Papers

We show that the best constant in the weak-type $(1,1)$ bound for the vector Riesz transform on $\mathbb{R}^n$ is at most $2$, independent of the dimension $n$. The proof relies on a new decomposition of the input data involving an obstacle problem for the fractional Laplacian and an associated Lewy-Stampacchia type estimate on an unbounded domain. This settles a problem posed by E. M. Stein at the 1986 International Congress of Mathematicians.

We resolve the admissibility problem for scattering transforms: every Parseval filter bank on $\mathbb{R}^d$ whose low-pass filter is nonvanishing at the origin induces a norm-preserving scattering transform, without requiring any analyticity, frequency-localization, or geometric covering assumptions. Furthermore, for any Sobolev input $f\in H^s(\mathbb{R}^d)$, we establish a universal decay bound of $\mathcal{O}(N^{-\min\{s,1\}/d})$ on the depth-$N$ residual norm. Finally, we construct a filter bank of band-limited Schwartz functions and a band-limited Schwartz input for which the low-pass multiplier equals one near the origin but the propagated energy decays subexponentially. This demonstrates that no universal exponential rate can hold under these assumptions alone.

We give a two-player zero-sum repeated game between a learner and nature whose value identity generates Bayesian updating and an exact accounting of exponential-weights regret at once, and supplies the comparator-class variational form that a wide class of concentration phenomena share. The terminal payoff is the most a comparator can gain at fixed relative entropy from the prior, and the one-step constraint is an information budget on nature's move under the learner's mixed action. With the learner's move otherwise unrestricted, Gibbs/Bayes weights emerge as its unique Bellman equalizer – the mixed action that makes the per-round loss independent of which direction nature moves – with log-partition functions playing the role of value functions. The regret decomposes exactly into three parts: a per-round information loss reflecting the variation in observed outcomes, an additive retempering drift that accounts exactly for any change of measurement scale between rounds, and the information the comparator carries relative to the prior. The variance and bounded-range proxies that drive standard regret bounds are looser relaxations of this decomposition, which holds generally and governs them all. Both players' strategies are read off from the decomposition term by term, and repeated play yields an information-theoretic ledger of self-play in place of the usual quadratic-variation surrogate. The same comparator-class geometry accounts for the classical large-deviation bounds, and methods across bandits, posterior sampling, aggregation, and boosting are specializations of the one regret decomposition.

We study how fast experimental designs can approach the semiparametric efficiency bound in finite samples, as measured by the excess variance of unadjusted treatment effect estimation. We prove an impossibility theorem: under weak conditions, no design can approach the variance bound uniformly over smooth outcome models unless covariate dimension $d \ll \log n$. Even in experiments with thousands of units, this permits only a handful of covariates. Motivated by this, we propose new designs based on discrepancy minimization that instead attempt to control imbalances over restricted-complexity nonparametric function classes. Such designs achieve fast rates to their corresponding restricted efficiency targets, permitting $d \ll n$ covariates in an additive nonparametric specification. They can also be combined with matching to protect against unmodeled outcome variation. In simulations calibrated to 12 published experiments, our designs reduce variance relative to matched pairs randomization in every empirical setting.

We disprove Sato's weak \(F\)-equivalence conjecture for nonsingular projective toric weak Fano varieties in every dimension \(d \geq 3\). Our counterexamples are smooth projective crepant models of centered reflexive simplices. The key input is a rigidity property of ray polytopes: if \(X_\Sigma\) is nonsingular and complete and \(-K_{X_\Sigma}\) is nef, then every nonzero lattice point of \(P_\Sigma=\operatorname{Conv}(G(\Sigma))\) is a primitive ray generator. For our models, this rules out every weak-Fano-preserving equivariant blow-up and blow-down throughout the flop class. We then introduce Gorenstein weak \(F\)-equivalence, generated by projective toric birational zigzags through normal projective Gorenstein toric weak Fano varieties, and formulate a corresponding refinement of Sato's conjecture. We prove this refined conjecture in dimensions \(d \leq 3\), as well as for the family of counterexamples constructed above in every dimension. Finally, we show that the refined conjecture implies the inclusion-connectivity of reflexive \(d\)-polytopes modulo unimodular equivalence, which is known for \(d \leq 4\) and remains open for \(d \geq 5\). The results were developed with the assistance of GPT-5.6 Sol.

