Papers

The Mahalanobis distance is a fundamental covariance-adapted metric for multivariate data and plays a central role in recovering latent geometry from nonlinear observations. We extend this principle from vector-valued data to probability measures by introducing a Wasserstein Mahalanobis distance. Our construction replaces Euclidean displacement vectors with optimal transport displacement fields and local covariance matrices with covariance operators defined on Wasserstein tangent spaces. We show that this construction inherits the geometry-recovery property underlying nonlinear independent component analysis. In particular, for Gaussian measures with common covariance transformed by a smooth nonlinear pushforward, the proposed Wasserstein Mahalanobis distance approximates the classical Mahalanobis distance between the transformed latent means. The correspondence is exact for affine transformations and holds up to controlled higher-order error terms for general smooth transformations. These results establish a distribution-valued analog of classical Mahalanobis geometry and provide theoretical support for covariance-adapted learning directly in Wasserstein space. Numerical experiments confirm the theoretical predictions and demonstrate accurate recovery of latent geometric structure.

This paper evaluates the Linked Autonomous Interplanetary Satellite Orbit Navigation (LiAISON) method in near two-body asteroid systems. While LiAISON's usefulness has been proven in multibody environments such as the cislunar region, the application of a two-body system with gravity perturbations has not been sufficiently demonstrated. Vesta and Ceres are good candidates to test LiAISON, because the asteroids have sufficiently strong perturbations to use LiAISON method. The study utilizes a Multi-Model Adaptive Estimation (MMAE) framework integrated with Unscented Kalman Filters (UKF) to estimate the spacecraft states and perturbations using only range and range rate measurements. Results from the Monte Carlo analysis indicate that the proposed framework makes accurate and consistent estimates of position, velocity, and the perturbations for cases of Ceres and Vesta.

The holomorphic foliations of degree $d$ on $\mathbb{CP}^{2}$ with isolated singularities are determined by their singular subschemes, which correspond to elements in the Hilbert Scheme of $N=d^{2}+d+1$ points. In this paper, we analyze the affine varieties parametrized by a partition of $N=7$ that cover the Hilbert Scheme of seven points, with the property that the Hilbert function of the elements in these affine varieties corresponds to that of holomorphic foliation of degree $d=2$. Moreover, we present some results for degree $3$.

We prove that, under mild assumptions, the limit of a cofiltered diagram of $\infty$-categories is a reflective (resp. coreflective) localization of its oplax (resp. lax) limit. This gives rise to a number of exceptional 'stability' results for categorical properties under such limits, in particular one for adjunctions. As an application, we recover a push-pull formula describing certain filtered colimits in the $\infty$-category $\mathrm{Pr}^L$ of presentable $\infty$-categories.

This paper studies information-theoretic inaccuracy measures for record values and their application to testing symmetry. We study the Kerridge inaccuracy measure, cumulative residual inaccuracy, cumulative past inaccuracy, and extropy-based inaccuracy measures to the distributions of nth upper and lower k-record values. Explicit expressions are derived for several common lifetime distributions (exponential, Pareto, Weibull, and uniform), and the monotonic behaviour of these measures with respect to the record order n, the parameter k, and the distributional parameters is analysed. Building on the known characterisation that equality of upper and lower record-based inaccuracy implies symmetry, we develop a nonparametric goodness-of-fit test for symmetry. The test statistic is the difference between the estimated Kerridge inaccuracy measures for upper and lower records, and its null distribution is obtained by a bootstrap procedure that enforces symmetry. A simulation study demonstrates that the test maintains its nominal level for a variety of symmetric distributions and achieves high power against skewed alternatives, while a real-data illustration on annual maximum temperatures confirms its practical usefulness.

