Papers

We study the truncated trapezoidal rule, a family of A-stable second order multistep methods parametrized by an integer $J \ge 2$, a compromise between BDF2 and the trapezoidal rule. We obtain a closed-form expression for the coefficients that minimize the principal error constant under the A-stability constraint and we derive an explicit formula for the corresponding principal error constant. The latter decreases to the optimal Dahlquist value $1/12$ as $J$ increases, and is strictly smaller than the BDF2 constant for every $J \ge 2$. We apply the truncated trapezoidal rule within the convolution quadrature framework. Its analyticity in a neighbourhood of the closed unit disk yields milder regularity and perturbation requirements than those of the trapezoidal rule, while the error constant can be made arbitrarily close to the optimal one by increasing $J$. Numerical experiments show that convolution quadrature based on the truncated trapezoidal rule remains stable under symbol perturbations, where the trapezoidal rule fails, while achieving a smaller error constant than BDF2.

We construct an example of a random dynamical system on the circle, formed by maps that are only locally invertible, which possesses an atomic stationary measure $\nu$. Moreover, this measure is physical: for Lebesgue-almost every initial point $x_0$, the Cesàro averages of its random trajectory almost surely converge to $\nu$. This shows that the Hölder regularity of stationary measures, known for (non-measure-preserving) random dynamical systems formed by diffeomorphisms, cannot be generalized to this class of systems. We also provide some related examples, including ones where a stationary measure charges a proper submanifold, despite the absence of a closed common invariant submanifold.

In this paper, we address the uniqueness of surfaces of constant spacetime mean curvature in an asymptotically Schwarzschildean lightcone of mass $m>0$. We prove that there exists a unique asymptotically flat foliation by surfaces of constant spacetime mean curvature for a fairly generic notion of asymptotic flatness. This foliation has Bondi energy $m$ and vanishing Bondi linear momentum. The authors have already established the existence of such a foliation in previous work, but proven uniqueness only in a very restrictive class of surfaces. Although this restrictive class of surfaces was necessary for the construction, here we show that the foliation is a posteriori unique under significantly weaker assumptions.

The goal of this paper is to systematically develop the fundamental arithmetic and combinatorial properties of the sector renormalization operation on rigid rotations. We employ the specific framework of modified continued fractions appropriate for sector renormalization and analyze their properties. By allowing infinite first return times, this framework yields a dynamical compactification of the space of irrationals called the parabolic compactification; we show that it is characterized by some universal properties. We also discuss the corresponding natural extension and introduce the continuant group, the time semigroup, and topological cascades. For example, we demonstrate how a bi-infinite tower of sector renormalizations of irrational rotations can be packaged within a single dynamical plane as a cascade of translations. This will serve as a foundational combinatorial tool for studying the geometric properties of sector renormalizations of holomorphic maps with irrationally indifferent fixed points, particularly neutral quadratic polynomials.

Links between physicals systems and computability properties have been an active field of investigation in recent years. Inspired by a previous work in the context of positive temperature Gibbs measures, we prove here that in the context of zero-temperature ergodic optimisation, for a computable potential and provided with several reasonable assumptions, the maximum ergodic average is a computable real number, and the set of maximising measures is a $\Pi_1$-computable compact set. Then, in the more specific context of symbolic dynamics, with finite-range interactions on subshifts of finite type, we provide an explicit algorithm to compute both the maximum ergodic average and the set of maximising measures in finite time, with a matching code repository.

Multicomponent multiple-precision arithmetic is constructed from existing floating-point operations using error-free transformations (EFTs). Its performance can be improved by employing branch-free algorithms that eliminate conditional branches. Zhang and Aiken proposed branch-free addition and multiplication algorithms for double-word (DW), triple-word (TW), and quad-word (QW) arithmetic. We implemented their TW and QW algorithms, for which substantial performance improvements over conventional algorithms were expected, and demonstrated their effectiveness. In this paper, we propose branch-free fused multiply-add (FMA) algorithms for DW, TW, and QW arithmetic. The proposed algorithms integrate multiplication and addition into a single computational network and require fewer arithmetic operations than separately performing branch-free multiplication and addition. The anchor-relative error bounds, the preconditions of all FastTwoSum operations, and the non-overlapping properties of the outputs are mechanically verified using FPANVerifier, and input-relative error bounds are subsequently derived analytically. Benchmark results on CPUs and GPUs show that the proposed algorithms provide performance improvements in many compute-intensive cases, including division, square root, and basic linear algebra kernels, while maintaining accuracy comparable to that of the existing branch-free algorithms.

