Papers

Given a functional dependent on the spectrum of a differential operator, we address the problem of finding a domain which optimizes this functional. PDE solvers might be used to tackle this optimization. It is however computationally expensive. We propose two neural network models which learn the spectrum directly from the geometry of the domain and can be used to optimize the domain from one or more eigenvalues. We investigate two representations. The first encodes the domain through Fourier coefficients and a light MLP, which is efficient on star-shaped geometries, achieving a precision of 0.2%. Through a rescaling of the coefficients the designed models satisfy the scaling law of the eigenvalues. Additionally, averaging the outputs of the trained surrogates over rotations and reflections induces invariance for these transformations. The second is a model that takes the landscape function, the indicator function and the gradient of the landscape function. A Gram-Schmidt process produces orthogonal eigenfunctions as output of the model along with the associated eigenvalues. The landscape model reaches 1% mean relative error on the first ten eigenvalues, compared with 4% for an FNO model. Replacing the landscape by an SDF worsened both prediction and optimization errors. The trained model also generalizes from synthetic shapes to domains given as classical image dataset. The resulting surrogates of both approaches recover classical spectral optima such as the disk for the first eigenvalue or the conjectured minima of higher eigenvalues. This confirms that our models produce accurate differentiable estimates of eigenvalues, which can be used in shape optimization problems involving spectral quantities.

We study a $p$-adic rigidity phenomenon suggested by rank two attractors. Near an ordinary special fiber of a Calabi–Yau family, the excellent Frobenius of Beukers and Vlasenko acts on the parameter disk. Achinger–Zdanowicz construct a canonical $W_2$-lift of the Frobenius twist, while Brantner–Taelman construct a canonical formal lift. We conjecture that the Dwork and Achinger–Zdanowicz first-order obstruction maps agree and that the fixed points of the excellent Frobenius are precisely the parameters of the Brantner–Taelman canonical formal lifts. Our main theorem gives a cohomological criterion in arbitrary dimension. Under its hypotheses, a Frobenius-stable rank two factor of the $p$-adic cohomology containing the holomorphic class forces that class to span a Cartier-stable line and gives supercongruences modulo $p^{2s}$ for every $s\geq1$. Moreover, if the hypotheses of the Beukers–Vlasenko excellent lift theorem hold, then the excellent Frobenius fixes the corresponding parameter. We apply the criterion to the Hulek–Verrill $A_4$ family at $t_*=-1/7$, a parameter predicted to be a rank two attractor. Assuming Dummigan's Conjecture~1.1, we prove that the excellent Frobenius fixes $t_*$, the supercongruences hold, and the fiber with parameter $t_*\bmod p^2$ is the Achinger–Zdanowicz canonical lift at every prime where the required Dwork ordinarity conditions are satisfied. A CM fiber of the mirror quartic K3 pencil gives a second application. We also prove that the $q$-power map on a split formal torus has the unit section as its unique periodic point in the identity residue disk over every finite extension of $W(\mathbb{F}_q)[1/p]$, where $q$ is the cardinality of the residue field.

We present formulas for the higher Alexander polynomials of satellite links.

We develop a general criterion for establishing absolute continuity of convolutions of fractal measures on the line, and more generally of prescribed line projections of fractal measures in the plane. The crucial new ingredient is exact scaling covariance of the \(L^1\)-norm of Littlewood-Paley pieces of the projected measure, under affine renormalization. We apply this criterion in three key settings. First, we show that the convolution of two measures on the line is absolutely continuous whenever their dimensions sum to more than one, provided one is a self-conformal measure whose defining IFS is not \(C^2\)-conjugate to linear, and the other is either self-conformal or Ahlfors–David regular. Second, we show that every line projection of a planar complex-analytic self-conformal measure of dimension greater than one is absolutely continuous, under natural nonlinearity and nondegeneracy assumptions. Finally, we show that every line projection of a planar self-affine measure is absolutely continuous, under natural irreducibility and proximality assumptions, whenever the correlation dimension of the measure and the Frostman dimension of its Furstenberg measure sum to more than two.