Distributionally robust Markov decision processes provide a principled framework for sequential decision making under model uncertainty. We study how many samples are necessary and sufficient to learn an $\varepsilon$-optimal robust policy under the average-reward criterion. A generative model provides samples from the nominal transition kernel, whereas policy performance is evaluated over $(s,a)$-rectangular total-variation uncertainty sets of radius at most $\sigma$. Let $H_0$ and $H_\sigma$ denote the nominal and robust optimal bias spans, respectively. We identify $\sigma H_0$ as the perturbation scale separating high- and low-tolerance regimes. Our matching upper and lower bounds show that, up to logarithmic factors, the minimax total sample complexity is $$ NSA \asymp \frac{SA}{\varepsilon^2}\begin{cases} \min\{H_0,H_\sigma\}, & \varepsilon\gtrsim\sigma H_0,\\ \min\{H_0,H_\sigma\}+\sigma H_\sigma^2, & \varepsilon\lesssim\sigma H_0. \end{cases} $$ Here $S$ and $A$ are the numbers of states and actions, and $N$ is the number of samples per state-action pair. The sample complexity consists of a linear-span term that resembles the nominal AMDP results and a robustness-specific term that appears only in the low-tolerance regime. We attain these rates using reduction-based plug-in procedures that select the reduction—nominal or robust—and its discount factor: a span-informed procedure that makes these choices using known span parameters, and a span-agnostic procedure that calibrates both choices from data.

Tuza conjectured that every finite simple graph $G$ satisfies $\tau(G) \leq 2\nu(G)$, where $\nu(G)$ is the maximum number of pairwise edge-disjoint triangles and $\tau(G)$ is the minimum number of edges whose deletion makes $G$ triangle-free. Puleo proved the conjecture for every graph of maximum average degree less than $7$; this covers maximum degree at most $6$ but no $7$-regular graph. We prove the conjecture for maximum degree at most $7$. The proof uses Puleo's reducible-set framework. At average degree seven his discharging step no longer forces a reducible configuration. In a minimal $7$-regular counterexample every vertex link is a connected seven-vertex graph outside the weak Konig-Egervary class. An exhaustive census of such links supplies, at every vertex, an incident edge lying in four, five or six triangles. We prove that its endpoints form a reducible pair: codegrees five and six use a packing and covering template and Fano-plane witnesses, while codegree four uses an explicit catalogue of 1,144 machine-checked local certificates. We do not provide a human-readable proof of that catalogue; the certificates and their verifiers accompany the paper. The constant $2$ is sharp already at maximum degree three.

For any $(p,q)$, we proved the existence of a rational unicuspidal curve with one Puiseux pair $(p,q)$ in an algebraic surface, with a self-intersection which realizes the upper bound $m_{p,q}$ conjectured by the first-named author in \cite{C}, thus strengthening the symplectic version of the result in \cite{C}. These “optimal" curves were obtained by examining two infinite families of rational bicuspidal curves, one in $CP^2$ and one in $CP^1\times CP^1$. In order to facilitate the computations, we derived certain recursive identities, and as a byproduct, we obtained a new formula for the bound $m_{p,q}$, which is more amenable to computations and gives us better insight concerning the nature of the bound. As a byproduct of this investigation, a formula for the last entry of the multiplicity sequence of the singularity was also found.

We propose and analyze the random batch vortex blob method for the 2D Navier–Stokes equation in vorticity form on the whole plane. The vortex blob method is based on an interacting particle system of $N$ particles with computational complexity of $O(N^2)$, which is reduced to $O(N)$ by the random batch method \cite{JinLiLiu2020}. Our main result is a quantitative law level mean field error estimate whose dependence on the blob radius remains algebraic. We treat the two main error mechanisms separately. The random batch error is controlled through a locally coupled auxiliary partition, a symmetric law comparison, and Fisher-information dissipation. The mean field fluctuation is estimated by exploiting the oddness and divergence-free structure of the Biot–Savart kernel. For smooth, strictly positive initial vorticity, we prove on every finite time interval a normalized relative-entropy bound of order $ O\!\left(\varepsilon^{-4}\tau^2+N^{-1} \right), $ with constants independent of $N$, $\tau$, and $varepsilon$. Here $\tau$ is the batch refreshing interval and $\varepsilon$ is the blob radius. As a consequence, the fixed-particle marginals converge strongly in $L^1$ to tensor products of the regularized vorticity solution when $N\to\infty$ and $\varepsilon^{-2}\tau\to 0$.

