Papers

Problems developed for a mathematics circle conducted throughout the school year in a primary school class, involving all pupils without any selection, thus working with a heterogeneous group of children.

We determine the exact number, up to rotations, of nonconstant positive $C^2$ solutions to the planar isotropic $L_p$ dual Minkowski problem for every $(p, q) \in \mathbb{R}^2$. We obtain a new parametrization of the associated period integral, characterize the regions where the period is strictly increasing or strictly decreasing, and prove that in the remaining nonmonotone region it has a unique nondegenerate maximum. As a consequence, we have a complete classification.

We study a support-collapse version of graph pebbling on paths. A configuration is stackable if a sequence of legal pebbling moves can produce a nonzero configuration supported on a single vertex. On the path P_n, we choose a configuration uniformly from all weak compositions of total n times mu_n, where mu_n is a positive integer. We prove a two-sided fixed-offset transition for the logarithmic density. The transition is centred at sqrt(log_2 n) - (1/2) log_2 log_2 n + log_2(3e). For every fixed epsilon greater than zero, the stackability probability tends to zero when log_2 mu_n is eventually at most the centre minus epsilon, and tends to one when it is eventually at least the centre plus epsilon. No assertion is made at zero offset. The proof uses an exact recursive stackability score on trees, a one-dimensional path-message recurrence, binary-partition asymptotics for rare dyadic deficit excursions, a constant-cost regeneration argument, and an exact deep-message necessity theorem. Conditioning independent geometric occupancies on their sum returns the uniform fixed-total model. The finite deterministic necessity theorem and its exact fixed-total corollary are formalised in Lean and registered with Palomar; the full probabilistic asymptotic theorem is not part of that registration.

Korenblum's maximum principle states that if $|f| \le |g|$ near the boundary, then $\| f \| \le \| g \|$. The optimal size of the region of domination has been studied in Bergman spaces since 1991 and in Fock spaces since 2006, but only bounds have been obtained. We determine the optimal size for the Fock space $F^2$ of entire functions that are square integrable with respect to $e^{-|z|^2}\,dA(z)$: if $f, g\in F^2$ and $|f(z) \le |g(z)|$ for all $|z|>1$, then $\| f \| \le \| g \|$. The radius $1$ is optimal and we determine all pairs for which equality holds. To our knowledge, this is the first space of Bergman or Fock type in which the optimal radius in Korenblum's principle has been determined. The proof combines a Schwarz-Pick estimate centred at infinity with a moment-duality argument, and requires no numerical computation. The same method gives a moment criterion for weighted Fock-type spaces. For every non-increasing radial weight, the optimal radius equals the upper bound given by the pairs $f\equiv c$, $g(z)=z$. In particular, the upper bounds of Wee and Le for such weighted Fock spaces are sharp in the Hilbert space case. We also show that the optimal radius is $\sqrt{(\beta+1)/\alpha}$ for the weight $|z|^{2\beta}e^{-\alpha |z|^2}$, whenever $-1<\beta\le5.44$.

We establish the Type I estimate for the scalar curvature of the Kahler-Ricci flow on compact Kahler manifolds, for arbitrary smooth initial metrics, whenever the flow develops finite time singularities. This estimate is a building block for the analytic minimal model program with Ricci flow.

A partition $\pi=\{V_{1},V_{2},...,V_{k}\}$ of the vertex set $V$ of a graph $G$ into $k$ color classes $V_{i},$ with $i\in\{1,...,k\}$ is called a quorum coloring if for every vertex $v\in V,$ at least half of the vertices in the closed neighborhood $N[v]$ of $v$ have the same color as $v.$ The maximum cardinality of a quorum coloring of $G$ is called the quorum coloring number of $G$ and is denoted by $\psi_{q}(G).$ A sub-quorum coloring of $G$ is an onto partial function $f:V\rightarrow\left\{1,2,\ldots,\ell\right\}$ having the property that for every vertex $v\in V,$ if $f(v)$ is defined, then at least half of the vertices in $N[v]$ having an image by $f$, have the same color as $v.$ The sub-quorum coloring number $\psi_{sq}(G)$ equals the maximum value $\ell$ in a sub-quorum coloring of $G.$ In this paper, we determine the exact value of the sub-quorum coloring number for some infinite families of caterpillars including complete $n$-tuple caterpillars and complete caterpillars with minimum spine-vertex degree three.