We investigate aspects of the relationship between global sections of divisors and birational geometry. We show that for varieties $X$ of klt Calabi-Yau type, global sections are governed by the same birational data that control the $D$-minimal model program ($D$-MMP). As a consequence, global sections of arbitrary big divisors can be systematically reduced to Euler characteristics $\chi(Y,\mathcal O_Y(D))$ on suitable birational models $Y$ using vanishing theorems, extending classical positivity-based methods beyond the movable cone. The reduction proceeds by removing fixed divisorial components, which do not contribute to the space of global sections, either directly by subtraction or birationally by contraction to a suitable $D$-minimal model. We obtain piecewise quasipolynomial formulae for global sections on the big cone, whose domains are the Mori chambers of the $D$-MMP. Restricting to the subclass of Fano type varieties extends such formulae to the entire effective cone. These formulae can be constructed from just the small birational class of $X$, yet encode properties of all birational contractions of $X$. Alongside the computation of global sections from birational geometry, we study the inverse process, wherein finite computations of $h^0(X,\mathcal O_X(D))$ determine aspects of the chamber decomposition and the associated birational models.

Among distinct optimal acceleration mechanisms for deterministic monotone root-finding problems and fixed-point problems, dual-anchoring has recently been shown to admit a more robust direct stochastic extension than standard anchor acceleration. However, without additional strong monotonicity, the existing stochastic dual-anchoring guarantee has two limitations: first, it requires cocoercivity in expectation, and second, it attains only $O(\epsilon^{-3})$ oracle complexity, leaving a gap to the near-optimal $\tilde{O}(\epsilon^{-2})$ complexity achieved by other methods. In this work, we address both of these limitations by combining dual-anchoring with stochastic resolvent approximation and optimized variance control. For unbiased stochastic oracles with variance bounded by $\sigma^2$, where sample operators are monotone and uniformly $L$-Lipschitz, our algorithm finds a point with $\epsilon$-residual with a near-optimal oracle complexity of $O ( (LD / \epsilon) \ell + (\sigma^2 / \epsilon^2) \ell^2)$, where $\ell = \log (1 + LD / \epsilon)$ and $D$ is the initial distance to a solution. This result improves the best known oracle complexity in the noise-dominated regime under these samplewise assumptions, reducing the poly-logarithmic factor from cubic to quadratic.

Uniform random rotations are an effective preprocessing step for quantization: they make normalized coordinate distributions approximately Gaussian, enabling the use of codebooks optimized offline. We introduce TORQUE, a framework that improves on previous quantization works that use random rotations by jointly optimizing how many and which coordinates to preserve at high precision both before and after rotation, under a fixed overall expected bit budget. Intuitively, before rotation, preserving large input coordinates at high precision can reduce overall error by preventing the rotation from spreading their values across many coordinates. Likewise, after rotation, preserving a small fraction of the largest-magnitude coordinates at high precision allows the remaining values to be quantized more accurately using codebooks optimized offline for the resulting truncated Gaussian distribution. We derive a quantization error upper bound and prove that top-$k$ pre-rotation retention minimizes it for each $k$. This reduces the search over coordinate subsets to an optimization over $k$, enabling a fast optimizer that uses offline codebooks and parallel parameter selection for practical implementation. We demonstrate an improved tradeoff between reconstruction accuracy and storage cost through numerical evaluation under the Gaussian model and experiments on nearest-neighbor retrieval, KV-cache compression, and activation compression.

The lack of rigorous safety and performance certificates remains a key bottleneck to the deployment of modern learning-based methods. Sample compression has recently emerged as a powerful tool for deriving such certificates, with particularly sharp bounds available for algorithms satisfying a so-called preference property – also known as stability in the learning theory literature. These bounds find direct application across domains as different as the Scenario Approach, Pick-to-Learn, and Support Vector methods. However, whether they are tight has remained an open problem. In this paper we resolve this question affirmatively and show that the state-of-the-art bound for preferent compressions is provably tight. We establish this by exhibiting an explicit construction based on the uniform distribution and order statistics that attains the bound in the limit. Along the way, we also provide a considerably shorter and more accessible proof of this bound, requiring only elementary counting arguments and no infinite-dimensional duality.

