Papers

We study constrained smooth optimization problems on Hadamard manifolds with closed geodesically convex feasible sets. We analyze two projected gradient schemes: one with a constant stepsize and another with a backtracking line search. The constant-stepsize scheme is analyzed under the assumption that the objective function has a Lipschitz continuous Riemannian gradient, whereas the backtracking variant does not require this assumption to establish stationarity of accumulation points. For both schemes, we prove that every accumulation point of the generated sequence is first-order stationary under the respective assumptions, without requiring compactness of the feasible set; compactness is needed only to ensure the existence of accumulation points. When the objective function has a Lipschitz continuous Riemannian gradient, we derive iteration-complexity bounds of order \(O(1/\sqrt{N})\) for projection-based stationarity measures for both schemes, together with the corresponding \(\varepsilon\)-complexity estimates. For the backtracking scheme, the complexity analysis additionally requires the trial line-search stepsizes to be uniformly bounded away from zero. Under the same respective assumptions, the generated sequences are also asymptotically regular. Finally, we illustrate the practical performance of the methods by solving constrained Karcher mean problems on the manifold of symmetric positive definite matrices.

We show an explicit version of the Van der Corput truncated Poisson summation formula (B-process). By using refined explicit exponential sum estimates, this improves the error term of previous explicit results by Patel and Yang (2024) and Arias de Reyna (2024). As an application, we obtain fine explicit estimates for the error terms in approximate functional equations for the Riemann zeta function, improving on previous explicit results of Simonič (2020) in certain ranges.

For a finite field $\F_q$, consider the triangular Cholesky map $\Gamma_{n,q}(U)=U^TU$ from upper triangular matrices to symmetric matrices. Generalized Cholesky theory describes the regular locus on which all leading principal minors are nonzero, but it does not determine the multiplicities or root ranks in a fiber over a singular target. We develop a fixed-target fiber theory for this singular boundary and answer three questions posed by Cooper and Whitlatch. Our principal results concern the zero fiber. Over $\F_2$ we construct an explicit, invertible, rank-preserving recursive bijection between square-zero upper triangular matrices and upper triangular matrices satisfying $U^TU=0$. Over every finite field we determine the entire rank distribution of this zero fiber: in even characteristic the square-zero recurrence, and hence the rank-refined equinumerosity, persists over every $\F_{2^e}$, whereas in odd characteristic its failure is measured by an explicit quadratic-character correction. We place these results in a uniform framework by proving an exact first-pivot recursion for the fiber cardinality over an arbitrary symmetric target. Its regular specialization gives the constant fiber sizes on leading-principal-minor cones, while its zero-pivot branch explains the new singular behavior. For binary diagonal targets we further compress the rank-refined count to $O(n^2)$ integer-arithmetic transitions and show that it depends on the order, not merely the number, of the diagonal entries. Executable implementations and exhaustive low-order checks are collected in an appendix.

BCH codes are among the most important classes of cyclic codes and have played a central role in coding theory and its applications. One of the fundamental problems in the study of BCH codes is to determine their exact minimum distances, which directly govern their error-correcting capability. Although the BCH bound provides a general lower bound, determining the exact minimum distance is often difficult, and many parameter families remain unresolved. In this paper, we investigate three conjectures and an open problem on binary BCH codes proposed by Chen, Xie, and Ding in \cite{Chen59}. We settle these conjectures on the exact minimum distances of three families of binary BCH codes by constructing codewords attaining the BCH bound. Beyond these conjectures, we further study the more difficult family codes and determine its minimum distance for some cases. We further study Open Problem 8.4: affirmative answers are obtained for the first two length families, while for the third family a sufficient condition is established and a counterexample shows that the unrestricted assertion does not hold in general.

We establish a whole-space solution framework in augmented spectral Barron spaces for uniformly elliptic parabolic equations with state-dependent principal coefficients. A frozen-symbol parametrix yields bounded Green and terminal operators and a one-derivative smoothing estimate on arbitrary finite horizons, without requiring the spatial variation of the principal coefficient to be perturbatively small. We then apply this linear framework to a finite-horizon Hamilton–Jacobi–Bellman equation. A semi-explicit gradient iteration converges on sufficiently short horizons at a Gamma-factorial rate in a whole-space augmented Barron norm, and its limit is identified with the stochastic-control value function. Finally, temporal Jackson approximation and spatial spectral Barron approximation give joint shallow cosine-network approximations in space and time for the value function and the optimal feedback, with quantitative error and neuron-count bounds. Taken together, these results connect variable-coefficient parabolic well-posedness and smoothing in augmented spectral Barron spaces with nonlinear HJB solvability and quantitative neural-network approximation.