In this paper, we introduce a deep truncated forward-backward stochastic differential equation (FBSDE) method for high-dimensional partial differential equations (PDEs). Compared with existing deep-learning solvers for fully coupled FBSDEs, where strong coupling may lead to numerical instability, our approach exhibits improved stability. The proposed method combines gradient-truncated iterative decoupling with fictitious-play averaging to separate the forward and backward processes in a coupled framework. This preserves the coupled dynamics while reducing the unstable feedback induced by parameter-dependent forward paths during optimization. Furthermore, we incorporate a pathwise consistency term to create explicit local gradient shortcuts, thereby providing a structural mechanism that may mitigate gradient vanishing. We also derive a residual-based error estimate and establish conditional convergence of the fully discrete numerical approximations under suitable conditions, in which the pathwise consistency loss is not required. Our approach is particularly effective for convection-dominated equations, where the coupled formulation provides a stable representation of nonlinear transport without introducing singular terms into the BSDE. Numerical experiments demonstrate improved accuracy and stability in both low- and high-dimensional problems and robust performance for strongly coupled problems.

The weak, stable, and ordinary Lusternik–Schnirelmann categories of a finite $T_0$-space $X$ satisfy $\operatorname{cat}_w(X)\leq \operatorname{cat}_s(X)\leq \operatorname{cat}(X)$. We give a general construction answering the simultaneous-strictness question of Cárdenas, Flores, Quintero, and Villar-Liñán. If $P$ is weakly contractible but noncontractible and the deletion of one point makes $P$ contractible, then adjoining $m\geq 2$ incomparable maximal points produces a connected finite space with category triple $(1,2,m)$. Moreover, after any positive number of barycentric subdivisions its ordinary category is equal to $2$. Applying the construction to a nine-point space yields examples on $m+9$ points and, in particular, a twelve-point example for which both inequalities are strict. Using additivity under disjoint unions, we also realize every triple $(a,b,c)$ with $1\leq a<b\leq 2a$ and $c\geq b$, as well as every triple $(a,a,c)$ with $2\leq a\leq c$. We conclude with questions concerning connected realizations and the minimum cardinality of a finite space exhibiting simultaneous strictness.

For estimating the mean vector of a $p$-variate normal distribution ($p \ge 3$) under quadratic loss, the positive-part James–Stein estimator dominates the original James–Stein estimator, but it possesses a non-smooth thresholding boundary. In this paper, by truncating the infinite series representation of the positive-part function to a finite sum of degree $m$, we propose a new class of smooth shrinkage estimators. We prove that the proposed estimator dominates the James–Stein estimator for any dimension $p \ge 3$, provided the truncation degree satisfies $m \ge 0.95\sqrt{p-2}$.

We construct the complete spectrum of gravitational total-transmission modes (TTMs) of Kerr black holes and find four globally continuous families, $n_\infty=1,2,3,4$, each with two complex-conjugate frequency branches. Compared with the three-family classification of Cook and Lu, our global continuation resolves the mirror-related sectors of their $n=2$ family into separate families without introducing additional symmetry-unrelated roots. This organization distinguishes complex conjugation at fixed $m$ from the mirror symmetry connecting the $m$ and $-m$ spectra. The $n_\infty=3$ family approaches the Schwarzschild algebraically special frequency, whereas the $n_\infty=1,2,$ and $4$ families diverge as $\omega\propto a^{-4/3}$ along lower-half-plane directions $-150^\circ$, $-90^\circ$, and $-30^\circ$, respectively. High-precision data up to $\ell=32$ confirm the Cook–Lu small-spin asymptotics and reveal how the divergent branches are embedded in the global four-family spectrum, with a family-dependent relation between the spherical-limit angular labels and the large-$|a\omega|$ ordering. For axisymmetric perturbations, the $n_\infty=2$ and $n_\infty=3$ branches coalesce at genuine exceptional points, where the scattering spectral function develops a second-order zero and the TTM excitation factors are strongly enhanced, offering a different interpretation of the structure previously described by Cook and Lu as an overtone-multiplet splitting. Finally, we identify a previously unreported anomalous proximity between the $n_\infty=3$ TTM family and an unconventional Kerr quasinormal-mode sequence, with frequency separations reaching order $10^{-8}$ in the representative case studied.