Let $d \geq 5$, $0 < \gamma < d - 2$, and $\Omega_N$ be the binomial random subset of $Q_N = [-N,N]^d \cap \mathbb Z^d$ with retention probability $p_N = N^{-\gamma}$. We prove that, with failure probability of optimal exponential order, every subset $B \subseteq \Omega_N$ of fixed positive relative density realizes, at each scale $p_N^{-2/(d - 2)} \lesssim \lambda \lesssim N^2$, a squared distance of the form $q^2 \lambda$, where $q$ belongs to a fixed finite set depending only on the dimension and the density. The lower scale $p_N^{-2/(d - 2)}$ is sharp. As a consequence,the squared-distance set $D^2(B)$ of $B$ has maximal order $N^2$ and contains affine copies of every fixed finite subset of $\mathbb Z$. The main new input is a finite multidilate supersaturation theorem for dense subsets of $Q_N$, which, together with boundedness estimates for the associated spherical distance graphs down to the sharp scale, allows us to apply Schacht's transference theorem.

Let $R$ be a commutative Artinian ring. We consider two graphs associated to $R$, namely the compressed zero-divisor graph $\Gamma_E(R)$ and the associate class graph $\Gamma_A(R)$. Partitioning the vertex set of a zero-divisor graph into its core and its boundary, we count the core vertices that dominate the core. This count is a graph invariant, and we estimate it for $\Gamma(R)$, $\Gamma_A(R)$ and $\Gamma_E(R)$. We prove that the count for $\Gamma_A(R)$ is bounded below by the count for $\Gamma_E(R)$, and that the lower bound is attained precisely when $R$ is Gorenstein. As a consequence we obtain that $R$ is Gorenstein if and only if $\Gamma_A(R)\cong\Gamma_E(R)$ as graphs, the isomorphism being an arbitrary one and not merely the natural compression map. Using the same counting technique we then answer, for Artinian rings, a question of Anderson and LaGrange by showing that $\Gamma(R)\cong\Gamma_E(R)$ if and only if $R\cong \mathbb Z_2^{\,n}$ for some $n\ge2$, or $R\cong\mathbb Z_4$, or $R\cong\mathbb Z_2[x]/(x^2)$.

In this article, we introduce a new rewriting method for computing hom-bases of diagrammatic categories. Our approach is based on stratified presentations, obtained by partitioning an equational presentation into strata according to the rewriting behavior of its relations. We establish modularity principles for termination and confluence in stratified rewriting systems with coefficients in a ring. We then introduce a completion procedure that transforms a stratified presentation into a normalized one, in which mixed interactions between successive strata are absorbed into the normalized rewriting rules, thereby inducing modularity of confluence. This leads to a stratified hom-basis theorem for diagrammatic categories. We illustrate the method by applying it to the $q$-Schur category, for which we construct a new hom-basis.

We prove that the relative log canonical algebra of a semi-dlt pair $(X,\Delta_X)$ projective over a scheme $T$ is finitely generated, equivalently, that $(X,\Delta_X)$ admits a relative stable model, provided that the normalization admits a log canonical model over $T$, that $(X,\Delta_X)$ admits a stable model over an open dense subscheme $T^0\subset T$ whose exceptional locus does not contain any stratum of the conductor, and that the log centers contained in the conductor have images meeting $T^0$. Our main contribution is a direct proof that avoids Kollár's gluing theory entirely. As a consequence, we recover demi-normal versions of results of Hacon–Xu and Birkar. Furthermore, we derive the existence of certain MMP steps for slc pairs, recovering results of Ambro and Kollár. Finally, these results lay the groundwork for a streamlined proof of the properness of the Kollár–Shepherd-Barron–Alexeev moduli space of stable pairs, to be completed in forthcoming work.

Several physically interesting Lindbladians, such as those describing generic open quantum system models as well as engineered state preparation processes, are often rapidly mixing, that is, they relax to steady states in time scaling polylogarithmically with the system size. Owing to the short mixing times, it is physically expected that local perturbations do not substantially increase the mixing time. In this work we question this expectation and ask whether rapid mixing survives. We provide counterexamples showing that rapid mixing is not stable in general, even when the perturbation does not change the steady state. We also show that arbitrarily weak local Hamiltonian perturbations can destroy rapid mixing. Complementing these results, we develop an array of sufficient conditions which imply stable rapid mixing, including several classes of quasifree fermionic systems, commuting Lindbladians, and local spin systems. We also identify conditions under which modified logarithmic Sobolev inequalities (MLSIs), a stronger form of convergence, lead to stable rapid mixing. As a further implication of our results, we show that rapid mixing does not imply a positive MLSI constant.

