Papers

For a uniformly locally finite coarse space $X$, the uniform Roe algebra $C_u^*(X)$ is the operator norm closure of the controlled operators on $\ell^2(X)$. The $K$-theory of uniform Roe algebras is known in asymptotic dimension zero, but it is not fully understood in higher dimensions. We compute $K_0(C_u^*(G,\mathcal E))$ and $K_1(C_u^*(G,\mathcal E))$ for every countable discrete abelian group $G$ and every finite-rank free abelian subgroup $H\leq G$, where $\mathcal E$ is the coarse structure generated by $H$. We use the Proietti–Yamashita spectral sequence to express the $K$-theory in terms of $H_*(H;\ell^\infty(G,\mathbb Z))$, which we then compute.

In this paper, first we give a new characterization of the cohomology of pre-Lie algebras using the Chevalley-Eilenberg cohomology associated to a morphism from the operad of Lie algebras to Hadamard product of the operad of pre-Lie algebras and its Koszul dual operad. Then we apply the same approach to study the cohomology of Novikov algebras, and give the cochain complex explicitly. The cochain complex of the underlying pre-Lie algebra is shown to be isomorphic to the quotient of the cochain complex of a Novikov algebra. Consequently, there is a long exact sequence connecting the cohomologies of a Novikov algebra and the underlying pre-Lie algebra. The cohomology of a Novikov algebra with coefficients in a representation is introduced using pseudo-tensor categories. As applications, we show that infinitesimal deformations and abelian extensions are classified by the second cohomology groups with different coefficients. Various examples are given to illustrate the difference between the cohomology of a Novikov algebra and that of the underlying pre-Lie algebra.

For a finite set $A$ of positive integers, let $H(A)$ be its set of subset sums, and let $g_3(n)$ be the least $N$ for which some $n$-element set $A\subseteq\{1,\ldots,N\}$ has $H(A)$ free of nonconstant three-term arithmetic progressions. Erdős and Sárkőzy asked whether $g_3(n)\gg 3^n$. We prove \[ \liminf_{n\to\infty}\frac{g_3(n)}{3^n}=0. \] More precisely, for every $\epsilon>0$ there is an integer $d\ge2$ such that $g_3(d\ell)\le \epsilon 3^{d\ell}$ for all sufficiently large $\ell$. The proof uses Korsky's characterization of the problem in terms of ternary coefficient sums and a consequence of an OpenAI construction that provides positive integer coefficients whose linear form is injective on large integer boxes. A base-three expansion then gives the result.

We prove that many curved Kakeya problems, originally motivated by Hörmander's oscillatory integral problem, lift to the classical Kakeya problem in higher dimensions. As a consequence, if the classical Kakeya set conjecture were true in all dimensions, the families of curves whose curved Kakeya sets have full dimension are dense among Hörmander-type families. This is in contrast to the nowhere dense Bourgain's condition, which is a necessary condition for best-case curved Kakeya maximal function estimates. Furthermore the linear Kakeya set conjecture would imply a fairly complete understanding of Kakeya sets of quadratic curves, as first studied systematically by Wisewell. These results give a better understanding of a question posed by Guo–Guth–Nadjimzadah–Shen–Zhang, and they more generally show that the classical Kakeya conjecture in high dimensions rests on a broad range of curved Kakeya problems in lower dimensions.

There is a remarkable variety of natural generalizations of the notion of scalar positivity to square matrices. The lists and tables indicate those that we consider here; they are practically exhaustive of those that arise in practice and theory. Since positive scalars are closed under taking arbitrary powers and roots, it is natural to ask which of these matrix generalizations enjoy analogous closure properties. Our purpose here is to examine and advance these generalizations with respect to their closure under integer powers and fractional matrix roots. In addition to surveying known results, we establish for several important classes of matrices that positivity is preserved under fractional powers, showing that key structural features are retained under matrix roots. The role of Pick functions in the analysis is emphasized throughout. Proofs, counterexamples, and related results are presented, and summary tables are included to provide a concise and useful reference.