For immersions of a closed $2h$-dimensional manifold $\Sigma$ into $\mathbb R^m$, we consider curvature energies that are invariant under ambient isometries and dilations, with the principal term $\int_{\Sigma} \big|\nabla^{(h-1)} \vec{\mathrm{I\!I}}\big|_g^2\,d\text{vol}_g$ and lower-order submanifold invariants. Following the four-dimensional approach developed in joint work with Bernard, Martino, and Rivière, we prove that every weak critical immersion in the natural Sobolev class $W^{h+1,2}$, whose induced metric and its inverse have $L^\infty$ coefficients, is real-analytic in harmonic coordinates. The proof combines geometric conservation laws, structural identities, and elliptic estimates with critical Sobolev coefficients to obtain Morrey decay and bootstrap to full regularity.

A metric space is plastic if every bijective nonexpansive self-map is an isometry. We prove that the hyperspace of nonempty compact subsets of every connected, locally finite, regular graph, equipped with the Hausdorff metric associated with the path metric, is plastic. The proof identifies the sets with smallest unit balls as the nonempty subsets of adjacent-twin classes and shows that every nonexpansive bijection induces an automorphism of the quotient graph preserving the sizes of these classes. Singletons are preserved when there are no adjacent twins, but need not be preserved in general. We also prove that $\mathcal{K}(K\times G)$ is plastic for every compact connected metric space $K$ and every connected, locally finite, regular graph $G$, with the supremum metric on the product, and give a more general criterion for such products. Further results include a plastic metric space whose hyperspace is not plastic, plasticity of every tree, and plasticity of the hyperspace of a connected, locally finite graph with only finitely many vertices of minimum degree.

We study preservation of finite gauge-energy flows under local network repair after Bernoulli edge failures. A macroscopic demand network records terminal pairs to be routed, while a microscopic physical network contains local backup routes, bypasses and shared corridors. We prove a deterministic gauge-energy repair theorem: if the usable demand network $\mathcal{B}^{\sharp}$ carries a finite $\Phi$-energy flow $\theta$, then the repaired physical network $H$ carries a lifted finite $\Phi$-energy flow $\Theta$ with $$ \mathcal{E}^\Phi_H(\Theta) \le L\,\beta_\Phi(K)\,\mathcal{E}^\Phi_{\mathcal{B}^{\sharp}}(\theta), $$ where $L$ bounds route length, $K$ bounds routing congestion, $\mathcal{E}$ marks energy, and $\beta_\Phi$ is the gauge dilation constant. We then convert this comparison into probabilistic repair criteria: finite-dependent local repair is handled via domination by product measures, and random repair lengths via a variable-cost formulation compatible with chemical-distance estimates. As a main application, we prove a finite-dependent local bypass theorem: any macroscopic network whose supercritical percolation cluster supports a finite gauge-energy flow remains gauge-energy stable after bounded-range local reinforcement, provided the local repair probability is sufficiently high. This yields reinforced lattice and wedge-type examples and provides a potential-theoretic framework for random network repair beyond tree-like or edge-disjoint constructions.

The radiative transport equation provides a powerful framework for describing wave propagation in scattering media. However, it cannot describe coherent waves whose incident wavefront is deliberately tailored. Here we establish a transport theory for wavefront-controlled waves based on a matrix transport equation. Unlike conventional radiative transport, which propagates a scalar radiance, our theory propagates a complex two-by-two matrix that retains phase information and obeys a nonlinear transport equation, despite the underlying wave dynamics being linear. We use this matrix transport equation to derive the spatial profiles of the energy and current densities of transmission eigenchannels, from closed to open channels, in arbitrary diffusive systems. Remarkably, these profiles can be expressed analytically in terms of the solution of the conventional radiative transport equation for a random wave in the same medium. The theory further extends to energy loss, including absorption and leakage into uncontrolled channels, revealing symmetry breaking of transmission eigenchannels in complex structures. Validated against numerical solutions of the wave equation, we establish matrix transport as a framework for describing and controlling coherent wave propagation in complex media.

We consider the two-dimensional generalized surface quasi-geostrophic equations in $\mathbb{R}^2$, given by \[\partial_t \theta+u\cdot \nabla \theta=0,\quad u=-\nabla^{\perp}\Lambda^{\beta-2}\theta,\,\beta\in [1,2).\] When $\beta=1$, the equation defines the SQG equation and for $\beta>1$, it defines a family of more singular active scalar equations. We prove that if the interval of existence of the smooth solution to the generalized SQG equations for some $\beta_0\in[1,2)$ is $[0,T]$, then with the same initial data, the interval of existence of the generalized SQG equations for $\beta$ close to $\beta_0$ also contains $[0,T]$. To prove these results, we develop estimates for a conservation law with flux modified around the generalized SQG equations. Furthermore, we also prove some new commutator estimates with bounds uniformly bounded as $\beta\to 1$ that may be of independent interest.

