Papers

We introduce frames generated by applying families of functions to a diagonal operator through functional calculus, with diagonal entries forming a Carleson interpolating sequence. First, for separated spectra contained in a Stolz domain and an angular sector, we prove that systems of fractional operator powers remain frames whenever the step size satisfies the natural angular restriction. This extends earlier results for positive real spectra and permits noninteger powers. We also show that the angular restriction is sharp in general. Second, we give an $\ell^2$-perturbation criterion for frames generated by more general continuous functions, assuming a density condition on a curve containing the spectrum. The results broaden the class of structured frames available in dynamical sampling and related operator-orbit problems.

Both generalized absolute value equations (GAVEs) and linear complementarity problems (LCPs) can be formulated as least-squares problems over the complementarity set. Because this feasible set is nonconvex, projected stationarity does not in general imply zero residual. We develop a safeguarded projected-gradient framework with a linear-system refinement on a selected polyhedral face. We establish matrix conditions under which every projected stationary point is a global solution and derive corresponding global convergence results. Specifically, when refinement on a correct active face returns a solution, we give explicit iteration bounds for active-face identification and prove finite termination. Numerical experiments on GAVE and LCP benchmarks demonstrate the high accuracy of the proposed framework across the tested settings.

We define via generators and relations an extended version of the matroidal Schur algebras introduced in earlier work of the author with Tom Braden. In the case of projective geometries, we relate the representation theory of the resulting algebras to the structure of the modular Steinberg representation of $GL(n,q)$. In the general case we show that the extended matroidal Schur algebras have a triangular, or Reedy, decomposition and are thus quasi-hereditary. We also obtain a new description of the original matroidal Schur algebras and clarify the connection to the flag spaces of Brylawski–Varchenko.

Higher-order Poisson equations involve an integer power of the Laplacian and a nonzero forcing term, and appear in areas of physics and engineering such as hydrodynamics, structural engineering, and image processing. We introduce a family of mean-value formulas for solutions to higher-order Poisson equations, given in terms of linear combinations of iterated means, with an exact remainder quantified by the oscillation of the forcing term. We also prove a regularity result and a strong converse to the mean-value property, in the sense that merely locally integrable functions satisfying our formulas are regular solutions of the higher-order Poisson equation. In contrast with the homogeneous case, where the mean-value property forces smoothness, the regularity attainable here is dictated by the regularity of the forcing term. Together, our results provide a mean-value characterization of solutions to higher-order Poisson equations.

A Legendre pair of length 333 would yield a Hadamard matrix of order 668, the smallest order presently unresolved by the Hadamard conjecture. We study the structured case in which both sequences are fixed by a common subgroup $H\leq(\mathbb Z/333\mathbb Z)^\times$ acting by coordinate multiplication. We prove that such a pair can exist only when $|H|\leq 6$. After a mod-3 compression reduces the problem to an order-108 kernel, there are exactly 30 subgroups. We exclude 21 of them, including all 19 subgroups of order at least 9. The final order-9 subgroup is eliminated analytically: its orbit structure restricts the 9-compressed entries to $\{\pm1,\pm17,\pm19,\pm35,\pm37\}$; the Legendre equations force a $+17,-17$ pair in one compressed sequence, and a single-shift autocorrelation bound then contradicts the required compressed correlation. The remaining exclusions use full-image compression, a row-sum congruence, exact meet-in-the-middle enumeration, and proof-carrying pseudo-Boolean encodings. The solver-assisted cases are accompanied by independently checked DRAT proofs or direct arithmetic certificates. The result constrains fixed common-multiplier symmetry only; the unrestricted existence problems remain open.

In the present paper, we identify all Jacobsthal numbers that may be expressed as a product of three Fibonacci numbers. More precisely, our main result shows that the only solution to the Diophantine equation \[ F_kF_lF_m=J_n \] for $2<k<l<m$ is \[ (k,l,m,n)=(5,7,8,12). \] The proof relies on techniques involving linear forms in logarithms.

