Papers

Many physical systems with or without nonholonomic constraints have a Lagrangian description. In the first case, the Lagrangian model can be represented by second-order ODEs that are constrained to a submanifold of velocities; in the latter case the ODEs are unconstrained. In this paper, using geometric techniques, we address the more general inverse problem: “When can a given constrained or unconstrained system of second order ODEs on a manifold be the representation of a Lagrangian model?”. We show that the constrained case has many more ambiguities and complexities than its well-understood, unconstrained counterpart.

Using degenerate Whittaker functions and explicit computations of Eisenstein series, we obtain explicit formulas for the Fourier expansions of theta lifts to the special orthogonal group $G={\rm SO}(3,n+1)$ over $\mathbb{Q}$, where $n\ge 3$ and $G$ splits at all finite places. The theta lifts in question are Hecke eigen, non-cuspidal, square-integrable automorphic forms of weight $l$ ($l\ge n+2$, even), arising from elliptic newforms for $\SL_2(\Z)$ of weight $l-\frac{n-2}{2}$ when $n$ is even and $2l-n+1$ when $n$ is odd.

We prove an arithmetic analogue of Kashiwara regularity for filtered strongly equivariant Berthelot arithmetic $\mathscr D^\dagger$-modules on formal flag varieties. Let $\mathcal G$ be a split connected reductive group scheme over a complete discrete valuation ring of mixed characteristic, let $\mathfrak X=\widehat{\mathcal G/\mathcal B}$ be the formal flag variety, and let $\mathcal K\subseteq\mathcal G$ be a smooth closed subgroup whose special fiber acts on $X_s$ with finitely many separable orbits. We introduce a filtered strong equivariance condition requiring infinitesimal equivariance to be realized on good finite-level models. This condition allows the principal symbols of the fundamental vector fields to be controlled at the level where Berthelot characteristic varieties are defined. We prove that the characteristic variety of every filtered strongly equivariant coherent $\mathscr D^\dagger_{\mathfrak X,\mathbb Q}$-module is contained in the union of conormal bundles to the $\mathcal K_s$-orbits; hence such modules are holonomic. In the Frobenius range, Caro's stability theorem for $F$-holonomicity over smooth projective formal schemes upgrades this to geometric overholonomicity. As a consequence, simple Frobenius filtered strongly equivariant coherent arithmetic $\mathscr D^\dagger$-modules are classified by pairs $(O,E)$, where $O$ is a $\mathcal K_s$-orbit and $E$ is an irreducible $\mathcal K$-equivariant overconvergent $F$-isocrystal on $(O,\overline O)$ whose intermediate extension satisfies the filtered strong condition. The result is a regularity theorem under a finite-level equivariance hypothesis, rather than an existence theorem for that hypothesis.

We study the long-time dynamics of many-body bosonic lattice systems under translation-invariant Gaussian quantum cellular automata. We formulate two sets of conditions on GQCAs which separately guarantee thermalization of any state on the local Weyl algebra to the infinite temperature state, whenever the state is locally normal and has uniformly bounded particle density. Our main intermediate result is a quantum many-body generalization of the classic Riemann-Lebesgue lemma which is a bound on expectation values of local Weyl operators involving their support and the state's particle density.