This study presents a three-phase suspension lava flow model with a Herschel-Bulkley rheology. The suspension contains crystals and gas bubbles, and two closures for the evolution of the crystal volume fraction are considered. The first closure minimizes the complexity of the system by treating crystal fraction as a transported quantity subject to relaxation towards an equilibrium state. This closure avoids a parametrization of the many heat transfer mechanisms a lava flow is subjected to. The other closure is on the lava temperature considering four heat transfer mechanisms and a prescribed temperature–crystallinity relationship. We deduce from this system and solve numerically a one-dimensional depth-averaged model. A comparison with a pre-existing model based on real lava flow data suggests that the prediction of flow parameters done with the multi-parametric evolution of temperature yields more accurate results than those obtained with the other closure. The transient nature of our model correctly predicts that confined lava traveling down an irregular steep slope yields a series of cascading fill-then-breakout lumps that causes the overall flow to be pulsatory. These pulses dominate the local dynamics and preclude a strict steady state to be reached. Theoretically, taking gas bubbles into account is best done with a general rheological relationship valid at any capillary number. In the conditions explored herein, bubbles modulate viscosity within a factor 2 with a shear thinning behavior, decelerating slow flows and accelerating fast flows. When a simplified rheology treating bubbles as hard spheres was used, the only dynamic parameter affected was bulk viscosity.

This paper compares two formats of online assessment in university Calculus: a standard Moodle-based step-by-step test and a STACK-based step-by-step test. Both tests assess integration by parts and divide the solution into consecutive response fields, but they differ in their validation mechanisms. The standard test relies on predefined scoring patterns, whereas the STACK-based test uses symbolic validation rules. The comparison is based on final manually verified scoring matrices from two student cohorts and follows a Classical Test Theory framework, including score distributions, reliability estimates, response-field-level indicators, and correlation-based measures. The results show high overall performance and ceiling effects in both formats. However, the STACK-based test required fewer manual corrections, showed higher internal consistency, and produced a more coherent relationship between solution steps and the total score.

Let (K(X)|K,w) be a valuation transcendental extension of rational function fields and take a minimal pair of definition (a, gamma). In this paper, we characterize those K-conjugates a' of a such that the monomial valuation induced by the pair (a', gamma) restricts to w on K(X). In particular, we show that a' satisfies this if and only if a' and a are conjugates over the henselization of (K,v). The second part of the paper concerns abstract key polynomials. We introduce the notion of a regular limit key polynomial, and more generally, that of a regular complete sequence of key polynomials. We prove that regularity is equivalent to the property that every root of every key polynomial determines the corresponding truncated valuation. This extends earlier work of Mahboub, Mansour and Spivakovsky by allowing both limit key polynomials and valuation algebraic extensions. As a consequence, we obtain that w always admits a regular complete sequence of key polynomials whenever (K,v) is dense in its henselization.

We study nonlinear filtering with misspecified initial distributions and dynamics, but a correctly specified observation model. Observations are treated as a random process generated by the true model, rather than as fixed realizations, and both the correct filter and the misspecified filter are analyzed under this common observation law. Our main result is a Bayesian total variation balance identity for the expected change in total variation distance under a Bayesian update: a covariance term minus a non-negative remainder. The identity characterizes conditional and averaged update non-expansivity. Combined with prediction contraction, it yields finite-time bounds separating the initialization error from accumulated dynamical discrepancies, while allowing expansive updates. When the averaged updates are non-expansive and a uniform joint prediction–update coefficient is less than one, a persistent dynamical misspecification yields an asymptotic error bound proportional to the one-step discrepancy. When the conditional updates are non-expansive, we characterize an accumulated joint contraction that implies almost-sure and \(L^1\) coalescence of the true and misspecified filters. Finally, we verify update non-expansivity for scalar Gaussian predictive laws under a variance-ordering condition and, separately, derive explicit prediction contraction bounds for truncated kernels and bounded transition functions with elliptically symmetric noise.

