Papers

A graph $G$ is list $k$-critical if $G$ is not $(k-1)$-list-colorable, but every proper subgraph of $G$ is $(k-1)$-list-colorable. In this paper, we study the function $f_{\ell}(n,k)$ denoting the minimum number of edges in an $n$-vertex list $k$-critical graph, as well as the function $g_{\ell}(k) = \liminf_{n \rightarrow \infty} \frac 2n (f_{\ell}(n,k) - k + 1)$. We show that for all $k \geq 4$ and $n \geq k+2$, every list $k$-critical graph on $n \geq k+2$ vertices has more than $(k-1+\frac 1{28}) \frac n2$ edges, which implies that $g_{\ell}(k) \geq \frac{1}{28}$ for all $k \geq 4$. This is the first result showing that $\liminf_{k \rightarrow \infty} g_{\ell}(k) > 0$. We also show that $g_{\ell}(k) \geq \frac{1}{24}$ for all $k \geq 352$. As a corollary to our result, we obtain the following improvement to Brooks' theorem: For all $d \geq 3$, if $G$ has no $K_{d+1}$ subgraph and has maximum average degree at most $d+\frac 1{28}$, then $G$ is $d$-list-colorable. All of our results hold in the setting of correspondence coloring (DP-coloring) as well. As a corollary of our correspondence coloring result, we also show that for each $d \geq 3$, a minimal unsatisfiable anti-functional constraint satisfaction problem (CSP) with variable domains of size $d$ has a primal graph either containing $K_{d+1}$ or with average degree at least $d+\frac 1{28}$.

A central theme in extremal graph theory is the supersaturation problem, which investigates the minimum number of copies of a target subgraph forced by prescribed edge conditions. This line of research goes back to Rademacher and Erdős for triangles, and was later extended to cliques by Lovász and Simonovits in the regime above the Turán threshold. Mubayi further extended this theory to color-critical graphs. Below the Turán threshold, a closely related existence-threshold phenomenon arises in the non-$p$-partite setting: a classical result of Brouwer shows that, for $n\ge 2p+1$, every $n$-vertex non-$p$-partite $K_{p+1}$-free graph has at most $e(T_{n,p})-\lfloor n/p\rfloor+1$ edges. Motivated by this threshold, we investigate a sharp clique-counting problem below the Turán threshold under the non-$p$-partite assumption. Let $p\ge 2$ and $s\ge 1$ be fixed integers. Let $Y_{n,p,s}$ be the graph obtained from $T_{n,p}$ by adding an edge inside a largest part and deleting all but $s$ of the edges from one endpoint of this new edge to a smallest part. Then $e(Y_{n,p,s})=e(T_{n,p})-\lfloor n/p\rfloor+s+1$. We prove that, for all sufficiently large $n$, every $n$-vertex non-$p$-partite graph $G$ with $e(G)\ge e(Y_{n,p,s})$ contains at least as many copies of $K_{p+1}$ as $Y_{n,p,s}$ does. The bound is sharp, as it is attained by the construction $Y_{n,p,s}$. Thus our result provides the exact clique-counting analogue of Brouwer's threshold for non-$p$-partite $K_{p+1}$-free graphs.

The panmagic square engraved in a temple in Khajuraho, India, and studied by Narayana Pandita in the 14th century, closely guards a secret: its group of symmetries is isomorphic to that of the hypercube, the four-dimensional analogue of the standard cube.

This paper introduces the extended $\mathcal{O}$-operators on Poisson algebras as a natural generalization of ordinary $\mathcal{O}$-operators, together with the extended Poisson Yang-Baxter equations. We show that $\mathcal{O}$-operators of weight $\lambda$ on Poisson algebras give rise to post-Poisson algebras, whose operad are the trisuccessor of the operad of Poisson algebras, and that extended $\mathcal{O}$-operators induce new Poisson algebra structures on module spaces.Equivalent characterizations of extended $\mathcal{O}$-operators are obtained via the symmetrizer-antisymmetrizer decomposition. The generalized Poisson Yang-Baxter equations are also introduced, and their connections with coboundary Poisson bialgebras and extended $\mathcal{O}$-operators are established. The tensor form of extended $\mathcal{O}$-operators leads to the notion of the extended Poisson Yang-Baxter equations, which generalizes the notion of the Poisson Yang-Baxter equations. Finally, the relationships among extended $\mathcal{O}$-operators, the extended Poisson Yang-Baxter equations, and the Poisson Yang-Baxter equations are studied in the framework of quadratic Poisson algebras and semi-direct product Poisson algebras.