We prove that $\mathsf{Cheq}$ is not finitely axiomatizable, resolving a longstanding open problem in intermediate and modal logics. We further prove that the undecidability of Medvedev logic implies the undecidability of $\mathsf{Cheq}$.

Spherical harmonic descriptors of closed 3D shapes depend on the parameterization, the pose and the scale of the surface, and the standard rotation-invariant reductions, the power spectrum and the bispectrum, discard the relative orientation of the harmonic bands and cannot distinguish a shape from its mirror image. We construct a descriptor that removes all three dependencies exactly and loses nothing else: a conformal parameterization normalized by its conformal barycenter, followed by polynomial invariants of the rotation group. Identifying each harmonic band with a binary form turns the rotation quotient into classical invariant theory and makes reflections visible as the sign of an invariant, so chirality is recorded. The descriptor is complete for the truncated expansion, stable in the orbit distance, and comes with numerical diagnostics. Benchmarks confirm the guarantees, and on bilateral anatomical structures the descriptor separates mirror-image pairs from asymmetric pairs, which parity-blind descriptors cannot.

The $\Delta$-unknotting number for a knot is defined as the minimum number of $\Delta$-moves needed to deform the knot into the trivial knot. In this paper, we discuss Montesinos knots whose $\Delta$-unknotting number is equal to one. We propose a conjectural characterization of Montesinos knots with $\Delta$-unknotting number one and provide examples admitting distinct $\Delta$-moves, each of which deforms the knot into the trivial knot.

We consider objects in the Fukaya category of a class of K3 surfaces defined by certain Lagrangian spheres. Assuming that homological mirror symmetry for K3 surfaces holds with sufficiently strong (expected) properties, we prove that, in this case, stability with respect to a suitable Bridgeland stability condition implies the existence of an isomorphic special Lagrangian sphere, as predicted by the general Thomas-Yau-Joyce conjectures. In particular this holds unconditionally for suitable quartic surfaces in $\mathbb{P}^3$ or sextics in $\mathbb{P}(3,1,1,1)$ and their mirrors. A variant holds on the Calabi-Yau threefolds obtained by taking the product of our K3 surfaces with an elliptic curve. These seem to be the first results of this type on compact manifolds.

We study the stability for a system of coupled second order evolution equations with indirect general memory-damping without the equal-wave-speeds-type hypothesis in a Hilbert space, where the damping only appears just in one equation, the memory kernel can not necessarily be nonnegative and nonincreasing, and the “wave speeds" implied by the system can be different. Taking advantage of new processing ideas and with the help of the properties of the Generalized Positive Definite Kernel, we overcome difficulties caused by the different “wave speeds", the lack of the decreasing and nonnegative property for the memory kernel and the system has only one equation with damping, and obtain an optimal polynomial stability result for the energy, which covers the previous related polynomial stability results for second order coupled equations (abstract or concrete) in the literature. Moreover, applications of the abstract result are given.

We reconsider the vacuum (Casimir) energy of a massive scalar field obeying Dirichlet boundary conditions inside a spherical cavity. Using zeta-function regularization, we carefully recalculate the analytic part of the zeta function via the uniform asymptotic expansion of the modified Bessel functions. We correct a sign error present in an earlier calculation (PR D56 (1997) 4896) that affected the large-mass behavior. The renormalized vacuum energy is shown to vanish for large mass, as required on physical grounds. Numerical results reveal that the energy changes sign several times as a function of the cavity radius.

Let $\Pi$ be a regular algebraic cuspidal automorphic representation of $\mathrm{GL}_{3}(\mathbb{A}_{\mathbb{Q}})$ of weight $(a, 0, -a)$, and let $p$ be a prime. Generalising work of Loeffler–Williams in the nearly-ordinary case, we construct $p$-adic $L$-functions for each small slope regular $P_{1}$-refinement of $\Pi$ at $p$ consistent with the conjectures of Coates–Perrin-Riou and Panchishkin.

