Papers

The containment of the code of the meet $G\wedge A$ in the hull and the identity $G\vee A-D=K-G\wedge A$ exchanging meet and join are known; imposing that $G\wedge A$ be principal constructs algebraic geometry codes with one-dimensional hull. We turn that construction into a measurement. For arbitrary divisors $G$ and $A$ we compute $C_L(D,G)\cap C_L(D,A)$ exactly: it is the code of the meet together with an excess $\varepsilon(G,A)$, canonically their quotient and a subquotient of $H^1(\mathcal O(G\wedge A))$. So $\varepsilon$ vanishes exactly when the meet is non-special; otherwise it certifies that $K-G\wedge A$ is linearly equivalent to an effective divisor, at degree zero the vanishing of a single class in the Picard group: the hull detects a linear equivalence rather than being built from one. For superelliptic curves $y^n=f(x)$ both sides can be computed: their weighted plane models in $\mathbb P^2_{(1,n/c,d/c)}$, $c=\gcd(n,d)$, identify codes $C_s$ of weighted forms of degree $s$ with those of $sD_\infty$ and turn hulls into lattice counts. The range on which $\varepsilon$ is blind is an explicit interval of degrees, where $\dim\operatorname{Hull}(C_s)=c\mu(s)-n\delta+1-g_X$, $\mu(s)=\min\{s,M-s\}$, depends only on its affine-point count. Outside it the meet and join are invariant under $s\mapsto M-s$ while $\varepsilon$ is not, so every asymmetry of the hull profile is excess and the threshold in $s$ refines the divisor class: two totally split curves of genus two, over $\mathbb F_7$ and over $\mathbb F_{11}$, present the same class at the same pair of degrees and are separated by the profile alone. If $0\leq\deg(G\wedge A)\leq2g_X-2$ the hull is at most $g_X+1$, so it is large only where it is blind, and over a prime field, under an explicit inequality on $(n,d,q)$, its maximum over the family is $\ell(\lfloor M/2\rfloor D_\infty)$, attained exactly on the totally split locus.

We introduce a new SVD-based tensor decomposition method for tensor networks with arbitrary graph topologies, extending classical hierarchical SVD-based techniques to networks with cycles and general connectivity. We also introduce addition and rounding procedures for traceable tensor graphs, enabling step-rounding time integration of high-dimensional PDEs directly in graph format, with rank truncation controlled to a prescribed tolerance at every time step. We demonstrate the new method on the decomposition of multivariate functions and on the numerical solution of the Fokker-Planck equation, and find that the graph-format representation attains comparable or better accuracy than the classical tensor train and hierarchical Tucker tensor formats, while using substantially fewer degrees of freedom at lower computational cost.

In this work we study a fractional wave equation with Balakrishnan–Taylor type damping posed on a bounded domain $\Omega\subset\mathbb{R}^n$. The model couples the fractional Laplacian with a nonlinear damping coefficient depending on the fractional energy, leading to a doubly nonlocal evolution equation that extends the classical wave equation with energy-dependent dissipation. By means of the theory of linear operators, we establish the global well-posedness of both mild and regular solutions. We then investigate the long-time dynamics of the associated semigroup and prove that it is gradient and asymptotically smooth. As a consequence, we establish the existence of a compact global attractor and show that it coincides with the unstable manifold of the set of stationary solutions. To the best of our knowledge, this provides the first characterization of the asymptotic dynamics for fractional wave equations with Balakrishnan–Taylor type damping.

We study normalized Ricci flow on simply connected homogeneous spaces G/H for which the isotropy representation splits into exactly two equivalent irreducible subrepresentations. We prove that every G-invariant metric evolves to one with positive Ricci curvature, and that the family of G-invariant metrics with positive Ricci curvature is forward-invariant under the flow. The proof relies on the fact that the phase portrait of the family of fixed-volume G-invariant metrics can be explicitly visualized.