In this paper, we study the axisymmetric Hall magnetohydrodynamics where $u = u^re_r+u^ze_z$ and $B = B^\theta e_\theta$. We construct small, smooth, compactly supported, and axisymmetric initial data $(u_{0},B_{0})$ and prove that there is a unique smooth axisymmetric solution with norm inflation in $H^{\sigma-1}(\mathbb R^3) \times H^{\sigma}(\mathbb R^3)$ for $1<\sigma<7/2$. Moreover, we study the case with fractional dissipation $(-\Delta)^\beta u$ and $(-\Delta)^\alpha B$, modifying the initial data slightly and proving that the solution inflates in $H^{\sigma-1}(\mathbb R^3) \times H^{\sigma}(\mathbb R^3)$ for $1<\sigma<7/2-2\max\{\alpha,\beta\}$ and $ 0 \le \alpha, \beta < 5/4$.

For an odd prime $p$, let $m(p)$ be the minimum cardinality of a set $A\subseteq \mathbb Z/p\mathbb Z$, with $|A|\geq2$, such that no sum in $A+A$ has a unique representation as an unordered pair from $A$, with repetition allowed. Bedert proved \[ m(p)\gg \log p\, \frac{\sqrt{\log^{(3)}p}}{\log^{(4)}p}. \] We prove the stronger lower bound \[ m(p)\gg \log p\,\log\log p. \] More generally, if $G$ is a finite Abelian group and $q(G)$ is the least prime divisor of $|G|$, then the same explicit estimate holds whenever $q(G)>2$, and in particular every subset $A\subseteq G$ with $|A|\geq2$ and no unique sum has cardinality $\gg \log q(G)\,\log\log q(G)$ as $q(G)\to\infty$. The proof has two structural inputs. First, a maximum subset of $A$ whose distinct-element subset sums of size at most four are all different has cardinality $\gg\log p$. This follows from a short-coordinate lemma and a collision-lattice determinant argument. Second, we refine Bedert's density increment. Alternative representations are oriented toward an uncovered endpoint, coalesced by their translation, and separated into wide, exposed, and recurrent batches. A load-sensitive entropy lemma codes the recurrent translations using their actual final fibre multiplicities. The resulting global shift-set complexity is $\exp(O(K))$, where $K$ is the ratio of $|A|$ to the level-four additive dimension. This forces $K\gg\log\log p$, and the theorem follows. All headline statements and the structural implications used to derive them have also been checked in Lean~4 with explicit integer constants. As a secondary and logically independent result, we construct weakly ternary-balanced sets and obtain \[ m(p)\leq \frac{(\log p)^2}{2(\log 3)^2} +\left(\frac{2}{\log 3}+o(1)\right) \frac{(\log p)^2}{\log\log p}. \]

We study deterministic Riemannian $p$-th-order oracle complexity (for $p\ge2$) on Hadamard manifolds, under strong horospherical ($h$)-convexity and strong geodesic ($g$)-convexity. The two notions agree in the Euclidean space. On a curved Hadamard manifold, $h$-convexity is a stronger notion than $g$-convexity and supplies global horospherical information. Writing the $p$-th-order condition parameter $Q_p=L_pR^{p-1}/\mu$, we obtain the Euclidean-optimal rate $Q_p^{2/(3p+1)}$ for strongly $h$-convex objectives on every Hadamard manifold, with a matching fixed-curvature lower bound. On hyperbolic space, the resulting horoball supports enable localization to the curvature scale. This replaces the condition parameter $Q_p$ by $Q_p \min\{1, 4/(\kappa R) \}^{p-1}$, subject to a logarithmic localization cost. Thus growing negative curvature ($\kappa R \rightarrow \infty$) can further improve the optimal Euclidean rate under $h$-convexity. For strongly $g$-convex objectives, matching upper and lower bounds recover the same Euclidean exponent when $\kappa R=O(1)$. In contrast, with growing $\kappa R$, we construct a hard family on the hyperbolic space with $Q_p\asymp_p(1+\kappa R)^p$ that requires $\widetilde{\Omega}_p(Q_p^{1/p})$ queries. This reveals a fundamental separation: the same hyperbolic divergence that sharpens horoball localization yields an information-theoretic obstruction for the full $g$-convex class.

