational clock converges to a $\beta$-stable subordinator, compressing finite activity-driven diffusion into vanishingly short physical time intervals. To overcome the noncommutativity of $\mathcal{L}_0$ and $\mathcal{L}_1$, we develop an extended-time parametrization paired with Khasminskii-type time discretization, establishing finite-dimensional convergence and strong semigroup convergence on $C_{0}(\mathbb{R}^{d})$ toward a jump process with nonlocal generator $\overline{\mathcal{L}}=\mathcal{L}_{0}-\kappa(-\mathcal{L}_{1})^{\beta}$. Under uniform ellipticity of $\mathcal{L}_{1}$, the fractional power $(-\mathcal{L}_{1})^{\beta}$ is identified as a $2\beta$-stable-like nonlocal operator. We also clarify physical-time path-topology obstructions and apply the framework to Poisson resetting pulses, Ornstein–Uhlenbeck diffusions, and multiscale stochastic volatility.

Our previous work established the mapping cone Chern character on the even-degree side, using the powers of the mapping cone curvature associated with the given mapping cone connection. In this paper, we carry out the odd-degree version using a trivial mapping cone connection and an automorphism on a trivial complex vector bundle. This version is based on the related odd-degree transgression.

Given a family of $N$ i.i.d. Gaussian points in a $d$-dimensional normed space, what is the probability that some linear transformation maps it to the unit sphere? The case of a Euclidean space is known as the ellipsoid fitting problem and has recently attracted major attention. We initiate a study of this fitting problem in a general normed space, both asymptotically and in a fixed dimension. We specifically focus on the space $\ell_\infty^d$, where a combinatorial reformulation allows us to show that $cd^2/\log d$ points can be fitted with high probability. We illustrate that the geometry of the normed space is reflected in the fitting problem by showing that in dimension $2$, the spaces $\ell_2^2$ and $\ell_\infty^2$ are characterized by having extremal fittability properties.

Visual structures are often characterized by directional geometry, local oscillation, and spatial–frequency coupling. To overcome the limitations of conventional spatial convolutions mainly operating on fixed Euclidean coordinates, this article develops the learnable linear canonical transform (LLCT) framework that learns parameters within the linear canonical transform (LCT) family. The framework treats the LCT family as parameterized analytical views spanning spatial, Fourier, and intermediate linear canonical representations. The framework has the following two complementary modes. The Linear Canonical Riesz Transform (LCRT) mode combines learned linear canonical coordinates with regularized Riesz-type directional responses at sites chosen for geometric and local-structural refinement. The learnable LCT mode provides the integrated responses for latent feature sites. We establish the parameter stability of the LCT representation under admissible perturbations of the canonical matrices. Both modes use chirp multiplication and the fast Fourier transform (FFT) to preserve feature dimensions and support automatic differentiation. Experiments on crack segmentation and single-image deraining evaluate the above two modes with multiple backbones. On four segmentation backbones, the full LCRT mode gives the highest reported point estimates of F1 and IoU for the crack class among the tested variants. The learnable LCT mode gives higher values in 30 of 32 paired PSNR and SSIM measurements across four deraining backbones and four datasets. In ablation studies, the full model has the highest reported PSNR and SSIM among five tested configurations.

A central goal in solving a polynomial eigenvalue problem is to compute eigenpairs that are exact for a nearby polynomial, with each coefficient perturbation bounded by a modest multiple of unit roundoff times that coefficient's norm. A backward stable eigensolver for a linearization does not automatically provide this coefficientwise relative guarantee. Scaling can improve accuracy, but a single scaling may not work across a widely spread spectrum. We construct a family of scaled companion linearizations that covers the nonzero complex plane. Each linearization is paired with a region and an eigenvector recovery rule for which the backward-error amplification factor is bounded at exact pencil eigenvectors, independently of coefficient norm disparities. Tropical roots guide the construction, and their multiplicities reduce the number of required solves. These results provide a basis for coefficientwise relative backward stability across the spectrum.

We determine the Leray–Schauder degree of the singular Dirichlet $N$-mean field equation on a smooth bounded domain in $\mathbb{R}^N$, $N\geq2$, for every positive nonresonant parameter and singular exponents $-1<\alpha_i\leq0$. The formula depends only on the Euler characteristic of the domain and the weights $(1+\alpha_i)^{N-1}$. We also identify the homotopy type of every sufficiently negative variational sublevel with the corresponding weighted barycenter space. We prove quantization of interior concentration masses and exclude boundary concentration and residual diffuse mass. The degree computation uses common descent fields for nearby parameters, bounds for their trajectories, and a finite-dimensional comparison with the Euler characteristic of negative sublevels. Annular averaging and dilation provide local contractions, which are extended to establish the sublevel homotopy equivalence. The arguments apply to the nonlinear, possibly degenerate $N$-Laplacian with singular weights, without requiring a global Palais–Smale condition or isolation and nondegeneracy of solutions.