We prove the inequality $R(k+1,s+1) \ge R(k,s) + 2k + 2s$ for classical Ramsey numbers, valid for all $5 \le k \le s$, and formalize the proof in Lean 4. As a consequence, we obtain the new lower bound $R(12,12) \ge 1641$.

We develop a foundational theory of soft quotient topology, providing a systematic approach to quotient constructions in soft topological spaces. We establish the universal property of soft quotient topology and investigate the relationship between global soft topology and its parametric slices, noting that slice-wise quotient behaviour is not sufficient to characterise soft quotients. The usefulness of the framework is illustrated through the construction of soft quotient spaces, together with a study of soft group actions and their orbit spaces. We propose a precise definition of soft covering maps that resolves inconsistencies found in the existing literature. Finally, to illustrate the applicability of our framework, we discuss a potential application in multi-agent motion planning, showing how it can significantly reduce combinatorial complexity under parametric uncertainty.

In previous work, we identified, for every $p\geq3$, a unique first critical temperature $\beta_1^p$ and proved that the Parisi measure is replica symmetric (RS) for $0<\beta\leq\beta_1^p$ and one-step replica symmetry breaking (1-RSB) on a nonempty interval immediately above $\beta_1^p$. In this sequel, we determine the rest of the phase diagram. We prove that there is a unique second critical temperature $\beta_2^p>\beta_1^p$ such that the Parisi measure is 1-RSB for $\beta_1^p<\beta\leq\beta_2^p$, while for $\beta>\beta_2^p$, $\operatorname{supp}\mu_\beta=\{0\}\cup[q_\beta,q'_\beta],$ for some $0<q_\beta<q'_\beta<1$, and is therefore full replica symmetry breaking (FRSB). Combined with our earlier results, this establishes the two transitions predicted by Gardner for the Ising pure $p$-spin glass.

The Karhunen–Loève transform (KLT) of the stationary AR(1) source is the classical optimality benchmark of transform coding. Its frequencies are roots of the transcendental equations of Ray and Driver, and the transform has therefore long been treated as an unstructured dense matrix. Rather than compute the exact KLT, the field turned to fast approximations of it, most famously the discrete cosine transform (DCT). This paper shows that the retreat was premature. The exact AR(1) KLT admits direct recursive factorizations, at every order $N\ge2$ and every correlation coefficient $\rho\in(0,1)$: at even orders, the transform reduces to a butterfly stage, two half-order copies of itself, and orthogonal corrections generated by rank-one boundary perturbations of the covariance; at odd orders, to two copies of the KLT of the one-sided prediction residual and an arrowhead stage absorbing the center sample. The residual KLT recurses through half order as well. All correction stages are Cauchy-structured, and applying them by the fast multipole method yields exact-KLT algorithms of $O(N\log N)$ complexity—the same order as the FFT and the fast sinusoidal transforms. The factorizations follow from the covariance matrix by the Sherman–Morrison identity and the classical secular eigenvalue updates, and degenerate, as $\rho\to1$, to the known parity splittings of the DCT-II. As secondary results, the factorization constants are shown to be algebraic functions of $\rho$—in radicals for all $N\le8$—and complete exact KLT modules for $N=2,\dots,8$ are given.