We study the Fujita phenomenon for semilinear heat inequalities generated by variable-speed Laplacians on infinite weighted graphs. Assuming that the graph carries a proper adapted path metric, we establish an integral volume-growth criterion forcing every nonnegative global classical supersolution on the open cylinder $(0,\infty)\times V$ to vanish, without assuming an initial value or trace. We prove that a nontrivial supersolution of this kind exists if and only if the equation has a positive global Cauchy solution for some nonzero point-source datum. A complementary heat-kernel construction gives global Cauchy solutions for all sufficiently small point-source data when the same volume integral converges and a matching anchored heat-kernel upper bound is available. This proves sharpness on integer lattices and on a family of logarithmically perturbed weighted half-lines; in the latter examples, even the exponent of an iterated logarithm can determine the existence–nonexistence alternative. A finer nonexistence criterion couples intrinsic volume growth with the capacity of intrinsic annuli. Its proof combines parabolic testing, a Laplace–resolvent reduction, and a pathwise decomposition of resolvent currents. The nonexistence results require no volume-doubling property, Poincaré inequality, heat-kernel bound, or stochastic completeness.

Equitable scoring functions have a long history in meteorological forecast verification and have recently gained renewed prominence through the use of the Stable Equitable Error in Probability Space (SEEPS) score for evaluating precipitation forecasts from numerical and machine-learning weather prediction systems. This paper provides a systematic analysis of equitable scoring functions through the optimal forecasts they induce. We introduce the generalized diagonal score, which defines a broad class of equitable scoring functions that includes, up to equivalence, the SEEPS, Gerrity, Peirce, Barnston and diagonal scores. Within this framework, we show that optimal single-valued and categorical forecasts are characterized by crossing points between the predictive and climatological cumulative distribution functions. Moreover, we show that the generalized diagonal score, when evaluating predictive distributions, is proper but not strictly proper, and is insensitive to substantial forecast misspecification away from crossing points. The framework also yields scoring rules that are both proper and equitable for probabilistic forecasts with categorical outcomes. For single-valued and categorical forecasts, the crossing-point characterization shows that optimal forecasts can vary across climatologies even when the underlying predictive distribution is unchanged. Consequently, identical predictive distributions may lead to substantially different optimal forecasts under equitable scores, with important implications for assessing the suitability of such scores for any given application.

We study graphs containing no cycle whose length is divisible by $3$ or congruent to $4$ modulo $6$. We prove that every such $n$-vertex graph $G$, where $n \ge 2$, satisfies $e(G) \le (11/8)n-7/4$. Moreover, equality holds if and only if $n=8k+2$ for some nonnegative integer $k$ and $G$ is isomorphic to the explicitly constructed graph $H_k$. We also construct, for every $n\geq2$, an $n$-vertex graph with $\left\lfloor (11/8)n-7/4 \right\rfloor$ edges satisfying the same cycle restriction. Consequently, this is the exact maximum number of edges for every $n \ge 2$.

We construct an explicit tilting equivalence between the incidence algebra of every rational Dyck staircase and a canonical idempotent corner of a higher Auslander algebra of type~$A$. In the coprime case, this corner identifies with the algebra $B_0$ introduced by Xing. The resulting Dyck-corner equivalence supplies the missing link in the previously known chain of equivalences and thereby proves the Chapoton-Ladkani-Rognerud conjecture for coprime positive integers. The Dyck-corner equivalence itself requires no coprimality hypothesis and is compatible with replicated algebras. Our main tool is a linear-categorical extension of the interval-tilting mechanism of Chapoton-Ladkani-Rognerud. The relative theorem applies to finite $\kk$-linear categories under finite-global-dimension assumptions on the total category and its fibers. In contrast with the incidence-category setting, it allows arbitrary finite-dimensional $\Hom$ spaces and zero composites of nonzero morphisms, and it does not require the diagonal endomorphism algebras to be semisimple. The tilting object is constructed from exact right Kan extensions of fiberwise representables. We compute its opposite indexed endomorphism category, including all forced-zero compositions, and hence its opposite endomorphism algebra. Iterating this construction one coordinate at a time yields an explicit derived equivalence between the incidence algebra of every finite coordinate staircase and an idempotent corner of a higher Auslander algebra of type~$A$. We further realize the resulting staircase derived categories as triangulated subcategories generated by product Lagrangians in partially wrapped Fukaya categories of stopped-disk symmetric products and, in the coprime Dyck case, as Fukaya-Seidel categories of symmetric Brieskorn–Pham singularities.