A pseudohemisphere hypergraph is a hypergraph $\mathcal{H}$ with an ordered set of vertices $V$ for which there exists an $ABA$-free hypergraph $\mathcal{F}$ on $V$ and a subset $X$ of $V$ such that the hyperedge set of $\mathcal{H}$ is a subset of $\{F\Delta X: F\in \mathcal{F}\cup \overline{\mathcal{F}}\}$. We prove that the chromatic number of pseudohemisphere hypergraphs is at most four.

Tilting mutation provides a natural way to construct derived-equivalent algebras by replacing an indecomposable summand of a tilting object. Holm, Schröer and Zimmermann~\cite{HSZ01} (see also the appendix of~\cite{AA08}) have described combinatorial transformations of the quiver and relations of a gentle algebra that yield derived equivalences via tilting complexes. In this paper, we develop a self-contained module-theoretic framework for tilting mutation via generalized $\mathrm{BB}$-tilting modules. For a gentle algebra $A=\mathbb{k} Q/\langle I\rangle$ and a vertex $k\in Q_0$, we first give a necessary and sufficient condition, expressed in terms of the arrows and relations incident with $k$, for the corresponding minimal left approximation of $P(k)$ to yield a tilting mutation. This criterion applies uniformly to vertices with or without loops. When the mutation exists, we determine the irreducible morphisms between the indecomposable summands of the mutated tilting module and use them to construct explicitly the Gabriel quiver and defining relations of its endomorphism algebra. In particular, we also provide a combinatorial mutation $(Q,I)\mapsto(Q',I')$ of gentle pairs such that $\mu_k^+(A)\cong \mathbb{k} Q'/\langle I'\rangle,$ so the resulting algebra is again gentle and derived equivalent to $A$. Finally, we formulate the dual cotilting mutation and its combinatorial description via opposite gentle pairs.

We study the problem of estimating the guidance that steers the distribution learned by a diffusion generative model toward a tilted target $q_0 \propto w\,p_0$ at inference time. Relying on the stochastic optimal control approach, we observe that the exact drift correction is the gradient of the logarithm of Doob's $h$-function, and we study the problem of estimating it from a sample. In the present paper, we assume that the score of the pretrained model is available, that the tilting weight is bounded and positive, and that the reference distribution has a bounded support, no smoothness of the weight is required. Introducing a penalized least-squares risk in which the penalty is the residual of the space-time harmonicity equation satisfied by the $h$-function, measured in a dual Sobolev norm, we derive high-probability bounds on the squared error of the resulting guidance estimate. Since the penalty vanishes at the target, the estimator is free of regularization bias, and in favourable scenarios its rate of convergence is faster than the minimax rate of estimating first-order derivatives of a smooth regression function. Assuming that $w$ is bounded and positive with $\mathbb{E}_{p_0}[w^{-\mathrm{s}}] < \infty$ for some $\mathrm{s} \in (0,\infty]$, and that the reference data are compactly supported, we prove that the guidance is estimable in squared $L^2$ at rate $\varepsilon_n^{\mathrm{s}/(\mathrm{s}+4)}$, where $\varepsilon_n = n^{-2(\beta-1)/(2(\beta-1)+d)}.$ We also transfer the obtained bounds to the total variation distance between the marginals of the estimated and the exactly guided samplers, and illustrate the performance of the suggested approach with numerical experiments.

In this article, we investigate the Assouad spectrum and Assouad dimension of graphs of fractal functions generated by generalized affine iterated function systems. We establish upper and lower bounds for the Assouad spectrum in terms of the scaling functions and the underlying partition of the generalized affine construction. If the scaling function is Lipschitz continuous and the partition is uniform, we obtain an explicit expression for the Assouad spectrum of the associated graph. These results provide a connection between the parameters defining the generalized affine fractal function and the local multiscale geometry of its graph. As applications, we consider two classic examples of generalized affine fractal functions, namely the Weierstrass and Takagi functions. For the classical Weierstrass function $W$, whose graph $\Gamma_W$ has the box dimension $2+\log_N\lambda$, we obtain \[ \dim_A^\theta(\Gamma_W) \leq \frac{2+\log_N\lambda-\theta}{1-\theta}, \qquad \theta\in \left(0,\log_N\frac{1}{\lambda}\right). \] and if $\lambda^2 N <1$, we get \[ \dim_A(\Gamma_W)\geq 1 +\log_N\left(\frac{1}{\lambda}\right). \] For the classical Takagi function $T$ with graph $\Gamma_T$, we show that \[ \dim_A^\theta(\Gamma_T)=1, \theta\in(0,1), \] and consequently its quasi-Assouad dimension is equal to $1.$ These results settle an open problem on dimension of graphs posed by Fraser.