We extend the explicit study of $G_4$-fluxes and general Hodge cycles from the ordinary Fermat sextic fourfold to tame Fermat-type hypersurfaces in weighted projective space. The main new feature in the weighted setting is that the Galois action on the cyclotomic period data need not preserve the $(2,2)$-subspace. As a consequence, the rational reconstruction of an integral self-dual class can involve additional middle-cohomology components, increasing the norm of the corresponding flux. We work at maximally symmetric Fermat points, where the period matrices and symmetry-invariant Hodge loci can be computed explicitly. Using Movasati's description of periods, cyclotomic period matrices, and Hermite/Smith normal form reductions, we construct the relevant integral lattices of symmetric self-dual classes in middle cohomology. This gives a controlled test of whether symmetric general Hodge cycles can satisfy the M2-brane tadpole bound. Our main conclusion is empirical. In the degree 12 example in $\mathbb{P}_{1,1,1,1,4,4}$, and in the degree 8 weighted example used as a comparison, the shortest computed symmetric general Hodge cycles overshoot the tadpole bound. In the degree 36 example in $\mathbb{P}_{1,1,1,9,12,12}$, which has $h^{1,1}=11$, the most general example we have, the tadpole conjecture is indeed verified at the maximally symmetric locus, although the computations get difficult and computationally expensive. These computations suggest that, in weighted Fermat examples, the relevant notion of a “symmetric flux” must take into account not only automorphisms of the variety but also the Galois action on the period field. Non-uniform Galois orbits provide a natural arithmetic mechanism by which symmetric Hodge classes can acquire large tadpole charge.

Classical spectral theory lacks a framework for understanding how spectra propagate through compositional systems like deep neural networks, feedback control loops, and quantum circuits. This paper develops a universal theory governed by three invariants: the operadic spectrum (local spectral data), spectral derivatives (perturbation sensitivity), and interaction residue (emergent interface-generated content). We prove three main theorems: the Spectral Propagation Theorem decomposes global output into propagated local spectra, residues, and derivative corrections; the Stability Theorem introduces the SOC stability radius and condition number; and the Universality Theorem shows any reasonable propagation rule is uniquely determined by the three invariants. These results provide a coordinate-free, representation-invariant language for spectral analysis of compositional operator systems.

Conformal prediction has been touted as a more formal, rigorous approach to adding uncertainty to a forecast. The sole objective of this note is to point out that rigor cuts both ways in the case of residual pooling, the technique used in the vast majority of conformal prediction applications. The fact that unconditional guarantee of coverage is provided is not in question, but we make clear, we believe for the first time, that there is an opposing guarantee too: a permanent gambit of logarithmic-score regret which no amount of data or tuning can subsequently reduce. We give the exact size of the sacrifice, and a financial reading of it as the growth rate of an oracle adversary betting against odds set by someone using residual pooling.

In this paper, we characterize the nilpotency and solvability of Gelfand–Dorfman (GD) algebras. In contrast with Poisson algebras and transposed Poisson algebras, we give examples show the nilpotency and solvability of a GD algebra are not determined by the nilpotency and solvability of its underlying algebras. To obtain more examples of special and non-special GD algebras, we give several construction methods and determine whether the resulting algebras are special. Futhermore, we study GD algebra structures on simple Lie algebras. We provide examples demonstrating that GD algebra structures on simple Lie algebras are not necessarily trivial, distinguishing them from Poisson and transposed Poisson algebras. Finally, we provide a complete algebraic classification of low-dimensional complex GD algebras, and determine their nilpotency, solvability and speciality.

Asymptotic statistical theory is a challenging domain for AI-assisted formalization: its central results mix convergence statements, asymptotic expansions, functional analysis, and regularity conditions that have a large gap from existing infrastructure in Lean 4 formalization. To address these challenges, we propose a hypothesis-disciplined Lean 4 formalization pipeline built from multiple agents: a manager that coordinates seven specialist roles for proof planning, skeleton scaffolding, Mathlib reconnaissance, proof construction, integration, independent review, and audit. The main methodological discipline is the hypothesis-disciplined audit, implemented by the Auditor agent: every main-theorem hypothesis and concept-layer field must be anchored in the source mathematical prose, justified as a Lean encoding adapter, marked as source-implied, or rejected as an unsupported strengthening. Using this workflow, we build a systematic formalization of asymptotic statistical theory, especially the parametric and semi-parametric models' asymptotic distribution and efficiency results. The resulting Lean development is axiom-clean and source-faithful, with Lean-checked and human-audited proofs of core parametric and semi-parametric theorems organized so that theorem-agnostic infrastructure and statistical concept definitions are separated from theorem-specific assembly. The formalization results are available at https://github.com/junwei-lu/Lean-Asymptotic-Statistical-Theory.