Complex Hadamard matrices encode perfectly balanced unitary transformations. Their classification is complete through order five, but order six – the first dimension in which several continuous families coexist with an isolated solution – has remained open for decades. Here, we give a complete and exact finite-incidence classification of order-six complex Hadamard matrices up to standard equivalence. We supply the global step missing from Szöllősi's dilation method, which allows us to prove an even stronger version of his conjecture: every complex Hadamard matrix of order six can be recovered algebraically from a suitable, dephased $3 \times 3$ corner defined by four initial phases. We then describe the geometry of the reconstruction from these phases and show that, except for Tao's isolated matrix and a single explicit Karlsson matrix, every class admits a representative obtained by solving one quadratic and one cubic equation in both the horizontal and vertical directions. Our work resolves the classification problem and provides a rigorous framework for further investigating order-six Hadamards, with applications to balanced six-mode interferometers and the study of mutually unbiased bases.

District heating networks are a central tool to achieve low-carbon heat supplies. In this realm, they face the challenge of dealing with increasingly heterogeneous, partially time-varying renewable sources, thermal storage, and meshed topologies. This paper examines global optimization of the operation of such district heating networks over multiple time steps, based on a stationary, yet realistical nonlinear network model. To accelerate the solution performance of a spatial branch-and-bound algorithm for one time step, the following new methodological ingredients are introduced: exclusion of cyclic flow, symmetry exploitation between supply and return networks, reduction of temperature mixing constraints, novel primal heuristics and branching rules. The proposed methods are evaluated on a set of generated and real-world benchmark network instances with cycles and several suppliers. On the generated benchmark instances, using these methods more than doubles the number of solved instances and more than halves the runtime. For the real-world benchmark instances, the resulting algorithm produces solutions with guaranteed quality in reasonable run time. For multiple time steps that are coupled by a storage, a time decomposition approach is investigated. Under assumptions that are reasonable in practice, this approach is shown to yield an optimal solution. On a small example network, this decomposition is able to compute optimal solutions in less than a second, while solving the complete time-coupled problem is not possible within one hour.

We study the supercritical regime $\beta>\beta_c$ of a large family of real-valued spin models on $\mathbb Z^d$, including the Blume-Capel and $P(\varphi)$ models. We consider their random cluster representations and prove that they are well-behaved, in the sense that local uniqueness of macroscopic clusters occurs with high probability, uniformly in the boundary conditions. This implies, among other things, a surface-order exponential bound for the (lower) large deviations of the empirical magnetisation. These results were previously known only in the cases of the Ising and $\varphi^4$ models.

Recent publications have started to explore the application of Geometric Algebra (GA) to the modeling, analysis and control of dynamical systems and, in particular, electrical circuits. Since a crucial element there is to transform the ordinary differential equations governing the dynamical system which models the systems' behavior from the real domain to the Laplace domain, a definition of the Laplace transform in GA is needed. In the present work, we extend previous works dealing with extension to some hiper-complex algebras by introducing a definition of the Laplace transform within the framework of Geometric Algebra (GA). In particular, our definition and its properties are applicable to geometric algebras with signature lower or equal than 5.

We study the convergence rates in periodic homogenization of general nonconvex, coercive Hamilton–Jacobi equations. We show that the optimal convergence rate is $O(\varepsilon^{1/2})$ in one dimension, $O(\varepsilon^{1/3})$ in two dimensions (up to a logarithmic factor), and $O(\varepsilon^{1/3})$ in dimension three or higher.