We give a complete unconditional description of the zero set of the Dirichlet series $\mathbf{D}(s) = \sum_{n \ge 1} \zeta(2n)n^{-s}$, which continues meromorphically to $\mathbb{C}$ with a single simple pole at $s = 1$. The series possesses neither an Euler product nor a self-dual functional equation, and descriptions with this level of completeness are exceedingly rare for such series. The key input is an exact functional equation of Hecke type, obtained from the Lipschitz summation formula, which expresses $\mathbf{D}$ in the left half-plane as a gamma factor times a dual series over the complex logarithms of the perfect squares; Riemann's functional equation appears as a single column of the dual series. The zeros fall into four families. The half-plane $\sigma \ge \sigma_0 = 1.5001\ldots$ is zero-free, and $\mathbf{D}$ has a unique real zero $\rho_0 = 0.2004\ldots$. The zeros in the critical strip are perturbed $a$-points of $\zeta$ for values of $a$ near $-(\zeta(2)-1)$, and their counting function obeys a Riemann–von Mangoldt law. The remaining zeros form two complex-conjugate strings that recede into the left half-plane along explicit rays, are eventually simple, and satisfy an asymptotic with geometrically decaying error. The string geometry is governed by interference between the two smallest frequencies of the dual series, $2\log 2$ contributed by the entire part of $\mathbf{D}$ and $\pm 2\pi i$ contributed by $\zeta$. No hypothesis of Riemann type is assumed at any point.

In two dimensions, we develop a virtual element method to solve the Darcy's problem coupled with a nonlinear heat equation. This coupled model may allow for thermal diffusion and viscosity as a function of temperature. Under standard discretization assumptions and appropriate assumptions on the data, we prove the well posedness of the proposed numerical scheme. We also derive optimal error estimates under suitable regularity assumptions for the solution and appropriate assumptions on the data. We conclude with a series of numerical tests performed on different families of meshes that complement the theoretical findings.

Fibonacci and Lucas numbers have been extensively studied in various algebraic and number-theoretic contexts, including their modular inverses and generalizations. Motivated by these developments, we study the inverses of Fibonacci and Lucas numbers with rational indices through the codenominator function $F$. Uludağ and Gökmen (2022) showed that the rational-indexed Fibonacci number $F_X$, where $X\in\mathbb{Q}_{>0}$, can be expressed using $F$, and that infinitely many such representations exist. In this paper, we extend their work by deriving a general explicit formula for $F_X$ through the codenominator function. We establish precise conditions under which $F_X$ coincides with a Lucas number or deviates from the classical Fibonacci sequence. Moreover, by means of these generalized formulas, we compute the units of Fibonacci and Lucas numbers, thereby proving the existence of multiplicative inverses for all Fibonacci and Lucas numbers within this framework. These results reveal new structural properties of Fibonacci- and Lucas-related sequences and suggest further directions for number-theoretic exploration.

Let $h \geq 2$, and let $\mathbf{b} = (b_1,\dots,b_h)\in \mathbb{Z}^h$ be a zero-sum vector with nonzero coordinates. For a set $A=\{a_1<a_2<\cdots\}\subseteq\mathbb{N}$, let $r_{A,\mathbf{b}}(n)$ denote the number of $h$-tuples $(x_1,\ldots,x_h)$ of pairwise distinct elements of $A$ satisfying $b_1x_1+\cdots+b_hx_h=n$. We study density restrictions on sets $A$ for which these representation counts remain small, obtaining analogues of the classical density theorem for infinite Sidon sets. In the case $\mathbf{b} = (c_1,-c_1,\dots,c_k,-c_k)$, we prove that if $A(x)/(x/\log x)^{1/2k}\to\infty$, then $\frac{1}{x}\sum_{|n|\leq x} r_{A,\mathbf{b}}(n)\to\infty$, whereas if $A(x)\gg x^{1/2k}$, then $\frac{1}{x}\sum_{|n|\leq x} r_{A,\mathbf{b}}(n)\gg\log x$. This recovers Chen's theorem on $B_{2k}$-sequences. For general zero-sum vectors $\mathbf{b}$, we prove analogous bounds under gap conditions: if $a_{n+1}-a_n=o(n^{h-1}\log n)$, then $\frac{1}{x}\sum_{|n|\leq x} r_{A,\mathbf{b}}(n)\to\infty$, whereas if $a_{n+1}-a_n\ll n^{h-1}$, then $\frac{1}{x}\sum_{|n|\leq x} r_{A,\mathbf{b}}(n)\gg\log x$.