We give a full list of translational nets which flex within their class of discrete surfaces of translation, by reducing the classification problem to the one of flexible complete bipartite frameworks on the sphere, for which the solution is known. We also obtained two novel classes which correspond to Bottema's spherical 16-bar mechanisms and the constant diagonal angle frameworks. Based on an algorithm for the construction of all flexible translational nets, we also discuss flexible translational tubes and toroids. Furthermore, we present novel results for both topologies which are implied by Bottema's spherical 16-bar mechanisms.

The geometrically decomposed bases of the polynomial differential form spaces $\mathcal{P}_r\Lambda^1$ and $\mathcal{P}_r^-\Lambda^1$ on a simplex are products of a barycentric scalar polynomial and a directional or Whitney 1-form. These bases depend on an ordering of the simplex vertices. On unstructured meshes, neighbouring cells need not agree on this ordering, making it non-trivial to achieve conformity in finite element software. Existing strategies compute degree-of-freedom transformation matrices numerically, or impose a global vertex ordering through face-bubble spaces or by preprocessing the mesh. We consider the standard basis of the $\mathcal{P}_r^-\Lambda^1$ (trimmed) space and introduce a new basis for the $\mathcal{P}_r\Lambda^1$ (full) space, for arbitrary polynomial order $r \geq 1$ and dimension $D \geq 2$, and prescribe their basis polynomials as shape functions. We show that the pullback of shape functions under the change of coordinates induced by an arbitrary vertex relabelling $\pi$ in the symmetric group $S_{D+1}$ admits a closed-form, combinatorial expression. For most basis functions this pullback is a single basis function of the relabelled ordering, up to sign. On an explicitly characterised "filter-hit" set, this pullback is a signed sum of at most $D$ (full space) or exactly two (trimmed space) relabelled basis functions. All coefficients are in $\{-1, +1\}$ for all $r$, $D$ and $\pi$. The inverse transformation is obtained by computing the formulas at $\pi^{-1}$, so no numerical inversion of a change-of-basis matrix is needed. Conforming assembly and evaluation on simplicial meshes with arbitrary vertex orderings reduce to index manipulation. Our results are compared with the study of relabelling-invariant bases of Berchenko-Kogan and Licht, and verified in an open-source Julia implementation in the Gridap.jl library.

We propose a general characterization of the hazard rate and reversed hazard rate stochastic orders for random variables that are absolutely continuous with respect to a common dominating measure. This framework is useful in giving a unified treatment of continuous, discrete, and mixed distributions without requiring ad hoc approximation techniques or restrictive integrability assumptions. Using this result, we provide direct proofs of the bivariate characterizations of both orders. Additionally, we introduce the class of $\overline{G}$-IFR and $\overline{G}$-DRHR distributions on additive groups, unifying aging properties across different domains and proving their closure under convolution. Finally, we revisit and extend the classic results of Shanthikumar and Yao (1991) on the preservation of hazard rate orders under random sums, simplifying the underlying conditions and accommodating discrete and mixed sum components.

Let $W_s=K_1+C_{s-1}$ denote the wheel on $s$ vertices. We give structural proofs that $R(C_{14},W_{11})=27$ and $R(C_{15},W_{11})=29$. Together with the theorem of Chen et al. for $n\ge16$, these equalities give $R(C_n,W_{11})=2n-1$ for every $n\ge14$. The two boundary values were included in an earlier survey announcement. We also give structural proofs of $R(C_8,W_7)=15$, $R(C_9,W_7)=17$, and $R(C_8,W_9)=15$. The common starting point is a Hamilton-connected core lemma. For the eleven-vertex wheel, bounds on vertex connectivity and on the matching number of a bipartite graph associated with a local cycle yield a vertex cut of order nine. Paths with prescribed endpoints then rule out every possible pair of orders of the two remaining vertex sets. For the smaller wheels, we use the structure of critical cycle colorings and local cycle-shortening arguments. We also give complete structural classifications of the $(C_8,C_6)$- and $(C_9,C_6)$-critical colorings, recovering the previously reported counts 24 and 26. All proofs are combinatorial and use no exhaustive graph enumeration.