Observation-driven filters typically use likelihood-score updates, corresponding to the logarithmic scoring rule. We generalise these updates to negative parameter derivatives of differentiable proper scoring rules, under a declared working family and predictable scaling. The rule determines the conditional risk projection and tail response, scaling converts its derivative into the update, and the autoregressive component determines the composite dynamic centre. Locally, risk curvature and scaling govern mean reversion, while the variance of the scaled innovation governs update noise. Under correct specification, unscaled curvature and score variance coincide for the log score by the information identity, but generally differ for other proper rules. For static parameters, we establish consistency and asymptotic normality under explicit stability and fixed-tuning conditions. In high-frequency scale models, centred updates converge to diffusions, whereas non-centred updates follow deterministic mean flows. For time-varying rule-specific projections, the local tracking error admits an Ornstein-Uhlenbeck approximation up to a stopping time, allowing correlated update and target shocks. Simulations and an international-equity application illustrate how criterion choice affects robustness, adaptation, variance-forecast loss, value-at-risk calibration and probability-integral-transform diagnostics. Predictive density and updating criterion are distinct design choices whose relative performance depends on the target and disturbance.

We show that, for every pair of convex bodies $K,L\subset\mathbb R^n$, $$ \operatorname{vr}(K,L)\leq C\sqrt{n\log(n+1)}. $$ The main point is to place $K$ and $L^\circ$ in isotropic position. We then consider a random orthogonal image of $L$ and control the corresponding operator norm by combining the isotropic mean-gauge estimate of Bizeul and Klartag with Letwin's recent dimension-free bound for the third-moment parameter appearing in their estimate. Our result improves the bound $ \operatorname{vr}(K,L)\leq C\sqrt n \log(n+1)$ proved by Giannopoulos and Hartzoulaki, which had remained the best general estimate for nearly two and a half decades.

Multiphysics initial-value problems couple processes with distinct stability properties, such as advection, diffusion, and stiff local reactions. Standard implicit-explicit (ImEx) additive Runge–Kutta (ARK) methods can treat these processes accurately, but require globally coupled implicit solves when diffusion is grouped with reaction; operator splitting avoids such solves but typically provides weaker coupling and no inexpensive temporal error estimate; and PIROCK is tied to a specific Runge–Kutta–Chebyshev super-time-stepping (STS) construction. We introduce extended super-time-stepping (ExtSTS) methods, a family of time integration schemes that combine super-time-stepping methods for diffusive terms with explicit, implicit, or ImEx Runge–Kutta treatment of the remaining terms. The coupling is based on multirate infinitesimal techniques, yielding solve-decoupled methods that retain localized implicit solves, support embedded error estimation for adaptive time stepping, and allow flexible use of modern STS methods. We present the ExtSTS method family, provide a robust technique for ExtSTS method creation, formulate the corresponding linear stability theory, and construct embedded ImEx, explicit, and implicit ExtSTS methods. Numerical experiments on one- and two-dimensional advection-diffusion-reaction problems show that ExtSTS methods are robust across parameter regimes and boundary conditions, and are often more efficient than ARK, Strang splitting, and PIROCK methods, especially when strong coupling between operators is important.

Continuing the study of mother-body measures with algebraic Cauchy transform, we associate with a positive algebraic germ $f$, its irreducible equation $P(z,w)=0$, and a compact convex set $K$ the functional \[ \mathfrak D_{P,K}(\mu)= \int_K\left|P\bigl(z,\mathcal C_\mu(z)\bigr)\right|^{1/d}\,dA(z), \qquad d=\text{deg}_wP, \] on the set of positive measures supported in $K$ and whose Cauchy transform has germ $f$ at infinity. We prove continuity of $\mathfrak D_{P,K}(\mu)$ and attainment of its minimum, characterize zero defect, and obtain the estimate \[ \mu(D(a,r))\le C_1r+C_2\mathfrak D_{P,K}(\mu)/r \] away from the zero set of the leading coefficient of $P$, where $D(a,r)$ is the disk of radius $r$ centered at $a$. We establish a dual formula for the defect in the rational case and construct positive algebraic Cauchy transforms by Herglotz theory, Fuss–Catalan and Raney laws, positive sums, polynomial pushforwards, and branch graphs. The passage from zero defect to a mother body is made under the planar-null hypothesis of the support.

We study the topology of learned representations in predictive coding networks (PCNs), a neuro-inspired bidirectional architecture, using a quantitative layer-wise persistent homology analysis. We train well-performing PCNs on a synthetic classification dataset ($\geq 99.9\%$ test accuracy) and on MNIST ($\geq 95\%$ test accuracy), and measure how topological features change across layers for different architectures and activation functions. We find that smaller PCNs collapse connected components across layers earlier than larger models (Spearman $\unicode{x1D70C} \in [0.72, 0.79]$ across activations), with model size measured as the sum of hidden-layer widths. We also observe a strong negative correlation ($\unicode{x1D70C} = -0.58$) between the depth at which simplification occurs and reconstruction error; i.e., architectures that simplify later reconstruct better. Finally, a seed-level bootstrap comparison across architectures and activations shows that PCNs consistently collapse connected components later than matched MLPs, with an average difference of $3.6$ layers. These results suggest that persistent homology offers a useful quantitative lens on the compression–reconstruction tradeoff in PCNs, and that both model capacity and the recurrent, bidirectional dynamics of predictive coding inference shape when this tradeoff is resolved across layers.