Given a coloring $c$ and an even $k\ge 4$, a nontrivial $k$-term arithmetic progression~($k$-AP) $a,a+d,\ldots,a+(k-1)d$ is called symmetrically colored if $c(a+(i-1)d)=c(a+(k-i)d)$, $\forall i\in[k/2]$. Deng, Tidor, and Zhao asked whether $[N]$ admits a coloring with $N^{o(1)}$ colors and no such 4-APs, and gave an $O(N^{\log_{22}3})$-coloring of $[N]$. We give an $O_k(p)$-coloring of $\mathbb Z/p^{k^2/4}\mathbb Z$ without such $k$-APs for every even $k\ge 4$ and every prime $p>k$, and hence an $O_k(N^{4/k^2})$-coloring of $[N]$, improving the exponent in the upper bound for $4$-APs from $\log_{22}3$ to $1/4$. The construction combines a carry-control coloring of base-$p$ digits with a layered field norm mapping. Together with Behrend-style product colorings, our result for $4$-APs gives $h(N)\leq N^{1/4+o(1)}$ in Erdős's Problem~160 on coloring every nontrivial 4-AP with at least three colors. This result also yields $\rho_4(\alpha)=O_\varepsilon(\alpha^{5-\varepsilon})$ for every $\varepsilon>0$, improving the bound toward Ruzsa's question. Our result for $k$-APs disproves Gowers' conjectured lower bound for all even $k\ge6$ for the first time.

In this work, we study the problem of testing conditional independence between random variables $X$ and $Y$ given a confounder $Z$. The local permutation test (LPT) offers a principled approach to this problem by partitioning the $Z$-space into pre-specified bins, and permuting the $X$ and $Y$ data within each bin, to assess the significance of an observed test statistic. However, when the partitions are pre-fixed, the resulting partition can be poorly balanced, as some bins may contain most of the samples while others contain only a few. This motivates the use of data-adaptive binning strategies, such as equisized bins with a fixed (typically small) number of points. We study this natural and practically important extension of LPT, providing finite-sample bounds on the Type I error for an arbitrary test statistic, providing stronger validity results than previously known. We also show that LPT attains power comparable to the oracle likelihood ratio tests derived from the Neyman-Pearson lemma. Within a linear confounder model class, we further analyze the effect of bin size and demonstrate that constant bin sizes can match the performance of partitions with growing bin-size. These results, further supported by extensive numerical simulations, position the proposed data-adaptive strategy as both practically implementable and statistically efficient.

A rotation of a permutation is a new permutation obtained by moving the first several terms of the permutation to the end of the permutation. A circular permutation is the set of all rotations of a permutation. The enumerations of permutations and circular permutations avoiding patterns of length three and four are well studied. In this paper, we provide exact formulas for the number of permutations whose first $k\geq 2$ rotations all avoid a given pattern of length three, as well as the number of permutations whose first three rotations respectively avoid the rotations of a given pattern of length three. In contrast to permutations and circular permutations avoiding patterns of length three, the Wilf-equivalence classes under study are entirely determined by complements and reverses. We also classify and enumerate permutations whose first two rotations avoid different patterns of length three.

