Papers

We study infinite-horizon time-inconsistent Markov decision processes with a countably infinite state space and unbounded reward functions. The reward is allowed to depend explicitly on the initial time and initial state, thereby accommodating general sources of time inconsistency. We seek relaxed feedback equilibria, and our approach is based on entropy regularization and weighted functional analytic methods. With entropy regularization, we characterize a regular relaxed equilibrium through a fixed-point operator. By introducing two weight functions with distinct roles, one controlling the growth of rewards and values and the other defining the ambient weighted space, we construct a compact invariant set under a product topology and apply the Schauder-Tychonoff fixed-point theorem to establish existence of regularized equilibria. Importantly, the invariant set can be chosen uniformly for small entropy weight $\lambda\in(0,1]$. We then let $\lambda\to0+$ and show, through compactness, concentration of Gibbs policies, and uniform-integrability arguments, that a subsequential limit is a relaxed equilibrium of the original unregularized problem. We further study a policy iteration algorithm (PIA) for the entropy-regularized equilibrium problem. Under a weighted-discounting structure and sufficiently strong discounting, we establish exponential convergence and uniqueness of the regularized equilibrium in a suitable weighted Banach space. Combining the policy-iteration error with a quantitative soft-max approximation bound, we show that the iterated policies constitute weighted $\varepsilon$-equilibria for the original unregularized problem and derive an explicit regret estimate. A numerical example illustrating the convergence of PIA under strong discounting and a counterexample demonstrating its failure under weak discounting are also provided.

We prove that any Busemann G-space such that every sufficiently small metric ball is convex is a topological manifold. The key ingredient in the proof is Ivanov's Helly theorem. The appendix contains a counterexample to a question of Berestovskii–Halverson–Repovš.

We study the propagation direction of bistable traveling fronts in the two-species Lotka–Volterra competition–diffusion system under strong competition. A complete characterization of the sign of the wave speed has remained a long-standing unsolved problem. We establish the first global necessary and sufficient criterion for zero wave speed by combining a Maxwell-type identity with a phase-plane rigidity argument. This criterion yields a unique zero-speed threshold, identifies its value in the symmetric case and its limiting values in two extreme regimes, and consequently determines the propagation direction throughout the strong-competition parameter region.

We report a Lean 4 formal verification accompanying the paper "Subregular Affine Cells and the Level -1 Vertex Algebra of Type D" (arXiv:2608.11997). The formalization kernel-checks substantial internal parts of the proof architecture, including the Section 4 membership/descent chain, the type-D norm-gap argument, zero-orbit energy and signed-permutation rigidity, node-weight and numerical rigidity calculations, the exhaustion logic, the passage to the candidate quotient classification, the simple-object count, the final additive Grothendieck-group comparison, and the coefficient-substitution layer of the uniform character formula. Higher representation-theoretic results whose foundational infrastructure is not presently constructed in the file are isolated as explicit semantic interfaces rather than introduced as Lean axioms. Thus the precise claim is a kernel-checked internal deduction from explicit representation-theoretic boundary inputs, not a from-scratch formalization of vertex algebras, BRST reduction, finite W-algebras, or affine Hecke theory inside Mathlib.

We establish a collection of sharp inequalities for real quadratic forms on the first-quadrant sector of a regular octagon. Starting from the complete extreme-point description of the associated polynomial unit ball, we compute the exact pointwise Bernstein function for the Euclidean gradient. One extreme curve controls the problem: its endpoint is active up to slope $1/2$, after which the maximizer follows an explicit Cardano branch. We obtain the sharp Markov constant $2\sqrt5$, the exact relative quadratic polarization constant $2$, the canonical unconditional constant $3$, and the body-relative Bohr radius $1/\sqrt3$. We also determine the optimal coefficient $\ell_q$-comparison for every $1\le q\le\infty$. The same norm-one polynomial is extremal for all these global constants. At $q=4/3$ the result is a sharp fixed-space coefficient inequality of Bohnenblust–Hille type.

Longo and Vigni extended a result of Greenberg by giving a relation between the cardinality of the Selmer group and the characteristic power series of its Pontryagin dual over the cyclotomic $\mathbb{Z}_p$-extension for a $p$-adic Galois representation. In this paper, we extend this result to the case of Artin representations, following the framework of Greenberg and Vatsal.