This paper provides formulas for Koornwinder polynomials in analogy with the creation formula, the alcove walk formula and the non-attacking fillings formula for the type $GL_n$ Macdonald polynomials. We state the creation formula in terms of the divided-difference operators used in Schubert calculus, and we use a box-greedy reduced word to reformulate the alcove walk formula in terms of uncompressed set-valued tableaux. Then two types of compression, “around-the-end compression” and “across-the-$0$-gap compression”, are used to derive a formula for Koornwinder polynomials in terms of compressed set-valued tableaux. Throughout we work in the full generality of relative Koornwinder polynomials, which are the analogues of the permuted basement Macdonald polynomials used in the type $GL_n$ case.

Let $M_\gamma=T^2\times_\gamma S^1$, where $\gamma\in\mathrm{SL}_2(\mathbb{Z})$ is hyperbolic. For every $N\geq2$, we compute the empty-skein, or quantum-torus, direct summand in Kinnear's decomposition of the $\mathrm{SL}_N$-skein module of $M_\gamma$, thereby answering his centralizer question for this summand in the hyperbolic case. Its dimension is expressed in terms of the periodic Nielsen numbers $N_k=|\det(I-\gamma^k)|$ and the visible Nielsen numbers $V_{e\mathbb{Z}^2}(f_\gamma)$ carried by torsion in the Weyl coinvariant lattices. Only moduli $e\mid N$ occur, so the rank-$N$ summand is determined by $N_1,\ldots,N_N$ together with the visible Nielsen numbers at the divisors of $N$. On the $\mathrm{GL}_N$ permutation lattice these coinvariants are torsion-free, so no such correction occurs; the observer corrections arise precisely upon passage to the $\mathrm{SL}_N$ character lattice. For $N=3$, we obtain an explicit formula with the single correction $V_{3\mathbb{Z}^2}(f_\gamma)$, and construct infinitely many pairs of non-homeomorphic hyperbolic torus bundles whose $\mathrm{GL}_N$-skein-module dimensions agree for every $N$, while their $\mathrm{SL}_3$ quantum-torus summands differ in dimension by six. These pairs also have identical periodic Nielsen data and finite-cover visibility profiles at every iterate. We do not compute the additional endomorphism-algebra summands of the full $\mathrm{SL}_N$-skein module.

For any oriented smooth $4$–manifold $X$ diffeomorphic to $(S^2\times D^2)^{\natural n}$ ($n\geq0$), the author establishes a natural isomorphism of abelian groups: $$\mathrm{Mod}(X,\partial X)\cong \mathrm{Mod}(D^4,\partial D^4)\times\wedge^2H_2(X;\mathbb{Z}),$$ concerning the (smooth) boundary-fixing mapping class group of $X$. For $n=2$, the Budney–Gabai barbell twist $\varphi\in\mathrm{Mod}(\mathcal{N},\partial\mathcal{N})$ is identified with a generator of the factor subgroup $\wedge^2H_2(\mathcal{N};\mathbb{Z})\cong\mathbb{Z}$. Up to boundary-fixing diffeotopy, the barbell spines of $\mathcal{N}$ are completely classified by the bases of $H_2(\mathcal{N};\mathbb{Z})\cong\mathbb{Z}^2$, forming a homogeneous set modeled on the group $\mathrm{GL}(H_2(\mathcal{N};\mathbb{Z}))\cong\mathrm{GL}(2,\mathbb{Z})$. Any barbell spine of $\mathcal{N}$ gives rise to an implanted barbell twist equal to $\varphi$ or $\varphi^{-1}$ in $\mathrm{Mod}(\mathcal{N},\partial \mathcal{N})$, according to the sign of the homological basis orientation.

