Papers

We investigate the distribution of zeros of the function $\zeta_2(s,s)$, namely the function defined by putting $s_1=s_2=s$ in the Euler double zeta-function $\zeta_2(s_1,s_2)$. Our theorems provide refinements of observations, obtained numerically in a previous paper by the first author and M. Shōji, with rigorous theoretical proofs.

We investigate forward self-similar solutions to the incompressible Navier–Stokes equations with homogeneous initial data $u_0(x)=\sigma(x/|x|)/|x|$, $x\in\mathbb R^n$, focusing on the critical four-dimensional case. For the existence theory in dimensions $3\leq n\leq5$, we develop a Galerkin scheme for the Leray profile problem, featuring a decomposition of the self-similar data and nonlocal heat kernel effects across spatial scales. Exploiting the structural properties of the data and the coercivity of the Leray operator, we derive uniform a priori estimates, pass to the limit, and obtain weak forward self-similar solutions. In the 4D case, we prove new Stokes-type estimates for the linear Leray system. The inverse Leray operator admits an integral representation as a singular integral operator; its $L^p$-boundedness follows from Calderón–Zygmund theory. Combined with a 4D compactness estimate for the convective term, this closes the regularity iteration. For spatial decay, we reformulate the problem in weighted spaces. The non-integrability of the low-frequency kernel causes a logarithmic loss under $\sigma\in W^{1,\infty}(\mathbb S^3)$. With $\sigma\in C^{1,\gamma}(\mathbb S^3)$, $0<\gamma<1$, frequency-localized estimates remove this loss and yield sharp decay.

One of the most important problems in computational algebra is the computation of real solutions of a polynomial system. Lasserre, Laurent, and Rostalski introduced a numerical method to compute the real radical of an ideal defining a finite set of real points. Their method is based on moment matrices and reduces the problem to solving a semidefinite program. In this work, we build on this method and on the work of Laurent and Mourrain to exploit the sparsity of the input polynomials. For this purpose, we extend the theory of moment matrices to the setting of affine toric varieties. We prove that, for sparse polynomial systems, this approach allows us to recover real points on affine toric varieties. Moreover, we show that a projection of the spectrahedron associated with the semidefinite program is, in fact, a polytope, and we propose a variation of the method to compute only one real solution of the system.

An invariant $v$ of $m$-component links is called "cableable" if there exists a $k$ such that whenever a link $L'$ is obtained from a link $L=(K_1,\dots,K_m)$ by replacing each knot $K_i$ with its $(p_i,q_i)$-cable for some $p_i$ and $q_i$, we have $v(L')=(p_1\cdots p_m)^kv(L)$. The following problem is implicit in a number of papers by P. M. Akhmetiev and originates from the Arnold-Moffatt program for finding topological lower bounds for the energy of a magnetic field: Does there exist a cableable finite type invariant of links in $S^3$ which is not a function of the pairwise linking numbers? According to Akhmetiev, he solved this problem himself, with the desired invariant being defined for $3$-component links as an analytic expression involving a magnetic field modeled on the given link. We offer a different solution, with the desired invariant for $m$-component links, $m\ge 3$, being a function of the coefficients of the Conway polynomial of the link and its sublinks; the function is given explicitly when $m=3$. Moreover, we show that the cables can be replaced by arbitrary satellites. Much of the proof is a study of low degree coefficients of the Conway potential function $\Omega_L(x_1,\dots,x_n)$ expanded as a formal power series in Conway's variables $z_i=x_i-x_i^{-1}$. We also discuss type $n$ invariants which are "cableable up to an invariant of type $n-1$", some cableable invariants which are not of finite type (particularly a certain modification of Milnor's $\bar\mu$-invariants), and applications to links of solenoids.

We prove a converse to the original Skoda $L^2$ division theorem, asserting that Skoda $L^2$ division estimates for a single nonzero holomorphic function force the weight to be plurisubharmonic. In addition, we establish corresponding results for higher-rank quotient bundles. Finally, we derive an optimal $L^2$ extension theorem directly from the division estimates.

