Papers

Our study connects root-mean-square ballistic transport to potential-theoretic recurrence for finite-range open quantum random walks. In homogeneous walks with primitive local channels, finitely many simple periodic Fourier peripheral eigenvalues, no nonzero stationary Fourier mode, and a nondegenerate quadratic spectral term, we prove a periodic uniform local limit theorem and recover exponential finite-set return bounds for nonzero drift. The centered case yields strong Green-function asymptotics in at least three dimensions, as well as potential-kernel asymptotics in one and two dimensions. According to these assumptions, the RMS ballistic speed equals the drift norm, nonzero drift indicates transience, and centered walks are recurring in effective dimensions one and two but transitory in higher dimensions. The low-dimensional finding shows a recurrence of the origin projection in TOM. For reducible walks, an explicit harmonic \(h\)-transform converts each absorption component to an OQRW, providing a detailed breakdown of Green occupation potentials. The squared RMS speed is the absorption-weighted mean of squared component drifts, but the reduced drift-dimension classification also needs specific component return estimations. A centered noncommuting family validates the fundamental spectral assumptions in all dimensions and provides explicit potential constants. Exact finite traps and sparse reflecting barriers provide a complementary nonhomogeneous method for zero speed and TOM recurrence.

We study nonconvex–strongly-convex bilevel optimization under a stochastic first-order oracle. We introduce MRT-FD, a single-loop first-order method that simultaneously tracks the upper-level variable, the lower-level solution, and the auxiliary response arising from implicit differentiation of the hyperobjective. MRT-FD performs one update of each variable per iteration and approximates the second-order derivative actions using order-$p$ finite differences. For any fixed finite smoothness order $p\ge1$ in the lower-level variable, MRT-FD finds an $\varepsilon$-stationary point using $\mathcal{O}(\varepsilon^{-4-2/p})$ stochastic gradient queries. We also prove a matching $\Omega(\varepsilon^{-4-2/p})$ oracle lower bound. The lower-bound construction starts from a hard nonconvex minimization chain with a stronger stochastic oracle, and lifts it to a bilevel problem through a sinusoidal coupling with a scalar lower-level variable. Consequently, the dependence on $\varepsilon$ is optimal for every fixed finite $p$, closing the upper–lower complexity gap in this stochastic first-order oracle setting.

In an earlier paper of P. Rossi and the first author, a construction of a wide class of quantum hierarchies was presented, with the input data given by an arbitrary cohomological field theory. A crucial role in the construction is played by the double ramification (DR) cycles on the moduli spaces of stable curves. As a special case, one gets a quantization of the hierarchy of the Intermediate Long Wave (ILW) equation. The ILW equation has two famous limits: the Korteweg–de Vries equation and the Benjamin–Ono (BO) equation, and for the latter one a quantum hierarchy was constructed by Nazarov and Sklyanin. In our paper, we propose a conjecture describing a relation between a limit of the quantum ILW hierarchy and the Nazarov–Sklyanin hierarchy and present various checks to support it.

We study the Lie algebra $L_{\ge2}$ of divergence-free polynomial vector fields on $k^n$, $n\ge3$, with coefficients of degree at least two, graded by coefficient degree. Over every field its derived algebra in degree $d\ge3$ is the space of exact fields, those whose contraction with the volume form is an exact form, and it is already spanned by brackets with quadratic fields. In characteristic zero this is the whole degree-$d$ component. In characteristic $p>0$ the abelianization is nonzero above degree two exactly in the degrees $d\ge(p-1)(n-1)$ with $d\equiv1-n\pmod p$, and Cartier descent identifies it with a Frobenius twist of a rational $\mathrm{GL}_n$-module, tensored with a power of the determinant. For $p\ge5$ each exact component is obtained from the previous one by bracketing with quadratic fields.