Score-based diffusion models are typically formulated using continuous-time stochastic differential equations and measure-theoretic stochastic calculus. In this paper, we develop a hyperfinite formulation of score-based generative modeling within the framework of Nonstandard Analysis. Starting from an internal diffusion process on a hyperfinite grid, we derive the associated infinitesimal generator and establish its correspondence with the classical Fokker–Planck equation. We then obtain a hyperfinite backward-mean identity that yields the reverse-time drift and provides a constructive derivation of the reverse-time SDE. Building on these results, we show that minimization of an internal score-matching objective recovers the score function required by the reverse-time dynamics, thereby connecting score estimation with generative sampling directly at the hyperfinite level. Under suitable assumptions, we further derive a hyperfinite Girsanov formula and establish a relationship between likelihood optimization and Fisher-divergence objectives. Finally, we analyze the second-order consistency of the hyperfinite dynamics and show that the leading correction term depends explicitly on the fourth moment of the increment distribution, with the Gaussian value $\kappa=3$ eliminating the leading dispersion contribution. Taken together, these results provide a unified hyperfinite framework for diffusion-based generative modeling–while laying foundations for further extensions–that links discrete grid dynamics, reverse-time diffusion, score matching, and likelihood-based formulations within a common nonstandard setting.

Polymer quantization program applied in Loop Quantum Gravity/Cosmology leads to nonseparable Hilbert spaces. Commonly, one sidesteps this problem by singling out and working with a separable superselection sector. This is however often no longer accessible in more involved models beyond isotropic ones. In an alternative approach one builds a separable Hilbert space as an integral over all available sectors. Here we test the dynamics following from the latter on the example of a flat isotropic Universe admitting negative cosmological constant and a massless scalar field. There, numerical evolution of (initially) semiclassical states shows that there is no relevant difference in their long term semiclassicality properties in comparison to those in a single sector approach. Further, the older (problem-specific) numerical methods are compared against an application of more common and more efficient standard tools (eigen library).

We show that the computational complexity of Reshetikhin-Turaev invariants of simply colored links is preserved when their underlying ribbon fusion categories are related by the zesting construction. Zesting modifies an $A$-graded ribbon fusion category $\mathcal{C}$ with additional algebraic data $\zeta$ to produce a new category $\mathcal{C}^{\zeta}$ whose link invariants are known to differ from those of $\mathcal{C}$ by an invariant of $A$-colored links $\mathcal{J}_{\zeta}$ depending only on $\zeta$. Building on this understanding and on earlier work on quantum braid group representations under zesting, our result suggests how zesting contributes to the organization of (2+1)D topological quantum field theories and topological phases into complexity-theoretic hierarchies. To prove our main result we develop a local formalism analogous to the Reshetikhin-Turaev construction to compute tangle invariants $\mathcal{J}_{\zeta}(T)$, which leads to a polynomial time algorithm to compute invariants of links $\mathcal{J}_{\zeta}(L)$. A byproduct of our construction is an identification (up to a sign) of the link invariants $\mathcal{J}_{\zeta}(L)$ as rack cocycle invariants, which may be of independent interest. Our formalism also extends to define invariants of closed $3$-manifolds with $A$-structure and we obtain similar complexity results for homotopy quantum field theories built from $A$-modular fusion categories.

We prove that every smooth two-component split sphere link $L\sqcup R\subset S^4$ admits infinitely many smooth splitting $3$-spheres that are topologically non-isotopic. This generalizes a theorem of Tatsuoka from the two-component sphere unlink to split links with arbitrarily knotted sphere components. In the course of the proof, we establish a general sufficient condition under which a connected sum of smooth $4$-manifolds admits infinitely many topologically non-isotopic splitting $3$-spheres. This criterion may be of independent interest; in particular, it applies to all previously known examples of nonuniqueness for splitting $3$-spheres of positive-genus surface links.

We study a Purcell three-link microswimmer whose joints are odd-elastic: the torsional stiffness is non-Hermitian, its antisymmetric part $k_o$ injecting mechanical work so that the internal elastic drift is non-conservative. Our main finding is a sharp separation between geometry and cost — odd elasticity is invisible to the control geometry of the swimmer and visible only in the energy of a manoeuvre. The mechanism is a single algebraic fact: the drift lies in the span of the two control vector fields, so the system is feedback-equivalent to a driftless one and the odd modulus enters the bracket structure only through the scalar $\det\textbf{K}=k^2+k_o^2$. From this we deduce that the abnormal extremals of the energy problem are unchanged by $k_o$, that the swimmer is globally controllable for every non-reciprocity with no threshold, and that its nilpotent model is the Cartan $(2,3,5)$ sub-Riemannian structure, deformed only by the metric scaling $g_\chi=(1+\chi^2)g_0$. The odd modulus acts solely on the cost: casting the optimal-steering problem in sub-Finsler (Randers) form, we prove that a prescribed reorientation is strictly cheaper for either sign of $k_o$. Full Resistive-Force-Theory simulations confirm the analysis and reveal that at isotropic drag the swimmer becomes a pure rotator, turning without translating.