We develop a harmonic-analytic method for constructing a generalized trigonometric system in noncommutative $L_p(\mathbb{T}^d_\theta)$ spaces arising from the strongly continuous representation of $\mathbb{T}^d$ and show that the generalized trigonometric system is a Schauder basis in $L_p(\mathbb{T}^d_\theta)$ for $1<p<\infty.$ In particular, we prove that this trigonometric system forms an RUC-basis in $L_p(\mathbb{T}^d_\theta)$ for $2<p<\infty.$ Our results provide a noncommutative counterpart of the classical trigonometric basis in $L_p(\mathbb{T}^d)$. Further, we obtain a weak $(1,1)$ type estimate of partial sum operators associated with noncommutative trigonometric systems. This allows us to study uniformly boundedness of partial sum operators between pairs of symmetric spaces that do not necessarily possess nontrivial Boyd indices, extending known results in this direction to the setting of quasi-Banach symmetric spaces.

This paper focuses on accelerating Markov chain Monte Carlo sampling in Bayesian inverse problems in which forward model evaluations dominate the computational cost. It builds on several established ingredients previously used in related scenarios: delayed acceptance, neural network surrogate models, Hamiltonian proposals, and proposal subchains. The main framework is the delayed-acceptance Metropolis-Hastings algorithm of Christen and Fox (2005). The first-stage proposal distribution is constructed from a subchain of Hamiltonian trajectories targeting the surrogate posterior. For each fixed surrogate model, the Hamiltonian subchain and delayed-acceptance correction define a kernel invariant with respect to the exact posterior. In the present work, the surrogate is updated only during a burn-in phase, after which the production run uses a fixed surrogate model. The sampling framework is implemented in Python using parallel processes. Several chains are generated in parallel and share a single surrogate model trained during burn-in on all collected data. The forward model is treated as a black box; therefore, the application area is broad. However, the main motivation is efficient solution of geotechnical inverse problems with material properties represented by Gaussian random fields. In this study, the sampling framework is applied to a geotechnical inverse problem in which hydraulic conductivity and porosity are modeled as non-stationary Gaussian random fields approximated using truncated Karhunen-Loeve expansions. Based on a precomputation, the truncation dimensions are chosen separately for hydraulic conductivity and porosity. The forward model outputs are pore pressure values at control points and selected observation times. These are compared with in situ pore pressure measurements collected over one year during the Tunnel Sealing Experiment in an underground laboratory in Canada.

We consider a dynamical bifurcation caused by a slow passage through a static bifurcation point: in a system depending on a parameter, the parameter changes slowly in time and passes through the critical value corresponding to the loss of stability of an equilibrium via a Poincaré–Andronov–Hopf bifurcation in the frozen system. If the system is analytic, then the loss of stability is inevitably delayed: phase points attracted to the equilibrium in the stability region remain near the equilibrium for a long time after entering the instability region, so that the parameter changes by an amount of order ~1 independently of how slow the variation of the parameter is. Remarkably, there exists a maximal delay: all phase points attracted to the stable equilibrium before a certain threshold value of the parameter leave a neighbourhood of the unstable equilibrium almost simultaneously near another threshold value of the parameter, known as a buffer point. A delay of stability loss beyond the buffer point is impossible unless the initial data have a very special form. We assume that, although the equilibrium is non-degenerate for real values of the parameter, one of its eigenvalues vanishes generically for some complex value of the parameter (a complex analogue of a saddle-node bifurcation), and that this complex singularity is, in a suitable sense, the closest one to the real Poincaré–Andronov–Hopf bifurcation point. We show that the value of maximal delay is determined by this complex singularity: the threshold values defining the maximal delay are the intersection points of the Stokes lines associated with this singularity and the real axis. We study these phenomena in the framework of slow–fast dynamical systems.