The paper studies a Jacobi-type method for the eigenvalue problem of general complex matrices with simple eigenvalues. The method applies elementary triangular similarity transformations in order to annihilate selected off-diagonal elements and, when convergent, produces highly accurate eigenvalues. We give a new proof of its asymptotic quadratic convergence and derive an explicit, verifiable bound that describes the region in which this convergence is guaranteed. To make the method applicable well beyond matrices already close to the diagonal form, we introduce a preconditioning strategy. We use two types of preconditioners, both based on theoretical convergence results. The preconditioner is computed at lower precision to reduce computational cost, the associated similarity transformation is applied either at working or at higher precision, to preserve spectral information, while the main algorithm performs at working precision. Numerical experiments demonstrate that the resulting algorithm is robust and produces very accurate eigenvalues.

Let $(M^3, g, \mathbf{k})$ be a smooth, connected, asymptotically flat initial data set with connected outermost past apparent horizon $\Sigma $. We prove the Penrose conjecture, namely that $m_{\mathrm{ADM}}(g) \geq \sqrt{\frac{|\Sigma |}{16 \pi }} $, under the assumptions of the dominant energy condition and the $2$-convexity condition that the sum of the two smallest eigenvalues of $\mathbf{k}$ is nonnegative. The main tool is the $\sigma $-inverse mean curvature flow, together with a monotonicity formula developed in \cite{Dong26SigmaIMCF}.

We identify the topological and dynamical conditions for a Hamiltonian system to possess port-Hamiltonian reformulations with increasing resolution of structural detail that remains hidden in the Hamiltonian theory. We find that the principally attainable resolution grows exponentially with the number of factor manifolds into which the configuration space of a canonical Hamiltonian system decomposes and is tempered only by degeneracies of the original Hamiltonian.

In large-scale machine learning, distributed training commonly involves multiple workers evaluating the gradients of the model on different dataset partitions. A common challenge is the presence of straggling workers, which may significantly slow down training. Traditional gradient coding (GC) addresses this by duplicating dataset partitions across workers, allowing for the replacement of missing gradients from stragglers. However, GC requires workers to evaluate gradients on multiple dataset partitions in each step, potentially increasing overall training time. In this paper, we propose to pipeline GC, such that gradient evaluation is segmented across multiple steps and each worker evaluates gradients on just a single dataset partition per step. We develop the pipelined version for fractional repetition (FR) and cyclic repetition (CR), two representative dataset placement schemes in GC, and prove convergence guarantees for both. Through extensive simulations and experiments on cloud infrastructure, our schemes not only significantly reduce training time but also accelerate convergence compared to GC and other baselines.

Let $G$ be a countably infinite discrete group and let $ \pi:(X,\mu,G)\to(Y,\nu,G) $ be a nontrivial relatively mixing extension, where $X$ is a compact metrizable $G$-space. We prove that there exists a constant $\delta>0$ such that, for every injective sequence $(s_i)_{i\geq 1}$ in $G$, there is a Cantor set $K_{(s_i)}\subseteq X$ whose distinct points $x,x'$ satisfy \[ \liminf_{i\to\infty}\rho(s_i x,s_i x')=0, \qquad \limsup_{i\to\infty}\rho(s_i x,s_i x')>\delta. \] The method also yields higher-order scrambled Cantor sets. As a principal application, for a sofic group $G$, positive topological sofic entropy implies the preceding conclusion, answering a question of Huang, Li, and Ye. The same conclusion also holds for actions of arbitrary countably infinite discrete groups admitting an essentially free invariant measure of positive Rokhlin entropy. In contrast, we construct a hereditary subshift over a residually finite group with positive naive topological entropy for which all points converge to a common fixed point along a prescribed injective sequence. This answers a question of García-Ramos and Li negatively even for sofic groups.