Matlis duality, first introduced by Eben Matlis in 1958, stands as one of the most elegant and powerful tools in commutative algebra, establishing a striking contravariant equivalence between Artinian and Noetherian modules over complete local rings. In this paper, we undertake a thorough and systematic investigation of Matlis reflexive modules — those modules for which the canonical evaluation homomorphism into the double Matlis dual is an isomorphism. Our exposition begins with a detailed historical narrative, tracing the intellectual trajectory from classical Pontryagin duality through Grothendieck's dualizing complexes to the modern formulation of Matlis duality. We then develop the core theory from first principles, demonstrating that the class of Matlis reflexive modules constitutes a Krull-Schmidt category — a result that generalizes the classical decomposition theorems for modules of finite length. Over complete Noetherian local rings, we prove that Matlis reflexive modules coincide precisely with the class of minimax modules (those possessing a Noetherian submodule whose quotient is Artinian). We establish full closure properties, including stability under submodules, quotients, extensions, and finite direct sums, with rigorous and self-contained proofs. The paper also surveys recent applications to generalized local cohomology, change-of-rings results, and connections to pure-injective modules and linear compactness, and concludes with a discussion of open problems and future research directions.

A communication system must choose error protection and decoding effort as channel conditions change. A Bayesian bandit encoder (BBE) uses receiver feedback to learn which transmission configuration to select. We study a receiver that decodes by guessing error patterns, using Guessing Random Additive Noise Decoding (GRAND). We extend BBE's selection component to 1,008 code and decoder configurations by learning shared performance patterns offline and updating their weights online. Decoder noise models remain fixed. On a six-configuration training-selected shortlist, sharing reduces accumulated utility loss by 33.5% relative to independent learning. A fixed training-selected configuration matches the shared learner that searches the full catalog. After channel changes, the pruned shared learner first meets a near-optimal selection criterion in 88.5% of events by 2,000 packets, compared with 54.2% for pruned independent learning with the same discounting. The results support combining sharing and pruning for configuration selection, although packet losses remain high for the tested codes under severe noise.

In this article, we apply Itô's excursion theory to find the joint law of minima of a Brownian excursion conditioned to go above level $1$ over the intervals defined by $k$ independent identically distributed (i.i.d.) points of intersection of the excursion with level $a \in (0,1]$. This approach offers an intuitive way to find the joint law of coalescent times of an i.i.d. sample of k particles alive at time $a$ in Aldous' continuum random tree of height $1$. This can be thought of as “sampling of the limit" of critical Galton-Watson trees conditioned to survive a large time, agreeing with some special cases obtained in [4] and [5] which instead consider “the limit of sampling" from critical Galton-Watson trees.

In 1966, McKean conjectured that the entropy production of Kac's caricature of a Maxwellian gas is nonincreasing in time \cite{McKean66}. We disprove this conjecture for the standard uniform-angle Kac equation. Our counterexamples are smooth, strictly positive, even mixtures of two Gaussians at different temperatures. We also show that every entropy derivative beyond the first violates its expected sign at some time along the same solution. The initial densities can be chosen with unit mass and second moment. mass and second moment.

Randomly pivoted Cholesky, QR, and LU are iterative algorithms that form structured low-rank approximations of a matrix by sampling columns, or rows and columns, from the residual. We give convergence rates for these methods that depend on the decay of the singular values of the original matrix. The rates hold, up to a constant, when pivots are drawn from an approximation to the exact distribution. This extra degree of freedom can be used to trade exact sampling for cheaper approximations. We use it to establish rates for variants of randomly pivoted LU, including one that picks pivots using a sketch of the input matrix.

Conditional dimension reduction (cDR) learns low-dimensional latent coordinates while accounting for observed covariates that represent known sources of variation in the data. Conditional Multidimensional Scaling (cMDS) is a cDR technique that works directly with dissimilarity data. Its standard squared-stress formulation, however, is sensitive to contaminated dissimilarity, since outliers can dominate the objective and distort the learned configuration. We proposed Robust Conditional Multidimensional Scaling (rcMDS) by replacing the squared-stress criterion with a Fair M-estimation objective. We developed a reweighted conditional SMACOF algorithm to optimize this objective. The proposed algorithm admits computationally tractable updates, and its stabilized objective values decrease monotonically and converge to a finite limit. Experiments on synthetic and real data show that the pro

We develop an AI-assisted search-and-certification workflow for constructing Carleman weights for wave equations on semi-infinite domains. Starting from a weighted cross-term identity used to derive the Carleman estimate, we derive explicit analytical screening conditions and use an AI system to propose symbolic candidate weights subject to those conditions. The search identifies a logarithmic weight structure with concrete parameter values; the subsequent human verification by the authors then retains this structure, derives the admissible parameter range, and rigorously certifies the selected weight by verifying the required quantitative conditions. Based on the identified and certified Carleman weight, the subsequent proofs, analysis, and applications are carried out entirely by the authors: we establish a global Carleman estimate for the wave operator on the half-line and derive a weighted conditional lateral Cauchy stability result. The proposed weight is further used in a finite-depth one-sided reconstruction problem for semilinear wave equations on the half-line, where the initial-time flux induces an endpoint graph stabilizer and a contractive frozen nonlinear reconstruction map. Numerical experiments support the predicted contraction behavior and show improved robustness to noisy data from the Carleman weighting, with the induced endpoint graph term providing further stabilization at higher noise levels.