We bound the error term in the counting function of cubic extensions of $\mathbb{Q}$ by $\mathcal{O}_\epsilon\left(X^{\theta+\epsilon}\right)$, with $\theta < 2/3$, improving upon results due to Bhargava, Taniguchi, and Thorne. The proof proceeds by relating a certain discriminant-reducing sieve, due to Bhargava, Shankar, and Tsimerman, to averages of Artin characters associated with the Artin $L$-function $\zeta_K(s)/\zeta(s)$, with $K$ a cubic field. Conditional on the Generalised Riemann Hypothesis (GRH) for Dedekind zeta functions $\zeta_K(s)$, we obtain further savings and show that the error term is $\mathcal{O}\left(X^{2/3-1/51+\epsilon}\right)$. Moreover, for the analogous smooth counting problem, we obtain a bound $\mathcal{O}(X^{1/2+\epsilon})$ for the error term, conditionally improving results of Shankar, Södergren, and Templier. Our arguments can also handle finitely many splitting conditions. The methods we use to study cubic fields are natural extensions of methods which are used to study the one-level density associated with the above family of Artin $L$-functions. In this direction, we improve the admissible support $[-\sigma,\sigma]$, from the previously known admissible value $\sigma < 2/5$, to $\sigma < 1$, conditional on the GRH. This also proves that at least $75\%$ of these $L$-functions are non-vanishing at the central point, conditionally improving upon results of Shankar, Södergren and Templier. Finally, one may study similar questions over a rational function field $\mathbb{F}_q(T)$, with $q$ coprime to $2$ and $3$. Here, we obtain a proportion of non-vanishing of at least $75\%$ for the analogous family of $L$-functions. Furthermore, we obtain a bound $\mathcal{O}(X^{1/2+\epsilon})$ for the error term in the counting function of cubic function fields, improving previous results of the author. Over function fields, our results are unconditional.

We compute the rational cohomology of the fine compactified universal Jacobians over the moduli spaces of stable curves of genus $2$ with $0$ and $1$ marked points.

Language model (LM) alignment broadly aims to perturb a given LM $Q$ into an aligned LM $q$ such that i) the outputs produced by $q$ and $Q$ are 'close' in probability, ii) $q$ has a higher expected reward than $Q$. Two common techniques for LM alignment are: KL-constrained RL, which requires knowledge of the LM distribution and is computationally expensive, and the best-of-$n$ algorithm, which requires only sampling from the LM. The work of Yang et al. established asymptotic closeness between the distributions produced by the two alignment methods for an $m$–length i.i.d. token sequence output by the LM, in the limit as $m$ increases to infinity. However, the i.i.d. assumption is not representative of practical LMs, whose output sequences often have memory. In this paper, we extend the asymptotic closeness result to the case when the $m$–length token sequence outputted by the LM is Markovian. Further, for finite-length output sequences – particularly, when $m=1$ – we provide a complete characterization of LM distributions and reward functions for which the KL-divergence between the distributions produced by the two alignment methods is zero – a question first posed in Yang et al.

When the system model is not fully known, measurement error and limited excitation can leave several models consistent with the same finite data. A command judged safe for one model may fail for another, while limited control authority can prevent the corrective action needed to preserve safety. To ensure safety under model uncertainty and asymmetric input limits, we develop a finite-data certificate that determines whether a command can enforce a prescribed safety inequality. For a linearly parameterized safety channel with exactly known regressors and bounded aggregate residual error, we derive a support formula for the worst-case safety contribution of all data-consistent models. The formula identifies the regressor directions that admit a finite bound, allowing rank-deficient records to contribute to safety certification. Using certified componentwise bounds on actuator tracking error yields an affine inequality with a necessary and sufficient test for pointwise command feasibility. The affine inequality reduces computation of the closest certified command to a scalar root-finding problem. It also yields a closed-form gate that selects the largest certified fraction of a prescribed command segment. The proposed certificate guarantees output safety within its operating domain, provided the feedback is locally Lipschitz and the uncertainty bounds remain valid. Domain retention and full-state continuation extend this guarantee to all time. A vehicle study demonstrates that output safety can be certified from finite measurements in a safety-critical setting with model and actuator uncertainty.

The predictively oriented posterior offers principled uncertainty quantification, even under model misspecification. However, it does not admit an explicit density and therefore cannot be computed using classical Monte Carlo sampling algorithms. We remedy this by deriving an approximation to the predictively oriented posterior whose density can be evaluated point-wise, and whose approximation error decays rapidly. These results are illustrated on case studies from epidemiology, spatial statistics, and low-energy nuclear physics.