We prove that the Hypergraph Neural Network, an invariant architecture with 3-body message passing, is a universal approximator for potential energy surfaces. Our main contribution is a multi-layer completeness theory. We show that $L$ layers of message passing on sparse, cutoff-based graphs achieve the same representational power as having access to the full $L$-hop neighborhood, provided the configurations are generic, satisfy an overlap condition and a connectivity condition. This provides the first rigorous justification for the common practice of using multi-layer message passing with a per-layer cutoff smaller than the physical interaction range, the setting used by virtually all practical graph neural network based machine-learned interatomic potentials. As immediate consequences, we show that both DPA3 and CHGNet architectures inherit universal approximation.

In the torn paper channel, a codeword is cut at random locations, and the resulting error-free fragments are delivered to the decoder as an unordered multiset. Although the capacity of this channel can be achieved using random code, decoding such codes generally requires exponential time. The interleaved-pilot construction of Shomorony and Vahid embeds a De Bruijn sequence among the symbols of a shifted erasure code and aligns only fragments that are sufficiently long through global statistical uniqueness. A subsequent local-alignment scheme by Liu and Raviv employs run-length-limited constraints and exclusive all-zero markers to identify pilot positions from local structure, substantially reducing the minimum fragment length that can be aligned. We further improve the method of Liu and Raviv by replacing its fixed pilot sequence with a multilevel successive local alignment and pilot-recycling procedure. In Liu and Raviv, the pilot sequence is chosen once and must simultaneously balance the length of the pilot sequence against the ability to align short fragments. Our construction removes this limitation by reusing pilot sequences across successive decoding levels. A pilot sequence with an independent random linear code is first used to align and decode the longest fragments. The information recovered at this stage is then recycled as a larger pilot sequence for the next stage.This process is repeated over multiple levels, so that progressively shorter fragments are recovered while the length of the pilot sequence remains small. Our construction depends on choosing a series of random linear codes, and we show that for any $\varepsilon>0$, there exists a choice of such codes which attains rate of at least $\varepsilon$ below the capacity, with high probability as the block length goes to infinity. Therefore, our construction achieves the capacity of the torn-paper channel.

Recently in the physics literature there have been a small, but growing, number of papers that have applied the Eynard-Orantin topological recursion procedure to spectral curves with ramification loci of infinite cardinality. However, there is a corresponding gap in the mathematics literature; indeed, it has not been checked rigorously whether the resulting infinite sums converge, or whether the resulting correlators have all the desired properties of the topological recursion. The present work aims to bridge this gap by defining topological recursion on a suitably broad class of spectral curves to cover the aforementioned physics applications and to rigorously prove that this definition has the properties one would intuitively expect. As a by-product, some interesting coincidences involving quantum curves are discovered, where two different spectral curves yield the same quantum curve, a situation that has, hitherto, not appeared in the literature.

We study the rational points on the star curve $X_0^*(N) := X_0(N)/W(N)$, the quotient of the classical modular curve $X_0(N)$ by the full group of Atkin–Lehner involutions, for squarefree levels $N$. Rational points on $X_0^*(N)$ parameterize $\mathbb{Q}$-curves, i.e.\ elliptic curves $E/\overline{\mathbb{Q}}$ that are isogenous to all of their Galois conjugates. Elkies conjectures that $X_0^*(N)$ has only CM or cuspidal rational points for all large enough $N$. We call any other rational points "exceptional". In this article, we provide new examples of exceptional points in genus 3 and 4, and we give evidence that no exceptional points exist in genus $g \geq 5$. Moreover, we investigate the underlying geometric reasons that might "explain" why these exceptional points arise in the first place, in the vein of Ogg and Mazur. In particular, we propose geometric explanations for Galbraith's exceptional points on $X_0^*(137)$ and $X_0^*(311)$.

For all $n \ge 2$, we obtain a homeomorphism of Cantor space that has a distributionally $n$-scrambled Cantor set containing a distributionally $(n+1)$-scrambled uncountable set, but does not have an $(n+1)$-scrambled Cantor set.

In this work, we study radically uncoupled learning in discounted general-sum Markov games. Assuming “$\mathsf{ETH}$ for $\mathsf{PPAD}$", we show that, for every fixed discount factor, there is no polynomial-time algorithm for computing inverse-polynomially accurate coarse correlated equilibria in discounted general-sum Markov games when players learn independently in decentralized settings. Complementing this hardness result, we provide what appears to be the first radically uncoupled algorithm with sub-exponential convergence guarantees to coarse correlated equilibria in discounted general-sum Markov games without imposing any structural restrictions on the game. Our algorithm is a layered variant of optimistic mirror descent with an increasing step-size schedule tailored to the multi-agent setting. Finally, we develop both full-feedback and partial feedback versions of the aforementioned algorithm and establish sub-exponential convergence guarantees for each case.