Many generative systems return multiple candidate responses and are evaluated according to the best one. Recent work shows that, when demonstrations are optimal, pass@$k$ can reduce the sample complexity of learning from demonstrations by a logarithmic factor in $k$. We ask what happens when the demonstrator is not assumed to be optimal. We find that multiple responses provide a qualitatively stronger benefit in this setting. In a finite reward-class model with no reward feedback, moving from pass@$1$ to any pass@$k$ with $k\ge2$ changes the worst-case dependence on target accuracy from $1/\varepsilon^2$ to $1/\varepsilon$, uniformly over demonstrator quality. Under standard evaluation, where an unknown reward is fixed before training, increasing $k$ provides an additional and distinct benefit: the optimal dependence on a reward class of size $N$ improves from $\log N$ to $\log N/\log k$. We further show that these two effects can be separated. Under robust evaluation, where one learned policy must compete with the demonstrator simultaneously for every reward in the class, the fast $1/\varepsilon$ dependence persists, while the $1/\log k$ improvement can disappear. We establish matching upper and lower bounds in the corresponding regimes and give a greedy multiplicative-weights learner achieving the upper bounds without any assumption on demonstrator quality.

We study weighted $L^2$ Bernstein-type inequalities for algebraic polynomials on a trapezoidal domain. Using a parametrization by the unit square, we construct a natural symmetric second-order Sturm–Liouville operator and derive exact directional Bernstein inequalities with explicit constants. We also obtain strengthened estimates for the ordinary partial derivatives. Finally, we relate the corresponding boundary factors to Dini derivatives of the Siciak extremal function of the trapezoid.

For each integer k >= 1, let sigma_k(n) = sum_{d|n} d^k, let sigma_k^{-1} denote its Dirichlet inverse, and let J_k denote the kth Jordan totient function. Using established prime-power values of sigma_k^{-1}, we first obtain the elementary criterion sigma_k^{-1}(n) + J_k(n) + 2 = 0 precisely when n is prime. At k = 1, this involves Euler's totient J_1 = phi and provides a prime-only comparison for the next equation. Our main result classifies the zeros of sigma_k^{-1}(n) + mu(n) + J_k(n) + 3: for k = 1, they are precisely the primes and 18; for k >= 2, they are precisely the primes. The proof separates integers according to their prime-exponent patterns. Sign and divisibility arguments exclude composite zeros outside the cube-free nonsquarefree case, where an additional divisibility condition isolates 18 at k = 1 and precludes composite zeros for higher k. Standard convolution and Lambert-series identities provide the arithmetic framework, while an intermediate-divisor relation explains the exceptional zero. These criteria are structural characterizations rather than efficient primality tests, since direct evaluation of the multiplicative formulas generally presupposes a factorization of n.

We classify the Classifying Grothendieck rings of pivotal fusion categories of rank four over the complex field. There are exactly fifteen, each admitting a unitary categorification. Central induction and Frobenius-Schur indicators give a uniform Frobenius-Perron dimension bound of 3600. The finite classification combines new arithmetic and twist obstructions with an exhaustive census and exact exclusion certificates.

We study robust reconstruction for the \(q\)-state Potts broadcast process on a Galton-Watson tree in the growing-alphabet regime \(q\to\infty\), with boundary depth growing sufficiently quickly in relation to \(q\). We show that, in the regime \(d\lambda >1\), where \(d\) is the expected number of offspring and \(\lambda\) is the nontrivial eigenvalue of the Potts transition matrix, the root posterior is asymptotically unchanged by a broad class of noise channels applied independently to the boundary labels. This remains true even when the probability of retaining the true label at an individual boundary vertex tends to zero as \(q\to\infty\). More precisely, under suitable moment and noise assumptions, the noiseless and noisy root posteriors converge to one another in expected total variation, and hence have the same asymptotic Bayes-optimal reconstruction accuracy.

In the planar $n$-body problem, Moeckel and Montgomery \cite{MM25} showed that there is no infinite spin for total collision solutions, when the reduced and normalized configuration converges to an isolated central configuration. Following their approach, generalizations have been obtained for partial collision solutions and parabolic solutions in the planar case, and for total collision solutions in the spatial cases by various authors, all under the same assumption that the reduced and normalized configuration converges to an isolated central configuration. Here we prove there is no infinite spin for any collision or parabolic solution in all dimensions without such an assumption.

In this paper, we mainly prove Hovey's conjecture: for any commutative coherent ring, there is a lattice isomorphism between the lattice of wide subcategories of the category of finitely presented modules and that of thick subcategories of the perfect derived category. We construct, for every finitely presented module, a bounded finite free complex whose zeroth homology is the given module and whose homology lies in its abelian closure. This realization gives an inverse to Hovey's correspondence without a closure operation. In addition, we study finite extension filtrations, their behavior under change of rings, and the limits of reconstructing wide subcategories of arbitrary modules from finitely presented modules.

