Papers

Characterizations of pointwise Gaussian upper bounds on graphs with possibly unbounded geometry in terms of localized functional inequalities contain errors depending on the vertex degree. We introduce a new space-time averaged form of the heat semigroup in terms of time-averaged $\ell^{q/(q-1)}-\ell^q$-estimates and obtain Gaussian upper bounds from large-scale Faber-Krahn inequalities which avoid such errors. A main analytic ingredient is an integrated version of Davies' method which yields off-diagonal estimates for this averaged quantity, and which tends to pointwise Gaussian bounds as time tends to infinity. Conversely, on large scales volume doubling and Gaussian bounds on this averaged heat semigroup norms imply relative Faber-Krahn inequalities for subsets of prescribed relative measure. Our proof is based on a lower bound for Dirichlet eigenvalues in terms of these averages. The Faber-Krahn dimension is scale-dependent, but converges to the doubling dimension for increasing radii. This gives a characterization of asymptotic Gaussian heat kernel behavior in terms of Faber-Krahn inequalities.

Consider a holomorphic vector bundle $E$ over a compact complex curve $C$ which decomposes as a sum of stable vector bundles. For the projectivization $\mathbb{P}(E)$, we prove that the existence of a compatible extremal almost Kähler (aK) metric of involutive type in the sense of Lejmi is equivalent to the existence of a Calabi extremal Kähler metric. This result rests on the Yau–Tian–Donaldson correspondence in terms of the moment polytope $\Delta$ for $\mathbb{P}(E)$, proved by the author and Yin in a previous work. The main advantage is that compatible extremal aK metrics of involutive type are solutions to a second-order linear PDE, rather than a fourth-order nonlinear PDE for Calabi's extremal Kähler metrics. As an application, we prove that when $E$ has rank $4$ and $C$ is an elliptic curve or the projective line, $\mathbb{P}(E)$ is a Calabi dream manifold, i.e. admits an extremal Kähler metric in every Kähler class.

We establish local existence and uniqueness of solutions for the ideal incompressible magnetohydrodynamics system posed on $[0,T]\times\mathbb{R}^n$, $n\ge2$, with a nonzero constant initial magnetic field $\mathbf{B}_0$ and arbitrary divergence-free velocity data $v_0\in H^s$, in the range $(n+1)/2<s\le n/2+1$. The proof uses a Lagrangian wave–Hodge reformulation and exploits an Alfvén null–structure hidden in the pressure forcing. In particular, the constructed Eulerian solutions are induced by a bi-Lipschitz measure-preserving flow map. Zhang first identified this null-structure in \cite{Zhang2024}; the present work provides a self-contained bridge from that Lagrangian theory to the Eulerian Cauchy problem.

Recently, Ehrenborg, Hetyei and Readdy expressed the toric $g$-polynomial of a simple polytope as a linear combination of a family of polynomials, called $g$-contribution polynomials, with coefficients given by the entries of its gamma-vector. They conjectured that these toric $g$-contribution polynomials are real-rooted. This paper proves this conjecture.

Fix $0<\theta\leqslant 1$. We prove that if $1\leqslant p \leqslant 2/\theta$, then the $\theta$-snowflake of $\ell_2^k$, namely, $\mathbb{R}^k$ equipped with the metric $((x,y)\in \mathbb{R}^k\times \mathbb{R}^k)\mapsto \|x-y\|_2^\theta$, embeds with distortion $O(1)$ into $\ell_p^m$ for some integer $m\lesssim_{p,\theta}k$, which is optimal as $k\to \infty$, as seen by comparing dimensions. However, for $p$ larger than the sharp threshold $2/\theta$ the following change in behavior occurs: If a $(1/\sqrt{k})$-dense subset of the Euclidean sphere $S^{k-1}$ embeds into $\ell_p^m$ with distortion $O(1)$, then necessarily $m\gtrsim_{p,\theta}( k/\log k)^{p\theta/2}$, which grows super-linearly in $k$ as $p\theta/2>1$, and this dimension bound is optimal as $k\to \infty$ up to lower order factors. We deduce from this statement that if $2<p<\infty$, then there exist arbitrarily large $n$-point subsets of $\ell_p$ with the property that if they embed with distortion $O(1)$ into $\ell_p^m$, then necessarily $m\gtrsim_p ((\log n)/(\log\log n)^2)^{p/2}$, thus demonstrating that the statement of the Johnson–Lindenstrauss dimension reduction lemma fails to hold for $\ell_p$