The type D ASEP is an asymmetric two–species interacting particle system on $\Z$, in which two separately conserved species hop, bind into a composite “bound pair”, and split. The model, along with its reversible measures and orthogonal polynomial duality, was constructed using the representation theory of $U_q(\so_{2n})$. The reversible measures and orthogonal polynomial duality are each a product of two copies of the single-species ASEP reversible measures and orthogonal polynomial duality. In this paper, we study the long-time asymptotics of the type D ASEP. In the fixed–$q$ regime, using an exact current–decoupling identity, we prove that the asymptotic hydrodynamic limit and Tracy–Widom fluctuations decouple, as predicted from the duality. In the weak–asymmetry (Edwards–Wilkinson) regime, when $q=1-c/N^2$, we prove that the two density fluctuation fields \underline{decouple}: each converges to a linear stochastic heat equation, with no cross–coupling in either the drift or the noise, the limiting noises having vanishing cross–correlation. More surprisingly, we then prove that the two limiting normal random variables are \underline{correlated} with a seemingly new correlation function. The correlation is exactly equal to $(1-e^{-4c})/(4c)$, with the positive parts of the normal random variables having correlations expressed by the Bessel–Struve function. This paper, with the exception of the abstract and introduction, was written entirely by Claude Opus 4.8 and Fable 5. The proofs were then formalized in Lean, using Aristotle by Harmonic AI. The human author of this paper verified the proofs manually.

The toughness of a non-complete graph $G$, denoted $\tau(G)$, is defined as \[ \tau(G) = \min\left\{ \frac{|S|}{\omega(G-S)} : S \subseteq V(G),\ \omega(G-S) \geq 2 \right\}, \] where $\omega(G-S)$ is the number of components of $G - S$. For a complete graph $G$, we define $\tau(G) = \infty$. A graph $G$ is $t$-tough if $\tau(G) \geq t$. For a positive integer $k$, a graph $G$ is $(K_2 \cup kK_1)$-free if it contains no induced subgraph isomorphic to $K_2 \cup kK_1$. Recently, Liu \cite{liu} showed that every $2k$-connected $(K_2 \cup kK_1)$-free graph $G$ with $\tau(G) > 1$ is Hamilton-connected. In this paper, we strengthen this result by proving that every $(k+1)$-connected $(K_2 \cup kK_1)$-free graph $G$ with $\tau(G) > 1$ and minimum degree $\delta(G) \geq 2k$ is Hamilton-connected. Moreover, by imposing restrictions to the independence number $\alpha(G)$, we prove that every $k$-connected $(K_2 \cup kK_1)$-free graph $G$ of order $n$ with $2k+1 \leq \alpha(G) < \frac{n}{2}$ and $\delta(G) \geq 2k$ is Hamilton-connected, and that the bounds on $\alpha(G)$ are sharp.

We introduce the notion of a double Lie bimodule, consisting of a Lie bialgebra L, a Lie L-bimodule P, and a double Lie bracket on P satisfying suitable compatibility conditions. We describe an algebraic construction that combines two double Lie bimodules into a Lie bialgebra. As a geometric application, we show that the Goldman-Turaev Lie bialgebra of a surface with boundary, together with the Kawazumi-Kuno intersection operations on the linear span of the set of homotopy classes of paths between pairwise distinct boundary points, may be assembled into a double Lie bimodule. We prove a decomposition theorem for the Goldman-Turaev Lie bialgebra of an oriented surface in terms of the double Lie bimodules corresponding to the two surfaces with boundary obtained by cutting along a simple separating closed curve.