The library librla is a randomized linear algebra library that is specifically designed for the intermediate-sized matrices (of dimension up to roughly 10,000) that arise in applications such as reduced order modeling, fast direct solvers, least squares solves and, in some settings, data compression. librla is the first software package that is both stable and efficient in several high-level languages: MATLAB, Python and Julia. It also provides increased functionality over existing software. Specifically, it allows the user to choose to create a factorization based on a fixed rank or a desired tolerance. The factorization options include QR, SVD and the interpolative decomposition. Additionally, the factorization can be generated either with access to the matrix or access to a matrix-vector multiplication routine. Numerical results compare the Python implementation with the available PyTorch and SciPy randomized factorizations. Performance of librla in the three languages is comparable.

Fuzzy-rule-based scalar objective functions provide a flexible way to encode qualitative preferences, reference regions, and interactions between criteria in multi-criteria optimization and decision making. However, the scalar preference landscape induced by such rules can differ substantially from the intended decision semantics. This paper investigates how membership placement, implicit single-criterion baseline rules, and explicit rule consequents affect the behavior of fuzzy scalar objective functions. Two analytically controlled bi-criteria Pareto fronts and an embedded two-dimensional dominated-reference formulation are used to separate front-selection mechanisms from reference improvement behavior. The study shows that apparent reference following can be caused by flat rule-activation plateaus, whereas genuine reference following requires localized minima with low tie ambiguity. In the dominated-reference setting, global memberships with competing consequents recover robust Pareto tradeoffs but do not necessarily improve each reference design. By contrast, a reference-based three-class rule set consistently improves dominated references, recovers the Pareto set, and avoids plateau-driven selection in the present tests. The results provide diagnostic metrics and practical guidance for constructing fuzzy scalarizations whose optimization behavior is consistent with the intended decision semantics.

Let $G=(V,E)$ be a simple graph, and let $k\geq 2$ be an integer. For an edge labeling $h:E(G)\to \mathbb{Z}_{k} \backslash \{0\}$, define the induced vertex label by \[ h^+(v)=\sum_{e \ni v} h(e) \pmod{k}. \] For $t\in \mathbb Z_k$, we say that $G$ is \emph{$t$-sum $\mathbb Z_k$-magic} if there exists such a labeling $h$ satisfying \[ h^+(v)=t \qquad\text{for all }v\in V. \] We say that $G$ is \emph{$\mathbb Z_k$-magic} if $G$ is $t$-sum $\mathbb Z_k$-magic for some $t\in \mathbb Z_k$. Similarly, if there exists an edge labeling $h: E(G) \to \mathbb{Z}_{k} \backslash \{0\}$ such that the induced vertex labeling $h^+(v)=\sum_{e\ni v} h(e)$ (mod $k$) is injective, then $G$ is called \emph{$\mathbb{Z}_{k}$-antimagic}. In this paper, we use the Combinatorial Nullstellensatz to analyze these two types of magic graph labelings.

Squeezed balls and spheres, introduced by Kalai, form a rich class of triangulated complexes arising from subcomplexes of cyclic polytopes, with well-understood shellability properties. Recently, Novik and Zheng introduced relative squeezed balls, obtained as differences of squeezed balls, and used them to construct large families of highly neighborly simplicial spheres. While these complexes are known to be constructible, their shellability has remained open. In this paper, we resolve this question by proving that both relative squeezed balls and their boundary complexes are shellable. We provide explicit shelling orders and characterize restriction faces, thereby establishing strong combinatorial structure for this new class of complexes.

Let $X(t)$, $t\in K$, be a centred Gaussian process with continuous sample paths on a compact metric space $K$, and let $M=\min_{t\in K}X(t)$. Let $\sigma_*^2$ denote the minimum covariance energy associated with $X$, and assume that $\sigma_*^2>0$. Motivated by the results of \cite{chakrabarty2018asymptotic} for smooth Gaussian processes, we show that, conditionally on $M>u$, the scaled overshoot $u(M-u)$ converges, as $u\to\infty$, to an exponential random variable with mean $\sigma_*^2$. Moreover, every weak subsequential limit of the conditional law of a measurable minimizer of $X$ is an optimal covariance-energy measure. In particular, if this measure is unique, then the conditional law converges weakly to it. The results are illustrated by stationary Gaussian processes, fractional Brownian motion, and fractional Brownian sheet.