Guruswami, Li, Mosheiff, Resch, Silas and Wootters conjectured that, for every fixed prime power $q$ and $p\in(0,1-1/q)$, random linear codes of rate $1-h_q(p)-\varepsilon$ have list size $\frac{h_q(p)}{\varepsilon}\cdot(1+o(1))$ as $\varepsilon\to0$, both for list decoding and average-radius list decoding. The conjecture was known for $q=2$ and remained open for larger alphabets. We prove the conjecture for list decoding for every prime power $q$, with the sharper bound $L=\frac{h_q(p)}{\varepsilon}+O_{p,q}(1).$ In contrast, we disprove the conjecture for average-radius list decoding. We show that the required list-size leading constant is at least $p\left(1+\log_q\frac{(q-1)(1-p)}{p}\right)$, which is strictly larger than $h_q(p)$ whenever $q^p(1-p)>1$. Moreover, for $p=1-\lambda/q$ with fixed $\lambda>1$, the required leading constant can be as large as $\left(\frac{\log\lambda}{e\lambda}+o_q(1)\right)\frac{q}{\log q}.$

Uniform-state discrete diffusion models update all tokens in parallel while keeping every position revisable. Even when the commonly used top-$p$ rule leaves only one candidate at a position, that choice affects only the current reverse step and can be revised at the next sampling step. We ask what changes when selected hypotheses instead become persistent context for later predictions. We therefore propose committed reveal sampling (CRS), a training-free sampler that stores selected argmax tokens and inserts them into subsequent model inputs. Our analysis separates persistent context from truncation. Within one reverse step, adaptive top-$p$, randomized or not, reproduces the update induced by a visible token only when a likelihood-ratio ordering holds, while in a simple latent-mode model, keeping the selected token visible helps later parallel predictions agree on the same sequence-level choice. Empirically, paired experiments on Duo-distilled then separate this persistent effect from single-step top-$p$ restriction and scalar temperature scaling. Under the same finalization rule, CRS without top-$p$ truncation reaches lower generative perplexity (GenPPL) than fixed $p=0.95$ and $p=0.9$ baselines across budgets of 8–64 function evaluations (NFE). At 64 NFE, the comparison at matched unigram entropy also gives lower GenPPL for CRS, shifting the GenPPL–entropy tradeoff under this metric. Base Duo shows the same direction in a descriptive comparison, while other diversity and continuation metrics can rank these operating points differently. A distributional distance to held-out text (SOL) under a fixed GPT-2-Large representation also decreases with persistent context at the same late top-$p$ threshold. These results identify support restriction and persistent context as distinct controls of that tradeoff.

Recursive LLM agents can broaden their search by spawning specialists. Some branches later request tools that send data or deploy code. When should a branch receive authority to act? We distinguish sandbox spawning, in which external controls prevent the specified harm, from capability activation, in which a selected branch crosses an irreversible-action boundary. Progressive Risk Vesting (PRV) holds a trajectory-level risk budget in escrow and debits it as branches are activated. We prove an anytime harm bound for adaptively generated trees. Branch outcomes may be dependent, but each local certificate needs to remain valid conditional on the full pre-activation history, including the information used to select the request. When activation gates, branch charges, and compute constraints are held fixed, delayed vesting preserves every policy available under irrevocable spawn charging. Marginal risk estimates can still fail after branch selection. In a stylized branching model, trajectory harm changes as the authority reproduction number $\mathcal{R}_A$ crosses one. As local risk $p$ approaches zero, trajectory harm is proportional to $p$ below criticality, proportional to $\sqrt{p}$ at criticality, and retains a positive floor above it. A finite-type occupancy model yields risk and compute shadow prices. For nested fanout modes with decreasing marginal value per unit risk, these prices produce a threshold rule. Branching calculations and a split-sample experiment illustrate the results. These synthetic studies do not estimate safety in deployed agents. The analysis suggests a design rule: search broadly in the sandbox and grant recursive authority sparingly, with an explicit risk charge.