Signomials—finite sums of generalized monomials with real exponents—are a natural machine learning model, combining parsimonious representations of power laws, inverse relationships, and multiplicative interactions with universal approximation and interpretable parameters. In fact, 45 of the 100 equations in the AI Feynman benchmark admit signomial representations. Moreover, signomial optimization is widely used in engineering design, communications, economics, and machine learning, but is computationally intractable in general. Convexity, the gold standard for efficient optimization, offers three routes to tractability for signomials: a signomial may be convex in its original variables, become convex after a logarithmic change of variables, or, when positive, become convex after additionally taking the logarithm of its value—the structure underlying disciplined geometric programming. We show that recognizing each form is strongly NP-hard, both on compact domains and globally. Our proofs start from gap-promise variants of polynomial convexity and develop polynomial-to-signomial reductions that preserve curvature gaps under the relevant logarithmic transformations.

We study continuous-time Galton–Watson trees whose offspring generating function takes the form \begin{align} \label{eq:gen} f(s) = s - (1-s)^\alpha L(1-s), \end{align} where $\alpha \in (0,1)$ and where $L:[0,1] \to [0,\infty)$ is slowly varying at zero. Processes with this offspring generating function explode in finite time. We observe that, conditional on explosion at time $T$, at each earlier time $t < T$ there is a unique particle $\xi_t$ alive at time $t$ who is an ancestor of all but finitely many particles at the explosion time. We call $(\xi_t)_{t \in [0,T)}$ the spine to explosion, and show that the births off the spine admit a Poissonian description, where they become both more frequent and larger as $t \uparrow T$. We undertake a careful study of the size of the population leading up to explosion, showing in particular that near explosion, the rescaled population is approximately gamma distributed. More generally, conditional on explosion at time $T$, define a stochastic process $Z^\varepsilon := (Z_t^\varepsilon)_{t \in \mathbb{R}}$ by setting \begin{align*} Z_t^\varepsilon := E(\varepsilon e^{-t})N_{T-\varepsilon e^{-t}}, \qquad E(t) = \mathbf{P}(N_t = \infty). \end{align*} We show that as $\varepsilon \downarrow 0$, $Z^\varepsilon$ converges in finite-dimensional distributions to a stationary continuous-state branching process with immigration whose branching mechanism is subcritical and whose immigration mechanism corresponds to spine events. Finally, we show that this limiting process has a Markovian time reversal, and that the coalescent process associated with it is a stochastically time-changed Beta$(2-\alpha,\alpha)$-coalescent.

We develop a variational framework for state-dependent diffusion on adaptive multiresolution representations in which the diffusion operator is generated by the adaptive state itself. The state consists of an admissible multiresolution tree, its active approximation space and basis, and the corresponding coefficient representation. It determines a symmetric nonnegative interaction form and an associated positive semidefinite intrinsic diffusion operator. In contrast to classical adaptive wavelet methods, where a prescribed operator is represented on an evolving approximation space, refinement and coarsening here modify simultaneously the representation, interaction graph, and operator. Because the adaptive hierarchy evolves through discrete topological changes, the coupled dynamics are formulated through a time-discrete variational principle rather than a differential evolution on a fixed space. We establish existence of the discrete updates, a discrete energy inequality, the coefficient-space null mode, and contractivity for frozen adaptive states. For regularized inverse problems, the construction yields Adaptive Multiresolution Diffusion Imaging (AMDI), combining data fidelity, intrinsic diffusion, coefficient sparsity, and tree complexity in a state-dependent energy. Numerical experiments verify the assembled operator identities, examine refinement-commutator decay, and confirm discrete energy dissipation. Adaptive Haar and higher-order multiwavelet calculations demonstrate localization of resolution on heterogeneous data. In denoising, AMDI retains high structural reconstruction quality with less than 10% of the full active representation, with stable behavior across held-out noise realizations.

We show that all closed geodesics in any arithmetic complex hyperbolic surface of second type are simple. These include the fake projective planes studied by Mumford, Klingler, Prasad-Yeung, and Cartwright-Steger.

Jurij Volčič conjectured that a noncommutative polynomial $g$ belongs to the two-sided ideal generated by $f_1,\ldots,f_l$ if and only if, for matrices of every size, the rank of the evaluation of $g$ is bounded by a constant times the maximum rank of the evaluations of $f_1,\ldots,f_l$. In this paper, we show this equivalence when the generators $f_1,\ldots,f_l$ are homogeneous. We then give an explicit counterexample in the non-homogeneous case.

Consider the circular beta-ensembles, Jiang and Matsumoto computed the moments of traces, and proved central limit theorems for polynomial linear statistics using Jack functions. We give a direct proof of the Gaussian limits for the traces of powers of circular beta-ensembles using the theory of orthogonal polynomials on the unit circle and Stein's method. The proof relies on an explicit coupling of the Verblunsky coefficients of circular beta-ensembles, which also allow us to obtain almost sure and $L^2$ convergence.