We study a deterministic family of sharpness-aware minimization methods for smooth nonconvex functions. The perturbation is $$ y_k=x_k+\rho\, \frac{\nabla f(x_k)}{\norm{\nabla f(x_k)}^\alpha}, \qquad 0\leq\alpha\leq 1, $$ so that its effective radius is $\rho\norm{\nabla f(x_k)}^{1-\alpha}$. For $0<\alpha\leq1$, we give an explicit complexity bound above the stationarity level $(L\rho)^{1/\alpha}$. A one-dimensional quadratic example reaches this level exactly, showing that the bound describes a real limitation of the constant-parameter rule. The unnormalized case $\alpha=0$ is treated separately and requires $L\rho<1$. We then introduce a clipped rule which agrees with the constant-$\rho$ rule away from stationary points and becomes proportional to the gradient near them. The clipped method has $\norm{\nabla f(x_k)}\to0$ and the usual $O(T^{-1/2})$ stationarity bound. Numerical tests on a quadratic function, the Rosenbrock function, and a five-dimensional nonconvex function illustrate the stationarity floor of the unclipped rule and the effect of clipping.

We develop a rigorous theory of discrete residual least-squares approximation for elliptic spectral equations $\mathfrak L_\beta u=f$ using linearized ReLU$^k$ neural networks on the sphere, where $\mathfrak L_\beta$ is a positive elliptic spectral multiplier of order $\beta$. Given a parameter set $\Theta_n=\{\theta_{j}^*\}_{j=1}^n\subset\mathbb S^d$, we approximate $u$ in the linearized network space $L_n^k(\Theta_n)$ by the discrete residual on the collocation points $\{\eta_i^*\}_{i=1}^m$ \begin{equation*} u_{n,m}\in\arg\min_{v_n\in L_n^k(\Theta_n)}\frac1m\sum_{i=1}^m\left(f(\eta_i^*)-\mathfrak L_\beta v_n(\eta_i^*)\right)^2. \end{equation*} With $k>\frac{d-1}{2}+\beta$, for antipodally quasi-uniform network parameter sets and any quasi-uniform collocation points with $m\gtrsim n$, we prove that \begin{equation*} \|u-u_{n,m}\|_{\mathcal H^{\beta}(\mathbb S^d)}\eqsim\|f-\mathfrak L_\beta u_{n,m}\|_{\mathcal L^2(\mathbb S^d)}\lesssim n^{-\frac{r}{d}} \begin{cases} \|f\|_{\mathcal W^{r,p}(\mathbb S^d)},&\frac{d}{p}<r\leq \frac{d}{2},~p>2,\\ \|f\|_{\mathcal H^r(\mathbb S^d)},&r>\frac{d}{2}. \end{cases} \end{equation*} We also establish a high-probability residual estimate, up to a logarithmic factor and an arbitrarily small smoothness loss, for i.i.d.\ uniformly distributed collocation points. The key analytical ingredient is a Bernstein inequality for linearized ReLU$^k$ network spaces. If $\underline h$ denotes the antipodal separation distance of the network parameters, then \begin{equation*} \|v_n\|_{\mathcal H^r(\mathbb S^d)}\lesssim\underline h^{-(r-s)}\|v_n\|_{\mathcal H^s(\mathbb S^d)},\qquad 0\leq s<r<k+\tfrac12. \end{equation*}

The weighted half-period cosine space has often been employed in the theory of quasi-Monte Carlo methods for multivariate integration and approximation of non-periodic functions. For integer-order smoothness, its norm equivalence to certain weighted unanchored Sobolev spaces has been established in the literature. In this work, we extend this equivalence to fractional-order smoothness up to $2$. By introducing an explicit representation via Slobodeckij-type seminorms, we prove a norm equivalence between the half-period cosine spaces and the corresponding weighted unanchored Sobolev spaces. Our Sobolev norm representation clarifies how the fractional regularity dictates the presence or absence of boundary constraints and (non-)periodic structures. Furthermore, we investigate the limiting behavior of these fractional spaces as the smoothness parameter approaches integer boundaries, establishing a continuous bridge to the classical integer-order Sobolev spaces. These equivalence results enable us to transfer the near-optimal error bounds and tractability results for multivariate integration and function approximation from the half-period cosine settings to our newly introduced fractional Sobolev spaces.