The central flow of Cohen et al. (2025) is an empirically accurate continuous-time model of gradient descent at the edge of stability in deep learning, However, its derivation is heuristic. We propose a perturbative regime in which the central flow is the limit of gradient descent: we assume that the loss decomposes as $f = g + \varepsilon h$; in the limit $\varepsilon \to 0$, the dynamics of gradient descent with learning rate $\eta$ converge to the gradient flow of $h$ constrained to the minimizers of $g$ of sharpness at most $2/\eta$. Our approach is formal rather than rigorous; it treats gradient descent as a singularly perturbed dynamical system in $\varepsilon$. Three timescales emerge: a fast timescale of oscillations along the sharpest direction, an intermediate timescale of the self-stabilization mechanism, and a slow timescale of the dynamics along the minimizers of $g$-the central flow. Using the method of multiple scales, a classical formal method from singular perturbation theory, we derive the expansion of the dynamics in $\varepsilon$: the central flow emerges as the leading-order term in the expansion, while the self-stabilization mechanism appears in the next-order term. We study this mechanism beyond previous analyses: with a single eigenvalue at the edge of stability, we compute the slow drift of the energy of the fluctuations; with several eigenvalues at the edge of stability, we derive the self-stabilization system and explain why fluctuations persist.

We develop a new class of inexact block triangular preconditioners for double saddle-point systems arising from PDE-constrained optimization. The proposed preconditioners are constructed through matrix factorization techniques while preserving the inherent block structure of the original systems. A comprehensive spectral analysis of the preconditioned matrices is provided, yielding explicit bounds for both real and nonreal eigenvalues. To enable efficient construction of the inexact preconditioners, randomized strategies are introduced to select the required subblocks. We establish high-probability bounds for the expected approximation error, with the error estimates explicitly characterized in terms of the eigenvalues of the associated matrices. Numerical experiments demonstrate the effectiveness, robustness, and scalability of the proposed preconditioners, and validate the efficiency of the randomized construction strategies.

In this article, we introduce a new distance function between hypersurfaces with free boundary. We show that our new quantity, which we call twisted Fermi distance, is monotone under mean curvature flow with free boundary. This overcomes the stumbling block that monotonicity of the usual distance function can fail for non-convex domains, and has several applications. Most importantly, we generalize the avoidance principle for free boundary Brakke flows, recently established by the first author for convex domains, to arbitrary domains. Using this, we then show that all results from our recent joint work, including the mean-convex neighborhood theorem and the uniqueness theorem for free boundary flows through cylindrical singularities, can be generalized to arbitrary domains without any convexity assumptions as well.

To establish a basis of the sheaf theoretical study of the Milnor fibers and monodromies of complete intersection varieties, we clarify the structures of the Verdier specialization sheaves associated to them. Assuming the Thom condition in a direction, some basic results on them will be obtained. Then we apply them to obtain formulas for the Euler obstructions of complete intersection varieties having non-isolated singular points.

We establish the conjecture by de Klerk and Pasechnik (2002), claiming that the semidefinite bounds $\vartheta^{(r)}(G)(r\geq 0)$ for the stability number $\alpha(G)$ are exact at $r=\alpha(G)-1$, by exhibiting an explicit sum-of-squares certificate. This certificate allows us to recover a known characterization of the minimizers of the Motzkin-Straus formulation for $1/\alpha(G)$. Additionally, we give sum-of-squares copositivity certificates for the matrices satisfying the Hoffman-Pereira sign condition, a crucial condition for characterizing copositive matrices with $\{-1,0,1\}$ entries.

OptiXDE is a matrix-free spectral operator framework for differential equations on uniform grids and embedded domains. Inspired by angular-spectrum propagation in Fourier optics, it maps transform-diagonal spatial operators to analytical modal multipliers and composes them with physical-space operators for nonlinearities, geometry and boundary enforcement. A common transform–operator–inverse-transform backbone is demonstrated across transient diffusion, periodic and embedded-domain Poisson problems, the cubic nonlinear Schr"odinger equation, viscous Burgers dynamics, the two-dimensional Allen–Cahn equation and incompressible flows from the Taylor–Green vortex to embedded-cylinder vortex shedding. Transform-compatible linear problems are recovered near the floating-point limit, whereas errors on the singular L-shaped domain remain localized near the re-entrant corner and regularized interface. Nonlinear benchmarks recover second-order temporal convergence and the expected conservative or dissipative behavior, while incompressibility remains near round-off level during long-time vortex shedding. The matrix-free updates require \(\mathcal{O}(N\log N)\) work and \(\mathcal{O}(N)\) memory. Device-resident transform workloads reach \(94.9\times\) GPU acceleration, and the complete embedded-cylinder solver achieves a \(42.1\times\) CPU–GPU speedup under matched numerical settings. These results establish OptiXDE as a deterministic and extensible operator-centric framework for structured and embedded-domain differential equations.