Moment-based relaxations of the positive partial transpose (PPT) criterion have been recently introduced, as a hierarchy of entanglement criteria involving only experimentally accessible quantities of a given bipartite state. The goal of this work is to study their typical detection performance on high-dimensional bipartite systems. Concretely, we investigate whether random bipartite mixed states on $\mathbb C^d\otimes\mathbb C^d$, obtained as the marginal over an environment $\mathbb C^s$ of a uniformly distributed pure state, generically satisfy or violate them. For each fixed level $m\in\mathbb N$ in this hierarchy of moment-based PPT criteria, we are able to identify a threshold environment dimension $s=\lambda_md^2$ at which the behavior of the associated random state switches from violating to satisfying it, with probability going to $1$ as $d$ grows. The proof combines combinatorics of permutations techniques to estimate the average value of moments of partially transposed random states and concentration of measure arguments to bound the probability of deviating from such average, when the underlying local dimension $d$ is large. We additionally need tools from the theory of Hankel determinant evaluation via orthogonal polynomials.

We introduce two first-order theories of hereditarily finite multisets: a schematic theory WF^- and a finitely axiomatized theory F^-, in the language with the empty multiset, singleton formation, multiset union, and a containment relation. We prove that WF^- is mutually interpretable with Robinson's theory R, and F^- with Robinson arithmetic Q; in particular, F^- is essentially undecidable. Multisets thereby join numbers, strings, trees, sets, and sequences in the mutual-interpretability classes of R and Q. The distinctive obstacle of the multiset case is the simultaneous failure of the standard devices for recovering ordered pairs: positional order, local order on immediate constituents, and idempotence-based Kuratowski pairing. We show that order is recoverable from bare multiplicity: the term pi(x,y) = <x> u <x> u <y> is provably injective in F^-, yielding a direct interpretation of the Kristiansen-Murwanashyaka tree theory T; conversely, F^- is interpreted in Q by arithmetizing a normal-form calculus for multiset terms within bounded arithmetic. Each structural axiom of F^- is shown independent of the others, with finite or Presburger-definable decidable witnesses, and the containment axiom is conservative. As an application, we identify Spencer-Brown's forms modulo commutative juxtaposition with hereditarily finite multisets and locate the boundary of essential undecidability within the calculus of indications.

We show that the generating curves of non-totally geodesic spherical rotational minimal hypersurfaces (catenoids, for simplicity) of the hyperbolic spaces $\mathbb{H}^{2n+1}$ are $p$-elastic curves for $p=(2n-1)/(2n)$. We employ this variational characterization, together with the Chern–Gauss–Bonnet formulas for locally conformally flat manifolds, to present an explicit expression for the renormalized area of catenoids in terms of hyperelliptic integrals. Further analyzing these special integrals, we show that the renormalized area of catenoids varies continuously from negative infinity to twice the renormalized area of the totally geodesic hypersurfaces $\mathbb{H}^{2n}\subset\mathbb{H}^{2n+1}$. Therefore, we conclude that the renormalized area is not bounded below and that, when $n$ is even, the renormalized area of minimal hypersurfaces in $\mathbb{H}^{2n+1}$ does not have a sign.

We study Schreier extensions of monoids and establish a Kaluzhnin–Krasner embedding theorem for Schreier extensions. First, we prove that the category of monoids is not locally algebraically cartesian closed (LACC) and that a monoid is algebraically exponentiable in the category of monoids if and only if it is a Dedekind-finite monoid. Second, we recall that the category of extensions of monoids is $S$-LACC with $S$ the class of Schreier extensions, which defines a wreath product $A \wr B$ for any two monoids. Finally, we prove a Kaluzhnin-Krasner embedding theorem for Schreier extensions that are not necessarily split, i.e. given any Schreier extension $A \hookrightarrow G \twoheadrightarrow B$ of monoids, there is a monomorphism $\phi_G \colon G \hookrightarrow A \wr B$, which is part of a morphism of extensions. The proof adapts the classical group-theoretic argument by replacing conjugation, which requires inverses, with a substitute made available by the Schreier property, namely, the unique factorization of elements in the fibers of the projection $p \colon G \twoheadrightarrow B$.