We make explicit and formalize a result of the author on squarefree numbers in short intervals, showing that for $0 < \varepsilon\le 1/90935 $, $X\ge \exp(10^{27}/\varepsilon^2)$, $H = X^{1/5 - 2/90935 + \varepsilon}$, we have that \[ \biggl|\sum_{X\le n\le X + H } \mu(n)^2 - \frac{6}{\pi^2}H\biggr| \le \frac{10^{450}}{\varepsilon} H X^{-\varepsilon/10^{25}}. \] This article gives an account of what went into making the exponent explicit. The Github repository linked contains the formalization in Lean 4 as well as an account of what went into the largely automated formalization.

An \emph{$n$-divisor set} is a finite set of positive integers containing a multiple of every integer from $1$ through $n$. Umans and Wang proposed, as the arithmetic-progression version of their Strong $(\alpha,\beta)$-Divisor Conjecture, an $n$-divisor arithmetic progression having at most $n^{2\beta}$ terms, each of magnitude at most $\exp(n^\alpha)$. We prove unconditionally that an $n$-divisor arithmetic progression of height $H$ with $\log H=o(\sqrt n)$ must have length \[ L\ge \left(\sqrt{\frac{8}{27}}-o(1)\right) \frac{n^{3/4}}{\sqrt{\log n}}. \] Consequently, the arithmetic-progression version is false whenever $\alpha<1/2$ and $\beta<3/8$. In particular, it is false at the proposed point $(\alpha,\beta)=(1/3,1/3)$, even if both bounds are relaxed by $n^{o(1)}$ at the exponent level. The proof uses primes in a fixed band below $\sqrt n$ to turn semiprime divisibility into a finite incidence structure. An elementary bounded-degree linear-space estimate then gives the result. This theorem concerns the one-dimensional arithmetic-progression version only; it does not disprove the higher-rank Strong Divisor Conjecture.

The existence and construction of Schwartz-class windows for lattice Gabor frames is a central problem in time–frequency analysis. For general time–frequency lattices, we characterize exactly those lattices that admit Schwartz-class windows and, additionally, windows in the Feichtinger algebra. Thereby we arrive at sharp classical and amalgam Balian–Low theorems for lattices in $\mathbb{R}^d$. For separable lattices and specified block-zero forms, we construct smooth windows that are compactly supported either in time or in frequency. These constructions are based on a characterization of pairs of lattices for which there exists a single set that tiles with one lattice and whose $\varepsilon$-neighborhood packs with the other.

We introduce and investigate a notion analogous to stable causality for Lorentzian foliations, considerably extending the framework of transverse causality initiated in a recent work. It is well-known that in spacetime geometry stable causality is characterized by several equivalent conditions, such as the stability of non-existence of causal loops under metric perturbations, the existence of smooth temporal functions, and relation-theoretic formulations via the so-called Seifert and $K^+$ relations. We define natural transversal analogues of these distinct formulations and establish partial logical implications between them in the foliation setting. Whether and to what extent the full equivalences can be established remains an open question for general foliations, partly due to the eventual non-Hausdorff nature and other topological complexities of arbitrary leaf spaces. Furthermore, we demonstrate that for the important class of simple foliations, which are defined by certain submersions, the equivalences of almost all the transverse analogues of stable causality are indeed recovered.

We discuss Bayesian inference on a low-dimensional targeted parameter in the presence of possibly highly complex nuisance components within the semi-parametric inference framework using an estimating function approach. We obtain a posterior distribution using non-parametric Bayesian methods through the Dirichlet process and the Bayesian bootstrap. We relax the commonly deployed notion of stochastic equicontinuity and develop a framework leading to posterior inference with good frequentist properties, specifically we demonstrate that the posterior distribution is asymptotically Normal and concentrates at the true value of the parameter. We emphasize the specific assumptions that are required to obtain these results, and how relaxing any of them alters the conclusions. We verify the analytical results in simulation.