We study the asymptotics as $n$ goes to infinity of $\mathbb E\left[\mathcal{N}(S_n,[0,2\pi])\right]$, the expected number of zeros in $[0, 2\pi]$ of a random periodic signal $S_n$ of the form \[ S_n(t)=\sum_{k=1}^{n}a_k f(kt), \] where $f$ is a non-analytic $2\pi-$periodic function and the coefficients $(a_k)$ are i.i.d. random variables, centered with unit variance. We show in particular that if $a_1$ admits a finite third moment and if the function $f$ is piecewise polynomials and of class $\mathcal C^{7}$, then we have the following universal asymptotics, independent of the particular law of the coefficients $(a_k)$ \[ \lim_{n \to +\infty}\frac{\Esp\left[\mathcal{N}(S_n,[0,2\pi])\right]}{n}= \frac{2}{\sqrt{3}}\sqrt{\frac{\|f'\|_{L^2([0,2\pi])}}{\|f\|_{L^2([0,2\pi])}}}. \] This result thus extends in expectation and at the scale of the whole period $[0,2\pi]$ the local universality property established {in [Angst-Poly, IMRN, 2019]}, in distribution and in shrinking intervals of size $1/n$. Moreover, it generalizes to a non-analytic context the global universality results obtained in the more classical frameworks of random trigonometric polynomials or random analytic functions. Our approach combines a new almost sure Central Limit Theorem à la Salem–Zygmund for the function $S_n$ when evaluated at a uniform random point in $[0, 2\pi]$, and as well as suitable uniform integrability and anti-concentration estimates.

Non-parabolic growth of reaction layers is frequently observed in solid-state diffusion couples, but the relationship between microstructural evolution and the resulting kinetics remains incompletely understood. This effect is commonly attributed to grain growth, which can progressively reduce the contribution of fast grain-boundary diffusion, causing the effective diffusivity of a polycrystalline product layer to evolve during reaction. This work develops a moving-boundary diffusion model to describe product-layer growth while accounting for the local grain-growth history of the continuously formed product, as well as both bulk and grain-boundary diffusion. Numerical simulations show that the reaction can pass through three distinct kinetic regimes: an initial approximately parabolic regime dominated by grain-boundary diffusion, a transient sub-parabolic regime associated with strong spatial variation of the effective diffusivity, and a subsequent approximately parabolic regime dominated by bulk diffusion. The magnitude and duration of the sub-parabolic regime depend on the relative grain-boundary and bulk diffusivities and on the kinetics of grain growth. The instantaneous growth exponent therefore evolves continuously and does not represent a unique kinetic constant. Nevertheless, fitting simulated layer-thickness data over finite experimental time intervals produces well-defined apparent exponents, demonstrating how a transient process can appear to obey a single power law.

We extend a $d$th-order Newton method for unconstrained optimization by Ahmadi, Chaudhry, and Zhang [Advances in Mathematics, 452:109808] to optimization with SOS-convex polynomial constraints. Consider the problem of minimizing a smooth function $f:\mathbb{R}^{n}\to\mathbb{R}$ subject to SOS-convex polynomial constraints. Given an iterate $x\in\mathbb{R}^n$, Ahmadi et al. define the next iterate $x^{+}$ as the minimizer of the $d$th-order Taylor expansion of $f$ at $x$ with a regularization term of degree $d^{\prime}$, where $d^{\prime}$ is the smallest even number greater than $d$, chosen such that this polynomial is SOS-convex, subject to the constraints. Constructing this polynomial and minimizing it subject to the constraints can both be reduced in time polynomial in $n$ to a semidefinite program (SDP). We prove that, if $f$ is strongly convex and the tensor of the $d$th-order partial derivatives of $f$ is Lipschitz continuous, then our method converges locally to the optimal solution $x^{\ast}$ with order $d$. We further prove that, under certain constraint qualifications, the set of active constraints at $x^{\ast}$ is identified locally in a single iteration. Next, we study the worst-case performance of the third-order Newton method in the unconstrained setting for two classes of univariate $f$ using performance estimation. Finally, we extend a globally convergent modification of the $d$th-order Newton method to the setting of SOS-convex polynomial constraints.

Among three-dimensional parallelohedra of fixed volume, the Archimedean truncated octahedron uniquely minimizes surface area, affirming a conjecture of K. Bezdek from 2006. The final scalar inequality is verified by an exact computer-assisted certificate.

