Papers

We construct explicit algebraic finite Gabor windows that are simultaneously full spark and phase retrievable, with quantitative control of the self-ambiguity function. In every cyclic dimension, a constant–Chu window has an exactly computable ambiguity minimum of order $N^{-3/2}$; for $N\ge25$, a two-site Chu repair has margin comparable to $dN^{-3/2}$ for every divisor $d\le\sqrt N$. A quantitative algebraic regularization, based on the high-degree specialization principle used in explicit full-spark Gabor constructions, imposes full spark while retaining a fixed proportion of a seed's ambiguity margin. These ingredients give explicit simultaneous constructions in every cyclic dimension and a factor-sensitive improvement when $N$ has a divisor near $\sqrt N$. Further consequences include exact Chinese-remainder tensorization, optimal-order squarefree families, near-SIC prime regularization, lifted stability, and a finite cyclic Schrödinger construction.

We study the structure of Gibbs states in weakly perturbed interacting fermionic systems. First, for a sparse Hamiltonian $H=H_0+V$ with a quadratic term $H_0$ and a non-quadratic perturbation $V$ of scale $\epsilon$, we show that the Gibbs state $\rho_{\beta}$ decomposes into a convex combination of Gaussian states whenever the inverse temperature satisfies $\beta \le O(\log(1/\epsilon))$. Moreover, we prove that this bound is asymptotically tight by establishing that $\beta \le \Theta(\log(1/\epsilon))$ is necessary for certain sparse Hamiltonians. This general framework applies directly to the weak-coupling (small-$\vert{}U\vert{}$) regime of the Fermi–Hubbard model with hopping $t$ and on-site interaction $U$ on any graph of maximum degree $D$. Complementarily, in the strong-coupling (small-$\vert{}t\vert{}$) regime, we show that the Gibbs state remains convex-Gaussian up to $\beta \le O\big(\vert{}U\vert{}^{-1}\log(\vert{}U\vert{}/(D\vert{}t\vert{}))\big)$, revealing a mechanism for convex-Gaussianity distinct from the weak-coupling setting.

Using a convex pentagonal monotile belonging to the Type 5 family, we investigate the relationships among the hat tile, turtle tile, and Tile$(1, 1)$. By applying Sugimoto's Perspective and Amfirifma's Perspective, we obtain four types of concave polygons, AH-tile, BH-tile, AT-tile, and BT-tile, each having Heesch number 1 under the conditions considered. We show that these polygons correspond to clusters used to generate the non-periodic tilings $\mathscr{T}_h$ and $\mathscr{T}_s$. We further discuss the possibility that tile sets consisting of pairs selected from these polygons may correspond to $\textit{ASPmr}\{\text{A-tile}, \text{B-tile}\}$.

Problem definition: The car resequencing problem involves rearranging the sequence of cars between two adjacent production shops via an intermediate buffer. This study explores a new variant that aims to minimize the resequencing makespan when the buffer features several forward lanes and a return lane (RL). Both types of lanes follow first-in-first-out rules, but operate in opposite directions. Through the RL, cars can circulate between forward lanes, thereby increasing resequencing flexibility and complexity. The goal is to position cars over time to complete the target sequence change in the shortest possible time. Methodology/results: We prove the problem is NP-complete and develop an exact two-stage branch-and-price approach integrated with constraint programming (BP-CP). Three versions of the master problem (MP) are proposed for the first stage, each paired with a subproblem defined on a specific graph. Using partial integer solutions from branching, a constraint-programming model built in the second stage yields either a feasible solution to the problem or a feasibility cut added to the MP. Experimental results indicate that BP-CP greatly increases the scale of solvable instances for a production batch, especially with a moderately relaxed MP. A comparative analysis shows that the RL markedly enhances resequencing flexibility, enabling smaller buffers to attain makespans comparable to those of larger buffers without the RL across varying resequencing complexities. Managerial implications: Expanding lane capacity alone yields limited gains in resequencing performance and may even reduce efficiency. The RL acts as several forward lanes, supporting a smaller and cost-effective buffer design while maintaining efficient just-in-time operations.