Given a reproducing kernel Hilbert space $\mathcal H$ and a bounded operator $T$ on $\mathcal H$, we say that $T$ obeys the strong reproducing kernel thesis (SRKT) if $\|T\|=\sup_k\|Tk\|$, where $k$ ranges over the normalized reproducing kernels in $\mathcal H$. Two problems are posed. One of these is to determine which Hankel operators on the Hardy space $H^2$ obey the SRKT. The other concerns the geometry of the unit ball of $BMO$ with respect to the Garsia norm.

We investigate the global solvability and the large-time asymptotic behavior of solutions to the Navier–Stokes–Maxwell system in two and three space dimensions. The system describes the interaction between a viscous incompressible fluid and an electromagnetic field through the Lorenz force and Ohm's law. The system exhibits a parabolic–hyperbolic coupling in which the dissipative Navier–Stokes system interacts with the hyperbolic Maxwell system. The Maxwell system is damped through Ohm's law with the Maxwell correction. For sufficiently small initial data with suitable Sobolev regularity, we establish the global well-posedness of strong solutions and derive optimal decay rates for the solution and its higher-order spatial derivatives. In addition, we show that the electric field decays faster by the extra factor $(1+t)^{-1/2}$ compared to the velocity and magnetic components. One of the key ingredients of our analysis is a detailed study of the linearized system, which reveals refined decay properties of the electromagnetic components. These linear estimates play a crucial role in the analysis of the two-dimensional case, allowing us to overcome the logarithmic loss arising from the borderline decay estimates. For the nonlinear system, our approach is based on a time-weighted energy method, specifically designed to capture the distinct dissipative properties of the fluid and electromagnetic components. A suitable compensating functional is introduced to exploit the coupling between the electric and magnetic fields and thereby recover the missing dissipation of the magnetic field. Combined with careful analysis of the nonlinear terms, these estimates yield our desired result.

The Forward-Forward Algorithm (FFA) replaces backpropagation (BP) with layer-wise local contrastive objectives, eliminating the backward pass and the need to retain intermediate activations, yet suffers a persistent performance gap with BP that worsens with depth. This paper diagnoses two structural failure modes: an optimization floor arising from concurrent local updates; and a geometric collapse of layer representations driven by the local update mechanism. On the optimization side, we prove that the FFA loss satisfies the Polyak–Lojasiewicz inequality at each layer; however, simultaneous layer updates induce inter-layer representation-distribution drift, so each layer optimizes against a moving input distribution and incurs an error floor. On the representational side, the pairwise similarity kernel of layer representations contracts exponentially toward rank one as depth increases, collapsing the diversity of per-layer error signals. This collapse bounds FFA's effective learning capacity, which measures the diversity of gradient information across layers, independently of depth, whereas BP's chain-rule signal preserves per-layer diversity, yielding a capacity that scales with depth.

Generalized smoothness assumptions have attracted growing attention in recent years, motivated in part by machine learning problems in which the gradient of the objective function may not be Lipschitz continuous. Among the most prominent of these is the $(\alpha, L_0, L_1)$-smoothness assumption. Existing methods that attain the best known complexity bounds require knowledge of, or upper bounds on, $\alpha$, $L_0$ and $L_1$, while the few parameter-agnostic methods available do not recover those bounds. In this paper, we show that trust-region methods attain the best known bounds for $(\alpha, L_0, L_1)$-smooth objective functions without prior knowledge of these parameters. Establishing these results requires new analytical tools, beyond those used in classical trust-region complexity analyses. We further show that our working model assumption allows the use of general model Hessian approximations and, in particular, accommodates limited-memory quasi-Newton updates. Finally, we show that our complexity bound is sharp in the nonconvex setting.

It remains an open question whether the modular data $(S,T)$ together with the Whitehead-link matrix $W$ determine a modular tensor category up to ribbon equivalence. We answer negatively by constructing two braided-inequivalent Dijkgraaf–Witten modular categories for $(\mathbb Z/p\mathbb Z)^3$ for prime $p\ge5$ with identical $(S,T,W)$ and, more generally, identical Reshetikhin–Turaev invariants of all framed oriented links with at most two components under a single bijection of simple objects. The construction varies an alternating $3$-cocycle while fixing a quadratic cohomology class, using restriction to subgroups of rank at most two to match link invariants and the quadratic and alternating classes together to obstruct braided equivalence. A complete classification of modular tensor categories up to ribbon equivalence therefore requires invariants beyond those of colored framed links with at most two components. We also analytically evaluate an uncolored partition function on a closed oriented three-manifold that separates all $(p-1)/2$ equivalence classes in the family.