Resistance curvature, introduced by Devriendt and Lambiotte, is a novel discrete curvature notion defined through effective resistance. A graph is called resistance nonnegative if there exists a choice of positive edge weights for which the resistance curvature is nonnegative at every vertex. This property has a notable combinatorial interpretation in terms of random spanning trees: a graph is resistance nonnegative if and only if it admits a distribution on its spanning trees under which every vertex has expected degree at most two. The resistance nonnegativity can also be characterized by the tree double matching polytope. These characterizations reveal strong connections among resistance curvature, effective resistance, spanning tree distributions, matching theory, and polyhedral combinatorics. Based on the sign of the curvature, Devriendt introduced the classes of resistance nonnegative (RN), resistance positive (RP), and strictly resistance nonnegative (SRN) graphs, and posed several questions concerning their recognition, polyhedral structure, and structural properties. In this paper, we first answer Devriendt's question on the computational complexity of recognizing RN, RP, and SRN graphs by proving that all three classes can be recognized in polynomial time. We then address his question concerning the tree double matching polytope $\Theta(G)$. Further, we characterize the vertices of $\Theta(G)$ in terms of full-rank systems of tight constraints. Whenever $\Theta(G)\neq\emptyset$, we also determine the least positive integer $k_G$ such that $k_G\Theta(G)$ is a lattice polytope. Finally, for every finite Cartesian product of paths, we explicitly construct an average point satisfying the condition for resistance nonnegativity, thereby obtaining that such graphs are RN. We further characterize the classes of such Cartesian product graphs that are RP or SRN.

We prove that any sufficiently slow uniform rotation stabilizes the supermassive star under axi-symmetric perturbations, and the stability persists for every adiabatic exponent in a left neighborhood of the mass-critical one. The key new ingredient of the linear analysis is a structural property of the self-similar family inherited from the mass-critical scaling: the associated self-similar direction becomes a reduced kernel direction of a modified linearized operator.

Transmission system operators often reconcile market-cleared DC dispatches with AC physics through power-flow solves (ACPF), yet the resulting setpoints can still violate voltage (Volt) and reactive-power (VAR) limits. Maintaining secure voltage profiles and adequate VAR support therefore requires fast corrective decisions that are implementable in operation. This paper presents a novel homotopy-based continuation method for discrete-control Volt/VAR Optimization (VVO) that coordinates switchable devices, such as on-load tap-changing transformers (OLTCs) and capacitor banks (CBs). Experiments on IEEE, PEGASE, and RTE systems show that the proposed VVO produces AC-feasible setpoints within practical runtime, while reducing voltage deviation, VAR dispatch, and generation cost. VVO also achieves comparable performance using ACPF-adjusted DC dispatches as inputs relative to AC-feasible dispatches, indicating that it can be integrated naturally into existing transmission dispatch practices to strengthen grid operation.

The trace of the average mixing matrix of a quantum walk measures the "laziness" of the walk: the higher the trace, the more likely that the walker returns home in the long run. In this paper, we develop tools to study this graph invariant arising from Laplacian quantum walks. It is known that the complete graph $K_n$ is the laziest connected graph on $n$ vertices. Using our machinery, we show that the star $S_n$ is the second laziest connected graph on $n$ vertices (and hence the laziest tree on $n$ vertices), the complete multipartite graph $K_{n-2,1,1}$ is the third laziest connected graph on $n$ vertices, and the double star $DS(n-3,1)$ is the second laziest tree on $n$ vertices. We also show that on the same number of vertices, more unbalanced double stars are lazier.

Let $n \geq 2$ be a positive integer, and $q$ be a prime power. We study the $number$ of additive decompositions of nonscalar matrices in the matrix ring $M_n(\mathbb{F}_q)$ as sums of elements from two prescribed conjugacy classes. Let $z \in M_n(\mathbb{F}_q)$ be nonscalar. We show that, except for the case $(n,q,\textrm{Tr}(z)) = (2,2,1)$, there exist conjugacy classes $X, Y \subset M_n(\mathbb{F}_q)$ such that the characteristic polynomial of $X$ is irreducible of degree $n$ and the characteristic polynomial of $Y$ is of the form $ (T- \lambda) h(T)$, where $\lambda \in \mathbb{F}_q$, $h$ is irreducible of degree $n-1$ and $h(\lambda) \neq 0$. These classes can be chosen so that $\textrm{Tr}(z) = \textrm{Tr}(X) + \textrm{Tr}(Y)$. For such $X$ and $Y$, let $$ N_{X,Y}(z) = \# \{ (x,y) \in X \times Y : x + y = z \}. $$ We prove the following estimate: $$ \left| N_{X,Y}(z) - q^{(n-1)^2} \right| \leq 42 q^{(n-1)^2 -1}. $$ Thus, for nonscalar matrices with matching trace, the number of such additive decompositions is $approximately$ the same, namely $q^{(n-1)^2}$, with an absolute error constant independent of $n$ and $q$.