We obtain sharp bounds for the third- and fourth-order Schippers functionals, $|\sigma_3(f)(0)|$ and $|\sigma_4(f)(0)|$, for subclasses of univalent functions associated with non-classical geometric domains. In particular, we investigate the lune-starlike class $\mathcal{S}_{\leftmoon}^*$ and the lune-convex class $\mathcal{C}_{\leftmoon}$ determined by the subordination \[ \frac{zf'(z)}{f(z)} \prec z+\sqrt{1+z^2}, \qquad 1+\frac{zf''(z)}{f'(z)} \prec z+\sqrt{1+z^2}, \] respectively, together with the bean-domain class $\mathcal{BT}_{\mathfrak{B}}$ associated with \[ \mathfrak{B}(z)=\sqrt{1+\tanh z}. \] Using Carathéodory coefficient parametrizations and extremal optimization techniques, we derive exact estimates for the higher-order Schwarzian derivatives at the origin and identify the corresponding extremal functions. In addition, geometric descriptions of the associated extremal image domains are provided to illustrate the sharpness phenomena. The obtained results further yield sharp bounds for the initial Grunsky coefficients $g_{1,1}$ and $g_{1,2}$. These findings provide a precise description of higher-order Schwarzian structures for univalent functions related to lune and bean shaped domains.

We give the first positive combinatorial rule for the multiplicities of irreducible $GL$-representations in the coordinate rings of type $A$ quiver orbit closures, valid for every orientation of the quiver. Previously, for the special case of varieties of complexes, work of De Concini–Strickland in the early 1980s gave an implicit description of these multiplicities. The combinatorial objects in our rule carry a crystal structure whose highest-weight elements compute these multiplicities.

Physics-Informed Kolmogorov-Arnold Networks have recently emerged as an effective class of neural solvers for partial differential equations, combining the expressive power of spline-based Kolmogorov-Arnold representations with physics-informed learning. However, existing PIKAN formulations are primarily developed for Cartesian domains and cannot naturally accommodate the geometric complexity introduced by curvilinear domains. In this work, we propose Coordinate-Transformed Physics-Informed Kolmogorov-Arnold Networks (CT-PIKAN), a geometry-aware framework for solving PDEs on arbitrarily shaped domains through coordinate transformation. A smooth mapping transforms the physical domain into a regular computational domain, while the transformed governing equations are enforced directly within the physics-informed loss. Unlike conventional transformed PINN approaches that require manually derived metric coefficients, CT-PIKAN employs automatic differentiation to evaluate Jacobians, metric tensors, and transformed differential operators directly from the coordinate mapping, eliminating analytical derivations and improving implementation flexibility. To establish the proposed framework, a data-free B-spline-based PIKAN is first constructed and validated on the two-dimensional advection equation. The CT-PIKAN methodology is subsequently assessed on representative elliptic, parabolic, and hyperbolic benchmark problems, including the Poisson, heat, and advection equations formulated in polar and wavy curvilinear coordinates. The proposed framework provides a general and extensible methodology for integrating differential geometry with physics-informed Kolmogorov-Arnold networks, enabling efficient and accurate PDE solutions on complex domains.

This work proves a theorem regarding injectivity of continuous flows that are defined via discrete vector fields on triangle meshes. In similar fashion to Tutte's embedding, the theorem shows a connection between harmonicity and injectivity. Specifically, a discrete harmonic function is computed over a spherical mesh, and its $90^\circ$-rotated gradients are taken as the vector field. We show that this leads to an injective flow over the mesh, i.e., when two points are advected along the vector field, their trajectories are guaranteed to not intersect. The core insight behind this theorem is that the characterization of the Poincaré-Hopf indices of discrete harmonic 1-forms is paralleled in the combinatorial rules for injective vector fields. Similarly to Tutte's embedding, the injective vector field can be computed by simply solving a sparse linear system. Furthermore, it is differentiably dependent on the entries of the Laplacian, enabling unconstrained gradient-based optimization of the vector field while preserving injectivity. The advection itself can be computed by a quick and accurate algorithm. We experimentally validate the injectivity of the results, as well as show examples of differentiable optimization, such as computing an injective approximation to a given non-injective vector field, and fitting an injective vector field to satisfy constraints.

This is a sequel to \cite{PaperI}, where it is shown that a discrete effective divisor on the open unit disk $\D$ is the zero divisor of a holomorphic function with integer Taylor coefficients if and only if it is invariant under complex conjugation. Building on that realization theorem and on the unit characterization established there, we study the prime and maximal spectra of the rings $R = \mathbb{Z}[i][[z]] \cap O(\mathbb{D})$ and $R_{\mathbb{R}} = \mathbb{Z}[[z]] \cap O(\mathbb{D})$. We give a trichotomy for maximal ideals, classify the prime ideals lying outside the class $\mathfrak{P}_1$ of primes containing a unit-constant-term element, attach to each element of $\mathfrak{P}_1$ a complete analytic invariant, and construct an injection from ultrafilters on admissible divisors into $\mathfrak{P}_1$ by an ultraproduct device. We prove the point-evaluation ideals $P_a$ are prime for all $a \in \D$ and maximal for real $a$, and deduce a conditional Bézout property for disjoint divisors. All results recalled from \cite{PaperI} are used only as cited; no proof of that paper is reproduced here.