We provide a complete classification of extensions of a trivial linear cycle set H by an abelian group I, under the assumption that both H and I are finite cyclic p-groups with p odd, or p = 2 and H has at most four elements. This yields an explicit parametrization of all possible extensions, offering a classification that is both comprehensive and computable. We also compute the socle and the center of all the linear cycle sets obtained.

We present a simple computer-assisted procedure to construct $SO(d_1+1)\times SO(d_2+1)$-invariant cohomogeneity one Einstein metrics, and use it to recover known Einstein metrics on $S^{10}$ and $S^{12}$, as well as find new ones on $S^{11}$, $S^{12}$, $S^{13}$ and $S^7\times S^3$.

A Boolean function in CNF format is of type Positive$\wedge$Negative} if each clause C is either positive (i.e. all literals of C are positive) or negative (i.e. all literals of C are negative). As is well known, deciding the satisfiability of such CNFs is NP-complete. We say that a CNF is of type DisjointPositive if its clauses are positive and mutually disjoint. Dually define DisjointNegative. It is shown that the satisfiability of CNFs of type DisjointPositive$\wedge$DisjointNegative can be decided in quadratic time. Moreover, the modelset can be output in polynomial total time. This is relevant since it affects not only the modelsets of CNFs of type Positive$\wedge$Negative, but more generally of type Horn$\wedge$AntiHorn. As to the latter CNFs, they e.g. occur in connection with the fixpoints of a Monotone Boolean Network. In another vein, the unsatisfiability of a Horn$\wedge$AntiHorn CNF can be demonstrated by means wholly different to the often used method of clausal proofs.

We determine the sharp Sadov constant for Shapiro–Diananda cyclic sums. Sadov proved the lower bound C >= log 2; we prove the matching upper bound by an explicit asymptotic construction, obtaining C = log 2. We also develop a local spectral stability theory for the equal point of the Shapiro–Diananda cyclic sums. The Hessian is diagonalized by Fourier modes, giving an exact local minimum/saddle/quadratic-degeneracy criterion for all n and k, periodic equality families, and explicit classifications for k = 2 and k = 3. The result determines the global infimum over all n and k, but does not solve the separate fixed-k asymptotic minimization problems.

We revisit the calculation of the spectrum of the bosonic ambitwistor string, understood as the BRST cohomology or, equivalently, as the semi-infinite cohomology of the $\mathrm{BMS}_3$ Lie algebra relative to the centre with values in a particular module. We work in momentum space, which allows us to work algebraically and interpret the BRST cohomology as inducing representations of the Poincaré group. In agreement with the existing literature, we find that all the cohomology resides in the massless sector, but a careful representation-theoretic analysis of the spectrum reveals, in addition to the usual massless sector of the closed bosonic string (dilaton, metric and Kalb–Ramond field), also a massless vector. We devote a large part of the paper to describing the cohomology at a massless momentum $p$ as a module over the stabiliser $H$ of $p$ in the Lorentz group, a task which is made difficult due to $H$ not acting reducibly when $p\neq 0$. This allows us to conclude that the spectrum is not unitary, forbidding the interpretation of the extra massless vector as a Maxwell field.

Let $X$ be a finite-area translation surface whose vertical flow is uniquely ergodic. Given a slit joining two nonsingular points of $X$, one can form a branched cyclic cover by gluing $\mathrm{N}$ copies of $X$ crosswise along the slit. We study when the vertical flow on the resulting cover is uniquely ergodic. We first prove a geometric criterion for unique ergodicity of the branched cover. We show that if, for a sequence of times along the Teichmüller geodesic, one endpoint of the slit is contained in embedded Euclidean disks of uniformly positive radius that avoid the other endpoint, then the branched cover is uniquely ergodic. The proof uses the special symmetry of the cover together with an analysis of forward and backward generic points for the vertical flow. We then show that this criterion applies for Lebesgue-almost every choice of slit endpoint under a natural geometric hypothesis on the Teichmüller orbit of $X$, namely a uniform lower bound for the embedded radius along a subsequence. Finally, we give sufficient conditions for such a lower bound in terms of the cylinder geometry of $g_tX$, introducing the notion of pipe cylinders and proving that embedded disks of definite size must exist. As a consequence, for the class of uniquely ergodic translation surfaces, almost every slit produces a uniquely ergodic branched $\mathrm{N}$-cover.