If $\boldsymbol{w} = (w_1,w_2,\dots)$ is a sequence in $\mathbb{N}=\{1,2,\dots\}$, the partial Bell polynomials based on $\boldsymbol{w}$ are $B_{n,k}$ for $k \in \mathbb{N}$ and $n\in\{k,k+1,\dots\}$. Let $F(z) = \sum_{n=1}^{\infty} (w_n/n!)z^n$ be the exponential generating function for $\boldsymbol{w}$, and assume the radius of convegence is positive $R>0$. Then $F(z)^k = \sum_{n=k}^{\infty} (k!/n!) z^n B_{n,k}$ for $|z|<R$. Alternatively, defining $Q_{k,n} = (k!/n!)B_{n,k}$, we have $Q_{1,n} = w_n/n!$, and $Q_{k+1,n}=\sum_{m=1}^{n-k} Q_{1,m} Q_{k,n-m}$ for $k\geq 1$. Let us say that the Cramér-type large deviation property holds if $$ \lim_{\substack{n \to \infty\\ k/n \to \kappa}} \frac{1}{n}\, \ln\left(Q_{k,n}\right)\, =\, \mathcal{G}(\kappa)\, ,$$ for every $\kappa \in (0,1)$, where $\mathcal{G}(\kappa)=\inf_{r \in (0,R)} (\kappa \ln(F(r))-\ln(r))$. The (Hardy-Ramanujan) Erdös induction argument suggests this should generally be true as long as two technical conditions are true: one an initial step, and the other a condition for small densities $\kappa$.

We introduce a notion of $2$-homological category modelled on pointed homological categories in the sense of Grandis, using bizero objects whose null $1$-cells are zero objects in the hom-categories for formulating $2$-dimensional pointedness. We first prove directly that the $2$-category of di-exact homological categories, which generalize Puppe exact categories, functors preserving all kernels and cokernels, and arbitrary natural transformations is $2$-homological. Its normal subcategories are saturated thick subcategories, and its exact quotients are constructed by a complete three-arrow fraction calculus similar to the one known for Puppe exact categories. We then adapt this proof to prove that the $2$-category of triangulated categories is also $2$-homological; here, instead of the ternary fractions we use the well-known Verdier fractions. Abstracting the common quotient structure of these proofs yields a general criterion, using which we further establish that the $2$-categories of pointed, additive and abelian categories are also $2$-homological. The criterion also applies to categories enriched in semimodules over a fixed commutative rig, with a zero object, and to their full sub-$2$-category with finite biproducts. Their quotients are linear congruence quotients. Combining enrichment with Puppe exactness gives further examples whose quotients are exact linear localizations, including linear abelian categories as the finite-biproduct case. In the abelian and triangulated cases the normal subcategories and quotients are, respectively, Serre subcategories and Serre quotients, and thick triangulated subcategories and Verdier localizations.

Optimal transport provides a common language for allocation, equilibrium, computation, and inference. Its primal problem assigns mass or agents, while its dual variables admit economic interpretations as utilities, prices, and scarcity rents. This review explains why this combination has proved unusually effective in economics. We first present the core results, including Kantorovich duality, integrality, cyclical monotonicity, the canonical distance and quadratic costs, and entropic regularization. We then trace the field's development from planning and operations research to modern analysis, statistics, and computation. The economic literature is organized around two roles for transport: as a model of matching, trade, hedonic equilibrium, and aggregate assignment; and as a tool for coupling distributions, measuring discrepancies, constructing multivariate ranks, solving inverse problems, and certifying economic conclusions. We conclude by examining extensions beyond transferable utility, one-to-one matching, static allocation, and unconstrained transport, together with open questions involving learning and identification across multiple markets.