We study subcategories of the module category of a finite-dimensional algebra that are closed under quotients or submodules. We propose the quotient–submodule equidistribution conjecture: over a representation-finite algebra, the number of quotient-closed subcategories of size $i$ is equal to that of submodule-closed subcategories of size $i$ for every $i$, where size is the number of indecomposable modules in the subcategory. We prove the following cases of the conjecture: (1) the five smallest and five largest values of $i$, hence all algebras with at most nine indecomposable modules; (2) Nakayama algebras; (3) algebras with radical square zero; and (4) representation-directed algebras. In the last case, we classify quotient-closed subcategories by a lower Bruhat interval in a Coxeter group constructed from the Auslander–Reiten quiver, extending the classification of Oppermann–Reiten–Thomas for Dynkin quivers. We also prove that quotient closure defines a finitary convex geometry and classify the functorially finite quotient-closed subcategories by Gen-minimal modules.

We introduce Ordered Diffusion Kernels (ODKs), a novel class of local kernels that can approximate the infinitesimal generator of an arbitrary Itô Stochastic Differential Equation (SDE). ODKs are designed to be applied to data sampled from dynamical systems where little dynamical information is available a priori. The Laplacian of classical diffusion kernels approximates the Laplace-Beltrami operator on the underlying manifold; adjusting the normalisation introduces an advection term that depends on the sampling density; recently, TMDmap generalised this normalisation to target an arbitrary measure, but at the cost of coupling advection to diffusion. More general local kernels can learn arbitrary second-order elliptic operators but are formulated in terms of known velocity fields — making the first step of any analysis a potentially ill-posed inference problem. To formulate ODK, we first relax the problem of potential estimation to the more tractable task of inferring an ordering of the data, which we represent through an ordering function. We prove ODK's Laplacian converges to the infinitesimal generator of a gradient-flow SDE with state-dependent isotropic diffusion, without coupling advection and diffusion. We provide various extensions of ODK to: arbitrary drifts via local ordering functions; anisotropic diffusions via a Strang splitting scheme; multiple ordering functions; and self-tuning bandwidths. In addition, we introduce two loss functions which exploit the structure of ODKs to solve a non-parametric inference problem. We validate this framework on synthetic data from deterministic and stochastic systems, demonstrating accurate recovery of operators, velocity fields, extrinsic curvature, and spatially dependent drift and diffusion coefficients.

For any finite group, we pose the following question: given an irreducible real representation, is the associated representation sphere non-orientable if and only if the representation is nontrivial of real type? We prove that the answer to this question is yes if the group has a normal Sylow 2-subgroup, but exhibit a 2-nilpotent group of order 112 for which the answer is no via elementary arguments. We also link the question to the unit group of the Burnside ring, where we recover a basis discovered by Bouc for 2-groups, and pose a conjecture about detection of non-orientability from solvable subgroups. The question is motivated by issues arising from the construction of permutation twisted cohomology for finite groups.

Let $E/\mathbb Q$ be an elliptic curve and for each prime $p$, let $N_p$ denote the number of points of $E$ modulo $p$. The original version of the conjecture of Birch and Swinnerton-Dyer asserts that $\prod \limits _{p \leq x} \frac{N_p}{p} \sim C (\log x) ^{\text{rank}(E(\mathbb Q))}$ as $x \to \infty$. In this paper, we formulate a similar conjectural asymptotic for smooth projective curves of genus at least 2, in which the contributions to the conjectured asymptotic come not only from the rank of the Jacobian but also from the Sato–Tate group of the curve. The key analytic input in formulating our conjecture is a conjecture due to Kurokawa (2012) on the convergence of Euler products of entire $L$-functions on the critical line. We also provide some numerical evidence for our conjecture in various cases.