Graphical small cancellation extends the classical small cancellation theory and provides a powerful method for constructing groups with prescribed subgraphs in their Cayley graphs. We prove that torsion subgroups of groups defined by possibly infinite C(4)-T(4) graphical small cancellation presentations are finite. We also prove the corresponding result for groups defined by C(6) graphical small cancellation presentations under the additional assumption that the presentation is torsion-essentially C(6)-free. Both results follow from a more general result on local-to-global fixed point properties for torsion groups acting by automorphisms on simply connected graphical small cancellation complexes.

Pilatte recently proved that there exists an infinite Sidon set of positive integers which is an asymptotic basis of order $3$, answering a problem posed by Erdős, Sárkőzy and Sós in 1994. In this paper, we strengthen this result by proving that for any $0<\eta<0.0527$, there exists an infinite Sidon set $\mathcal{S}\subset \mathbb{N}$ which is an asymptotic basis of order $3-\eta$; that is, every sufficiently large integer $m$ can be represented as \[ m=s_1+s_2+s_3 \] for some $s_1,s_2,s_3\in \mathcal{S}$ satisfying \[ \min\{s_1,s_2,s_3\}\leq m^{1-\eta}. \] To prove this, we develop a truncated version of Pilatte's construction and use a deep result of Sawin on sums of Dirichlet convolutions of the von Mangoldt function over function fields.

Tipping transitions are abrupt reorganizations between statistical regimes, usually anticipated through critical slowing down: recovery slows, autocorrelation and variance rise, and spectral power shifts toward low frequencies. In noisy, spatially extended systems, however, these signatures alone cannot distinguish weakening resilience, increasingly likely excursions toward a competing state, and spatial reorganization. We address these questions in the stochastic Ghil-Sellers energy balance model, whose ice-albedo feedback supports warm and snowball metastable regimes, through three viewpoints. Reduced Ruelle-Pollicott resonances and Kolmogorov modes diagnose relaxation and response in physically interpretable observables; Extreme Value Theory probes the accessibility and persistence of tail excursions; and Data-Adaptive Harmonic Modes diagnose the frequency-resolved organization of the temperature field. Near tipping, several reduced decay rates slow together and their modes become geometrically harmonized along a common transition direction, while Green functions show delayed recovery and enhanced low-frequency susceptibility only when the response residues are nonzero. Cold extremes become less sharply bounded and more clustered, and the full field becomes less compressible as its phase distribution broadens, even as a dominant low-frequency component emerges. Read jointly, these diagnostics reconcile apparently contrasting signatures and connect statistical indicators to the physical geometry of competing climate states. We further show that a small localized albedo change can create an additional pair of folds, so that a distance to tipping may carry structural as well as statistical uncertainty. The results provide a framework for interpreting early warnings that distinguishes changes in the system's dynamics from uncertainty in the model's bifurcation structure.

Dual-source trolleybuses alternate between overhead catenary supply and on-board battery operation, creating energy-use patterns driven by route attributes, high-frequency trajectories, and hourly weather. Existing models struggle to represent these heterogeneous inputs and rarely explain the causal drivers of consumption. This paper proposes a time-aware tabular deep learning framework for inter-stop energy management. Periodic time encoding is integrated into a parameter-efficient batch-ensemble backbone to jointly learn static and sequential features, while Bayesian optimization with tree-structured density estimation tunes hyperparameters. To move beyond prediction, a three-layer causal explanation pipeline combines feature attribution for marginal effects, a linear non-Gaussian acyclic model for causal direction discovery, and a meta-learner for net average treatment effects. Experiments on the Zurich trolleybus dataset enriched with meteorological records achieve a MAPE of 6.52% and R of 0.982, outperforming ten statistical, tree-ensemble, and deep learning baselines. Ablation results show that periodic time encoding contributes most to the accuracy gain. Causal analysis identifies regenerative braking ratio and average speed as the strongest energy-saving factors, while coasting distance is the main driver of excess consumption. The findings offer actionable thresholds for vehicle technology, driving behavior, capacity allocation, and catenary network planning.