In the 2004 collection Unsolved Problems in Mathematical Systems and Control Theory, Erik Verriest posed the problem of characterizing Riccati stability “without invoking additional matrices.” We give such a characterization through a scalar invariant of the resolvent family. The invariant is defined by covariance balances using at most $n^2+1$ frequency–direction pairs. A finite-dimensional separation argument shows that this covariance radius equals the optimal common ellipsoidal norm of the resolvent family. Combined with the strict bounded real lemma, this yields a necessary and sufficient condition for Riccati stability involving only the Hurwitz property of the first matrix and a single scalar inequality. The invariant reduces to the ordinary spectral radius for one matrix, lies between the pointwise spectral-radius and unscaled small-gain levels of the resolvent family, and coincides with the operator-space spectral radius of Shalit and Shamovich for the natural resolvent function space. The main results have been formally verified in the Lean proof assistant.

We investigate a special ansats that allows for an iterative solution of the constant Yang-Baxter equation. Testing this ansatz, we construct four sequences of the constant R-matrices. In each sequence the R-matrices act on the tensor squares of vector spaces of linearly growing dimensions. Each R-matrix also depends on a single complex parameter. By analyzing the spectra of the R-matrices, we conclude that the first and the third series are associated with the symmetric tensor representations of the quantum groups U_q(sl(2)) and U_q(sl(2|1)), respectively. The other two series appear to be related to representations of quantum groups in the case where q is a root of unity. The elements of the second series are associated with nilpotent representations of U_q(sl(2)). We also check the correspondence between these R-matrices and link invariants. Applying the Reshetikhin-Turaev procedure to the first, second, and fourth sequences yields, respectively, colored Jones polynomials, ADO invariants, and Alexander polynomials. To our surprise, a similar calculation for the third series yields a trivial result, identically equal to 1.

Appeals to typicality are common in physics, but it is often unclear what it means for a physical state to be typical relative to a probability measure, and correspondingly unclear what a law that appeals to typicality asserts. Here we consider how one might characterize physical typicality using ideas from the theory of algorithmic randomness. As a concrete example, we show how taking a physical state to be typical relative to a computable measure when it is Martin-Löf random allows one to formulate the distribution postulate in Bohmian mechanics as a statistical constraining law of the theory. Using a toy model, we show how this constraint guarantees the standard Born statistics for computable experimental protocols. Algorithmic Bohmian mechanics (aBM) thus illustrates how algorithmic randomness may be used to provide precise content to a statistical law.

We prove the Tate conjecture for abelian fivefolds over finite fields. The proof constructs correspondences for a residual motive using a Moret–Bailly family, Gross–Schoen heights, and monodromy. We also prove standard conjecture~$D_\ell$ over $\overline{\mathbf F}_p$ and independence of $\ell$ of rational cycle class kernels over algebraically closed fields of characteristic $p$. Over finite fields, rational and numerical equivalence agree with rational coefficients, and higher algebraic $K$-groups vanish rationally.

We prove that the number of curves of a fixed genus g with n marked points over finite fields is a polynomial function of the cardinality of the field if and only if g = 0 or 3g + 2n is less than or equal to 24. This confirms a conjecture of Canning, Larson, and the authors.

We study the dependence of parabolic solution operators on a finite signed measure describing the delay law. For a positive self-adjoint generator with compact inverse, a weighted dyadic spectral sum characterizes the norm of the semigroup-integrated memory perturbation. An exact realization identity transfers the matching lower estimate to two positive point delays in a fixed linear equation, using a common smooth, finite-spectral history. For the Dirichlet Laplacian on a nonempty bounded open set, the sharp worst-case modulus on a bounded ball of continuous $L^2$-valued histories is $d\sqrt{\log(e/d)}$, where $d$ is the bounded-Lipschitz distance between the delay measures. This is a rough-history endpoint result: logarithmic spatial regularity of order $\gamma>1/2$ restores resolution-uniform Lipschitz stability in the linear model. The critical order $\gamma=1/2$ retains a square-root double-logarithmic loss. More generally, a reciprocal-square summability criterion over occupied spectral bands gives the exact weighted threshold, including sparse spectra. Consequences include sharp finite-resolution Lipschitz constants, worst-case errors for prescribed midpoint quadrature of the memory measure, and semilinear upper estimates under Hilbert-space-valued local Lipschitz assumptions. The results distinguish sensitivity to the delay law from spatial approximation error and do not preclude uniform approximation of positive-time states on rough history balls.