Let $E/K$ be an elliptic curve over a field defined by the short Weierstrass equation and let $\mathbf{P}=(P_1, P_2) \in E(K)^2\setminus \{(O, O), (\pm P, \pm P)\}$ be a pair of non-torsion points, where $O$ denotes the point at infinity. We give a uniform upper bound for the quantity $\min(\nu(\Phi_{(v_1, v_2)}(\mathbf{P})), 2\nu(\Psi_{(v_1, v_2)}(\mathbf{P})))$, where $\nu$ is a normalised discrete valuation associated to primes where $E$ has bad reduction and $\Psi^2_{(v_1, v_2)}(\mathbf{P})$, $\Phi_{(v_1, v_2)}(\mathbf{P})$ are rational functions for describing $\mathbb{Z}$–linear combination of points $[v_1]P_1+[v_2]P_2$ on $E$, higher–dimensional analogues of division polynomials $\psi^2_n, \phi_n$ known as net polynomials. They are also elliptic nets, which are $n$–dimensional arrays with values in $K$ satisfying the same nonlinear recurrence relation that division polynomials do as well. Hence, they are useful in studying counting problems on elliptic curves.

Pseudo-Boolean optimization – minimizing a polynomial over $\{\pm 1\}^n$ – includes MAX-CUT, binary quadratic optimization, and Ising models. The moment-sum-of-squares (Moment-SOS) hierarchy provides certified semidefinite relaxations that are often empirically exact at low order. We ask: once a relaxation is exact, how difficult is the resulting semidefinite program (SDP) to solve with first-order methods? Exactness does not imply regularity. At every order $d \ge 2$, a unique rank-one moment optimum is primal degenerate; moreover, if one optimal SOS certificate is strictly complementary, then the certificate set is positive-dimensional and contains singular boundary points. Thus first-order methods may converge to singular endpoints, around which pointwise analyses that assume regularity at the limit cannot establish a local linear convergence rate. We show that, for exact Boolean Moment-SOS relaxations in their canonical quotient form, this degeneracy is nevertheless benign. We develop an abstract mechanism, which we call residual-field rigidity, that propagates regularity from the relative interior of the optimal set to its singular boundary. For Boolean Moment-SOS, unique rank-one exactness and the existence of one strictly complementary certificate suffice to trigger this mechanism, yielding a uniform error bound around the entire optimal set. Consequently, exact ADMM, exact sGS-ADMM, and fixed-step PDHG converge eventually linearly from every initialization, even when their limiting certificate is singular. Finally, outside a lower-dimensional semialgebraic exceptional set of Boolean objectives, these hypotheses hold simultaneously at every uniquely rank-one exact order.

We introduce a new model of parking functions, called $\rho$-Naples parking functions, in which each car has a certain number of spots to check preceding its preferred one before looking for available spots by moving forward. This notion, where each car has its own number of previous spots to check, generalizes that of Naples parking function, where this number is the same for all cars. We start by showing that, for a given sequence of preferences for each car, a successful parking strategy (that is, an assignment of how many backward spots each car is allowed to check) remains successful if we increase the number of backwards spots to check for some of the cars. This means that the set of successful strategies for a given parking preference is an up-set in the lattice of parking strategies, which raises the intriguing question of characterizing such up-sets. Though we are not able to completely solve this problem, we provide a characterization of those successful strategies which minimize the total number of allowed backward steps for a given parking preference, and we determine a lower bound for the number of cars which are allowed to move backwards (also showing that it can be attained). To conclude, we analyze the specific situation in which each car can take at most one backward step, and we completely characterize the set of successful parking preferences in this case.

Deep Gaussian processes (DGPs) use iterative sampling to create non-stationary random fields. This is notoriously expensive because the covariance at each non-initial layer depends on the realised field from the layer below, so it changes with every sample. Moreover, standard fast sampling techniques like circulant embedding or matrix compression fail since the underlying kernel usually lacks the necessary regularity and structure. We develop and analyse a wavelet compression method for convolution Matérn DGPs. At each layer the covariance matrix is compressed using wavelets and a Lanczos algorithm computes the random sample. For specific parameter ranges involving the smoothness of the Matérn kernel as well as the smoothness of the wavelets, we show that the accumulative compression and sampling errors through all layers decay exponentially with increasing truncation parameter. We obtain a practical DGP sampling algorithm by choosing a truncation that gives log-linear covariance storage and matrix-vector costs, enabling DGP sampling at nearly linear computational cost.