For origin-symmetric convex bodies in $\R^n$ invariant under $n-2$ orthogonal hyperplanes reflections, we establish the logarithmic Brunn–Minkowski inequality and classify all equality cases without regularity assumptions. We derive the logarithmic Minkowski inequality and rigidity for cone-volume measures, including an explicit classification for bodies of revolution and uniqueness for simplicial polytopes in the prescribed symmetry class.

We give negative answers to two tilting-completion questions for finite-dimensional algebras. We construct two finite-dimensional basic connected quasi-hereditary $\mathbb C$-algebras. The first admits a faithful basic full-rank pretilting module with no tilting completion; the second admits an almost-tilting module with no tilting completion. The full-rank example also yields counterexamples to two conjectures: Enomoto's Self-orthogonal Wakamatsu-tilting Conjecture and the Self-orthogonal Faithful Conjecture of Chen, Li, Zhang, and Zhao. We further show that the Self-orthogonal Wakamatsu-tilting Conjecture holds for all finite-dimensional algebras if and only if the Self-orthogonal Faithful Conjecture holds for all finite-dimensional algebras. The construction ultimately stems from Krah's non-full exceptional collection of maximal length on a rational surface and Kalck's associated full-rank presilting example.

We develop a parameterization of small amplitude finite gap solutions to the KP equation with prescribed spatial wavenumber vectors using classical Schottky uniformization. This parameterization is suitable for a Lyapunov-Schmidt reduction. Using this parameterization, we prove the persistence, under Hamiltonian perturbations, of periodic one-gap solutions for both KP-I and KP-II and of bi-periodic two-gap solutions for KP-I.

Let $A\in\mathcal{B}(H)^+$ and $B\in\mathcal{B}(\ell^2)^+$ be positive bounded operators. We investigate reduced weighted adjoints of densely defined closable operators and use them to develop frame-type systems in the semi-Hilbert spaces induced by $A$ and $B$. Closedness, boundedness, range behavior, and algebraic properties of the $A$-adjoint are established, with particular attention to sums, products, and double adjoints in the unbounded setting. We then study the associated $AB$-analysis, $AB$-synthesis, and $AB$-frame operators. Operator-theoretic characterizations of $AB$-Bessel sequences, $AB$-lower semi-frames, and $AB$-frames are obtained, and examples show that several natural weighted-adjoint identities may hold only as proper inclusions. A Douglas-type range characterization for $AB$-frames is also derived.

We study frames and related sequences in tensor products of separable Hilbert spaces from an algebraic perspective. For sequences $\{f_n\}_{n\in\mathbb{N}} \subset H_1$ and $\{g_m\}_{m\in\mathbb{N}} \subset H_2$, we consider the tensor sequence $\{f_n \otimes g_m\}_{n,m\in\mathbb{N}}$ in $H_1 \otimes H_2$. We characterize the lower semi-frame and Riesz–Fischer properties of tensor sequences in terms of the corresponding properties of the component sequences. These results extend to infinite tensor products. In addition, we study the action of tensor sums of operators on frame-related sequences in tensor product Hilbert spaces.

We construct representation-independent families of solutions of the non-braided scalene Yang–Baxter equation from quadratic noncommutative algebras. Beginning with two anticommuting generators, we obtain a continuous deformation in terms of quantum-plane algebras and extend the construction to an arbitrary number of generators. In the latter case the scalene Yang–Baxter relation fixes the multiparameter exchange matrix to an exact multiplicative form, thereby selecting a distinguished subclass of multiparameter quantum affine spaces. We construct finite-dimensional realizations using singular matrices and finite Heisenberg–Weyl operators, as well as infinite-dimensional realizations in terms of bilateral weighted shifts and multiplicative-shift operators. The explicit realizations are generically purely scalene: although the ordered triple satisfies the scalene Yang–Baxter equation, its individual constituent operators do not satisfy the ordinary non-braided Yang–Baxter equation. These results provide a representation-independent algebraic framework for constructing Yang–Baxter-irreducible scalene triples and a starting point for investigating their possible applications to quantum integrability.