This paper presents a regularized cyclic coordinate minimization method for solving nonconvex composite optimization problems having the objective function formed as the sum of two terms, one is twice continuously differentiable and the second term is simple and separable. We analyze the convergence behaviour of our coordinate minimization method, in particular we provide convergence rates to a first-order optimality criterion and objective residual depending on the assumptions on the problem. Then, we show that our algorithmic framework can be efficiently applied for solving quantized matrix factorization problems that arise in e.g., lossy image compression. More specifically, on KODAK and CLIC 2024 datasets, our method notably outperforms JPEG at low bit rates, achieving savings in bits per pixel without excessive degradation and remains comparable at higher bit rates. Moreover, we evaluate both the original and quantized compressed images in a classification task using 3 well-known convolutional networks AlexNet, ResNet50 and MobileNetV2 based on floating point and integer arithmetic representations. Remarkably, when comparing the two numerical representations of the convolutional networks, there is no substantially accuracy degradation on ImageNet dataset. The numerical results on image compression and classification using real data show the flexibility and efficiency of our algorithm compared to well-established methods from the literature.

We study the Lonely Runner Conjecture (LRC), conceived by Jörg M. Wills in the 1960's: Given positive integers $n_1, n_2, \dots, n_k$, there exists a positive real number $t$ such that for all $1 \le j \le k$ the distance of $t \,n_j$ to the nearest integer is at least $\frac{ 1 }{ k+1 }$. We prove that for any counterexample or tight instance $\mathbf{n}$ of LRC, $\mathbf{m} \cdot \mathbf{n} = 0$ for some $\mathbf{m} \in \mathbb{Z}^k$ with $0 < \| \mathbf{m} \|_1 \le \min(2k+3, \ \frac{ k+1 }{ k-1 } \mathrm{flt}(k))$ where $\mathrm{flt}(k)$ denotes Khinchin's (1948) flatness constant limiting the lattice width of a $k$-dimensional convex body without interior integer points. In other words, potential counterexamples to LRC lie on a finite set of hyperplanes in the parameter space. Our proofs use Fourier analysis and a geometric reformulation of LRC, and our results generalize to the situation of shifted lonely runners of varying measures of loneliness. Our results imply and generalize a theorem of Czerwiński (2012) that when we choose $\mathbf{n}$ at random then, with probability tending to 1, the measure of loneliness $\frac{1}{ k+1 }$ can be replaced by $\frac 1 2 - \epsilon$.

We study an interpolation problem in extended Gevrey classes defined by the sequences $M_n^{\tau,\sigma}=n^{\tau n^\sigma}$, $n\in\mathbb N$, $\tau>0$, $\sigma>1$. We show that, under a suitable growth condition on a divergent sequence of derivative orders, estimates of extended Gevrey type imposed only along this sequence imply corresponding estimates for all derivative orders. In particular, we explicitly quantify the change of the parameters in the resulting estimates. To this end, we first establish an equivalence between two definitions of the extended Gevrey classes, one involving the supergeometric factor $h^{n^\sigma}$ and another in which this factor is omitted. We then establish a corresponding interpolation principle for extended Gelfand–Shilov spaces, including a symmetric characterization and a formulation in terms of the associated function of the defining sequence.

A lattice Gabor frame consists of a window function and its time-frequency shifts along a lattice. We classify the lattices that admit Schwartz-class and, equivalently, Feichtinger frame windows. Except for symplectically irrational lattices of critical covolume 1,, we determine the lattice-dependent achievable window smoothness and decay using the scale of modulation spaces. For symplectically rational lattices, including the rational lattices typically used in digital applications, the classification is governed by an integer lattice parameter introduced herein, the symplectic index gap. Our results extend the classical and amalgam Balian-Low theorems. The companion paper "Tilings, Packings, and Smooth and Compactly Supported Gabor Windows" develops the associated tiling-packing theorem and compact-support constructions.

We propose a method for closing moment chains for the Vlasov-Poisson kinetic system (the Landau fluid model), which yields a hyperbolic system of equations at each subsequent step. This system is interpreted as a multicomponent medium, where each component has its own density, velocity, and pressure. The coupling between the components is mediated by an electric field. The principle of closing moment chains is that the final component of the fluid is assumed to be pressureless.