We study a family of questions in convexity in which size does not matter: one records a convex cross-section only up to translation and scaling, so that the data attached to a convex body $B$ and a direction $\rho$ is a path in the compact metric space $\A_{n-1}$ of aligned shapes. The object of interest is the asymptotic behaviour of this path as the cutting hyperplane approaches the last supporting hyperplane, encoded by an invariant $T(B,\rho)$ that we call the tail. We show that tails are always continua, that polyhedral and smooth support points are “boring” (the tail is a point), and that non-boring behaviour forces degenerate contact. We show that cross-section paths are locally rectifiable, that every locally rectifiable path is realisable approximately and a dense class exactly, and that exact realisation fails in general: a second-order obstruction of bounded-turning type produces a rectifiable path that is not a cross-section path. For tails, by contrast, no such restriction survives: every continuum of shapes occurs as a tail, on the nose rather than up to approximation. We construct bodies possessing nearly universal points, at which the renormalised cross-sections approximate every planar (more generally $(n-1)$-dimensional) convex shape arbitrarily well; such points can be made dense in the boundary, with arbitrary prescribed tails at the grafting sites. Every result below has been formally verified in Lean~4. We close with several optimisation questions and a higher-codimension variant.

We axiomatize the differential-comparison continuation method of [Zf] into a general machine for proving Mahler measure identities at CM points: a modular parametrization with rigorous tail bounds, a single-valued holomorphic Mahler differential on the complement of the critical image, a propagation lemma replacing all monodromy and universal-cover arguments, interval-arithmetic path certification, and an exact three-track CM evaluation. Applied to Samart's family $n_4(s)=4\mathrm m(x^4+y^4+z^4+1+s^{1/4}xyz)$, the machine proves twelve further entries of his 2015 table [Sa15, Table 6], including the first identities in this family proved at non-degenerate CM points of class number two (attached to $\mathbb Q(\sqrt{-6})$ and $\mathbb Q(\sqrt{-15})$) and of class number four (attached to $\mathbb Q(\sqrt{-21})$, with four newforms and the genus group $(\mathbb Z/2)^2$). New ingredients include exact twist levels via grossencharacter conductors (Hecke's theorem), with the interval-locked Fricke ratio as an exact root-number sign lock; a twist-level trap ($g_{16}\otimes\chi_8$ has level $64$, not $32$); a genus-character decomposition of the lattice sums in class number two; and the discovery that Samart's applicability boundary $\{\mathrm{Im}\,\tau=1/\sqrt2\}$ is not the topological boundary of the continuation region. Every identity is certified by an exact algebraic track, rigorous final-identity interval locks (half-widths $4.6\times10^{-53}$ to $5.1\times10^{-52}$), and independent 50–60-digit cross-checks; all scripts are public. The evaluation track also yields six new identities at Heegner points ($D=11,19,27,43,67,163$), stated as conjectures with 60-digit evidence; we close with an umbrella conjecture organizing the proved and conjectured identities, and a precise analysis of the five remaining open entries.

Boyd's 1998 tables of conjectural identities $m(P)=r\,|L'(E,0)|$ between Mahler measures of two-variable polynomials and $L$-values of elliptic curves begin with the smallest possible conductor, $N=11$. The third conductor-$11$ identity, (C3), concerns the polynomial $S_0=y^2+(x^2+1)y+x^3$, which vanishes on the unit torus, and asserts that a signed split integral $I_{\mathrm{split}}$ of $\log|y|$ around the branch cut equals $b_{11}=L'(E_{11},0)$. We prove (C3). The proof identifies the split integral with a regulator integral along Samart's signed open chain $\tilde\gamma$, closed by a small-branch arc $\beta_0$ into a closed anti-invariant cycle $C'$, and proves its homology class is $2\gamma^-$ with $\gamma^-$ generating $H_1(E,\mathbb{Z})^-$: the period ratio is a-priori integral, and ball arithmetic (Arb) pins it to $2$. A direct regulator computation via Brunault's proved Siegel-unit formula—the symbol $\{x,y\}$ being a pair of modular units on $X_1(11)$, so Bloch's diamond theorem is not needed—then yields $I_{\mathrm{split}}=b_{11}$; the sign is certified in interval arithmetic, and the identity agrees with the numerical value to $366$ digits. For the family $S_k=y^2+(x^2+kx+1)y+x^3$ we determine exactly the torus intersections and the modular-unit/tempered cases; further Boyd-type evaluations are recorded as conjectures. At $k=-1$ (conductor $53$) the mechanism provably fails; an appendix treats Samart's conductor-$17$ analogue conditionally. All computations are reproducible from the accompanying code; every certification step is carried out within interval arithmetic.