Transport resilience work is often split across separate data preparation scripts, network models, simulation tools, image inspection systems and reports. This fragmentation makes it difficult to move from an observation to a tested and reviewable decision. We introduce ResiliFlow, an open transport world model concept and an implemented platform for infrastructure resilience, response and recovery. The platform connects two workspaces. Disaster Transport Resilience Analysis provides six map-centred functions for critical-road and critical-area identification, recovery prioritisation, disruption routing, resilience testing and scenario simulation. AI-based Transport Infrastructure Perception and Decision Support organises street-level and satellite evidence, detects visible road, footpath and kerb conditions, and prepares these observations for human-reviewed intervention planning. Both workspaces share an eight-step cycle of perception, prediction, model development, verification, execution, decision, feedback and memory. Research Validation records assumptions and checks, while a local Assistant and an optional multi-provider large language model Copilot translate user questions into bounded calls to executable tools. We document the platform architecture, representative mathematical models, interface evidence and computer-vision learning results. Examples show accurate recognition across eight visible-condition classes, while compact error analysis demonstrates how difficult cases guide continued learning. ResiliFlow shows how transport models can become an inspectable, reusable and question-led system rather than a collection of disconnected analyses. The accompanying release is intended to support research collaboration, public scrutiny and extension under institutional review.

We study lattice vertex algebras of type ADE over an algebraically closed field $\mathbb{F}$ of prime characteristic $p>2$ and their representations. Let $L$ be a root lattice of type ADE, and $G_{L}$ the Gram matrix of $L$. When $\det G_{L}\not \equiv 0\pmod{p}$, we establish an isomorphism between the lattice vertex algebra $V_{L,\mathbb{F}}$ and the level-one simple affine vertex algebra of the same type. Via this isomorphism, we classify the irreducible $\mathbb{N}$-graded modules of $V_{L,\mathbb{F}}$ viewed as an $\mathbb{N}$-graded vertex algebra. We also consider the case where $L$ is of type $A_n$ with $\det G_L=n+1\equiv 0\pmod{p}$. We show that $V_{L,\mathbb{F}}$ is not simple and determine the simple $\mathbb{N}$-graded quotient of $V_{L,\mathbb{F}}$. Furthermore, when $n+1=ap$ for some $a\in \mathbb{Z}_+$ with $\gcd(a,p)=1$, we give the classification of the irreducible $\mathbb{N}$-graded modules for the simple quotient of $V_{L,\mathbb{F}}$.

Fix an arity $r\ge 2$, a group $G$, and a full ordered $r$-ary tree $T$ with at least two internal vertices. For an arbitrary operation $\omega:G^r\to G$, let $\omega_T$ denote the operation obtained by iterating $\omega$ according to $T$. We classify all $\omega$ for which there exists a bijection $F_T:G\to G$ such that $\omega_T(x_1,\ldots,x_n)=F_T(x_1\cdots x_n)$. We prove that $\omega$ must have one of the forms $ax_1\cdots x_r$, $x_1\cdots x_rb$, $d\,\psi(x_1\cdots x_r)$, where $a,b\in G$, $d\in Z(G)$, and $\psi\in\operatorname{Aut}(G)$, with explicit conditions on the parameters determined by $T$. We next reverse the problem. Fix an operation $\omega$ of one of these three forms and determine every full ordered $r$-ary tree $T$ for which there exists a bijection $F_T:G\to G$ satisfying $\omega_T(x_1,\ldots,x_n)=F_T(x_1\cdots x_n)$. The answer is governed by the order of $aZ(G)$ or $bZ(G)$ in $G/Z(G)$, or by the order of $\psi$ in $\operatorname{Aut}(G)$. We also ask, for each of the three solution forms above, how much associativity remains. More precisely, for two full ordered $r$-ary trees $S$ and $T$ with the same number of leaves, we determine exactly when $\omega_S=\omega_T$. Pairs of $r$-ary trees encode changes of parenthesization, and modulo simultaneous expansion they represent elements of the Higman–Thompson group $F_r$. The changes of parenthesization that preserve the iterated operation form a subgroup of $F_r$, which we determine explicitly.

We consider the isotropic-nematic interface problem for liquid crystals based on the Landau-de Gennes model. We show that when the anisotropic elastic constant $L_2$ is negative, then the surface tension strength of the isotropic-nematic interface is proportional to $\sqrt{L_1+\frac{2}{3}{L}_2}$, and the homeotropic alignment is more favored near the interface. This result gives a rigorous confirmation for de Gennes' formal derivation in \cite{deGen1971} on the isotropic-nematic tension strength and the anchoring alignment condition in the case of $L_2<0$.