Let $E$ be an $n$-dimensional rigid adelic space over a number field $K$. We study the minimum number $g_E(R)$ of proper $K$-subspaces needed to cover the projective height ball of radius $R$, together with the maximum cardinality $h_E(R)$ of a subset in linear general position. We show that, once $R$ is sufficiently large compared with the last Roy–Thunder minimum of $E$, both quantities have order $\Psi_E(R)^{[K:\mathbb Q]}$, where $\Psi_E(R)$ is an explicit expression in the Roy–Thunder minima. The comparison constants are effectively computable and uniform in $E$. For the standard adelic space $K^n$, this gives order $R^{[K:\mathbb Q]n/(n-1)}$.

We outline the contours of an emerging theory of Kähler metrics in derived noncommutative geometry. This is a refinement of the theory of Bridgeland stability conditions encoding underlying differential-geometric structures. We propose elements of such a structure in both Archimedean and non-Archimedean settings, including metrized objects, mass measures satisfying a BPS inequality, harmonic metrics, minimizing flows, and complexified Kähler potentials. We develop the framework through examples and constructions involving Fukaya categories, quiver representations and associated C$^*$-algebras, spectral networks, and comonadic adjunctions of stable $\infty$-categories.

Lipshitz and Sarkar asked in an ICM 2018 question whether their homotopy-theoretic refinement of Khovanov homology can detect mirror reflection when ordinary integral Khovanov homology cannot. We give an affirmative answer by presenting a prime knot whose integral Khovanov homology agrees with that of its mirror in every bidegree, including torsion, and yet over $\mathbb{F}_2$, the second Steenrod square has rank one for the knot and rank zero for its mirror in a specified bidegree. We also give a composite example. The construction, search and computations are done with the help of chat-GTP 5.6 sol Pro combing over the Regina Census of knots.

We work over a field of characteristic zero - primarily over an algebraic closure of the field of rational numbers. We prove a necessary and sufficient condition for a vector field whose coefficients are rational functions separating variables to be algebraically integrable, that is, the subring of rational functions killed by this vector field, its ring of first integrals, is of maximal possible dimension. We do it arithmetically by reducing the vector field modulo almost all primes. In particular, we verify the generalized Grothendieck–Katz p-curvature conjecture for foliations defined by these vector fields. Finally, we use the outcome of that verification to provide explicit formulas for coefficients of all algebraically integrable vector fields separating variables, and their first integrals.

We consider the Vlasov-Maxwell system with a repulsive point charge. Under a spherical symmetry condition, this reduces to the relativistic Vlasov-Poisson system with a point charge. For this system, the linear characteristics solve the equations of motion for the relativistic Kepler problem. We define action-angle variables for the linearized equation, and then study the nonlinear system in these coordinates. Within this framework, we prove global existence of classical solutions for small initial data and establish their modified scattering behavior. The proof makes use of the ideas developed in the non-relativistic case.

In his 2018 paper “On the unique predual problem for Lipschitz spaces”, N. Weaver published proofs that Banach spaces $\mathrm{Lip}_0(M)$ of Lipschitz functions on a complete metric space $M$ have strongly unique preduals whenever $M$ has finite diameter or is geodesic. A gap was recently noticed in the proof of a crucial lemma that claimed that the property of having a strongly unique predual passes to 1-codimensional weak$^*$-closed subspaces. In this note, we confirm that the lemma is actually false by providing an explicit counterexample. We also expand on some of Weaver's original arguments to provide a new, valid proof of the following particular case: $\mathrm{Lip}_0(M)$ has a strongly unique predual whenever $M$ is a convex subset of a finite-dimensional normed space.

In this paper we are interested in isoperimetric inequalities for plane n-gons inscribed in a circle. More specifically, we will examine a conjecture provided by Paul Levy that we propose to solve in particular cases.

Let $K$ be any field, $E$ any directed graph, and $L_K(E)$ the associated Leavitt path algebra. As described first by Chen, and subsequently generalized by Ara and Rangaswamy, for each cycle $c$ in $E$ one can build the simple left $L_K(E)$-module $V_c^E$, and then more generally $V_{p(x),c}^E$ (where $p(x)$ is an irreducible polynomial in $K[x,x^{-1}]$). A cycle $c$ is called exclusive in case none of the vertices of $c$ is the base of any cycle other than $c$. In our main result we provide an explicit description of the injective envelope of $V_c^E$, and then more generally of $V_{p(x),c}^E$, for each exclusive cycle $c$. Our method involves defining an $L_K(E)$-module structure on an appropriately-built $K$-vector space of infinite series. Our main result significantly generalizes previous work of the authors, in that the result holds for all graphs (finite or not), and all exclusive cycles.