This work focuses on optimal experimental design (OED) methods for passive imaging. We adopt a Bayesian inverse problem framework for passive imaging source problems, primarily focusing on spatially uncorrelated sources and systems governed by the Helmholtz equation. A major challenge in passive imaging is that the use of correlation data causes the observation dimension to grow quadratically with the number of sensor locations, compounding the computational difficulty of finding optimal designs. To overcome the computational bottleneck of repeated PDE solves in optimal design algorithms, we develop a two-level, low-rank approximation of the A-optimal design objective. This effectively decouples the problem into an offline and an online phase, enabling efficient evaluation of the design objective and its gradient without additional PDE solves. Our numerical results demonstrate that the proposed algorithm efficiently scales to large problems and that the resulting optimal designs significantly outperform random sensor placements in minimizing posterior uncertainty.

A subset of the vertex set of a graph is geodetically convex if it contains every vertex on any shortest path between two elements of the subset. The convex hull of a set of vertices is the smallest convex set containing the set. We study two games in which two players take turns selecting vertices of a graph until the convex hull of the jointly unselected vertices is too small. The last player to move is the winner. The achievement game ends when the convex hull of the jointly unselected vertices is not the vertex set. In the avoidance game, the convex hull of the jointly unselected vertices must always be the vertex set. We study the nim-values for several graph families, including cycle graphs, hypercube graphs, complete multipartite graphs, wheel graphs, generalized wheel graphs, and graphs with a unique minimal generating set.

We investigate the periodic $L_2$-discrepancy of infinite sequences $\S_d$ in $[0,1)^d$ and its analytic counterpart, the diaphony. We prove that infinite order-2 digital sequences over $\mathbb{F}_2$ attain the optimal order $L_{2,N}^{{\rm per}}(\S_d) \le C_d (\log N)^{d/2}/N$ for all $N \in \mathbb{N}\setminus \{1\}$, matching known lower bounds for infinitely many $N \in \mathbb{N}$. This confirms the conjectured optimality of order-2 constructions. By this result, we improve upon previously known constructions using order-5 digital sequences, and reduce the underlying dimension for the interlacing construction from $5d$ to $2d$, significantly improving practicality. We establish our bounds within a broader framework of quasi-Monte Carlo integration for periodic Besov spaces $S_{p,q}^rB(\mathbb{T}^d)$ with dominating mixed smoothness $r \in (1/p,2)$, where $p,q\in [1,\infty]$. Rules based on infinite order-2 digital sequences yield worst-case errors of order $(\log N)^{(d-1)(1-1/q)} / N^{ \min(r,1)}$ for $r \not=1$, and $(\log N)^{d(1-1/q)}/N$ for $r=1$, for all $N \in \mathbb{N}\setminus\{1\}$, while preserving extensibility in $N$.

We establish robust log-convex interpolation inequalities within the scale of Gagliardo seminorms. We achieve this by deriving some Chebyshev-type integral inequalities for general non-synchronous functions. Our primary motivation for establishing these robust interpolation inequalities stems from the study of the asymptotic nonlocal-to-local stability of weak solutions to the boundary Dirichlet problem associated with the regional fractional $p$-Laplacian. More precisely, if $u_s \in W^{s,p}(\Omega)$ weakly satisfies $(-\Delta)_{p, \Omega}^s u_s = f_s $ in $\Omega$ and $ \gamma^s_0(u_s) = g_s$ on $\partial\Omega,$ with $\frac{1}{p} < s \leq 1$ and $\Omega \subset \mathbb{R}^d$ is bounded Lipschitz, then, under appropriate convergence of the data $f_s$ and $g_s$ as $s \to 1^-$, we establish that $\| u_s - u_1 \|_{W^{\eta,p}(\Omega)} \xrightarrow{s \to 1^-} 0 $ for all $0 \leq \eta < 1$.