The familiar wave kinetic theory revolves around turbulent energy cascades characterized by random phases of the individual normal modes. By contrast, recent years have revealed turbulent cascades where the transfer of energy is mediated by phase-coherent dynamics. Such coherent cascades have been previously established in deterministic systems with a tightly controlled algebraic structure of the mode interactions, such as nonlinear Schrödinger equations. Here, we show that the presence of phase-coherent cascades is robust with respect to substantial alterations of the mode couplings. We prove that, in infinite-dimensional resonant Hamiltonian systems with fully resonant normal mode spectra and random nonlinear couplings, a structured subset of couplings — an infinitesimal fraction of the total — suffices to support coherent cascades characterized by phase alignment, the formation of power-law spectra, and either unbounded growth of Sobolev norms or a finite-time blow-up. Numerical simulations further demonstrate the emergence of coherent cascades from random initial data through a process of phase locking, suggesting a mechanism for their emergence in broader classes of initial data and systems. We thus show that disordered mode interactions can sustain, rather than block, coherent energy transport across arbitrarily distant scales.

We determine asymptotic global fidelities of cloning generic mixed states with simple (non-degenerate) full-rank spectra. In fixed dimension $d$, let $n$ inputs and $m_n$ outputs satisfy $m_n/n\to\gamma>1$. For a known simple full-rank spectrum $p_1>\cdots>p_d>0$, we prove that the optimal root fidelity between the entire output and the product target state is $$ F_\rm{orb}(\gamma,p) =\prod_{i<j} \frac{\sqrt{\gamma}+\sqrt{q_{ij}(\gamma-1+q_{ij})}} {\gamma+q_{ij}}, \qquad q_{ij}=\frac{p_j}{p_i}. $$ When the spectrum is unknown, one spectrum-independent cloner attains $$ F_\rm{univ}(\gamma,p) =\left(\frac{2\sqrt\gamma}{1+\gamma}\right)^{(d-1)/2} F_\rm{orb}(\gamma,p), $$ uniformly on compact subsets of the simple full-rank chamber and is minimax optimal. The two factors quantify spectral and eigenbasis uncertainty. Our channels combine classical processing of Schur–Weyl labels with a covariant Cartan-sector channel. Local asymptotic normality and a Gaussian fidelity bound against arbitrary channels yield the converses. For flat projector states $\rho=P/r$, with $P$ a rank-$r$ orthogonal projector, we separately prove the optimum $\gamma^{-r(d-r)/2}$. Furthermore, we estimate the exact asymptotic fidelity of purify–clone–trace (PCT), and show that our cloners strictly outperform PCT at every fixed gain for simple full-rank spectra and rank-$r$ projector states with $1<r<d$.

We study the topological band theory of quantum Hall systems in which the adiabatic flux parameters of an external gauge potential form a high-dimensional manifold. This situation arises both for strongly correlated matter on spatial Riemann surfaces of genus $g > 1$, as well as in the formal study of matter on compact oriented manifolds of dimension $4\ell+2 = 2p$ coupled to $p$-form gauge potentials. In these cases, the underlying topological invariants defined over the parameter space are significantly richer than the low-dimensional (i.e., two-dimensional) case and involve both the curvature (i.e., first Chern class) as well as higher order curvature invariants. These higher curvature invariants correspond to topologically protected contributions to Kubo-like formulae, constructed from correlation functions of the physical current operators. We illustrate these general considerations with an explicit example from hyperbolic band theory based on the genus-two Bolza Riemann surface, a case which has recently been simulated on a synthetic-dimension platform.