We present an exact analytical model of axisymmetric gravitational collapse leading to the formation of the Kerr black hole. The model depends only on the total mass ${\cal M}$ and a time dependent rotation parameter $a(v)$, and incorporates both the extremal Kerr black hole and a novel quasi extremal regime without requiring $a\approx{\cal M}$. The evolution develops curvature singularities induced by $a(v)$, which are expected to remain enclosed within a trapped region close to the evolving Kerr radius $r=h(v)$. It also predicts a transient anisotropic exterior curvature decaying as $1/r^2$, which may imprint observable signatures associated with the formation of a rotating black hole. The present construction should therefore be interpreted as an exact analytical description of the final stage of rotational gravitational collapse, immediately preceding the formation of the Kerr black hole.

Put \(n\) identical coins on a table with no two overlapping. Which arrangement makes the perimeter of the convex hull of the cluster as small as possible? Despite its elementary statement, the solution of this problem is known only up to four disks. We produce a calculus for hard disk clusters of arbitrary finite size, providing class criticality conditions, first order descent tests, and second order spectral criteria for perimeter minimisation. A central difficulty is that the perimeter formula changes with the hull combinatorics, while the admissible first order geometry changes with the realised contacts. Our approach is guided by the principle that the realised geometry intrinsically determines both the local form of the functional and the admissible motions. As an application, this article takes the first step beyond four disks by providing a solution for the five disk case. The minimum perimeter is \(10+2\pi\), attained in exactly three realised classes. Two admit perimeter preserving flexes, of dimensions one and two, while the third is rigid modulo rigid motions. The first order theory provides pruning criteria, and the reduced admissible space together with its intrinsic Hessian distinguish rigidity, second order instability, and perimeter flat degeneracy.

Latent subgroup analysis is central to fields such as genomics, precision medicine, and social science, where the goal is to identify heterogeneous populations with distinct covariate structures or response behaviors. Mixture models provide a natural probabilistic framework for this task, representing the data-generating distribution as a weighted combination of subgroup-specific laws with unobserved labels. In high-dimensional regimes, these analyses face significant challenges. Often, only a small subset of covariates drives meaningful subgroup separation; the remaining variables may introduce noise or redundancy. Standard clustering methods typically treat all dimensions as equal, but in high-dimensional spaces, irrelevant coordinates can distort distances and obscure the low-dimensional structures defining latent classes. This paper introduces a Bayesian distilled clustering framework for high-dimensional mixture models. We propose that clustering should occur within a statistically justified subspace rather than the full ambient space. Our method utilizes a Bayesian variable selection model to estimate posterior inclusion probabilities, quantifying the evidence that each covariate contributes to subgroup separation or response behavior. A "distilled" covariate set is then identified by controlling the expected false-discovery proportion. Clustering is performed on this reduced subspace, followed by conditional independence diagnostics to examine subgroup-specific dependencies among selected variables. Critically, this framework is model-based: the distillation step is tied directly to the mixture structure and response model rather than a generic dimension-reduction criterion. This ensures the resulting subspace remains aligned with the scientific objective: identifying latent subgroups that differ in both distributional structure and behavior.

Recent advances in combinatorial dynamical systems that generalize the classic discrete Morse theory have prompted algorithmic studies of combinatorial vector fields. In this regard, authors in [7] recently proposed the concept of Conley-Morse persistence barcode that summarizes the continuation of invariant sets in an evolving vector field through homological persistence. They proposed an algorithm to compute this barcode using a filtration of the so called index pairs on a poset called transition diagram. The algorithm becomes costly due to multiple runs of zigzag persistence it executes on filtrations of `unwieldy' structures of index pairs. We overcome this difficulty by replacing the index pairs with blocks, which are structurally much simpler. These replacements need reversal of certain relations in the transition diagram resulting in a much simpler algorithm. The algorithm works by updating matrix decompositions akin to computing `vineyard' in standard persistence.