We show that, at a fake exponent of an $A$-hypergeometric system and for an ordered negative support family, intrinsic perturbation within $\ker_{\mathbb{Z}}(A)$ can produce a strictly smaller coefficient space than ambient perturbation, which answers a question of Okuyama–Saito. We measure the failure by an intrinsic-perturbation obstruction module, a quotient of two colon ideals whose graded dual is the ambient coefficient space modulo the intrinsic one, and we realize this module by two right-exact sequences involving $\operatorname{Tor}_1$ and present it finitely by two antichains. We give configurations of lattice rank 1 and 2 with nonzero obstruction, including one with an ordered distinguished collection consisting of all negative supports attained only by lattice shifts of nonnegative weight, and thereby settle the case left open by Okuyama–Saito. We give, for every integer $q\geq 1$, a configuration of lattice rank 2 with a connected column matroid and with a normal affine semigroup generated by the reduced Gröbner basis vectors, whose obstruction module has dimension $q^2$, so this dimension is unbounded at fixed lattice rank.

Toric varieties can be constructed from rational polytopes, and several invariants of toric varieties can be expressed in terms of the combinatorics of the corresponding polytope. Barthel-Brasselet-Fieseler-Kaup (BBFK) introduced combinatorial intersection cohomology for convex polytopes, which agrees with the intersection cohomology of the associated toric variety when the polytope is rational. Maxim-Schuermann computed the intersection cohomology signature of a projective toric variety, corresponding to the case of a polytope with rational vertices. Using the combinatorial framework of BBFK, we show that the Maxim-Schuermann formula extends to arbitrary convex polytopes. Finally, we discuss a version of the Hodge index theorem for polytopes.

For $m=2,3$, we prove that every smooth immersion $F:\mathbb{R}\mathbb{P}^m\looparrowright\overline{\mathbb B}^{,N}(1)$ satisfies $\kappa(F)^2\ge 2m/(m+1)$, with equality only for the Veronese embedding, up to congruence. We also prove that a closed, connected, orientable three-manifold admitting an immersion into a Euclidean unit ball with $\kappa(F)\le\sqrt{3/2}$ is diffeomorphic to $S^3$, $\mathbb{R}\mathbb{P}^3$, or $S^2\times S^1$. All three possibilities occur, while $\kappa(F)<\sqrt{3/2}$ forces $X\cong S^3$. These results answer a question of Petrunin and prove a conjecture of Chodosh–Li concerning the normal curvature of three-manifolds. The key intrinsic input is the strict scalar–systolic inequality \[ (\min_Y R_g)\text{sys}(g)^2<6\pi^2 \] for every spherical three-space form $Y$ with $|\pi_1(Y)|>2$. Its proof uses systolic monotonicity along Ricci flow with surgery. This strict inequality complements the sharp scalar–systolic inequality for $\mathbb{R}\mathbb{P}^3$ of Bray–Brendle–Eichmair–Neves.

Transport-dominated problems remain challenging for data-driven methods, which often exhibit severe numerical oscillations near shocks or steep gradients due to globally supported basis functions or overly smooth hypothesis spaces. Gegenbauer reconstruction has shown promise in mitigating such oscillations, but its effectiveness critically depends on the reconstruction parameters, particularly the weight parameter $\lambda$ and truncation order $m$. For data-driven models, variations in governing problems, training data, and model architectures make systematic parameter selection particularly challenging. To address this issue, we propose a physics-informed machine-learning framework that predicts probability distributions over candidate Gegenbauer parameter pairs, enabling probabilistically weighted reconstruction while accounting for parameter uncertainty. A two-stage strategy is adopted, in which a general predictor is first pre-trained and then fine-tuned for target problems to balance accuracy and computational cost. The framework is evaluated for reduced-order and neural operator models, represented by POD-Galerkin and DeepONet, respectively. Numerical experiments on one- and two-dimensional transport-dominated problems show that the framework learns effective spatially adaptive parameter distributions. Compared with conventional reconstruction strategies, it reduces numerical errors by up to one to two orders of magnitude and achieves a more favorable accuracy–cost trade-off than problem-specific model retraining.