We study distributed testing of $\mathrm{Ber}(\alpha)$ versus $\mathrm{Ber}(\beta)$ in the broadcast, or shared-blackboard, model. For protocols with constant advantage, we characterise up to universal constant factors the information complexity under either hypothesis for every pair $\beta<\alpha$. The characterisation shows that the two information costs can be quite different and identifies three parameter regimes, with optimal protocols based respectively on clean samples, a noisy binary symmetric channel, and an asymmetric $Z$-channel. The lower bounds rely on a novel mixed Hellinger–Jensen–Shannon inequality that may be of independent interest. We also characterise the constant-advantage information complexity of testing arbitrary discrete distributions via an optimisation problem over channels, and show that binary-output channels suffice. We obtain bounds for bounded likelihood-ratio distributions, and give general upper bounds in terms of $\chi^2$ divergence. As applications, we recover the broadcast-model set-disjointness lower bound, and derive stronger lower bounds in the multi-pass streaming setting for some problems considered in prior work.

This work introduces an algebraic framework yielding explicit, closed matrix differential-difference equations for eighteen models in the exactly solvable sector of the KPZ universality class across four scaling regimes. By organizing Fredholm determinant data into an overdetermined linear problem on a directed lattice graph, we derive a compatibility system termed the diamond equations. Elementary seed data extracted from the shift structure of the Fredholm kernel provides simple solutions to this system. We then construct a Darboux transformation to compress the infinite-dimensional Fredholm data into a finite-dimensional matrix observable. We show this dressing procedure preserves the diamond equations; consequently the resulting matrix observable obeys the same nonlinear structure as the initial seed data. Verifying a closed nonlinear equation for any specific model thus reduces to checking a handful of linear conditions on its kernel data. Under a scalar reduction, the framework produces variable-coefficient Hirota-Miwa equations for Fredholm determinants, recovering the one-point bilinear equations of the author's earlier work as specializations. To supply the necessary seed data, a product graph construction with admissible propagators builds multipoint data in the fully discrete regime, while Euclidean division in a polynomial quotient algebra handles vertex and polymer models. Finally, we demonstrate the diamond equations are a gauge-equivalent reparametrization of the non-abelian Hirota-Miwa system, a central system in classical integrability theory.

Multimodal supervised learning seeks to leverage multiple heterogeneous data sources to improve predictive performance. A central challenge is determining the fusion granularity across modalities: over-integration may amplify noise while under-integration fails to exploit cross-modal dependence. Existing approaches rely on pre-specified fusion architectures, from early to late fusion, that may not adapt to the underlying dependence structure among modalities. We propose DAIF, a data adaptive intermediate fusion framework that combines random matrix theory and non-parametric dependence measures to learn fusion structure directly from data. We operate under a Bayesian multimodal factor model where the prior on the latent factors determines the cross-modal dependence. Our method clusters modalities based on estimated intermodal dependence, then performs clusterwise empirical Bayes estimation of the priors. These estimated priors are used to construct denoisers within an approximate message passing (AMP) framework, yielding denoised low-dimensional features that borrow strength across related modalities while preserving modality-specific signal. The resulting embeddings are used for downstream supervised prediction. We evaluate the framework through simulations under varying dependence structures and signal regimes, comparing against several benchmark methods, and demonstrate its practical utility on two multimodal datasets, namely a trimodal TEA-seq dataset (Swanson et al., 2021) and TCGA-BRCA dataset (Goldman et al., 2020). In the first example, we predict the expression level of a T-cell differentiation marker protein and in the second case we analyze patient survival prediction based on multimodal information. Our method competes with or outperforms the state-of-the-art techniques in both prediction problems, demonstrating its versatility across diverse supervised learning tasks.