A sequence of sparse weighted graphs $(G_N:N\ge 1)$ indexed by the number of vertices $N$ is said to be left-convergent if all (suitably weighted) subgraph counts converge to a limit as $N\to\infty$. This notion generalizes in a natural way Benjamini-Schramm's definition of local weak convergence. A broad research agenda aims at determining which `global' graph properties are determined by left or local weak convergence (in other words, which of these properties are in fact local). We prove that, under a spectral condition on the weighted adjacency matrix ${\boldsymbol A}_N$ of graph $G_N$, the free energy density of the Ising model on this weighted graph is continuous in the left convergence topology (and hence is a local function). Our proof uses the decomposition of the Ising measure as a log-concave combination of product measures, and of the rapid mixing of Langevin dynamics for log-concave measures. As applications, we derive new limit theorems for the free energy density of spin glasses, antiferromagnets, and magnetization constrained ferromagnetic models, on locally tree-like graphs.

Two nonradial scattering criteria are established for the three-dimensional focusing energy-critical generalized Hartree equation below the ground-state threshold, under a bounded-scale concentration–compactness reduction. The first criterion is formulated in terms of fixed-center occupation windows and quantitative bounds on the motion of the concentration center. The second is based on uniform low-frequency \(L^2\)-decay, which yields finite mass, precompactness in the inhomogeneous energy space, a zero-momentum normalization, and sublinear center drift. The proofs combine a localized Hartree virial identity with a critical Hardy–Littlewood–Sobolev estimate controlling the nonlocal tail.

For separable $C^*$-algebras $A$ and $B$, the $E$-theory group $E(A,B)$ carries a natural, generally non-Hausdorff, second-countable group topology, Hausdorff exactly when the closure of zero $\overline{\{0\}} \subseteq E(A,B)$ is trivial. The nonzero elements of $\overline{\{0\}}$ are the phantom classes, invisible to every continuous Hausdorff-valued invariant. We identify $\overline{\{0\}}$ as a derived inverse limit over any shape system $(C_n)$ of $SA \otimes \mathcal{K}$, giving a natural Milnor sequence $0 \to {\varprojlim}^1 [C_n, S^2B \otimes \mathcal{K}] \to E(A,B) \to \varprojlim [C_n, SB \otimes \mathcal{K}] \to 0$ for all separable $A$ and $B$, with no UCT or nuclearity hypothesis. For nuclear $A$ it agrees with the Willett-Yu controlled-$KK$ Milnor sequence, and when $A$ satisfies the UCT, nuclear or not, its ${\varprojlim}^1$ term is the pure-extension group $\mathrm{Pext}^1_{\mathbb{Z}}(K_{*+1}(A), K_*(B))$, as in Schochet's fine-structure description of the Kasparov groups.

We study holomorphic and formal first integrals for germs of codimension-one holomorphic foliations on normal complex analytic spaces. In dimension two, under the assumption that the dual graph of the exceptional divisor of a resolution is a tree, we prove that the foliation admits a holomorphic first integral if and only if its leaves are closed outside the singular point and only finitely many leaves accumulate at that point. This extends a classical integrability theorem of Mattei and Moussu to singular ambient spaces. We also prove a holomorphic prolongation theorem for normal quotient germs admitting a smooth quasi-étale cover and a smooth connected lift of a generic two-dimensional section. We record, in addition, a conditional formal prolongation statement under depth assumptions on the conormal powers and an injectivity condition for the corresponding differential-form obstruction modules. Under the quotient-prolongation hypothesis, and with a reduced tangent cone where formal restriction must be detected, the higher-dimensional integrability results follow from their surface counterparts. We give a reduced nonnormal example satisfying both dynamical conditions but admitting no holomorphic first integral, showing that normality is essential. Our arguments combine resolution of singularities, holonomy techniques, formal completion, and extension properties of holomorphic functions on normal analytic spaces.

Artificially introducing a delay in the observations of a system can be an effective mechanism to mask the system itself, with the goal to increase its opacity and thus its security. This work investigates opacity in discrete event systems with delayed observations. We focus on two questions: how to verify opacity under delayed observations, and how to synthesize sensor activation policies that guarantee opacity under such delayed conditions. To address these questions, we first introduce the definition of opacity under delayed observation and develop a corresponding verification method. We then extend such analysis tool into a synthesis tool by proposing an optimization approach for designing sensor activation policies guaranteeing opacity under delayed observations. An example is used to illustrate the analysis and synthesis procedures.