We study pseudoreflections of geometric origin on Prym varieties of étale double covers. We prove that if the genus of the base curve is $g \geq 4$ then every such pseudoreflection has order 2. We use this result to show that, for $g \geq 5$, a non-trivial finite group $G$ of automorphisms of geometric origin acting faithfully on the Prym $P$ with $P/G$ smooth must be isomorphic to either $\mathbb{Z}/2\mathbb{Z}$ or $(\mathbb{Z}/2\mathbb{Z})^2$. We also show that the latter case can occur only for $g \leq 7$. This sharpens results of Auffarth, Lahoz and Naranjo.

We study $n$-party fault-tolerant consensus against general (non-threshold) adversaries. We describe an infinite family $Z_\mathsf{proj}^{n,d}$ of $Q^d$-satisfying $n$-party adversary structures based on finite projective geometry which cause error-free $R$-round protocols for interactive consistency on $L$-bit inputs to require $\Omega(Ln^{2+1/d})$ bits of expected communication. Likewise, $Z_\mathsf{proj}^{n,d}$ causes error-free byzantine agreement and broadcast to cost $\Omega(Ln^{1+1/d})$ bits. In every case, the lower bound is $\Omega(L_{\mathsf{out}} \cdot n^{1+1/d})$ bits, where $L_{\mathsf{out}}$ is the output length. The family $Z_\mathsf{proj}^{n,d}$ also causes reliable broadcast and byzantine agreement to cost $\Omega(Ln^{1+1/d})$ bits of expected communication in asynchronous networks. Moreover, there exists a related family $Z_\mathsf{2\textsf-proj}^{n,d}$ of $Q^d$-satisfying adversary structures that make core set agreement cost $\Omega(Ln^{2+1/d})$ bits. These asynchronous lower bounds hold against send-omission adversaries, even if the protocol uses cryptography. Their basis is that if a quorum of non-faulty parties agree on an output and terminate, then the messages they sent before terminating must suffice for the parties outside the quorum to also terminate with the same output. Surprisingly, if we do not require the parties to terminate (stop sending messages) after they output, then these bounds no longer hold. We show this by designing a non-terminating omission-tolerant reliable broadcast protocol that can for any parameter $\delta > 1$ be tuned to cost $(1 + \frac{1}{\delta - 1})Ln + O(\delta n^2\log(\delta n))$ bits, which is of independent interest. Lastly, we show how to get termination with $O(Ln^{1+1/d} + n^2\log n)$ bits (assuming the $Q^d$ condition), and thus prove our asynchronous lower bounds tight.

Lehel's conjecture states that every 2-edge-colouring of K_n admits a partition of its vertex set into two monochromatic cycles. It was proven for sufficiently large n by Łuczak, Rödl, and Szemerédi in 1998, later improved by Allen in 2008, and fully resolved by Bessy and Thomassé in 2010. In this paper, we consider a rainbow analogue of Lehel's conjecture in the setting of properly edge-coloured complete graphs. We prove that, for sufficiently large n, every properly edge-coloured Kn admits a partition of its vertex set into two vertex-disjoint rainbow cycles

We study the action of the Fourier transform on the Stokes data of irregular connections on the complex affine line with symmetric irregular classes at infinity, both from the point of view of Stokes filtered local systems and of Stokes local systems, and we show that it is governed by a rich combinatorial structure: (1) Observing that, in this setup, a Stokes filtration is fully determined by the data of either its recessive or subdominant solution spaces, and making the link with results of T. Mochizuki, we show that the Fourier transform amounts to exchanging recessive and subdominant solutions via the Gale transform of configurations of points in projective spaces. (2) We show that the equivalence between recessive solutions and Stokes local systems is deeply connected with the triality relating point configurations, superperiodic linear difference equations and frieze patterns obtained by Morier-Genoud-Ovsienko-Schwartz-Tabachnikov: Up to signs, the coefficients of the difference equations and friezes coincide with the nontrivial Stokes matrix entries. It follows from this Stokes-frieze correspondence that the Fourier transform of Stokes representations is given by their combinatorial Gale transform, leading to explicit closed formulas.