These are the slightly edited notes from a 10-hour course I taught at Institut Henri Poincaré in January-February 2025 in the framework of the thematic semester "Representation theory and noncommutative geometry".

We establish that, for a binary operad Var, the Hadamard product of Var and the operad Nov of Novikov algebras is a binary operad (i.e., coincides with their Manin white product Nov $\circ$ Var) if and only if the quadratic part of Var governs a 2-variety. The condition Nov $\circ$ Var $=$ Nov $\otimes$ Var is known to be equivalent to the existence of an easily verifiable characterization of those differential Var-polynomials which lie in the free derived Var-algebra. For a binary quadratic operad, these conditions are also equivalent to the Dong property of formal distributions. We distinguish the white product defined as a generated sub-operad from its quadratic approximation and give an example in which the canonical epimorphism between them is not injective.

A dominating set $D$ of a graph is called $1$-limited if every vertex of $D$ has at most one neighbor outside $D$, while a $(1,2)$-dominating set is a dominating set in which every vertex of the set has at least two neighbors within the set. These two notions coincide on cubic graphs. We prove that the decision problem 1-Limited Dominating Set is $\mathsf{NP}$-complete even when restricted to $2$-connected planar cubic graphs, thereby completing the known complexity results for $k$-Limited Dominating Set for all fixed positive integers $k$. We also determine the exact $1$-limited domination number of the entire Goldberg family. This provides a further infinite family of cubic graphs supporting several open conjectures and proposed bounds concerning $(1,2)$-domination and induced cycles.

Let $p$ be an odd prime, let $m\geq1$ be an integer, and let $Q$ be a nondegenerate quadratic form on $\mathbb F_p^{2m}$. For a nonempty set $E\subseteq\mathbb F_p^{2m}$, write $\Delta_Q(E)=\{Q(x-y):x,y\in E\}$. We prove that, whenever $|E|>p^m$, \[ |\Delta_Q(E)|\gg \min\left\{\frac{p}{L_E},\, \frac{|E|}{p^{m-1}L_E^2}\right\}, \qquad L_E=\log\left(2+\frac{p^{m+1}}{|E|}\right), \] with an absolute implied constant independent of $p$, $m$ and $Q$. This confirms the Erdős–Falconer distance conjecture up to a logarithmic factor. In the anisotropic planar case $m=1$, for small sets, we also prove that \[ |\Delta_Q(E)|\gg\frac{|E|}{\log(2|E|)}, \qquad (1\leq |E|\leq p), \] which is also optimal up to a logarithmic factor.

We prove that if the product of two class transpositions has finite order, then its order divides 840. This gives a complete affirmative answer to Question 18.48 in the Kourovka Notebook. We also obtain a criterion for the product of two class transpositions to have infinite order. In addition, we construct a pair of class transpositions whose product has order 840, showing that the bound is sharp.

Quantum key distribution (QKD) links are provisioned from security analyses of stationary channels, whereas the devices that determine the channel drift between recalibrations. Whether an eavesdropper who cannot alter the channel's own noise gains by following that drift has not been quantified. Adaptive eavesdropping is posed here as a constrained Markov decision process in which the attacker selects one circuit per round while the noise level follows an Ornstein–Uhlenbeck process and the abort condition is a budget over each block of rounds. The value of adaptation is bounded by the best fixed circuit and a dynamic-programming upper bound. The actions are learnt attacks. Whereas Decker et al. trained a parametrised circuit on a fixed gate template against a fixed channel, here the gate structure and rotation angles are searched jointly. This yields circuits compact enough to form a discrete action set, extending the construction to noise models lacking a known template, including the amplitude damping channel. On device-independent E91 under bilateral depolarising noise, a reinforcement-learning attacker raises her Holevo information from $0.135$ for the best fixed circuit to $0.348$ at zero detection, $98\%$ of the upper bound. On BB84 under a drifting bit-flip channel, she exceeds a conservative noise-indexed rule by $0.024$ in fidelity, reaching $99\%$ of the upper bound. Under stationary noise, the attacker's gain from basis asymmetry changes sign between an averaged and a per-basis error-rate constraint. The search, started from random gate sequences, recovers the analytical cloners and the collective-attack key rate, and meets the lower bound of the Winick–Lütkenhaus–Coles objective from above.