In this article, we explore the integrable aspects of generalized non-linear fluid equation possessing mixed spatial and time evolutions. This non-linear model has been acknowledged as integrable tool for the non-linear dynamical character- ization of dense material such as in optical fibre, metallic ionic fluid. This model widely has been applied in physical interpretations of electromagnetic propaga- tion through perturbed charged plasma. Here, we use few integrable approaches and attain a broad range of solitary solutions involve deep analytical analysis. These results are furnished with computational simulations, which enrich under- standing about the accurate pictorial flow of energy. We also present a bifurcation analysis for canonical coordinates to examine the impact of parameter variations on non-linear behavior of the systems under consideration. Subsequently, we per- form stability analysis to evaluate the robustness of optical solitons features in the vicinity of equilibrium points. These results establish the reliable and accurate framework for analyzing non linear response of materials for energy propagation by varying physical parameters associated with this non-linear equation model.

We prove that two finite-dimensional linear representations of the braid group are isomorphic. One representation comes from the monodromy of the boundary Casimir connection, and the other comes from the $\iota$quantum Weyl group. Both representations are defined for any split symmetric pair $\mathfrak{k}\subset \mathfrak{g}$ and for any integrable representation of $\mathfrak{k}$. Our proof uses $(O_m,\mathfrak{so}_{2n})$ spin Howe duality, so it is only for the pair $\mathfrak{so}_m\subset \mathfrak{sl}_m$.

For a convex body $K \subset \mathbb R^d$ let $\Delta(K)$ be the expected distance between two independent uniform points of $K$, and let $\theta(K)$ be the corresponding expectation for normalized surface measure on $\partial K$. The Zaporozhets-Tarasov conjecture asserts $\Delta(K) \le \theta(K)$. We prove this conjecture in case $d=2$. In addition, we give a six-vertex convex polytope in $\mathbb R^3$ for which the reverse strict inequality holds, and obtain counterexamples in every dimension $d \ge 3$ by taking products with segments. Finally, we show that $\theta(K)> \frac{\operatorname{per}K}{6}$ for every planar convex body.

Families of $\mathrm{Sp}(1)\simeq\mathrm{SU}(2)$ gauge theories with eight supercharges are found to have a Higgs branch which is an isolated symplectic singularity. These are, in some sense, the most “minimal” gauge theories as they only Higgs to a trivial theory. The matter content is $N$ fundamental half-hypermultiplets and one half-hypermultiplet in the $\mathrm{Sym}^k$ representation where $k=1,3,5,7$. The cases of $k=1,3$ reproduce the known Kraft–Procesi construction for minimal nilpotent orbit closures of $\mathrm{SO}(N+1)$ and the $g\mathrm{SO}(N)$ singularities of \cite{Bourget:2025wsp}, respectively. The cases $k=5,7$ are new isolated symplectic singularities which are termed $h\mathrm{SO}(N)$ and $i\mathrm{SO}(N)$, respectively. The classification of these isolated symplectic singularities is argued for through the Higgs mechanism, with Hilbert series and highest weight generating (HWG) functions computed for some cases. Each of the $g\mathrm{SO}(N)$, $h\mathrm{SO}(N)$, and $i\mathrm{SO}(N)$ families has a (quaternionic) one-dimensional member; these are the Klein $A_3$, $E_6$, and $E_8$ singularities, respectively. Our construction hence provides realisations of these Klein singularities as Higgs branches (hyper-Kähler quotients) of $\mathrm{Sp}(1)$ gauge theories, complementary to Kronheimer's construction \cite{Kronheimer:1989zs} using the $\widehat A_3$, $\widehat E_6$, and $\widehat E_8$ affine quivers. The Klein $E_7$ singularity is also realised as a Higgs branch (hyper-Kähler quotient) of an $\mathrm{Sp}(1)\times\mathrm{O}(1)$ gauge theory.