Existing classifications of real plane cubic curves rely on sophisticated tools and require a lengthy exposition. This paper provides a concise yet elementary treatment of this classical topic. Using the centers of polynomials, we establish a correspondence between the algebraic structure of these centers and geometric properties of binary cubic polynomials, which yields a novel approach to Newton's canonical forms.

We construct Ulrich bundles on smooth intersections of two quadrics. We determine all possible ranks and prove the existence of indecomposable Ulrich bundles of every allowable rank. To our knowledge, this gives the first such construction for a family of nonhomogeneous varieties of arbitrarily large dimension. We also study the moduli spaces of these bundles and relate them to moduli spaces of vector bundles on curves. For intersections of two even-dimensional quadrics, our results prove a conjecture of Eisenbud and Schreyer.

We present an intrinsic finite element framework for the numerical approximation of scalar- and vector-valued partial differential equations on general manifolds described by atlases of charts. Weak formulations and their discretisation are expressed exclusively in the flat parametric space of the manifold, where the exact geometry enters only through the metric tensor that is evaluated at quadrature points. Conforming finite element spaces of arbitrary order for the whole de Rham complex are obtained on multi-chart atlases through purely topological inter-chart gluing. We develop a nodal change of basis construction for vector-valued spaces on multi-chart atlases, where nodal degrees of freedom couple charts with incompatible coordinate systems through transmission maps. Inter-chart continuity holds at the nodes with jumps of the order of the approximation error, and the resulting discrete field is exactly tangent to the manifold. Optimal convergence rates for this method are observed numerically for the surface Stokes problem; no a priori error analysis is provided. Numerical experiments on the cubed sphere manifold in two and three dimensions for the Hodge Laplacian problems demonstrate the intrinsic framework is free of the geometric consistency errors for a sufficient quadrature degree, and that intrinsic assembly is substantially cheaper than its extrinsic counterpart. Further considering the rotating shallow water equations that include orography as a perturbation of the geometry demonstrates that mass is conserved exactly, and energy is conserved up to the time discretisation error.

Signal correlation and PnL correlation are correlations over different index sets - across assets at each date versus across dates for scalar payoffs - and practitioners often treat the first as a proxy for the second. We give an exact decomposition that shows what that proxy sees and what it discards. We recall that at each date a normalized signal's projection onto the realized demeaned return direction is its realized cross-sectional Pearson IC, so that fixed-signal PnL is return dispersion times IC (Qian and Hua, 2004), and we identify the normalized similarity of two signals in the orthogonal complement as their partial correlation controlling for realized returns; the residual rotational freedom is an orthogonal gauge whose invariants are the transverse Gram matrix. Signal correlation therefore equals an uncentered IC cross-moment plus transverse similarity, while Pearson PnL correlation centers and dispersion-weights the IC series alone. Our main result is a non-identifiability theorem: absent constraints on transverse geometry, neither correlation bounds or orders the other, which sharpens the simulation finding of Sorensen, Qian, Schoen, and Hua (2004) into an exact statement. For normalized ensembles the transverse Gram matrix re-enters through the normalization denominator; the ex post optimal combination under Euclidean and general covariance-risk metrics is the classical multiple-correlation bound, and we show that the two rules agree for every longitudinal exposure if and only if the risk metric is isotropic on the signal span. The classical exact null law of partial correlation under isotropy, a synthetic mechanism illustration at weak-IC scales, and inference guidance under temporal dependence complete the treatment.

Using pinching antennas to enhance a user's connection to its own base station (BS) is intuitive and has been well investigated in the literature. This letter focuses on a less intuitive advantage of pinching antennas from the interference-suppression perspective. In particular, using pinching antennas ensures that all BSs serve their users with low transmit power, i.e., BSs can “whisper", rather than “shout", to their users. As a result, by experiencing less interference, a user's data rate can still be improved with pinching antennas, even if its link quality to its own BS remains unchanged. A stochastic geometric study is carried out in the letter, where BSs employ fractional power control. Analytical and numerical results are presented to reveal the significant impact of this interference-suppression capability offered by pinching antennas.