We study the one-dimensional linear-potential system as an affine extension of the conformal triangle formed by the free particle, harmonic oscillator (HO) and inverted harmonic oscillator (IHO). Unlike these homogeneous quadratic Hamiltonians in the $sl(2,\mathbb R)$ sector, the linear-potential Hamiltonian involves the Heisenberg ideal of the Schrödinger algebra. Its direct relation to the free particle is a regular accelerated-frame transformation, developed here at the levels of the classical action, canonical transformation, wave-function intertwiner and propagator; its HO and IHO realizations instead arise through singular displaced-oscillator limits. The Airy energy eigenstates follow from a cubic-phase transform of free-particle momentum eigenstates, from condensation of highly excited harmonic-oscillator levels, and from the limit of a subdominant parabolic-cylinder scattering branch of the inverted oscillator. We also develop a planar extension in homogeneous crossed electric and magnetic fields, where uniform acceleration generates the electric interaction, while uniform rotation produces the Landau coupling and the centrifugal inverted-oscillator term; the guiding-center dynamics yields the Hall drift.

Incorporating an absolute refractory period into the population density approach for spiking neurons remains an open problem, despite evidence that refractoriness can strongly affect nonlinear transfer functions and network stability. We develop a rigorous operator-theoretic framework for neuronal population dynamics with a finite refractory time by augmenting the state space to include refractory history and formulating the problem as a non-self-adjoint boundary eigenvalue problem for the Fokker-Planck operator. This yields a complete spectral characterization of the generator, proves dissipativity and the existence of a contraction semigroup, and identifies defective eigenvalues as exceptional points where oscillatory modes emerge from coalescing relaxational modes. Within the framework of linear response theory, we also derive an exact transfer function that accounts for boundary conditions modulated by external input, correcting previous heuristic derivations and revealing additional threshold-noise contributions. Using this transfer function under a mean-field approximation, we further show that refractoriness in populations of interacting neurons can facilitate the onset of limit cycles, that is, stable oscillations in the firing rate. These results provide a rigorous foundation for spectral decomposition methods in computational neuroscience, opening the way to their further rigorous mathematical analysis.

We prove the existence of positive weak solutions for a quasilinear equation with exponential nonlinearity on arbitrary stratified Lie groups for \emph{horizontal $q$-Laplacians,} also called $q$-sub-Laplacians on these groups. The nonlinearity combines a concave term with an exponential growth that can be subcritical, critical, or supercritical with respect to the Trudinger–Moser inequality for Lorentz spaces on stratified Lie groups. We prove a version of this inequality in this paper. Since the problem is not variational in the natural energy space, classical minimisation or critical-point techniques do not apply directly. Positive solutions to the resulting semilinear equation are then obtained via a carefully designed Galerkin method applied to this setting. Our main theorem recovers previous known results on $\mathbb{R}^n$ in the complete range $q\in (1,\infty),$ and the previous known result on the Heisenberg group $\mathbb{H}^n$ for $q=2n+2,$ and is extended here in the full range $q\in (1,\infty)$. Other particular and fundamental cases, including the Engel group, are discussed.