Multiverse analysis makes explicit how empirical conclusions depend on alternative, defensible analytical specifications. Standard approaches usually generate the multiverse first and then summarize it through decision tables, specification curves, model weights, or scalar outputs such as estimates and p-values. This stagewise view is useful, but it can hide how inferential uncertainty is arranged across specifications. We propose a distributional-geometric framework in which each admissible specification is represented by a probability distribution on a common target-output space. After defining a suitable distance between these distributions, the induced geometry allows the multiverse of analyses to be studied through local neighbourhoods, diameters, Fréchet barycentres, and dispersion measures. Numerical examples alongside a real case study illustrate how the approach complements existing multiverse summaries by retaining both effect variation and uncertainty variation.

In this work, we consider the identification and control of nonlinear systems with finite action spaces. The unknown dynamics are estimated from finite samples with Koopman operator regression in a reproducing kernel Hilbert space, yielding a linear switching predictive model, the switches governed by the value of the control variable. In order to perform control in closed-loop, the learned dynamics are employed in an infinite-horizon optimal control problem with time-varying stage cost, which is solved by means of model predictive control. In a theoretical analysis, we derive learning rates for the Koopman dynamics approximation. We further quantify, under suitable assumptions, the sub-optimality of the model predictive control strategy, both in the case of exact Koopman dynamics, and in the case of learned ones. Numerical simulations on the Duffing oscillator complement our theoretical findings.

We study a continuous-time portfolio optimization problem in which an investor is evaluated relative to a non-replicable benchmark and seeks to control the persistence of benchmark-relative underperformance. We introduce a benchmark-relative drawdown-duration criterion that penalizes the expected discounted time spent in unfavorable benchmark-relative performance states. Despite the path dependence induced by benchmark-relative drawdowns, we show that the problem admits a one-dimensional Markovian representation and derive the associated Hamilton-Jacobi-Bellman equation. We obtain an explicit projection-based characterization of the optimal feedback control, establish a verification theorem, and identify geometric settings under which the associated closed-loop reflected diffusion admits a unique strong solution. Our results provide a tractable downside-risk-oriented alternative to classical benchmark-tracking formulations and reveal a novel projection-based control structure for benchmark-relative risk management.

Inspired by recent advances in notions of spectral approximation of digraphs [Ahm+20], we study spectral algorithms for finding periodic structures in digraphs via the spectrum of a class of rotated Laplacian matrices. This class of Laplacian matrices was previously studied by Lange, Liu, Peyerimhoff, and Post [Lan+15]. We consider a notion of periodicity ratio that generalizes the bipartiteness ratio of Trevisan [Tre09], and show that it is closely related to the spectrum of rotated Laplacian matrices. In particular, if the digraph is strongly connected and represents a Markov chain, this periodicity ratio for a given $p \in \mathbb{N}$ is a quantitative measure of how close this Markov chain is to having periodicity $p$. We propose and analyze a periodicity-ratio variant of the spectral algorithm by Louis, Raghavendra, Tetali and Vempala [Lou+12]. We show that the algorithm runs in randomized polynomial time and can find many nearly periodic components (i.e, components with small periodicity ratio). This also implies a new higher-order Cheeger-type inequality for periodicity in the spirit of that in [Lou+12; LOT14]. As part of our analysis, we prove a new theorem that upper bounds the probability that the largest magnitudes of two sequences of coordinate-wise correlated complex Gaussian random variables occur at different indices, which may be of independent interest. Previously, an analogous result was known only for real Gaussian random variables.