We prove that the Gibbs state of any spin chain can be exactly decomposed into a mixture of product states over contiguous blocks of qubits with bounded length, at any constant temperature. As a consequence, any such Gibbs state can be prepared by local unitary circuits of constant depth, with efficient classical preprocessing. We place strong limits on the spatial structure of thermal entanglement: both the entanglement depth and entanglement width (measures of multiparticle entanglement) are bounded by constants, while the localizable entanglement (the bipartite entanglement that can be generated with the aid of local measurements and classical communication) vanishes exactly beyond a constant threshold distance. Our result also has implications for quantum thermalization: for any translation-invariant spin-chain Hamiltonian with nondegenerate spectral gaps, typical low-complexity pure states drawn from a maximally entropic ensemble at any constant effective temperature become locally indistinguishable from the Gibbs state upon unitary evolution. Additionally, we obtain an analogous decomposition for fermionic systems: the Gibbs state of any 1D local fermionic Hamiltonian is exactly a mixture of constant-depth local fermionic circuits acting on Gaussian states.

We introduce Solver Agent, an AI framework based on large language models for calculations and proofs in mathematics and theoretical physics. The solution process is tracked through a persistent ledger that records assumptions, derivations, and computations. A central agent delegates tasks to specialized sub-agents, while independent agents verify both intermediate steps and the final result. This setup improves the traceability, reproducibility, and verification of computer-assisted calculations. Applying Solver Agent, we study global F-theory uplifts of Type IIB orientifolds and their S-fold generalizations. We establish sufficient conditions for Weierstrass models over projective threefolds with terminal $\mathbb{Z}_k$ quotient singularities ($k\in\{2,3,4,6\}$) to give $\mathbb{Q}$-factorial projective elliptically fibered Calabi-Yau fourfolds with isolated Gorenstein terminal quotient singularities. These geometries realize O3-planes and S-folds, where local D3-brane probes of the latter yield four-dimensional $\mathcal{N}=3$ superconformal field theories. Using stringy invariants, we derive fixed-point contributions to Hodge data and Euler characteristics, and show that these Euler corrections determine the localized D3-brane charges required for tadpole cancellation. We illustrate these results using toric hypersurface constructions, where a single three-dimensional polytope determines both the Type IIB Calabi-Yau threefold and the F-theory base; here, the orientifold double cover naturally forms a bisection of an alternative genus-one-fibered uplift with discrete $\mathbb{Z}_2$ gauge symmetry. Finally, we provide methods for toric computations and four-form flux analysis in four-dimensional $\mathcal{N}=1$ compactifications with non-abelian gauge sectors.

Let $\chi$ be a Dirichlet character mod $D$ with $L(s,\chi)$ its associated $L$-function, and let $\psi(x,q,a)$ be, as usual, Chebyshev's prime-counting function for the primes of the arithmetic progression $a$ (mod $q$) with $(a,q)=1$. Let $\chi$ be a primitive character modulo $D$, and let $\nu>0$ be small. We prove that if $L(s,\chi)$ has a Siegel zero at $s=\beta=1-\frac{1}{\eta\log D}$ with $\eta>\eta_0(\nu)$ for some large $\eta_0(\nu)$, there exists a range of $x$ for which the asymptotic $$\psi(x,q,a)=\frac{\psi(x)}{\phi(q)}\left[1+O(\varepsilon_{\eta_0})-\chi\left(\frac{aD}{(q,D)}\right)\right]$$ holds for $q<x^{\frac{30}{59}-\nu}$. We also show slightly better bounds for $q$ if we take an average over a range of $q$, finding an Elliott-Halberstam-type result for $q\sim Q$ on the range $Q<x^{\frac{16}{31}-\nu}$. This improves on a 2003 result of Friedlander and Iwaniec that requires $q<x^{\frac{233}{462}}$ and builds on recent work of Sachpazis.

We develop accelerated gradient-free methods for stochastic convex optimization with constraints defined by expectations. Our batched primal-dual sliding method uses two-point evaluations sharing a random sample and guarantees expected objective error and expected maximum constraint violation at most $\varepsilon$. It achieves $O(\varepsilon^{-1/2})$ sequential oracle rounds for smooth data and $O(d^{1/4}/\varepsilon)$ for nonsmooth data in dimension $d$, recovering the accuracy and dimension dependence of the corresponding unconstrained accelerated methods. The smooth rate is optimal in accuracy. Each round collects all samples needed for its inner primal-dual updates. Total evaluations retain quadratic dependence on inverse accuracy, with only polylogarithmic dependence on the number of constraints under vector feedback. Complementary lower bounds distinguish the dimension cost of gradient estimation from the unavoidable logarithmic cost of estimating noisy constraint levels. For smooth strongly convex objectives, restarts give logarithmic round complexity and objective evaluation cost linear in inverse accuracy, while constraint-level estimation necessarily remains quadratic. Known affine constraints require no constraint-oracle calls.