We prove stability of geodesic spheres under volume-preserving mean curvature flow in the simply connected space forms under genuinely first-order geometric assumptions. The initial hypersurface is required to lie in a sufficiently thin annulus around a geodesic sphere and to have outward normal making a small angle with the radial direction. No convexity, mean convexity, initial curvature bound, or Hölder control of the normal is assumed. The main ingredient is a quantitative Alexandrov reflection argument that converts these initial assumptions into a strong reflection property preserved by the flow. The resulting radial and angular control allows us to derive estimates for the mean curvature and the full second fundamental form, and hence long-time existence, eventual convexity, and smooth exponential convergence to a geodesic sphere. In hyperbolic space, the eventual convexity can be strengthened to horospherical convexity. The reflection threshold also yields a quantitative estimate for the displacement of the limiting centre. For Lipschitz initial hypersurfaces, we develop a separate positive-time curvature estimate that is uniform along smooth approximations. This produces an instantaneously smooth evolution with the same eventual convexity and smooth exponential convergence.

This paper develops a natural extension of the multipoint spherical semidefinite method of Liao and Yu~\cite{LY}, studying the interaction between semidefinite programming and strongly regular graph counting. Starting from the positive semidefinite kernels furnished by Gegenbauer polynomials, we express weighted kernel sums and counting identities in the same labelled induced-subgraph counts. Eliminating the counting equations gives a semidefinite feasibility problem. Numerical dual solutions guide the construction of rational certificates, whose contradiction identities are verified exactly. Our aim is a systematic method that treats many parameter sets through the same construction. For least eigenvalue $-3$, it gives new counting certificates for $(351,140,73,44)$, $(550,162,75,36)$, $(703,182,81,35)$ and $(1344,221,88,26)$, previously excluded by Liao and Yu~\cite[Theorem~1]{LY}, and for six additional parameter sets: \[ \begin{array}{@{}lll@{}} (375,102,45,21), & (405,132,63,33), & (441,88,35,13),\\ (476,133,60,28), & (540,147,66,30), & (575,112,45,16). \end{array} \] Seven certificates use Gegenbauer positivity and counting information alone at the displayed orders. The cases $v=375,441,575$ additionally use proved short-vector restrictions, unique vertex extensions, and integer common-neighbour inequalities. The proofs end in displayed inequalities with incompatible signs. Their justification is given by the dual identities and the proved positivity and counting relations; accompanying code automates the longer rational arithmetic.

Let $S=(\N,\max,+)$ be the max-plus algebra of the nonnegative integers, and let $\overline S$ be obtained from $S$ by adjoining $-\infty$ as a semiring zero. We first prove that the varieties generated by the full matrix semirings $\Mat_n(S)$ and by the upper triangular matrix semirings $\UT_n(\overline S)$ form strictly ascending chains. Every interval between distinct members of either chain has cardinality $2^{\aleph_0}$ and contains a chain and an antichain of that cardinality. The proof uses finite syntactic semirings associated with prime-cycle languages and an independent family of identities. We then construct finite commutative ai-semirings from languages of monomials closed under divisibility and prove their bounded-variable property by cancellation, explicit integer evaluations and independence in a free abelian group. A loop-block identity for upper triangular matrices and dimension-independent cubic-padding identities for full matrices show that every variety in each of the above intervals has infinite axiomatic rank. In fact, every variety between $\V(S)$ and the variety generated by all full matrix dimensions has infinite axiomatic rank. These varieties are therefore nonfinitely based, also relative to the ai-semiring axioms. Finally, for every fixed full matrix dimension, we prove equational equivalence over all six positive or nonnegative integer, rational and real coefficient domains. The original convex-ideal countermodels and their full-matrix transfer proof are recalled separately for comparison; they are not used in the algebraic proofs. Throughout, the signature consists of two binary operations and contains no constant symbols.

An old conjecture dating back at least to Hildebrand's work in 1989, asserts that the Möbius function exhibits cancellation on shifted primes; that is, $$\sum_{p\le x}\mu(p+2)=o(\pi(x)),\quad \text{or equivalently}\quad \sum_{n\le x}\Lambda(n)\mu(n+2)=o(x).$$ This is also a well-known model case for the parity problem in sieve theory. In this paper, we prove this conjecture. As an application, we show that each of the two sets $\{p:\, \mu(p+2)=1\}$ and $\{p:\, \mu(p+2)=-1\}$ has asymptotic density $1/2$ among the primes $p$ with $\mu^2(p+2)=1$: \begin{equation*} \lim_{x\to\infty} \frac{\#\{p\le x:\ \mu(p+2)=\kappa\}} {\#\{p\le x:\ \mu^2(p+2)=1\}} =\frac12, \quad \kappa\in\{-1,\,1\}. \end{equation*}