Understanding how thermal fluctuations modify or suppress quantum-enhanced sensing is a central problem in quantum metrology. Closely related questions arise in quantum thermodynamics, particularly regarding the relation between the work induced by a signal and the information acquired by a sensor. Here, we establish general relations among the sensitivity of thermal quantum states, as quantified by different quantum Fisher information metrics, their temperature dependence, and the work induced by a unitary signal. We further show that, at high temperature, all monotone QFI metrics coincide to leading order with a universal quantity that can be expressed as the variance of a simple observable and itself defines a lesser-known QFI metric. This emergent uniqueness is reminiscent of the uniqueness of classical Fisher information in information geometry. In continuous-variable systems, the infinite-temperature limit is finite and generically non-zero for quadratic Gaussian signals, while it diverges for higher-degree signals. Remarkably, for any purely quadratic signal generator, the infinite-temperature QFI is at least twice the zero-temperature SLD QFI or, equivalently, at least eight times the ground-state variance of the generator. Consequently, the Cramér–Rao lower bound on the estimator variance is reduced by at least a factor of two. We illustrate these results using harmonic chains of bosonic modes relevant to trapped-ion platforms and quantum field-theoretic models.

We show that if $\varphi$ is a colour-permuting automorphism of a connected, finite Cayley graph, and the order of the Cayley graph is either odd or square-free, then $\varphi$ is the composition of a group automorphism and a colour-preserving graph automorphism. Some analogous results are also established for isomorphisms between two different Cayley graphs.

We establish a density property for the obstacle problem of a linearly elastic elliptic membrane shell required to remain confined in a prescribed half-space. For a bounded Lipschitz reference domain, and under the sole assumption that the reference configuration lies at a strictly positive distance from the confining plane, we prove that smooth admissible displacements for which the confinement constraint is inactive (satisfied with a strictly positive margin) are dense in the set of all admissible displacements. This removes the additional condition on the unit normal to the mid-surface required in previous work, and the approximation scheme is elementary and adaptable to other unilateral obstacle problems.

This is the second in a series of papers on Connes' integration in weak Lorentz ideals. Building on the Dixmier trace theory, Birman–Solomyak perturbation theory, and strong measurability developed in Part 1, we establish a spectral form of Pietsch's correspondence for traces on these ideals, extending to this setting results of Semenov–Sukochev–Usachev–Zanin for the weak trace-class. The correspondence describes the positive normalized traces in terms of Banach limits and yields a complete spectral characterization of strong measurability. We also introduce hypermeasurability (measurability with respect to every normalized trace). Unlike the ambient ideal, it depends on the specific regularly varying function chosen, and we characterize it spectrally by means of eigenvalue sums. We further show that hypermeasurability and spectral measurability are incomparable. Finally, we apply these results to examples arising from nonclassical Weyl laws in the sense of Simon, including the logarithm of the Laplacian on a closed manifold, the double Laplacian, multi-tensor products of Laplacians, and an example arising from Connes' approach to the Riemann Hypothesis.

We prove that if $\varphi$ is an isomorphism between two connected Cayley graphs of abelian groups, and $\varphi$ respects the natural edge-colourings of the Cayley graphs, then $\varphi$ is the composition of a group isomorphism and a colour-preserving graph automorphism. This implies that if every colour-preserving automorphism of a connected abelian Cayley graph $Cay(G;S)$ is an affine map, then the same is true for every colour-permuting automorphism. We also show that this property holds if and only if the subgroup generated by $\{\, s \in S \mid 2s \neq c \,\} \cup \{c\}$ has index $\le 2$ for every element $c$ of order $2$ in $G$.