In the $D$-dimensional Vicsek graph, we prove that the Riesz-like inequality $ \|\nabla f\|_p \leq C \|\Delta^\gamma f\|_p $ holds for every $p\in(1,\infty)$ and every $ 0<\gamma<\gamma^*(p):=\frac{1}{D+1}+\frac{D-1}{D+1}\,\frac{1}{p}, $ while it fails whenever $p\in(1,\infty)$ and $\gamma^*(p)<\gamma<1$. Thus, the validity of the inequality remains open only at the critical exponent $\gamma=\gamma^*(p)$. This provides the first example of an $L^p$-bounded “super-Riesz transform”, namely an operator of the form $\nabla \Delta^{-\gamma}$ with $\gamma$ strictly larger than the Euclidean threshold $\frac12$. To achieve this, we establish a more general result linking the diffusion escape rate and a Poincaré inequality on balls to the validity of the reverse Riesz-like inequality $\|\Delta^\gamma f\|_p \leq C \|\nabla f\|_p.$

A grand Motzkin path with air pockets is a non-empty lattice path in the first and fourth quadrant of $\mathbb{Z}^2$, starting at the origin $(0,0)$, ending on the $x$-axis, and consisting of up-steps $(1, 1)$, horizontal steps $(1, 0)$, down-steps $(1, -k)$ where $k \geq 1$, and with no consecutive down-steps. A {grand Dyck path with air pockets} is a grand Motzkin path with air pockets that uses no horizontal steps. We present the first known 2-Gray codes for grand Motzkin paths with air pockets. Setting the number of horizontal steps to zero in our algorithm yields the first known 2-Gray codes for grand Dyck paths with air pockets. Our three-stage algorithm generates each path in constant amortized time per string, using $O(n^2)$ memory. We also provide enumeration formulae for grand Motzkin paths and grand Dyck paths with air pockets.

Many nonlinear iterative systems generate high-dimensional trajectories whose early behavior is informative but difficult to compare directly. This paper derives a fixed-dimensional F-transform coordinate representation for early trajectories of iterated Pearson correlation matrices. The construction is defined on the first five-point post-transient signal window, which is the shortest sampled window that simultaneously places the three symmetric fuzzy nodes at observed positions and supports a nondegenerate centered first-degree F-transform coefficient, thereby providing the earliest feasible local level–trend characterization within this sampled geometry. The representation combines two logarithmic observables of the post-transient dynamics: step size and contraction ratio. Applying this same four-coordinate construction to the step-size and contraction-ratio signals yields the eight-dimensional descriptor $\Psi=(v_1,v_2,v_3,s_2,u_1,u_2,u_3,r_2)$, with a common coordinate form across matrix sizes. For the fixed construction, $\Psi=M(q_2,\ldots,q_7)^{\top}$, $q_k=\log\delta_k$, with $\operatorname{rank}M=6$. Thus, the descriptor is an injective, overcomplete representation of the six logged step sizes underlying the two channels. The representation is Lipschitz stable, and the centered first-degree coefficient recovers affine trends exactly. Convergence-length approximation is used as a downstream test of retained dynamical information. Across 22 matrix dimensions and 22,000 trajectories, repeated train–test evaluation shows predictive performance comparable to raw two-channel and statistical-summary representations. PCA shows that the first two principal components explain on average $84.47\%$ of the descriptor variance. Clustering reveals reproducible coarse organization, with the strongest mean silhouette at $k=2$ and high stability for smaller numbers of clusters.

We show that the continuous functional calculus is computable. As consequences we obtain the computable compactness of the spectrum of any computable normal element of a computably presented $\mathrm{C}^*$-algebra, the existence of effective approximate units for computably presented $\mathrm{C}^*$-algebras, and an effective version of the Spectral Theorem for compact operators on separable Hilbert spaces.