In this paper, we introduce a new approach for constructing multivariate fractal interpolation functions and $\alpha$-fractal functions associated with multivariate functions. Unlike the existing methods that rely on the Banach contraction principle, here the construction is based on Matkowski and Rakotch contractions. While numerous methods for constructing multivariate fractal interpolation functions have been explored in the literature, the approach given in this paper is distinct in the sense that it generalizes all previously known techniques and provides a broader framework for such constructions. We propose a technique to develop nonlinear iterated function systems using the generalized contractions and establish that the attractors of such systems are the graphs of continuous multivariate functions interpolating theoretical data points. Furthermore, for the Rakotch contractions, the existence of an invariant Borel probability measure supported on the graph of the associated multivariate fractal interpolation function is explored.

The cluster graphs on $n$ vertices, the disjoint unions of complete graphs, have the integer partitions of $n$ as their isomorphism classes, and the quotient edit distance $q^*(\lambda,\mu)=\min_{\sigma\in S_n}|E(G_\lambda)\triangle\sigma E(G_\mu)|$ makes that set a metric space. Its geometry and its complexity both issue from one identity: $q^*$ is an affine function of the maximum of $\lVert X\rVert_F^2$ over the contingency tables with margins $\lambda$ and $\mu$. Our main result is an explicit optimal embedding. The weighted dyadic sums of the Ferrers staircase, taken at the critical exponent $\frac14$, give a map $F_n$ into $\ell_2^{\,<4n}$ that acts on a single partition and is computable in $O(n)$ time, and its distortion is $\Theta(n^{1/4})$. That order is optimal, since $c_2(\mathcal K_n)=\Theta(n^{1/4})$: the lower half follows from a $\Theta(\sqrt n)$-dimensional Hamming cube of partitions and Enflo's theorem, so the determination needs no other external input. The analytic core is a scale-free inverse inequality for every integer sequence with $v(1)=v(N+1)=0$ and $v(s)-v(s+1)\in s\mathbb Z$: its critical dyadic energy is at least $\lVert v\rVert_1^2/(63504\sqrt{\mathrm{TV}(v)})$. Combinatorially the same identity yields two explicit $\ell_1$ models, the vertex-mass metric on sorted degree sequences with $\frac12\delta_1\le q^*<\frac32\delta_1$ and the block-energy metric with $q^*\le B\le2q^*-1$, both constants optimal; hence $c_1(\mathcal K_n)\le2$, and an $O(n\log n)$-time algorithm returns an alignment of cost below $2q^*$ carrying the certificate $q^*\in[\lceil(B+1)/2\rceil,B]$. Computationally, deciding $q^*(\lambda,\mu)\le Q$ is strongly NP-complete and admits no FPTAS, while the farthest alignment is polynomial-time solvable. The best constant in the inverse inequality remains open; an exactly solvable chirp family caps it at $\frac23$.

Let $\mathcal H^\infty$ be the space of all Dirichlet series that admit a bounded holomorphic extension to the open right half-plane $ \{s\in \mathbb C: \operatorname{Re} s >0\}, $ and let $$ \mathcal S_N: \mathcal H^\infty \to \mathcal H^\infty; \sum_{n=1}^\infty a_n n^{-s} \mapsto \sum_{n=1}^N a_n n^{-s}. $$ be the $N$-th partial sum operator. This note establishes the asymptotic lower bound $$ \liminf_{N\to \infty} \frac{\|\mathcal S_N\|_{\mathcal H^\infty \to \mathcal H^\infty}}{\log N} \geq \frac{1}{2\pi}. $$ Together with the upper bound of R. Balasubramanian, B. Calado, and H. Queffélec, this shows that the growth of $\|\mathcal S_N\|_{\mathcal H^\infty\to \mathcal H^\infty}$ is of sharp logarithmic order.