Computation of the Frobenius closure of ideals in rings of prime characteristic is a difficult problem. For a given ideal, its Frobenius test exponent provides a valuable degree of uniformity in performing the calculation. Hence, it is desirable to find uniform upper bounds on the Frobenius test exponent of ideals. Even in nice rings of low dimension, Brenner showed considering the collection of all ideals is generally hopeless, but for Cohen-Macaulay rings, Katzman-Sharp there are uniform upper bounds on the Frobenius test exponent for the class of parameter ideals. Subsequent efforts in controlling the Frobenius test exponents have typically involved studying the Frobenius action on local cohomology and the degree to which this action is nilpotent. The goal of this survey article is to examine the history of the Frobenius test exponent problem and its relationships to singularity types for local rings defined in terms of the Frobenius action on local cohomology. We also explore several generalizations of these prior results to a setting where less nilpotence in the Frobenius action is assumed.

Neural networks can leverage feature superposition to encode more concepts than dimensions, but cross-feature interference constrains the linear accessibility of simultaneously active features. By framing linear accessibility as a compressed sensing problem, we derive high-probability bounds for fixed supports under subgaussian noise, proving the sufficient dimension scales linearly ($d=O_{\varepsilon}(k \log m)$) rather than prior worst-case quadratic limits. We characterize the asymmetry between active and inactive interference and the trade-off between interference and observation-noise budgets. We then validate these bounds across system parameters through Gaussian-tail approximations. We also introduce IHT-SAE, which uses learned iterative refinement to improve feature recovery beyond the limits of linear availability. These results quantify the geometric constraints of the linear representation hypothesis, providing a framework for evaluating sparse autoencoders, compositional generalization, and neural interpretability.

We study the linearized Boltzmann collision operator for three-dimensional hard spheres and reorganize the complete operator so that its angular and radial structures are explicit before any Sonine truncation. From the Carleman representation, the collision integral becomes a superposition of orthogonal translations; in Hermite/Fock and Bargmann form these generate a single analytic coherent-state kernel for all Hermite collision brackets. Rotational covariance separates fixed angular-momentum sectors, each reducing exactly to a one-variable radial Bargmann space with an $\mathfrak{su}(1,1)$ structure. The radial operator is $\mathfrak A_\ell=\kappa\pi G(\mathcal J_\ell)-\mathfrak C_\ell$, where $\mathcal J_\ell$ is an explicit Laguerre–Jacobi operator and $\mathfrak C_\ell$ is compact. A unitary intertwiner identifies this realization with the classical Burnett/Sonine coordinate, recovering the Burnett matrix and standard stress and heat-flux corrections. For the untruncated operator, the compact gain gives only compact corrections, while the loss sets the common essential spectrum $[\nu_{\min},\infty)$. Radially squeezed $\mathfrak{su}(1,1)$ coherent packets yield threshold Weyl sequences in every fixed angular sector. With $\ell\to\infty$ and squeezing scaled as $\ell^{-1}$, they furnish full-space Weyl sequences throughout the essential band interior, with $\langle j\rangle\sim\ell^2$, revealing quadratic continuum corridors in the $(\ell,j)$ plane. At fixed $\ell$ and $j\to\infty$, the matrix tail approaches a universal Toeplitz operator whose zero at $\theta=\pi$ is the local signature of the same threshold escape. Thus the Fock reformulation provides exact radialization and constructive control of the global spectral geometry.

We extend the convergent reaction-diffusion master equation (CRDME) framework on unstructured grids to include two-body interaction potentials between particles. Such interactions are essential for modeling systems in which particles experience electrostatic repulsion, volume exclusion, or van der Waals attraction. Our approach discretizes the weak form of the forward equation for a particle-based stochastic reaction- drift-diffusion model to obtain a new CRDME by combining two complementary techniques: an edge- averaged finite element method with mass-lumping quadrature for spatial transport terms that include potential interactions, and a mix of finite volume and mass-lumping quadrature approximations for reaction terms. This yields transition rates for a spatial jump process of particles hopping between dual mesh voxels and undergoing reactions. The resulting spatially discrete CRDME retains key physical properties of the continuum system in closed domains. It preserves the Gibbs-Boltzmann equilibrium distribution in the absence of reactions, and satisfies detailed balance of both drift-diffusion and reactive fluxes for systems with reversible reactions. Numerical comparisons with Brownian dynamics simulations validate our method, and we demonstrate empirical second-order convergence as the mesh is refined.