In the sequence reconstruction problem, a codeword $\x$ is transmitted through several identical channels where each channel produces a noisy read of $\x$, and the problem is to analyze how to uniquely reconstruct $\x$ based on these noisy reads. Levenshtein has studied the minimum number of reads which guarantees unique reconstruction of $\x$, which is one sufficient condition for unique reconstruction. In this paper, we move on to a different perspective and propose a new framework for unique reconstruction. Our new sufficient condition for unique reconstruction takes both the number of reads and the distances among the reads into consideration. We offer both theoretical analysis and corresponding efficient reconstruction algorithms for our reconstruction framework.

The article addresses the construction and geography of negatively curved metrics on hyperbolic complex manifolds. We introduce a mechanism for constructing complete Kähler metrics with negative bisectional curvature. This applies to some product complex manifolds, thereby resolving a longstanding problem attributed to N. Mok. We then construct projective Kobayashi hyperbolic surfaces with negative holomorphic sectional curvature whose Chern slopes $c_1^2/c_2$ realize any $s \in \mathbf{Q} \cap \left( \frac{2}{7}, \frac{2}{3} \right)$. For slopes $s\in \mathbf{Q}\cap \left( \frac{2}{7},\frac{1}{3} \right)$, the corresponding surfaces admit a Hermitian metric with $\text{HSC}<0$, but their Kähler–Einstein metric cannot have $\text{HSC}<0$. We finally construct, for every $s \in \left( \frac{1}{2}, 3 \right)$, a sequence of projective Kobayashi hyperbolic surfaces that do not admit a Hermitian metric of nonpositive holomorphic sectional curvature, whose Chern slopes $c_1^2/c_2$ converge to $s$.

There are $2^{n-1}$ ways to tile a $1 \times n$ rectangle with rectangular tiles (of any length, of course they all must have width $1$), but in how many ways can you tile a $100 \times 100$ checkerboard with such tiles? Neither humankind, nor computer-kind, will (most probably) ever know the exact number. But it is possible to compute these numbers for $m \times n$ rectangular grids, if $m$ is not too big, while $n$ can be as big as one wishes. This was initially done in 1988 by David Klarner and Spyros Magliveras, and beautifully extended, around 2006, by, at-the-time, first-year LSU undergraduate Joshua Smith, in collaboration with his faculty mentor, Helena Verrill. Here we extend this to weighted-counting, also keeping track of the number of tiles (that ranges from $1$ to $mn$), and the number of participating grid-edges (that range from $2m+2n$ to $2mn+m+n$). This quickly leads to statistical analyses (mean, variance, and higher moments) of these quantities. While we admire the clever approaches of Klarner-Magliveras and Smith-Verrill, we use two alternative approaches to the original problem, that are more amenable for deriving these generalizations. At the same time, we illustrate the power and beauty of experimental-yet-rigorous enumerative combinatorics.

We study the deterministic first-order oracle complexity of finding \(\epsilon\)-stationary points in smooth nonconvex optimization when the objective satisfies higher-order smoothness assumptions. While the classical \(\epsilon^{-2}\) rate is optimal under only Lipschitz gradients, higher-order smoothness leads to accelerated first-order upper bounds, most notably the \(\epsilon^{-7/4}\) rate under Lipschitz Hessians and the \(\epsilon^{-5/3}\) rate under Lipschitz third derivatives. The matching lower bounds, however, have remained open. We resolve this gap by proving a new dimension-free first-order lower bound for higher-order smooth nonconvex functions, valid for every finite smoothness order. In particular, our construction gives a matching \(\Omega(\epsilon^{-7/4})\) lower bound in the Hessian-Lipschitz case and a matching \(\Omega(\epsilon^{-5/3})\) lower bound in the third-order-smooth regime. The hard instance is based on a block-chain mechanism that enforces blockwise oracle revelation while preserving the smoothness structure needed for the scalar hard instance. The lower-bound construction was discovered with the assistance of ChatGPT 5.5 Pro and subsequently verified by the authors.