We study the spatially homogeneous Boltzmann equation with continuous mass exchange on $X=(0,\infty)_m\times\mathbb R^d_v$, with a Grad cut-off hard-potential collision kernel. For bounded continuous symmetric mass-exchange rates, every nonnegative initial datum with finite number, mass, and kinetic energy admits a global nonnegative $L^1$-integral weak solution in $W^{1,\infty}(0,\infty;L^1(X))$ with number and mass conserved, kinetic energy dissipated. If $\int_X (m|v|^2)^{1+\delta} f_0(x)dx<\infty,$ for some $\delta>0$, this higher-energy moment propagates on every finite time interval and kinetic energy is conserved through the constructed solution. Moreover, every energy-dissipating solution propagates any such moment. Under the additional $1+\gamma$ moment assumption, the solution is unique among all energy-dissipating $L^1$-integral weak solutions with the same initial datum. We also establish a local theory for a linearly growing mass-exchange rate. With $H_p(f)=\int_X (1+m+m|v|^2)^pfdx,$ every datum with $H_p(f_0)<\infty$, where $p\ge1+\gamma$ admits a conservative local $H_p$-solution. Moreover, an $H_p$-solution continues across every finite time $T$ for which $H_{1+\gamma}(f)\in L^1(0,T).$ This proof requires no detailed-balance or relative-entropy structures. It is based on a new two-stage bootstrap method. The first stage rules out mass concentration at $m=0$, while the second stage combines this control with collision geometry to establish uniform integrability. These estimates provide the compactness needed for the global solution and for the identification of the nonlinear collision form.

In this paper, we construct counterexamples to the Schauder estimate for kinetic Fokker–Planck equations, disproving \cite[Conjecture~1.3]{HW}. The conjectured estimate would control second velocity derivatives using only velocity Hölder regularity of the coefficients and forcing, with no spatial regularity. We also give a zero-forcing variant, where the same {conditions are imposed on} a bounded zeroth-order coefficient. Finally, in dimensions $d\geq 2$, we give endpoint examples showing that bounded velocity-independent forcing may produce bounded distributional stationary solutions whose second velocity derivatives are not locally bounded.

Despite more than six decades of research, a tractable closed-form distribution for the typical Poisson–Voronoi cell volume remains unknown beyond one dimension. Building on the classical complementary-theorem structure for the Poisson–Voronoi fundamental region, we develop an explicit configuration-space factorization of the Palm-typical cell volume for a tessellation generated by a stationary Poisson point process of intensity \(\lambda>0\) in \(\mathbb R^d\), \(d\geq1\). Conditional on the number \(k\) of effective facets, the Voronoi flower volume is a \(\operatorname{Gamma}(k,\operatorname{rate}=\lambda)\) scale variable independent of the effective-neighbour configuration normalized to have unit flower volume. Mapping this normalized configuration to its cell-to-flower volume ratio \(A_k\) gives the conditional cell-volume representation as the product of the classical Gamma scale and a bounded geometric shape factor. From this representation, we derive exact mixture formulae, transform and moment identities, and criteria characterizing when the conditional cell-volume laws are Gamma. We also distinguish shape-factor variability within facet-number strata from mixing across strata as two sources of departure from a single Gamma law and recover the universal bound \(0<A_k\leq2^{-d}\). For the lower tail, we reduce critical negative moments of the shape factor to inverse-volume integrals over normalized configuration spaces. Under explicit critical-integrability and higher-facet summability assumptions, the density of the intensity-normalized cell volume has leading order \(y^d\) as \(y\downarrow0\). The framework recovers the one-dimensional distribution, admits explicit planar coordinates, and supports numerical evaluation of the mixture representation in dimensions two through four.

In arXiv:2306.01837, it was conjectured that one can engineer a large class of quantum field theory states for which the modular flow on a spacelike slice is "instantaneously local." Here we show that on null slices, such flows are highly constrained; ultraviolet universality essentially requires null-local modular flow to be unique. Concretely, we study massless Klein-Gordon theory in Minkowski spacetime, and construct, for any sufficiently regular future-directed vector field on the past null boundary of a causal diamond, a state that has this vector field as its instantaneous modular flow. We then show by explicit computation that no two distinct states in this class can be realized in the same local Hilbert space. Using a more abstract argument, we also show that in the vacuum sector of the theory, the only null-local modular flow in a causal diamond is provided by the vacuum state itself. We also comment on the construction of null-local modular flow for massive scalars, free Maxwell fields, free gravitons, and in curved backgrounds.