We prove that every convex body $K\subset\mathbb{R}^3$ of constant width $w>0$ satisfies \[ \operatorname{Vol}(K)\ge \frac{1651}{4000}w^3=0.41275\,w^3. \] This improves the certified lower bound $0.411040473721188\ldots\,w^3$ announced by Hyra and the earlier spectral bound $4\pi w^3/33$ of Nishioka. The main new ingredient is a contact-tetrahedron estimate near equality in Jung's circumradius inequality. Four balanced contacts of a minimum circumscribed ball form a near-regular tetrahedron; constant width then forces, beyond the tetrahedron itself, spherical regions over its faces, spindle sectors along its edges, and additional face-edge sectors. These regions can be selected with pairwise disjoint interiors and quantified by the total squared-edge deficit. A sharp convexity argument relates that deficit to the circumradius, and a symmetrization argument controls the four face contributions with only quadratic loss. This gives the analytic bound $\operatorname{Vol}(K)>0.41275318$ near the Jung endpoint. For the complementary circumradius range we use Hyra's radial transport/tangent-majorant mechanism together with closed-form contact and antipodal profiles. The antipodal contribution has an elementary primitive. For the contact contribution, a second-order concavity estimate converts the angular Jacobian into an explicit rational lower density; that density is monotone in the circumradius. The whole intermediate range therefore reduces to a single endpoint calculation, giving $\operatorname{Vol}(K)>0.41275168$. For smaller circumradius the tangent-majorant inequality with its nonnegative gap discarded already gives $\operatorname{Vol}(K)>0.41308109$. The proof is entirely analytic and requires no computer-assisted certificate.

We compute quasi-derivations of Block Lie algebras $\mathcal{B}(q)$. In particular, we prove that $\mathcal{B}(q)$ has various nontrivial quasi-derivations, i.e., quasi-derivations different from derivations and $\delta$-derivations. As corollaries, we obtain a complete description of $\delta$-derivations, $\delta$-biderivations, and transposed $\delta$-Poisson structures on $\mathcal{B}(q)$ for $\delta\notin\{0;1\}$. In particular, for $\delta\notin\{0;1\}$, $\mathcal{B}(q)$ up to isomorphism admits only one nontrivial transposed $\delta$-Poisson structure if $q\in\mathbb Z$, and $\mathcal{B}(q)$ does not admit nontrivial transposed $\delta$-Poisson structures if $q\notin\mathbb Z$.

We introduce this class of multivariate Kappa-Gaussian distributions by maximizing a type of Kaniadakis $\kappa$ entropy subject to certain first- and second-moment constraints. This differs slightly from the approach of Rényi and Tsallis, because the $\kappa$-entropy arises from a relativity-consistent deformation of the exponential and logarithmic functions and yields a family of distributions whose tails follow a power law, a family that actually reverts to the standard Gaussian distribution as $\kappa \to 0$. We then create a $k$-nearest-neighbor (NN) estimator for $\kappa$-entropy and describe it using the difference between two NN functionals involving sums of powers. Next, we establish $L^2$-consistency and asymptotic normality, using the well-known subadditive Euclidean functional and moment expansion framework for NN entropy estimators. Furthermore, using a scaling identity for the $\kappa$-exponential, we derive a closed-form formula for the entropy of the $\kappa$-Gaussian distribution itself. From this, we determine the exact threshold value $\kappa < 2/(m+2)$ at which the distribution preserves finite variance; this is analogous to the “degrees of freedom $>2$” condition in the Student's-$t$ distribution. With these components, we construct a non-parametric goodness-of-fit test and then verify it through simulations. Finally, we mention the Sharma–Mittal entropy, which combines the Rényi and Tsallis entropies as a natural direction for future work.