We study holomorphic functions on permutation-invariant subsets of an infinite-dimensional sequence space that are invariant under all finite permutations of coordinates. Working on domains modeled on the space $\czero$ of null sequences, we show that permutation symmetry forces the first differential to vanish along every constant tail: the coordinate derivatives agree there and an averaging argument annihilates their common value. This yields rigidity—constancy—on domains of constant sequences, and more generally on any $\cc$-chain-connected domain on which the differential vanishes on the finitely supported directions. The method has a sharp boundary: a precise obstruction prevents the elementary argument from reaching the finitely many exceptional coordinates of a non-constant point, so unconditional rigidity on general eventually-constant domains remains open. In the other direction, an explicit example—any Banach limit on $\ell^\infty$—shows that the chain-connectedness and boundedness hypotheses cannot be dropped: mere topological connectedness admits non-constant symmetric entire functions. We record the vector-valued corollary and discuss the motivating application to holomorphic cross-sections of the zero-set map, isolating the lifting problem that remains open.

A finite group-valued temporal reference frame defines a correlated random-unitary process rather than a list of independent channels. We identify a regime in which its complete trajectory law is an exact operational coordinate. If the carrier contains every irreducible representation of the finite group $G$, one ancilla-assisted input resolves all group elements, and for arbitrary laws $\mu,\nu$ on $G^n$ the strategy half-distance equals $d_{\mathrm{TV}}(\mu,\nu)$. Optimal output-side causal post-processing likewise reduces to classical convolution on $G^n$. This converts temporal hiding into a $q$-ary coding problem. Uniform coset laws of an $[n,k]_q$ code are identical on every set of fewer than $d(C^\perp)$ slots and perfectly distinguishable globally. For $M$ sectors with leakage at most $\eta$ from $t$ selected slots and global decoding error $\epsilon$, we obtain $(1-\epsilon)\log M \leq (n-t)\log q+\eta+h_2(\epsilon)$. For nested codes $C\subset D$, the payload $R=\dim D-\dim C$, invisible depth $t=d_{\mathrm{rel}}(C^\perp,D^\perp)-1$, and sector distance $d_{\mathrm{rel}}(D,C)$ obey $R+t+d_{\mathrm{rel}}(D,C)-1\leq n$ and $R+t+2e+f\leq n$, where $e$ and $f$ are adversarial errors and known erasures. Nested generalized Reed-Solomon codes attain the bounds in their existence range. We also give the exact projection-rank leakage profile and show that hiding arbitrary coherent superpositions from $t$ slots is precisely quantum erasure correction, yielding $\log_q K+2t\leq n$. The results separate classical trajectory privacy from coherent temporal privacy and quantify the robust payload hidden from reduced process tomography.

In this paper, we address the following question: if a flat torus $\mathbb{T}^n$ is isometrically and minimally embedded into a sphere $\mathbb{S}^N$, must its translation group extend to the isometry group of the ambient sphere? As shown by Robert Bryant, for $n=2$ the answer is positive. Furthermore, while Ying Lu, Peng Wang, and Zhenxiao Xie recently demonstrated that the answer is negative for immersions when $n \geq 3$, the question for embeddings remained open. This problem is deeply tied to the work of Mikhail Gromov and Anton Petrunin concerning optimal curvature bounds. Petrunin proved that any immersion of a torus into a unit ball must have a maximum normal curvature of at least $\sqrt{\frac{3n}{n+2}}$. This bound is attained, for example, by families of tori constructed by Gromov. We call the tori that attain this optimal bound "Gromov tori". In this work, we first demonstrate that any Gromov torus is intrinsically flat, lies within a sphere, and is minimal inside it. We then establish the necessary and sufficient conditions for defining these tori. Finally, we present our main result: for dimensions $n \ge 3$, there exists a non-equivariant embedded Gromov torus, which provides a definitive negative answer to the question above.