We numerically implement full quantum circuits of optimal tapered Quantum Phase Estimation (in the form of an implementable approximate bandwidth-limited DPSS taper) for the purpose of minimum eigenvalue computation, where the Hamiltonian time-evolution unitary is implemented using standard Trotterization, and the taper state is constructed using an optimized bandwidth-limited taper. We illustrate, on representative quantum Hamiltonians, with tractable exact classical numerical quantum circuit simulations, how tapered QPE performs with respect to optimal phase sampling rate, absolute error, Shannon entropy of the optimal phase distribution, when the tQPE algorithm parameters are changed. Those algorithm parameters are Trotter error, finite sampling, imperfect initial states, total evolution time, and digitization error (number of phase qubits). The reported numerical experiments include up to $\ell=10$ phase qubits of precision with $m=3$ additional phase-register error suppression qubits, on a $4$-qubit Heisenberg quantum magnet model Hamiltonian, using up to 12th-order Trotterization. We numerically show i) the steady-state optimal phase sampling rate is determined by the initial state overlap with the ground-state, ii) with respect to phase sampling probability, or overall eigenvalue sampling error rate, there is no strong evolution time dependence for tQPE unless there is high Trotter error, iii) the approximated time-evolution unitaries in tQPE, like in standard QPE, can result in substantial ringing, which leads to non-physical eigenvalue estimates.

We characterize the arithmetical categories - the exact Mal'tsev categories whose lattices of congruences are distributive - by a simple condition on some suitable pushouts and a finite limit: a regular category is exact arithmetical if and only if, for every triple of regular epimorphisms with common domain, the pairwise pushouts exist and the comparison morphism to the limit of the diagram they form is a regular epimorphism. For a regular category, the validity of this condition for pairs and for triples is equivalent to its validity for $n$-tuples for all $n \ge 2$, and characterizes arithmetical categories among regular ones. For pairs alone the condition reduces to the characterization of exact Mal'tsev categories among regular ones, due to Carboni, Kelly and Pedicchio. For $n$-tuples with $n \ge 4$ nothing further is obtained: arithmeticity is the third and last rung of that ladder. The proof rests on a categorical form of the Chinese Remainder Theorem; in the exact Mal'tsev context this form of the theorem lies close to results of Hoefnagel on majority categories. As a direct application, we give a new proof of Pixley's characterization of arithmetical varieties using a suitable diagram of free algebras.

Horn conjectured that whenever $K$ is a nonempty compact convex subset of a Banach space and $F,G:K\to K$ are continuous commuting maps, there exists $x\in K$ such that $F(x)=G(x)$. We give a counterexample in a separable Hilbert space. The construction uses an exactly commuting ladder of coincidence-free maps between finite trees, due to Oversteegen and Rogers.

The $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive codes are subgroups of $\mathbb{Z}_2^{\alpha_1}\times\mathbb{Z}_4^{\alpha_2}\times\mathbb{Z}_8^{\alpha_3}$. A $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code is a Hadamard code which is the Gray map image of such a code. A recursive construction of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive Hadamard codes $\mathcal H^{t_1,t_2,t_3}$, with all $\alpha_i\neq0$, $t_1\geq1$, $t_2\geq0$, and $t_3\geq1$, is known, as are the linearity and kernel of the corresponding codes $H^{t_1,t_2,t_3}$ of length $2^t$. Their rank, however, was not known in closed form. In this paper, we determine $rank(H^{t_1,t_2,t_3})$ for every admissible triple and obtain an explicit closed formula. The proof computes the dimension of the space spanned by the binary coordinate functions of the code, viewed as Boolean functions of the binary digits of the message. We then use the rank formula, together with the known kernel dimension, to classify the family as far as these two invariants allow. The only pairs of distinct types of the same length sharing both invariants are $\bigl(H^{1,2,t-6},H^{2,0,t-5}\bigr)$ for $t\geq7$ and $\bigl(H^{2,2,t-9},H^{3,0,t-8}\bigr)$ for $t\geq10$. Consequently, the number of distinct pairs $(r,k)$ among the codes of length $2^t$ is $\lfloor(t^2+6)/12\rfloor$ for $3\leq t\leq6$, $\lfloor(t^2+6)/12\rfloor-1$ for $7\leq t\leq9$, and $\lfloor(t^2+6)/12\rfloor-2$ for $t\geq10$. Thus, rank and kernel dimension classify the family completely if and only if $3\leq t\leq6$. We also determine all coincidences with the $\mathbb{Z}_4$-, $\mathbb{Z}_2\mathbb{Z}_4$-, and $\mathbb{Z}_8$-linear Hadamard codes of the same length: there is exactly one infinite family with the first two, namely $H^{2,1,t-7}$ and $H^{5,t-9}$ for $t\geq10$, and exactly five families with the $\mathbb{Z}_8$-linear codes.