For a finite real alphabet $T$, let $g(T)$ be the supremum of the lengths of words over $T$ containing no two adjacent nonempty blocks of equal length and equal sum. We classify all four-element alphabets with $g(T)<139$. Up to affine equivalence, these are the known endpoint-sum family and seven isolated classes. Exhaustive finite searches give the isolated values, while 77 explicit words prove that every remaining real alphabet admits a word of length 139. The latter argument uses exact intersections of finite unions of lines, so it covers irrational parameters as well. The cutoff is sharp because $g(\{0,2,3,7\})=139$. We also obtain a sharp cutoff of 167 within the family $\{0,1,x,x+2\}$, $x>1$. The required supplement prints the word certificates and specifies the complete-search evidence.

We study online vector balancing with $N$ random vectors in $\mathbb{R}^M$ revealed sequentially, where each vector must be assigned an irrevocable sign upon arrival. The goal is to minimize the expected $\ell^\infty$ norm of the final signed sum. For i.i.d. entries with mean zero, variance one, and a finite fourth moment, we prove that, as $M/N\to\alpha\in(0,\infty)$, the optimal value divided by $\sqrt N$ converges to a limit $R_\alpha$ independent of the entry distribution. This limit is the stochastic control value identified for Gaussian inputs by Fiedler, Jackson, Lacker, and Niles-Weed. In particular, it determines the exact asymptotic optimum for Rademacher inputs. For every $\kappa>R_\alpha$, we construct a randomized online algorithm whose final signed sum has $\ell^\infty$ norm at most $\kappa\sqrt N$ with high probability; for $\kappa<R_\alpha$, every online algorithm has vanishing success probability. Consequently, the online threshold of the symmetric binary perceptron is universal at every positive margin. The main step is a coupling that transfers Brownian controls to non-Gaussian inputs, while truncation controls rare large entries.

For every $d\geq2$, we prove that $d-1$ independent finite tiles in $\Z^d$ admit a fully periodic joint co-tile whenever they admit a joint co-tile. Here independence means that the tiles contain the origin and every choice of one nonzero vector from each tile is linearly independent. This removes property $(\star)$ from the periodic existence theorem of Meyerovitch, Sanadhya and Solomon and recovers Bhattacharya's theorem when $d=2$. More generally, we prove periodic existence for any finite family admitting a joint co-tile whose indicator is a fully periodic real function plus finitely many bounded real functions, each invariant under a subgroup of rank $d-1$. Extending the Boolean normal-form method of Greenfeld and Tao, we obtain a configuration in the translation orbit closure whose restrictions to cosets of a finite-index lattice are either invariant under a rank-$(d-1)$ subgroup or given by a three-term affine fractional-part formula with density one half. The tiling equations pair the restrictions given by this formula and determine a finite bipartite graph. A two-coloring replaces those restrictions by constants while preserving every tiling equation. We deduce a characterization of periodic tilability by independent companions and a decision algorithm for families containing an independent $(d-1)$-subfamily, with the dimension as part of the input.

Let $M_{\ell,m}=\ell\,\mathbb{HP}^{2}\# m\,\overline{\mathbb{HP}^{2}}$ be the connected sum of $\ell$ copies of $\mathbb{HP}^{2}$ with $m$ copies of $\mathbb{HP}^{2}$ endowed with the opposite orientation. We characterise, in terms of Pontryagin classes, the real vector bundles over $M_{\ell,m}$ admitting a stable complex structure, and deduce that $M_{\ell,m}$ is stably almost complex if and only if $\ell-m$ is even. We then determine when an oriented real vector bundle of even rank over $M_{\ell,m}$ admits a complex structure. Consequently, $M_{\ell,m}$ admits an almost complex structure if and only if $\ell=2m+1$ and $m$ is odd. This corrects an assertion made by Sato and Suzuki in 1974, according to which $M_{\ell,m}$ is never almost complex. We also identify the unjustified step in their argument.