In this paper, building on the work of Dai–Sasahira–Stoffregen, we derive constraints on the intersection forms of smooth, compact, indefinite, spin $4$-manifolds bounded by AR homology spheres using two methods. First, we use the constraints on the existence of $\operatorname{Pin}(2)$-equivariant maps between representation spheres, established by Hopkins–Lin–Shi–Xu in their proof of the $10/8 + 4$-inequality. Second, we compute Lin's $\kappa o$-invariant for AR homology spheres. Combined with Lin's version of the relative $10/8$-inequality, this computation yields constraints different from those based on Hopkins–Lin–Shi–Xu. As applications, we obtain new lower bounds on the minimal genus of surfaces bounded by knots whose double branched covers are AR homology spheres in various $4$-manifolds, and provide an obstruction to smooth $4$-manifold decompositions related to the $11/8$-conjecture.

Consider a community of agents who are seeking consensus on a set of parameters. The agents agree to use the same Bregman-type divergence to quantify disagreement between their individual estimates of the parameters but have varying confidence in each other's abilities. Each agent is happy to revise their estimate by moving to the weighted barycenter of all individual estimates with higher weights applied to more trusted agents. We show that such revisions naturally lead to an iterative algorithm which converges to a unique consensus estimate of the parameters. Furthermore, since the consensus estimate is itself a barycenter with computable weights, the group emerges as a collective super-agent with a well-formed opinion regarding the ability of each individual agent.

The problem to establish a lower bound for $\mbox{AC}^0[p]$-Frege refutations of a system of polynomial equations over ${\bf F}_p$ was in K. (2024) reduced to the existence of a pseudo-solution (a notion defined there) for the system. Here we reduce this further, for the system expressing the negation of the PHP, to a property of search trees querying values of linear maps on the vector space of low degree polynomials over ${\bf F}_p$.

We ask whether transposition changes the stationary heavy tail of the co-moving recursion $\mathbf x_{t+1}=Q_tBQ_t^\top\mathbf x_t+Q_t\boldsymbol\eta_t$ in $d=3$. The model has independent Haar rotations, Gaussian body-frame noise with covariance $\Sigma$, and an explicit positive metric $\mathsf g$. The tail index $\alpha_\star$ depends only on singular values, so $\alpha_\star(B)=\alpha_\star(B^\top)$. The structure theorem and polar-twist mechanism are unconditional. In the $\mathsf g=I$ chart, write $B=S+\widehat{\boldsymbol\omega}$: $\Delta_6(B)=-16\det[\boldsymbol\omega,S\boldsymbol\omega,S^2\boldsymbol\omega]$. Then $B\not\sim_{O(3)}B^\top$, $\Delta_6(B)\ne0$, $\operatorname{rank}[\boldsymbol\omega,S\boldsymbol\omega,S^2\boldsymbol\omega]=3$, and controllability of $(S,\boldsymbol\omega)$ are equivalent; intrinsic operator chirality is a Kalman rank condition. Unlike the exponent, the tail level need not be transpose-invariant. The exact polar-twist identity rotates the anisotropic noise frame under mirroring, showing how level asymmetry can enter. A nonzero asymmetry is established only conditionally and numerically, not by a closed-form example: under Assumption R on the SPD chart, $\Delta\log C=\langle A,G\rangle+o(\|A\|)$ as $A\to0$, with $G=2H\circ[\Sigma,\partial_\Sigma\log C]$; finite-threshold simulations are consistent with this first-order law. Strict-Haar averaging makes every lagged second-order cross-moment transpose-blind; a fourth radial moment retains an odd channel. No finite scalar-weighted radial-moment combination uniformly cancels the even quadratic form while retaining the odd covector.