For a self-map (f\colon X\to X) of a finite connected CW complex, we introduce visible Nielsen numbers by pushing the Reidemeister trace to finite regular covers compatible with (f). Optimizing over such covers gives a visibility profile and three resolution thresholds, including the observer complexity (\operatorname{oc}(f)), the least cover degree that separates all essential Nielsen fixed-point classes. We identify finite-observer indistinguishability with twisted conjugacy in the profinite completion and determine the associated blind subgroup. We also realize nonzero Reidemeister traces that vanish for every finite observer, and show that observer complexity is unbounded on a fixed finite complex even when the induced maps on the fundamental group and homology, the Lefschetz number, and the Nielsen number are fixed. For toral endomorphisms we compute the complete visibility profile as an exact divisor staircase. For compatible maps of principal torus bundles, in the finite-Reidemeister regime, we identify the unique maximal resolving kernel and express (\operatorname{oc}(F)) in terms of the Nielsen number and the radical of the bundle characteristic class modulo the fibre map. This yields exact formulas for scalar dilations of integral Heisenberg nilmanifolds, together with observer entropy and a rational observer-complexity zeta function. Finally, we construct toral and Heisenberg systems with identical Nielsen and Lefschetz sequences, Nielsen zeta functions, differential eigenvalues, and topological entropy, but exponentially different observer-complexity sequences.

Polar codes achieve channel capacity as block length increases, but this asymptotic advantage comes at the cost of decoding speed: the conventional successive cancellation (SC) algorithm is inherently sequential, which limits its practicality for high-throughput applications. This work addresses that limitation by introducing the Speculative Successive-Cancellation (Spec-SC) framework, which incorporates speculative execution into the SC decoding process. While node-based decoders with special nodes enable partial parallelism for specific, predefined structures, Spec-SC extends this capability by permitting parallel execution in general nodes that do not match any special-node pattern. A built-in verification mechanism ensures that this speculation does not compromise decoding correctness. Simulation results show that Spec-SC achieves identical BER and FER performance to standard SC decoding, while reducing the average number of sequentially traversed nodes by up to $69\%$ for a polar code of length $N=4096$. These results demonstrate that Spec-SC enables additional algorithm-level parallelism beyond what special-node approaches alone can achieve, paving the way for speculative hardware decoder implementations.

Spectral transitions in non-Hermitian lattices often arise from the competition between non-reciprocal skin accumulation and inter-component hybridization. In short-range systems formed by two coupled chains, this competition conventionally leads to the logarithmic critical-length law $N_c\sim\ln D$, where $D$ is the transverse separation between the chains. Here we show that long-range hoppings fundamentally reorganizes this critical behavior, producing a hierarchy of distinct scaling laws. When only the hybridization couplings are power-law decaying with exponent $\alpha$, the onset becomes algebraic, $N_c\sim D^{\alpha/3}$. When the hoppings within each chain are themselves also power-law decaying, in addition to the hybridization couplings, the system enters a scale-covariant regime for $\alpha<2$, in which the criticality threshold equation depends only on the system aspect ratio $N_c/D$. At $\alpha=2$ and beyond, this regime is followed by a marginal logarithmically corrected and algebraically corrected regimes, respectively. We identify two new non-local mechanisms that enable this unconventional critical hierarchy: a nonanalytic band-edge dispersion from long-range intra-chain hoppings, and parity-mixing hybridization induced by non-reciprocity. Our results show that nonlocality systematically removes the physical length scales i.e. skin depth underlying conventional critical non-Hermitian skin behavior, offering a platform-independent framework testable in programmable topoelectrical circuits, photonic lattices and digital quantum simulators.

Universal hallmarks of quantum chaos—such as the ramp and plateau in the spectral form factor—and the ramp's gravitational duals involving wormholes are widely interpreted as consequences of spectral or ensemble averaging. In this paper, following an earlier proposal of~\cite{Liu25c}, we develop an alternative approach: these phenomena arise as macroscopic smooth structures hidden within erratic microscopic data, which can be isolated through a smooth filter projection. Using the semiclassical Gutzwiller trace formula as a paradigmatic example, we illustrate how many features characteristic of random matrix models—including the ramp, the plateau, the spectral curve, and single-eigenvalue instantons—can be derived in the semiclassical limit without invoking ensemble or explicit spectral averages. We postulate the existence of a minimal Gutzwiller-like structure in the large-$N$ limit of holographic systems and explore its consequences. Beyond deriving the ramp and the plateau, this Gutzwiller-like structure predicts universal rapid macroscopic oscillations in the density of states and the possible existence of hyper-instantons, both of which involve double exponentials in $1/N^2$. On the gravity side, we demonstrate how spacetime wormholes enable the construction of emergent hyper-non-perturbative objects—such as baby-universe and wormhole condensates—which yield double exponential effects in $G_N$. This mirrors the postulated boundary Gutzwiller-like structure and provides a dual gravitational derivation of the universal rapid macroscopic oscillations in the density of states and the spectral plateau.