The last two decades have witnessed a growing trend towards proving sparse random analogues of combinatorial theorems. One unified approach to proving such theorems, formalised by Conlon and Gowers [Ann. of Math. 2016], involves establishing a 'transference principle' which allows one to translate between robust properties in the dense setting and the sparse $p$-random setting, provided $p$ is not too small. Our results provide a more general transference theorem, extending the results of Conlon and Gowers and also those of Schacht [Ann. of Math. 2016]. Among a variety of other applications, we use this to obtain a sparse counting lemma for graphs and hypergraphs which are not necessarily strictly balanced. Our method achieves asymptotically optimal bounds on the probability $p$, and the probability of success.

Let $(M^n,g)$ be a complete Riemannian manifold with $\Ric\ge-kg$ and uniformly positive $m$-intermediate curvature in the sense of Brendle–Hirsch–Johne. We prove that the Fisher eigenvalue $\lambda_{n-m+1}$ is small, after heat averaging, below the curvature scale $k^{-1}$. Consequently, balls have polynomial volume growth of order $R^{m-1}$ for $R\le k^{-1/2}$. At larger scales we obtain the corresponding estimate with an exponential factor $\exp(C\sqrt k R)$, and an example shows that this factor is necessary.

Gray elements and material overlaps at the interfaces limit density-based multi-material topology optimization. This article extends the level set-based negative-mapping interpolation to the multi-material proportional topology optimization of macro-scale structures in two and three dimensions. An alternating active-phase algorithm decomposes M-phase problems into two-phase sub-problems described by level set functions; an evolutionary strategy removes the need for sentitivity analysis; and the negative-mapping interpolation eliminates the overlaps of the phases in the interface elements. Cantilever and half-MBB beams in two dimensions and a cantilever beam in three dimensions, with three and four material phases, give smooth boundaries, no gray elements and overlap-free designs. The compliance lies below the SIMP value in four of the eight two-dimensional cases and 0.3%, 0.4%, 4.8% and 12.7% above it in the other four. The influence of the material properties, of the interface treatment and of the algorithmic parameters is also examined.

We develop a general class of non-reversible Hamiltonian Monte Carlo dynamics on discrete state spaces. The method augments the discrete state with a continuous momentum variable and does not require a continuous embedding of the discrete state space. We establish conditions for invariance of the target distribution and study the resulting processes in terms of ergodicity, exponential contractivity, asymptotic variance and relaxation time. We then derive scaling limits on increasingly fine lattices and high-dimensional hypercubes. In both settings, suitable rescalings converge to Hamiltonian dynamics in continuous space, revealing a diffusive-to-ballistic speed-up over reversible samplers, even for heterogeneous target distributions where standard non-reversible methods become diffusive. The proposed dynamics can be simulated exactly in continuous time, given access to the target distribution at all neighbours of the current state. To reduce computational cost, we provide approximation schemes based on splitting, $\tau$-leaping and gradient approximations. We illustrate the proposed framework numerically on examples involving high-dimensional hypercubes, mixture models and permutations.

Let $W_N$ be an exact length-$N$ block from a strictly positive i.i.d. source $\mathbf p$ on a fixed finite alphabet. We prove that the canonical T-complexity $c_T$ satisfies \[ \frac{c_T(W_N)}{e^{-\gamma}h(\mathbf p) N/\log N}\longrightarrow1 \] in probability and in $L^r$ for every fixed $1\le r<\infty$, where $h(\mathbf p)$ is the source entropy in nats and $\gamma$ is the Euler-Mascheroni constant. The proof combines an exact length budget for canonical recovery, a critical-scale $E_1$ estimate for an ideal backward chain, and an exact finite-block boundary representation. An exact Doob-transform identity expresses the finite-boundary law relative to the ideal law conditioned at each step to avoid the current history-dependent successor codeword. A history-uniform renewal estimate then makes the telescoping endpoint density uniformly asymptotic to one, so no one-step approximation errors accumulate.