In this thesis, quantum channels are studied from the point of view of Mathematics. We studied the CPTP (completely positive trace preserving) and UCPTP channels, the latter being the unital channels. In both cases, the sets are compact convex, so they are the closure of their extreme points. We constructed extreme CPTP maps for every possible rank. We also constructed extreme UCPTP maps with rank 2 for each possible dimension. For ranks of at least 3, we constructed extreme UCPTP maps for many dimensions and ranks. We also proved that the tensor product of extreme CPTP maps is an extreme CPTP map. For UCPTP maps, we proved that the same is not always true, with many counterexamples. It is known that the category of CPTP maps is semicartesian. We add to this a classification of binary products. We proved that products only exist in trivial cases. For binary coproducts, we obtained the dimensions they should have, if they exist. In both cases, topological techniques were used. For this reason, we also investigated the topology of the closure of the set of extreme points of the CPTP maps. Topological invariants can be computed from decompositions of a space such as CW structures. These are a type of decomposition in which the space is expressed as a gluing of balls. In the search for a CW structure for the closure of the set of extreme CPTP maps, we obtained a partial decomposition that resembles a CW structure of the complex projective space.

Let \(\F\) be an algebraically closed field of characteristic zero, \(\g=\s\oplus\z\) a finite-dimensional reductive Lie algebra over \(\F\), and \(V\) an arbitrary finite-dimensional \(\g\)-module. We classify all 2-local derivations of \(\g\) on \(V\), and show that every 2-local derivation is a derivation if and only if \(\dim\z\leq1\) or \(V^\g=0\). If \(\dim\z\geq2\) and \(V^\g\ne0\), the nonlinear homogeneous maps give all exceptional 2-local derivations.

Let $f:(X,\Delta)\rightarrow Y$ be a surjective holomorphic map between two normal Kahler varieties, where (X,\Delta) is a log canonical pair, -(KX+\Delta) is nef and Y is Q-Gorenstein. In Fu-Guo-Song-Wang [18], it is proved that -KY is pseudo-effective if f is a projective morphism. In this paper, prove that -KY is pseudo-effective without assuming that f is projective

We prove Lipschitz, sub-Hölder and Hölder stability estimates for recovering the low Fourier modes of an unknown potential from the Dirichlet-to-Neumann (DN) map. We study three different cases, depending on the regularity of the difference $q_1-q_2$. First, we consider \[ (-\Delta - \lambda^2 + q)u=0 \quad \text{in} \quad \Omega\subset \mathbb{R}^n. \] We show that the difference $q_1-q_2$, assumed to be $M$-bandlimited, can be recovered in a Lipschitz stable way from the difference of the corresponding DN maps. This holds whenever $\lambda$ is sufficiently large relative to $M^{n/2}$. The proof involves real geometrical optics solutions. Secondly, we consider \[ (-\Delta + q)u=0 \quad \text{in} \quad (-\pi,\pi)^n. \] We show that the low Fourier coefficients of the difference $q_1-q_2$, assumed to be real-analytic and periodic, can be recovered in a sub-Hölder stable way from the difference of the corresponding DN maps. The number of recoverable Fourier modes grows as the DN maps become closer. Finally, we consider the case where the Fourier coefficients of the difference $q_1-q_2$ decay at a super-exponential rate $e^{-c|k|^{n/2}}$. We prove that the low Fourier modes can be recovered with Hölder stability, with the number of recoverable modes tending to infinity as the DN maps become closer. In all cases the $L^\infty$ potentials themselves do not need to satisfy additional assumptions or belong to a finite dimensional space. The constants in the stability estimates are uniform in the number of recovered Fourier modes.

Remote attestation is a procedure by which a system (such as a Trusted Execution Environment) proves its own configuration to a remote peer. Here, the configuration includes the measurements (hash digests) of software components. When multiple nodes perform mutual attestation, each node must maintain the reference values for its peers' measurements. We refer to this as the reference-value bootstrapping problem for mutual attestation. In this paper, we formulate the problem as a fixed-point equation and solve it directly using Kleene's second recursion theorem. In our approach, nodes can reconstruct one another's source code, thereby allowing each node to derive the reference values at runtime. When a source code itself is deployed, as with a Python script, a node derives the peer's reference value directly from its reconstructed source. When a build artifact is deployed, as with AWS Nitro Enclaves, a node instead reproducibly builds the peer's artifact from its reconstructed source and derives the reference value from it. As a proof of concept, we develop transpilers in Python and Nix to generate mutually referencing programs. Using these transpilers, we demonstrate the mutual derivation of reference values for the two deployment models mentioned above. Our solution is architecture-independent, requires no trusted third party, and works with existing attestation stacks without modification.