Let $K$ be any field and $E$ any directed graph. We characterize up to isomorphism the simple (i.e., minimal) left ideals of the Leavitt path algebra $L_K(E)$. Then, for each simple %(i.e., minimal) left ideal $I$ of %the Leavitt path algebra $L_K(E)$, we explicitly construct the injective envelope of $I$. This result generalizes to all graphs $E$ and all simple left ideals in $L_K(E)$ the construction presented previously by the three authors for the specific case of the Jacobson algebra $R=K\langle X,Y | XY=1\rangle$ and the simple left $R$-ideal $R(1-YX)$. Our method involves defining an $L_K(E)$-module structure on a $K$-vector space of infinite series. We conclude the article by showing how our construction directly gives a description of the injective envelope of simple $L_K(E)$-modules arising from two types of infinite emitters in $E$.

Conditional independence tests (CITs) test for conditional dependence between two random objects $X$ and $Y$ given a third random object $Z$. Existing CITs have limited applicability to high-dimensional data, especially multimodal data like text. However, we show that such tests are of interest for large language model (LLM) outputs, where we test whether an output $X$ generated from a source text $Z$ carries information about an attribute $Y$ beyond $Z$ itself. For this purpose, we propose embedded CITs (eCITs), which embed $X$ and $Z$ and apply an existing CIT to the resulting representations and to $Y$. We show that, provided the embedding of $Z$ is sufficient, i.e. retains the information $Z$ carries about either $Y$ or the representation of $X$, the null hypothesis transfers from $X$ and $Z$ to their representations, so that a CIT valid for the embedded hypothesis is valid for the original one. We further give conditions for equivalence of the two hypotheses, and show that sufficiency weakens to mean sufficiency when the embedded test targets conditional mean independence. We propose a semi-synthetic simulation design to assess type I error (T1E) control and power of the eCITs for given embedding maps on a specific dataset and task, and use it to evaluate them on our application. Applying the eCITs to German Parliament speeches, we find for all combinations of embedding maps considered that the summaries of two LLMs contain information about the speaker's faction and gender beyond the speech they were generated from.

A longstanding fundamental problem of Burr, Erdős, Faudree, Rousseau and Schelp (Trans. Amer. Math. Soc., 1982) is to determine the exact value of the least integer $f(m)$, for odd $m\ge3$, such that every tree $T_n$ on $n\ge f(m)$ vertices satisfies $R(T_n,C_m)=2n-1$. We settle this problem for all sufficiently large odd $m$. Indeed, we establish $$f(m)=\left\lceil \frac{2m-1}{3} \right\rceil$$ for all such $m$, where the lower bound follows from a result by Faudree, Lawrence, Parsons and Schelp. This also confirms a conjecture of Huang, Zhang and Chen for all such $m$.

Pole-skipping occurs at special complex frequencies and momenta where the retarded Green function of a black-hole or black-brane background is not uniquely defined. The local near-horizon mechanism is well known: at resonance, a Frobenius recurrence matrix loses rank and the space of smooth horizon solutions enlarges. We address the global question of how two boundary-normalized solutions enter this resonant horizon solution space. For a general second-order scalar radial equation on a nonextremal background analytic near the horizon, the Frobenius recurrence at the resonant order yields a solvability condition. We prove that its vanishing is equivalent to singularity of the horizon recurrence matrix, absence of the logarithmic Frobenius term, and existence of two independent smooth horizon solutions. Away from resonance, the source zero is equivalent to horizon smoothness of the response-normalized solution, while the response zero is equivalent to horizon smoothness of the source-normalized solution. We then analyze the parameter-dependent simple pole of the nonresonant ingoing solution as resonance is approached. After removing this singularity, its resonant limit is proportional to the larger-root Frobenius solution, with proportionality factor given by the same solvability function. Continuing to the boundary shows that the same condition is equivalent to simultaneous vanishing of the two regularized boundary connection coefficients. This establishes a horizon-to-boundary dictionary between local Fuchsian resonance and the boundary source-response 0/0 structure of pole-skipping. Finally, different directions of approach can select different resonant solutions, with the selection governed by the first parameter variation of the same solvability function.