We study one-dimensional binary cellular automata with a five-slot radius-two local rule of exact algebraic-normal-form degree two, acting on periodic rings of length n. We prove that every such rule is non-injective whenever 4 | n and n >= 8. The proof begins with the four-cell collapse, where the two extreme formal slots coincide. A structural classification of the resulting four-variable maps separates the 65,472 exactly quadratic rules into 63,456 rules with an immediate ring-four collision and 2,016 exceptional lifts. The latter split into layers of sizes 480 and 1,536. Their remaining finite obligations are represented by 136 parameter-region constructions and 768 per-lift records, respectively. Each certificate supplies differentiating closed walks of lengths 8 and 12 with a common pair-graph base vertex. Concatenation then gives lengths 8a + 12b, which are exactly the multiples of four from eight onward. The load-bearing finite certificate core therefore contains 904 = 136 + 768 independently replayable objects checked by standalone, non-searching programs. The complete checker CLIs additionally reconstruct expected populations and execute coverage, complement, and regression/guard checks; 904 is not a count of total checker operations. Periodic extension also yields full-shift non-injectivity; that consequence is used here only as a corollary.

Fishburn's latent-subset conjecture, proposed in 1987 and revisited in 1988, asserts that for every dual intersecting family $\mathcal F\subseteq 2^{[n]}$, there exists $i\in[n]$ such that $|\mathcal F^L(i)|\geq|\mathcal F(i)|$. Here $\mathcal F^L$ is the family of the subsets of members of $\mathcal F$ that do not belong to $\mathcal F$. In this paper, we prove the conjecture using the recent weighted star inequality of Chang, Liu, and Liu.

High-resolution remote sensing imagery (RSI) transmission is constrained by satellite-ground bandwidth and channel impairments, yet existing methods struggle to simultaneously achieve extreme compression and robust transmission. To address this, we propose a dual-branch vector-quantization aided satellite digital semantic communication (DVQ-SDSC) framework for RSI transmission over affine frequency division multiplexing (AFDM), whose bandwidth savings arise from two interrelated aspects. First, at the source-coding level, a dual-branch framework is developed to unify deep joint semantic coding, VQ-aided index transmission, channel estimation and adaption in an end-to-end architecture; departing from symmetric encoder designs, the codec is recast as an asymmetric dual-branch architecture that separately processes the high-frequency residuals and the low-frequency structural semantics, with gated fusion and channel-adaptive reconstruction jointly restoring the semantic content. Second, at the index-coding level, we develop a principal component analysis (PCA)-aided codebook reordering to align index topology with latent correlations, and devise group differential pulse-code modulation (G-DPCM) to encode prediction residuals rather than absolute indices, lowering the index bitrate while locally isolating clipping and channel errors. A two-stage training strategy further decouples channel impairments from the semantic codec. FAIR1M experiments over 3GPP NTN-TDL-D demonstrate that DVQ-SDSC with G-DPCM index coding outperforms the conventional JPEG-LDPC scheme at the base rate of 0.0625 bits per pixel (BPP), and that G-DPCM applies directly to the trained codec without retraining, yielding additional index compression at no extra cost.

We prove that free additive convolution preserves the $k$-convex order for every $k\ge2$, assuming finite $(k-1)$st absolute moments for the compared measures and a finite $\max\{2,k-1\}$th absolute moment for the convolving measure. In particular, the case $k=4$ gives an affirmative answer to a question of Heinävaara concerning fourth-order convolution comparison.

We prove that Rademacher type $s$ coincides with Talagrand type $(s,\psi_{s,s/2})$ for every $1<s<2$.

We show that if an $(n,d,\lambda)$-graph satisfies $\lambda\leq (1-\delta)d$ and $d\gg \delta^{-6}(\log n)^{3}$ for some $\delta>0$, then it is Hamiltonian. A qualitatively similar result was recently proven by Bradač and Janzer. Our proof here is shorter and gives better quantitative bounds. \par In our proof, as in earlier work of Ferber, Han, Mao, and Vershynin, we use a random matrix inequality to show that a mild spectral gap is typically preserved after randomly sampling an appropriate proportion of the vertices. This allows us to deduce that typical balanced bipartite subgraphs of pseudorandom graphs contain perfect matchings. To convert a collection of perfect matchings into a Hamilton cycle, we use a variant of the sorting network method.