We compute the Beilinson–Bloch height of the generalized Ceresa cycle $\mathrm{Ce}(V)=\frac 1 2(V-[-1]_*V)$ for any cycle $V$ of pure dimension $d$ on an abelian variety $A$, in terms of an arithmetic intersection number of a certain adelic line bundle. As an application, we prove that this height is given by an adelic line bundle over algebraic families.

We study the tensor nuclear norm minimization with general linear constraints, a fundamental model for low-rank tensor optimization that includes many practical applications in computer vision, recommendation systems, revenue management, transportation research. Although the tensor nuclear norm is a convex surrogate for the tensor rank, optimizing it remains computationally challenging because evaluating the nuclear norm itself is NP-hard. We propose a semidefinite programming framework that exposes a finite structure hidden in this problem, even though it appears to possess infinitely many constraints. In particular, we show that the tensor nuclear norm admits an exact SDP representation supported by a finite number of points on the unit sphere. Based on this representation, we develop two complementary solution approaches. The first replaces the unknown support set by a finite sphere discretization which yields a solution with a provable worst-case approximation guarantee. The second selects spherical points adaptively, returns an exact optimum upon finite termination, and otherwise converges asymptotically to an optimal solution. Numerical experiments on synthetic tensor completion instances and real highway traffic data demonstrate that the adaptive method achieves competitive, and often substantially improved, recovery accuracy relative to existing methods.

We propose an approach using matrix symmetry to accelerate hierarchical low-rank compression for wave transmission scattering. For Helmholtz transmission problems, we introduce a symmetric Müller-like boundary integral equation formulation. Although not of the second kind, this formulation creates no difficulties for direct solvers. A one-point corrected quadrature in a Nyström discretization preserves the complex symmetry of the discretized system. This formulation is integrated into a variant of a hierarchical semiseparable direct solver. Within this framework, the computational workload of the low-rank approximation in the proxy method is halved by exploiting the symmetric relationship between the left and right coefficients. Furthermore, we extend this symmetry-exploiting approach to elastic transmission problems by using a symmetric Galerkin boundary element method. Numerical experiments on a multi-node computing cluster demonstrate that the symmetric Müller-like formulation for Helmholtz scattering achieves on average a 1.41-fold speedup compared to the non-symmetric baseline while maintaining third-order convergence. Furthermore, the symmetry-aware solver for elastic scattering is more than 1.5 times faster.

Pricing European options under local volatility requires repeatedly solving a PDE whose coefficients change with recalibration, while practitioners need both prices and sensitivities across spot-time surfaces. Neural surrogates can amortize these solves, but near expiry the solution loses regularity, making curvature difficult to learn and allowing violations of no-arbitrage bounds. We propose a finance-informed Deep Operator Network (FI-DeepONet) that decomposes the pricing operator into an input-dependent Black-Scholes carrier and a learned correction. The carrier uses strike-line integrated variance, capturing the leading near-expiry curvature singularity, while a smooth monotone Financial Admissibility Layer enforces pointwise price bounds. We derive a short-maturity estimate and differentiated asymptotics for the exact correction, together with identities linking correction error to price, sensitivity, and PDE-residual errors. On in-distribution tests, FI-DeepONet reduces global relative price error by nearly an order of magnitude versus vanilla and physics-informed DeepONet baselines. On parameter-shift OOD tests within the same local-volatility family, generalization is moderate. On index-option data, the frozen model remains accurate without market-specific retraining, though it does not outperform the same-input analytic formula. We also give a quantum-compatible realization in which selected linear maps are exactly compiled or approximated by a restricted diagonal-orthogonal family before supervised adaptation.