We provide a new presentation for a cylinder object inside of the category of co-globular $\infty$-groupoids. We prove that the cylinder construction that Lanari provides is isomorphic to the one we build in this paper.

Let $M=\widetilde H^*(MO(2);\mathbb F_2)$ and let $\mathcal A(1)$ be the subalgebra of the mod $2$ Steenrod algebra generated by $\operatorname{Sq}^1$ and $\operatorname{Sq}^2$. In this work, we determine the quotient $\mathbb F_2\otimes_{\mathcal A(1)}M$ in every internal degree $h\geq0$, where internal degree $h$ corresponds to cohomological degree $h+2$. To overcome the limitations of finite experimental extrapolation, our proof is strictly certifying: in each degree, we exhibit a determinant-one minor of the binary hit matrix and a complementary family of linear dual functionals annihilating every hit column, mathematically guaranteeing the exact rank for all degrees. This rigorously establishes the monomial generating families and module decomposition proposed in Shih's 2012 report. Furthermore, we derive a closed binomial formula for every Steenrod square from the length-two symmetric-polynomial model, correct a boundary omission in the $\mathcal A(0)$ calculation, compute the exact Hilbert series of the $\mathcal A(1)$-cohit, and prove an intrinsic direct-sum decomposition of $M$ into two stable $\mathcal A(1)$-submodules.

Paraconsistent Annotated Logic with two values (PAL2v) annotates a proposition with a pair $(\mu,\lambda)$ of degrees of favorable and unfavorable evidence, drawn from a bilattice whose four extreme points are the values of Belnap and Dunn. In applications, however, the second coordinate is almost always taken to be the complement of the first, and several such annotations are then fused. We show that this practice determines the sign of the degree of contradiction in advance, independently of the data: for aggregation functions $F$ and $G$ the fused degree of contradiction equals $F-G^{d}$, where $G^{d}$ is the De Morgan dual of $G$, so that it vanishes identically when the two are dual, and has a constant sign otherwise. In particular the componentwise minimum used throughout the literature yields only consistent annotations, and the conjunction of annotated logic only exact ones; five of the twelve logical states, contradiction among them, are essentially unreachable by construction. We also give a decision procedure for the twelve states that is provably total, in place of the inequality lists in current use, which assign two states to about half of the non-extreme region. We then propose a construction in which the two coordinates are computed from disjoint families of evidence, show that every logical state is attainable. Educational assessment provides a natural instance, since summative and formative instruments are institutionally distinct: mapping examinations to $\mu$ and disengagement to $\lambda$ makes the degree of contradiction equal to the gap between what a student demonstrates and what that student invests. Simulations over three cohort profiles illustrate the classification.

For any graph $G$, we associate a family of real symmetric matrices, $S(G)$, where for any $A \in S(G)$, the location of the nonzero off-diagonal entries of $A$ are governed by the adjacency structure of $G$. Let $q(G)$ represent the minimum number of distinct eigenvalues over all matrices in $S(G)$. In this work, we provide an alternative technique to establish that $q(G) \leq 4$ for any threshold graph $G$ as presented in [L. Emilio Allem, C. Hoppen, J. Lazzarin, L. Siviero Sibemberg, F. Colman Tura, The minimum number of distinct eigenvalues of a threshold graph is at most 4, Linear Algebra and its Applications, 726 (2025) 32 to 53]. In addition, we show that all connected threshold graphs admit a matrix having any four distinct eigenvalues. Further

OpenAI recently announced a proof of the Cycle Double Cover (CDC) Conjecture. Most media reports have characterized it as a 50-year-old open problem. In reality, according to a 1987 letter from Tutte to Fleischner, the Cycle Double Cover Problem has been open for at least 80 years. Two early results regarding the CDC conjecture were established in one of Tutte's 1949 publications.