Laface and Ugaglia recently constructed an irrational one-point Seshadri constant on the blow-up of $\mathbb{P}^2$ at nine very general points by combining a dihedral orbit on $\mathbb{P}^1\times\mathbb{P}^1$, a sequence of de Jonquières transformations, and a reflection argument along a $(-4)$-curve with balanced normal bundle. We show that the same mechanism extends uniformly to every odd integer $n\geq 5$. For $n=2k+1$ we prove $$ \varepsilon\bigl(\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(n-4,1);p_1,\ldots,p_{2n}\bigr)=\sqrt{\frac{n-4}{n}} $$ at $2n$ very general points, and already at a very general free orbit of a fixed dihedral group of order $2n$. For every odd $n\geq 7$ this produces an explicit ample line bundle on the blow-up of $\mathbb{P}^2$ at $k+7=(n+13)/2$ very general points whose one-point Seshadri constant at a very general point equals $$ 2\sqrt{n(n-4)}. $$ More precisely, after one quadratic transformation we obtain the ample divisor $$ L_n=(3n-4)H-nE_1-(n-2)(E_2+\cdots+E_5)-4(E_6+\cdots+E_{k+5})-2(E_{k+6}+E_{k+7}), $$ with $L_n^2=4n(n-4)$ and $\varepsilon(L_n;x)=\sqrt{L_n^2}$. We also isolate an abstract balanced-reflection principle underlying the construction: a nef class on a special fiber can be reflected across a rational curve of square $-2a$ whenever the curve has normal bundle $\mathcal{O}_{\mathbb{P}^1}(-a)^{\oplus 2}$ in the total space, and the reflected class is nef on very general fibers.

We prove non-uniqueness for the advection-diffusion equation with a drift in a logarithmically near-critical class below $L_t^2L_x^\infty$. The first construction is a diffusion-assisted inverse cascade of scalar energy that diverges as $t \to 0$. The second uses compressed inverse mixing, with diffusion acting perturbatively, and produces a bounded parabolic solution with the energy jump at the initial time. In both constructions the drift and scalar are smooth for positive time. We also consider the drift in $L^{2,\infty}_tL^\infty_x$ and solutions satisfying the energy inequality starting from almost every time. In this setting, we prove uniqueness of solutions with bounded energy, and construct non-unique solution with energy blowing up as time goes to zero.

We show that not every symplectic Lefschetz fibration over the $2$–sphere admits a smooth section, settling a long-standing open problem. To prove this, we develop a method for studying sections via lifts of monodromy factorizations with point-pushing maps and the induced handle decomposition of the total space, and apply it to an infinite family of Lefschetz fibrations. In contrast, we show that every Lefschetz fibration over $S^2$ admits positive multisections. Some of these examples admit no multisections with spherical components. We also construct signature-zero Lefschetz fibrations of every odd genus $g\geq 7$, which in turn give symplectic Lefschetz fibrations of any prescribed signature in each of these genera.

Let $\mathfrak{g}$ be a basic classical Lie superalgebra and let $\widehat{\mathfrak{g}}$ be the corresponding affine Lie superalgebra. In this paper, we first prove that a Cartan subalgebra acts semisimply on ordinary modules for the simple affine vertex operator superalgebra $L_{\widehat{\mathfrak{g}}}(k,0)$ at boundary admissible level $k$. Then we prove that the category $\mathcal{O}_{k}^{ord}(\mathfrak{g})$ of ordinary $L_{\widehat{\mathfrak{g}}}(k,0)$-modules is finite, semisimple and $\mathcal{O}_{k}^{ord}(\mathfrak{g})$ is exactly the category $KL_k(\mathfrak{g})$ of finite-length generalized modules for the affine vertex operator superalgebra $L_{\widehat{\mathfrak{g}}}(k,0)$.Thus $\mathcal{O}_{k}^{ord}(\mathfrak{g})$ is a braided tensor supercategory. Furthermore, we obtain the rigidity of the supercategory $\mathcal{O}_{k}^{ord}(\mathfrak{g})$ and thus it is a ribbon supercategory. Finally, we conclude that $\mathcal{O}_{k}^{ord}(\mathfrak{g})$ is a ribbon fusion supercategory.