We design adaptive overlapping temporal decomposition (AOTD), a parallel sequential quadratic programming (SQP) method for long-horizon nonlinear optimal control problems (OCPs). As designed, AOTD is the first overlapping temporal decomposition (OTD) algorithm that adapts the overlap size online, eliminating the need to fix a suitable overlap a priori. Specifically, at each iteration, it partitions the long horizon into overlapping nonlinear subproblems, performs one SQP step on each in parallel, and adaptively selects both the overlap size and the accuracy to which the resulting local KKT systems are solved. Two conditions govern the accepted step: an adaptive residual condition bounding the distance from the concatenated inexact direction to the exact Newton direction of the full problem, and a descent condition on the adaptive penalty parameters that certifies descent for a carefully chosen exact augmented Lagrangian. The residual condition is solver independent and accommodates both deterministic iterative solvers and randomized sketching solvers. Under standard regularity assumptions, we prove that the KKT residual converges to zero from any initialization and establish a local linear rate for sequences converging to a strict local minimizer. We validate AOTD on a power-system frequency regulation OCP and a Burgers PDE OCP, attaining a reduction in estimated FLOPs of approximately $2.5\times$ to the best fixed-overlap variant while converging across multiple horizon lengths with a single parameter setting.

In this paper we fully categorize all hybrid-variable (HV) discretizations of linear advection equations according to their stability property and establish the stability barrier of these methods. In the HV discretization framework, we find numerical approximations to both cell-averaged solutions and nodal solutions and evolve them in time simultaneously. We prove that the HV methods have a similar stability barrier as the classical finite difference schemes, that the semi-discretized HV scheme is stable if and only if it uses a stencil such that the number of unknowns in the upwind direction is precisely that in the downwind direction plus one or two, with only one exception when the two numbers are three and zero, respectively. It is also proved that all central HV schemes are neutrally stable, in the sense that the L2-norm of the solution remain uniformly bounded at all times. The theoretical predictions are verify by extensive numerical tests.

We prove that the five-column recursive-line Zarankiewicz number satisfies $z_{RL}(m,5)=3m+5$ for every integer $m\ge 12$. The proof is based on an explicit $12\times5$ seed configuration combined with a recursive one-row extension scheme. The seed attains the five-column cell bound and contains no unoccupied cells. Two selected pairs in the seed are opened and replaced by parallel paths, ensuring that each inserted row increases the total number of augmented edges by three while preserving the required simple configuration. The main challenge lies in verifying the strengthened recursive-line condition ${\rm (RW3+)}$ uniformly across all extension lengths. To this end, we develop a distance-reducing rectangle lemma that transfers certified inner-product relations along extension paths, reducing relations between distant labels to those at smaller path distances. This confines the verification for arbitrarily long extensions to a finite collection of seed and interface certificates. The resulting construction successfully satisfies pair identification, preserves distinct selected-edge classes, and certifies the orthogonality of distinct edge representatives. Furthermore, combining this constructed lower bound with the parameter hierarchy yields $z_2(m,5)=z_{RL}(m,5)=3m+5$ throughout this range.

For an integer $r\geq 3$ and real numbers $a_1,\ldots,a_r\in(0,1)$, let $a=a_1\cdots a_r$. We show that \[ \prod_{i=1}^r(a_i+a_i^2+\cdots+a_i^r)\geq \prod_{i=1}^r(a_i+a_i^2+\cdots+a_i^{r-1}+a). \] This inequality enables us to bound the measure of $r$-cross $t$-intersecting families.

Let $\cX$ be a curve of genus $g\ge2$ and $p$-rank zero over an algebraically closed field of odd characteristic $p$, and let $G\le\Aut(\cX)$ satisfy $|G|>24g(g-1)$. We prove that $G$ fixes a point unless $(\cX,G)$ belongs to an explicit list. For $g\ge4$, the curve is a cyclic quotient of a Hermitian curve, of a Giulietti–Korchmáros curve, or of Skabelund's cover of a Ree curve. We determine the equations and all the groups $G$ that occur, including their central kernels. The exceptional $A_7$ action in characteristic $5$ is on the Hermitian curve of degree $6$; genera $2$ and $3$ are treated separately. The equations are recovered from the group action by two rigidity lemmas for equivariant one-point $p$-group covers of the affine line. If $G$ has no common fixed point and its Sylow $p$-subgroups are noncyclic, then, unless $G$ induces ${}^2G_2(3)$ on the orbit of Sylow fixed points, the bound $|G|>\max\{84(g-1),2g^2\}$ gives a rational quotient with exactly one wild and one tame short orbit, and a rational Sylow quotient.