We study the BRST quantization of the null string after completing its local gauge symmetry by Carroll-Weyl transformations. The resulting worldsheet theory possesses three first-class constraints, $C_1 = P^2$, $C_2 = P \cdot X'$, and $C_3 = P \cdot X$, whose modes realize a Weyl-BMS algebra. The additional Carroll-Weyl constraint qualitatively changes the quantum gauge complex: its scalar $s$-ghost is intrinsically coupled to the BMS $bc$-ghost sector, and the anomaly analysis involves three independent cocycles rather than a single Virasoro-type central charge. Starting from the gauge-fixed action, we derive the complete Faddeev-Popov complex, construct the matter and ghost currents and the BRST charge, and evaluate their equal-time operator products in the flipped, equivalently highest-weight, representation. The matter and ghost anomaly coefficients are $(c_{LL},c_{LS},c_{SS})_{\mathrm{matter}}=(2D,-D,-D)$ and $(c_{LL},c_{LS},c_{SS})_{\mathrm{ghost}}=(-54,6,4)$. Because the corresponding central terms multiply linearly independent ghost bilinears in $Q_B^2$, BRST nilpotency requires the three conditions $D=27$, $D=6$, and $D=4$, respectively. These conditions are mutually incompatible. Consequently, there is no target-space dimension in which the minimal flat Carroll-Weyl matter-plus-ghost complex is anomaly-free in the highest-weight representation. The familiar $D=26$ condition of the ILST null string is recovered only after truncation to the two-constraint BMS subsector, which defines a different quantum gauge complex.

Let $G$ be a connected reductive group over a non-archimedean local field. The main result of Fargues and Scholze [FS21] for the geometrization of the local Langlands correspondence is the construction of a “spectral action” on the category of $\ell$-adic sheaves on $\text{Bun}_{G}$, the stack of $G$-torsors on the Fargues-Fontaine curve. The goal of this paper is to prove a conjecture of Fargues which says that one can extend this construction to the larger stack $\text{Bun}_G^e$ of $G$-torsors on the Kaletha gerbe over the curve, as introduced by Fargues [Far22]. This “extended spectral action” allows for a version of the categorical local Langlands conjecture for an arbitrary connected reductive group $G$, and is the first such statement for those $G$ which are not extended pure inner forms of a quasi-split group, such as non-trivial inner forms of $\mathrm{SL}_{n}$. Finally, we prove this conjecture for tori, following the original argument of [Zou24] and, under the same assumptions as [Zou26] (including connected center), we reduce the “extended” version of the categorical conjecture to the one in Fargues–Scholze.

For integers $N\ge0$ and $n\ge2$, let \[ \Delta_N^{(n)} =\left\{\alpha\in\mathbb Z_{\ge 0}^n: \alpha_1+\cdots+\alpha_n=N\right\}. \] We study discrete unique continuation for functions on this lattice simplex. For an integer $R\ge1$, let $g:\Delta_{nR}^{(n)}\to\mathbb R$ satisfy \[ \sum_{i=1}^n g(\beta+e_i)=0, \qquad \beta\in\Delta_{nR-1}^{(n)}, \] where $e_i$ is the $i$th standard basis vector. We prove that nonvanishing at the center implies \[ |\operatorname{supp} g|\ge c_n R^{\lceil n/2\rceil}. \] Here $\operatorname{supp} g$ is the set of points where $g$ is nonzero, and $c_n>0$ depends only on $n$. The exponent $\lceil n/2\rceil$ is optimal. The proof represents the values of $g$ as polynomial coefficients and uses a Pascal uncertainty principle, which bounds from below the total number of nonzero coefficients of a one-variable polynomial and its unit translate in terms of their degree.