We study three recurrent double sequences whose colored diagrams encode mirrored versions of the finite-ordinal construction. Two sequences use three letters and realize variants with two and three initial symbols. A four-letter sequence realizes the von Neumann construction from one initial symbol. For each sequence we describe its cells explicitly and identify the ordered nesting of its rectangles. For four letters, we distinguish ordinary and virtual boxes and prove their assembly rules. The proofs proceed by induction along antidiagonals. For the constructions with one and three initial symbols, we reduce the recurrence to finite sets of local configurations and verify those sets by computer. We derive formulas for the numbers of nested boxes, including arithmetic-term representations, and we prove that none of the three sequences is $k$-automatic for any integer $k \geq 2$.

In this paper, we study the number of points of $\mathbb{Z}$-schemes $X$ of finite type over the finite rings $\mathbb{Z}/p^m\mathbb{Z}$ for positive integers $m$ and prime numbers $p$. Motivated by Igusa's conjecture for exponential sums and the Lang-Weil estimate, we establish an estimate of $X(\mathbb{Z}/p^m\mathbb{Z})$ in terms of $m$, $X(\mathbb{Z}/p^{m-1}\mathbb{Z})$ and poles of the $p$-adic Igusa local zeta function of $X$ uniformly in $m\geq 2$ and $p$. Our estimate might be viewed as an inductive version for counting points of schemes over finite rings of the Lang-Weil estimate. Our estimate agrees with the expected bound of Igusa's conjecture for exponential sums if $X_{\mathbb{Q}}$ is not a locally complete intersection having only rational singularities. This can be regarded as an analogous version of the estimate of exponential sums modulo powers of primes of polynomials with non-rational singularities given by Cluckers, Mustaţǎ and the author in 2019. In the case that $X_{\mathbb{Q}}$ is a locally complete intersection having only rational singularities, our estimate is very close to the expected bound of Igusa's conjecture for exponential sums. To prove our estimate, beside using the usual tools such as resolutions of singularities and Denef's formula for Igusa local zeta functions, we also use the Hermit interpolation formula to study the numerators of Igusa local zeta functions. By the same method, we are able to approach Igusa's conjecture for exponential sums in the case of rational singularities. Lastly, to have further applications of our result in the future, we study heavily on $p$-adic analysis to give an upper bound of the real parts of non-trivial poles of Igusa local zeta functions in terms of suitable generalized log canonical thresholds.

Two integer partitions $t$-intersect if they have at least $t$ common parts, counted with multiplicity. We study the largest $t$-intersecting families of integer partitions of $n$ into exactly $k$ positive parts. The canonical $t$-star consists of the partitions containing at least $t$ ones. Applying Kupavskii's weak-spread theorem, we prove that this star is largest whenever $n\ge Ak^3$, for every fixed $A>24$ and all sufficiently large $k$, uniformly over $1\le t<k$. We also give counterexamples to Borg's conjecture at every intersection scale: for all sufficiently large $k$ and for every $1\le d<k$, one may choose $d/4\le t\le d$ and $n=\lfloor tk^2/3\rfloor$ so that a $t$-intersecting family is strictly larger than the canonical star.

We show, under a certain assumption, that the Cheeger constant of a connected Cayley graphon is bounded from above in terms of its bipartiteness ratio, and also in terms of the gap between the top of the spectrum of its Laplacian and $2$. Using this result, we prove that if a connected Cayley graphon $C(G,\gamma)$ is bipartite, then there exists an index two subgroup $H$ of $G$ that avoids the support of $\gamma$.

We develop Toeplitz infinite GMRES for solving large sparse analytic parameterized systems $A(s)x(s)=z$ at many parameter values. The method exploits a block upper triangular Toeplitz structure in the companion Krylov sequence to construct the Arnoldi process and recover solution approximations without storing full Arnoldi vectors. We derive incremental recurrences requiring $\mathcal{O}(np^2+p^3)$ arithmetic and $\mathcal{O}(np+p^2)$ storage for $p$ Arnoldi steps, excluding factorization setup and assuming linear-cost coefficient actions and triangular solves. A dynamic generator-refreshing strategy addresses cancellation in the basic recurrence. For matrix functions admitting a fixed separated representation, we further develop an implicitly rebased method based on a compact, contractive nilpotent matrix. Both variants preserve the Arnoldi process in exact arithmetic, with exact matrix-function actions required for the rebased method. A residual lower bound for square-summable Taylor coefficients explains a scaling obstruction outside the normalized unit disk. Theoretical analysis and numerical experiments illustrate the computational efficiency of the proposed methods.