We consider the two-dimensional Dirac operator with constant magnetic field that is perturbed by a combination of electrostatic and Lorentz-scalar delta interactions with variable coefficients supported on a smooth closed curve. Self-adjointness is studied in the so called non critical and critical cases. In the non-critical case the essential spectrum is unchanged - it remains to be the set of the Landau-Dirac levels, the eigenvalues of infinite multiplicity of the unperturbed operator - while in the critical case an additional interval of essential spectrum emerges in the spectral gap containing zero. Our main result concerns the discrete spectrum in the non-critical case: using the pseudodifferential properties of the involved boundary integral operators, we show that the eigenvalues accumulate at each Landau-Dirac level at a rate governed by the logarithmic capacity of the curve. A novel and surprising phenomenon is the change in the side of the accumulation depending on the position relative to the critical value. As a byproduct, clusters of eigenvalues for a family of exterior boundary value problems are obtained via confining couplings; the infinite-mass boundary condition arises as a special case.

Surface hopping is an attractive mobility mode for small-body exploration, but designing fuel-optimal hops on asteroid 433~Eros requires solving a nonconvex optimal control problem with an irregular polyhedral gravity field, thrust–mass coupling, and collision-avoidance constraints. We show that a physics-informed Bayesian Optimization (BO) warm-start—a Gaussian-process search over a single Bézier control point that requires no offline training—provides a more reliable initialization for Sequential Convex Programming (SCP) than the standard straight-line guess. The straight chord penetrates the asteroid on every inter-site transfer considered here and degrades convergence. The physics-informed BO reference feeds an SCP stage in which a log-mass change of variables convexifies the thrust–mass coupling and nearest-facet half-spaces enforce collision avoidance, and an automated Pareto time-of-flight sweep selects fuel-priority solutions without introducing bilinear terms. Applied to all 20 ordered transfers among five representative surface sites and validated under 10 random seeds, the framework tracks every trajectory to meter-level terminal accuracy in closed-loop Monte Carlo simulation, completes a five-site tour for one fifth of the propellant budget, and reduces mean $\Delta V$ by roughly one third relative to an idealized two-impulse ballistic trajectory estimate. A straight-line ablation credits the warm-start with cutting the mean SCP iteration count from 8.8 to 5.9 and removing the one convergence failure. A target-perturbation analysis further shows that the solutions vary smoothly with the landing target, with no jumps between local basins.

Let $p\ge 5$ be a prime with $p\equiv -1\pmod 3$, let $q=p^r$, and consider \[ \cC:\qquad y^3=x^{(q^2+1)/2}+x \] over $\F_{q^6}$. We prove the exact formula \[ \#\cC(\F_{q^6})=q^6+1+(-1)^{r+1}(q^2-1)q^3. \] Since $g(\cC)=(q^2-1)/2$, the curve is maximal when $r$ is odd and minimal when $r$ is even. The proof uses a birational Kummer model and an explicit Jacobi-sum point count. A congruence together with Frobenius invariance reduces the relevant Jacobi sums to cubic Gauss sums, whose sign is determined from the Fermat cubic. In particular, the maximality of $y^3=x^{13}+x$ over $\F_{5^6}$ appears as the first case of an infinite family.

Jin recently claims that the distinguished level $-1$ vacuum block of $L_{-1}(D_\ell)$ realizes the specialized dual affine left-cell module attached to the subregular cell containing $s_0$. We show that the proof of this Grothendieck-group statement contains a gap. We give an explicit counterexample over the discrete valuation ring $\Bbbk[[t]]$, showing that the categorical implication used in the proof is false in general. Consequently, Jin's Theorem~6.3 does not follow from the argument given in his Section~6.2. We then supply a complete replacement proof for the grading-restricted level-$-1$ vacuum block. Combining the classification of simple objects, we prove finite length, construct an injective signed normalized-character map into a completed singular orbit module, and identify its image with the specialized dual subregular left-cell module. Thus the cell-module realization claimed is established for the grading-restricted vacuum block.