Chartrand and Zhang asked whether there exists a graph $G$ without isolated vertices whose target Ramsey number satisfies $\mathrm{TR}(G) > \mathrm{R}(G)$. We answer the question affirmatively, even under strong structural restrictions. If $\Gamma_t = P_t \square P_t$ is the square $t \times t$ grid, then the elementary counting bound of Chartrand and Zhang gives $\mathrm{TR}(\Gamma_t) \geq 2t(t-1)+1$, whereas a theorem of Mota, Sarkozy, Schacht and Taraz gives $\mathrm{R}(\Gamma_t) = (3/2 + o(1))t^2$. Consequently, $\mathrm{TR}(\Gamma_t) > \mathrm{R}(\Gamma_t)$ for all sufficiently large $t$, so there are infinitely many connected, planar, bipartite counterexamples of maximum degree four. Higher-dimensional grids show that the ratio $\mathrm{TR}(G)/\mathrm{R}(G)$ is unbounded on connected bipartite graphs, while each fixed-dimensional witnessing family has bounded maximum degree. We also record an independent construction of disconnected counterexamples from the theorem of Burr, Erdos and Spencer on Ramsey numbers of multiple copies: if a fixed graph $H$ satisfies $e(H) > 2v(H) - \alpha(H)$, then $\mathrm{TR}(qH) > \mathrm{R}(qH)$ for every sufficiently large $q$.

For integers $1\le m\le n$, let $s(n,m)$ denote the number of $m$-tuples $(k_1,\ldots,k_m)$ of nonnegative integers satisfying \[ n=\sum_{j=1}^{m}\frac{k_j}{j}. \] We obtain a complete asymptotic expansion for $\log s(n,m)$, uniformly for all $n\ge m$ as $m\to\infty$. The coefficients are given explicitly in terms of limiting prime-block functions arising from the residue structure of the problem. We also determine the full hierarchy of multiplicative corrections in the sparse regime, identifying explicit constants at every fixed order and, in particular, the first correction constants $1/4$ and $(1-\log 2)/4$. The results give uniform two-parameter asymptotics as both the target and the number of allowed reciprocal parts grow.

For every odd integer $t\ge17$, we prove that an explicit circulant graph on $5t-10$ vertices is doubly saturated $R(3,t)$-good. The graph is triangle-free and has independence number $t-1$. Adding any nonedge creates a triangle, whereas deleting any edge creates an independent set of order $t$. This settles Conjecture 2 of Przybocki, Mackey, Heule, and Subercaseaux. A cyclic sumset identity and explicit witnesses prove the local saturation properties. Writing $t=2m+1$, a five-layer reduction proves the independence bound via a uniform affine certificate for $m\ge30$ and an exhaustive checker for $8\le m\le29$. The checker soundness and the complete argument are formalized in Lean 4.32.2. Consequently, $2t-1\le \operatorname{DS}(3,t)\le 5t-10$ for odd $t\ge17$.

At the end of 2025, a record computation of $\pi$ reached 314 trillion decimal digits, providing the largest numerical dataset ever generated for this constant. We exploit this unprecedented dataset to investigate whether the digits of $\pi$ themselves can serve as a source of pseudorandom numbers for estimating $\pi$ through the simplest Monte Carlo method. Our results go beyond the normality hypothesis by providing empirical evidence of a high degree of statistical randomness in the available digits, although not of digit independence, which cannot hold for a deterministic sequence. By optimizing the mapping of the digit sequence into Monte Carlo samples, we obtain the highest precision allowed by the dataset. As predicted, the method successfully reproduces the first sequence of decimal digits, demonstrating that the largest available dataset of $\pi$ digits can be used to recover $ \pi \approx 3.141593 $ through Monte Carlo simulation.