Analytic calculations of zero- and single-scale quantities in perturbative quantum field theory result into special numbers and functions, the first of which have been revealed during the last decades. These are generalizations of the polylogarithm in form of Kummer-Poincaré iterative integrals over special alphabets and extensions thereof.With growing order in the coupling constant, the polylogarithms, Nielsen integrals, the iterated integrals over linear denominator terms, cyclotomic letters, letters induced by quadratic forms, square-root valued letters, and more general functions contribute. For the nested sums we consider nested harmonic sums, generalized harmonic sums, nested sums implied by quadratic forms, cyclotomic harmonic sums, and nested sums containing central binomials. We construct the $q$-extensions of these special functions and of the nested sums, which are associated to them by the series expansion at $x=0$, and their Mellin transform in the $q$-free case. These functions are expected to play a role in perturbative calculations in the case of $q$-deformed commutation relations. For the simpler function spaces closed form solutions are presented. For more involved alphabets we present the algorithmic steps leading to the $q$-extension for the individual cases. We also derive the determining differential and difference equations of these higher transcendental functions. The $q$-extended special functions are quite different form the corresponding $\mu$-extended functions.

For each countable ordinal $\alpha\geq 2$, Filipów, Kowalczuk and Kwela introduced an ideal $\mathsf{conv}_\alpha$ on the countable compact ordinal space $\omega^\alpha+1$. Kowalczuk later proved that, for each countable limit ordinal $\lambda$, the ideal $\mathsf{conv}_{<\lambda}$ is the greatest lower bound of $\{\mathsf{conv}_\beta:\beta<\lambda\}$ in the Katětov order. At the first uncountable level, let $A\subseteq[2,\omega_1)$ be uncountable and let $D$ be a countable dense subset of $X_A=\prod_{\alpha\in A}(\omega^\alpha+1)$. The coordinate ideal $\mathsf{Conv}(A,D)$ on $D$ consists of those $B\subseteq D$ with $\pi_\alpha[B]\in\mathsf{conv}_\alpha$ for every $\alpha\in A$. For a pair $D\subseteq E$ of countable dense sets, call $\alpha$ non-small if $\pi_\alpha[E\setminus D]\notin\mathsf{conv}_\alpha$. In ZFC, if at most countably many coordinates are non-small, then $\mathsf{Conv}(A,D)\equiv_K\mathsf{Conv}(A,E)$. Under CH this countability bound is sharp: for every $A\subseteq[3,\omega_1)$ with $|A|=\aleph_1$, there are countable dense sets $D\subseteq D^*\subseteq X_A$ such that $\mathsf{Conv}(A,D^*)\leq_K\mathsf{Conv}(A,D)$ but $\mathsf{Conv}(A,D)\not\leq_K\mathsf{Conv}(A,D^*)$, and in particular $\mathsf{Conv}(A,D)$ and $\mathsf{Conv}(A,D^*)$ are not Katětov equivalent. The non-reduction is obtained, under CH, by diagonalizing along $\omega_1$ coordinates against the elements of $\omega^\omega$ that code retractions $D^*\to D$.

We develop a unified framework for analyzing the complexity of quantum property testing through functionals of the form $\mathcal{L}_{\phi}(\rho) = \operatorname{tr}(\phi(d\rho))/d$, where $\rho$ is an unknown $d$-dimensional quantum state and $\phi$ is a given function. A master theorem is established that derives sample complexity lower bounds for estimating $\mathcal{L}_\phi(\rho)$ from properties of $\phi$, combining Haar-random moment encoding with moment matching and best polynomial approximation. Corresponding query complexity lower bounds follow from quantum sample-to-query lifting. The framework yields (nearly) tight bounds for a broad class of problems, including entropy estimation (von Neumann, Rényi, and Tsallis), closeness estimation (trace distance and Uhlmann fidelity), spectrum estimation, rank testing (operator rank, Schmidt rank, and matrix product states). In particular, combined with known upper bounds, we show that the sample complexity of von Neumann entropy estimation is \[ \Theta\!\left(\frac{d^2}{\varepsilon\log^2(d)\max\{1, \varepsilon\log^2(d)\}} + \frac{\log^2(d)}{\varepsilon^2}\right), \] and the sample complexity of spectrum estimation is \[ \Theta\!\left(\frac{d^2}{\varepsilon^2\log^2(d)\max\{1, \varepsilon^2\log^2(d)\}}\right). \] In summary, our results resolve several open problems and establish the optimality of 37 quantum algorithms since 2015, up to constant or polylogarithmic factors.