Strict unimodality of Ooms spectra fails for Frobenius parabolic Lie algebras beyond the maximal-parabolic class. We construct explicit Frobenius parabolics whose principal multiplicity sequences contain arbitrarily long outward rises: for every $L\ge1$ there are $L$ consecutive negative first differences, each of quadratic size in $L$. In a simple one-parameter subfamily the transition to nonunimodality occurs in $\slie_{136}$, where two adjacent multiplicities are $541<542$. We also give a representation-theoretic criterion for unimodality. If $F$ is a Frobenius functional with semisimple integral principal element $H$ and $\g=\bigoplus_j\g_j$, the Kirillov form makes the centered grading symplectic and its character has the unique expansion \[ \Theta_\g(x)=\sum_{m\ge1}(\dim\g_m-\dim\g_{m+1})\,\chi_{2m-1}(x) \] in irreducible $SL_2$ characters. Thus Ooms unimodality is equivalent to effectivity of this virtual $SL_2$ character, or equivalently to extension of the centered principal cocharacter to a symplectic $SL_2$ action. For Frobenius seaweeds the unbroken-spectrum theorem says that, after centering, every odd weight in the support occurs; strict unimodality asks for the stronger positivity of every irreducible coefficient. A companion paper proves that stronger property for every Frobenius maximal parabolic. The families constructed here show that beyond that class there can be arbitrarily long blocks of negative irreducible coefficients, with unbounded total negative mass.

We construct a class of global naked singularity solutions for the $(3+1)$-dimensional spherically symmetric Einstein–Maxwell-charged scalar field equations. The solutions arise from $C^{1,\alpha}$ asymptotically flat one-ended characteristic initial data with $0<\alpha\ll1$ and develop a curvature singularity that is not enclosed by an event horizon. The resulting spacetimes are genuinely charged, with a nonzero Maxwell field that satisfies a quantitative positive lower bound up to the singularity and admits an explicit asymptotic profile. As part of the construction, we develop a general framework for constructing naked singularity exteriors, namely, the causal future region of the past light cone emanating from the singularity. The framework accommodates a broad range of singular characteristic data and, in particular, allows for singular behavior of the Maxwell field near the singularity.

In these notes, we view control as a dynamically constrained sampling problem on the state-space of a control system. With this viewpoint, we extend methods from generative modeling, such as flow matching, normalizing flows and denoising diffusions to control problems. Concepts such as controllability, optimal control and trajectory planning play an important role in guiding the extension and understanding well-posedeness of the corresponding algorithms, with application to steering systems to target states or distributions and sampling from reachable sets. The notes are intended as an accessible introduction for readers with a background in control theory and robotics.

Let \(X\subset \mathbb P^n_{\mathbf w}\) be a weighted hypersurface over \(\mathbb F_q\), with associated stack \(\mathfrak X\). For \(\mathbf x\in X(\mathbb F_{q^r})\), let \(g_{\mathbf x}(r)=\gcd(k_{S(\mathbf x)},q^r-1)\), where \(k_S=\gcd(w_i:i\in S)\), and define \(\Theta_r(s)=\sum_{\mathbf x}g_{\mathbf x}(r)^{1-s}\) and \(Z_H^{\mathrm{can}}(\mathfrak X,s;t)=\exp(\sum_{r\ge1}\Theta_r(s)t^r/r)\). We prove the finite sector factorization \[ Z_H^{\mathrm{can}}(\mathfrak X,s;t)=\prod_{e\in E_{\mathbf w}} Z(X^{(e)}_{\mathbb F_{q^{o_e}}},t^{o_e})^{J_{1-s}(e)/o_e}, \] where \(X^{(e)}\) is the closed isotropy sector, \(o_e=\operatorname{ord}_e(q)\) for \(e>1\) and \(o_1=1\), and \(J_{1-s}\) is the Jordan totient function. Hence the specializations \(s=1,0,-1,-2,\ldots\) are rational in \(t\): \(s=1\) gives the Hasse–Weil zeta function of the coarse space, \(s=0\) the twist zeta function, and \(s=1-k\) the masses of the \(k\)-fold inertia stack. The factorization yields a functional equation when the nonempty sectors are self-dual and equidimensional; if they are also pure and geometrically irreducible, equidimensionality is necessary for every integral \(s\le0\). For weighted diagonal hypersurfaces of degree \(D\) with \(p\nmid D\), a finite Fermat cover gives purity and self-duality. We also prove a Lang–Weil asymptotic for isotropy entropy, with periodic leading term governed by isotropy periods and Frobenius orbits, and identify the function-field height with the stable stack height of Ellenberg–Satriano–Zureick-Brown, while the degree-based height remains a coordinate-complexity statistic.