Pairs of diffeomorphic Riemannian manifolds are shown not to be quasiconformally equivalent, while no degree one invariant (extremal lengths, conformal parabolicity,...) can distinguish them. The obstruction arises from conformal cohomology in degree 2. Similar examples are constructed in the contact subRiemannian category

Let $K$ be a number field with ring of integers $\mathscr{O}_K$, and let $\mathfrak{B}$ be an Erdős family of ideals in $\mathscr{O}_K$. We prove that the associated $\mathfrak{B}$-free subshift $(X_{\mathfrak{B}},(S_a)_{a\in\mathscr{O}_K})$ is intrinsically ergodic: it carries a unique measure of maximal entropy, which we identify explicitly as a relatively independent extension of the Haar rotation on $\prod_{\mathfrak{b}\in\mathfrak{B}}\mathscr{O}_K/\mathfrak{b}$. This is the first proof of intrinsic ergodicity for $\mathfrak{B}$-free systems beyond dimension one, and relies on the work of Araújo–Dymek–Kułaga-Przymus. Via their reductions, we also settle the $k$-free and $\mathfrak{B}$-free lattice-point cases and the $k$-free number-field case. We give two independent proofs of the underlying rigidity statement: one through a single-site relative-entropy argument, and one through an exact-tiling realisation of Peckner's induce-and-split scheme.

Universes are central to dependent type theory, and they are notoriously difficult to handle in a way that is both correct and usable. We propose a new "fuss-free" generalised algebraic presentation for polymorphic cumulative universes that dispenses with the intricate theory of coherent universe coercions in favour of a simpler formulation, which we prove equivalent by means of a normalisation theorem for the former. We study the fuss-free formulation by means of (1) an abstract specification of its bidirectional elaboration algorithm, and (2) a concrete implementation in Haskell. We also describe and implement an extension of the fuss-free universe hierarchy with a judgemental notion of datatype description from which prior notions of cumulative inductive type may be derived.

Phylogenetic networks are used to represent the evolutionary history of species. Due to biological interpretations and computational advantages, researchers have focused on restricted classes of phylogenetic networks, such as tree-child, orchard, and tree-based. These classes capture different notions of tree-likeness: tree-child networks require every internal vertex to have a taxon reachable by a tree path, orchard networks are trees with horizontal arcs (for modelling histories rife with horizontal gene transfers), and tree-based networks are trees with additional (not-necessarily horizontal) arcs. A natural question to ask is “how far is a given network from belonging to a particular class?” This motivates the study of proximity measures, which measure the minimum number of graph modifications required to transform a network into one belonging to a particular class. In this paper, we consider three proximity measures based on leaf addition, valid arc deletion, and arc deletion. We study pairwise comparability of the proximity measures, prove complexity results, and derive extremal bounds for the classes of tree, tree-child, orchard, and tree-based networks.

We construct an explicit dual-feasible point for the Cohn–Elkies linear program in dimension $36$, built from the space of weight-$18$ modular forms for $\Gamma_0(24)$ following the method of Cohn and Triantafillou. The certificate shows that the two-point linear programming bound on the sphere packing density in dimension $36$ exceeds the density of the best packing currently known – the Kschischang–Pasupathy packing, of center density $2^{18}/3^{10}$ – by a factor of at least $32.91$. In particular, no Cohn–Elkies auxiliary function can certify the best known packing in dimension $36$ as optimal. To our knowledge this is the first such dual bound in any dimension above $32$, extending the table of Cohn–Triantafillou ($d=12,16,20,28,32$), Li ($3\le d\le 13$), and de~Courcy-Ireland–Dostert–Viazovska ($d=6$). The certificate is rigorous and machine-checkable, with exact rational data and certified interval bounds: the dual point is a rational vector, coefficient nonnegativity is verified by exact arithmetic up to $n=800$, and eventual positivity of the two relevant $q$-expansions is proved via an explicit Deligne-type tail bound whose constant is certified with outward-rounded interval arithmetic. Two methodological points may be of independent interest: a constraint-generation (cutting-plane) formulation of the exact rational LP, which is what makes an exact vertex whose Eisenstein data supports the tail argument reachable; and a sharpened, lift-aware form of the Deligne bookkeeping constant that discounts deep oldform lifts, without which the finite verification in dimension $36$ fails (the crossover moves past the verified window).