We prove uniqueness of finite-time geometric endpoints for the Möbius-invariant Willmore flow in $\mathbb{S}^3$ under uniform quantitative nonumbilicity. The multiplicity-counting varifolds converge, without reparametrization or Möbius renormalization, to a unique integral two-varifold. An intrinsic transport estimate gives quantitative total-variation convergence of the induced area measures on the fixed domain and bounded-Lipschitz Cauchy control of their pushforwards. Together with Allard compactness and rectifiability, this upgrades subsequential compactness to full-trajectory varifold convergence. The limit has generalized Euclidean mean curvature in $L^2$ with the natural endpoint lower-semicontinuity bound. For finite maximal trajectories with initial energy at most $8\pi$, Jakob's subsequential alternative becomes sequence independent: the limit is zero, or it has unit density and embedded Lipschitz support of genus zero or one. For Hopf-torus trajectories under the same energy bound, nonumbilicity is automatic and the anchored constant-speed profiles converge weakly in $W^{2,2}$ and strongly in $W^{1,2}$ and $C^{1,\alpha}$ for every $\alpha<\frac{1}{2}$. At infinite time, the same method yields a unique limit under an additional finite-dissipation-length condition.

Let \[1\longrightarrow N\longrightarrow G\longrightarrow D\longrightarrow1\] be exact, with $N\neq1$ free pro-$p$ of arbitrary rank and $D$ Demuškin. Writing $W=N/\Phi(N)$, we prove \[H^3(G,\mathbb{F}_p)^\vee\simeq W^D,\qquad \textrm{cd}_p G=3\text{ if and only if }W^D\neq0.\] Thus dimension three is detected on the first Frattini layer of the free kernel. Finite-rank kernels always satisfy the criterion, so a dimension drop requires an infinite-rank kernel, while the criterion remains stable under passage to open subgroups. On the fixed-point-free locus $W^D=0$, the remaining dimension-one/two alternative is governed by $H^1(D,W^\vee)$ together with the Hochschild–Serre transgression. Dualizing the latter produces a relation-defect class in the topological coinvariants of $W$, which admits an explicit description in terms of a lifted defining relation of the Demuškin quotient. This yields a complete criterion for the lower-dimensional branch and separates the top-degree fixed-point obstruction from the extension data controlling dimensions one and two.

We present a proof classifying continuous isometry invariant valuations on polytopes in hyperbolic space $\mathbb{H}^n$ as linear combinations of the intrinsic volumes. The argument relies on an adaptation of the spherical case recently established by the author that circumvents an approximation step which does not generalize to the hyperbolic setting.

The complexity of time-domain simulation of modern power systems has increased significantly because converter-based resources introduce control dynamics that must be simulated alongside slower system-level and fast electromagnetic dynamics. The resulting wide range of timescales may force classical time-domain solvers to use small timesteps, complicate the solution of nonlinear equations at each timestep, and reduce solver reliability under strongly nonlinear and multi-timescale transient conditions. This paper introduces RAPTOR, a first-of-its-kind time-domain simulation framework for power system dynamic simulations. At its core, RAPTOR introduces a new integration technique that represents the unknown trajectory of hybrid differential-algebraic equations (DAEs) over a time interval using a physics-informed random-projection neural network (PIRPNN) built from fixed Gaussian radial basis functions (RBFs). A nonlinear solver, e.g., Newton-Raphson, determines how to combine the fixed RBFs so that the resulting trajectory satisfies the governing equations. By combining RBFs with broad and localized shapes, RAPTOR can represent complex multi-timescale behavior over comparatively long time intervals. This enables larger effective simulation advances, fewer overall nonlinear solver iterations, and faster integration over long time horizons. Numerical studies on stiff RMS models, IEEE 9-, 14-, 39-, 57-, and 118-bus RMS simulation benchmarks, and EMT test cases show that RAPTOR accurately finds solutions with substantially fewer sequential simulation advances while offering superior accuracy-runtime performance. With speedups of over 10x in certain cases, these results showcase RAPTOR's potential to outperform long-standing and widely used integration methods, such as the Radau method and the trapezoidal rule.

We study operators acting on functions defined on the half-space, with methods inspired by sparse domination. Using the specific link between dyadic grids on the boundary and Whitney regions on the half-space, we obtain a very precise form domination with model operators of Hardy type. We introduce mixed-norm off-diagonal estimates that measures both tangential and transversal decay without requiring pointwise control in the transversal variable, substantially weakening the assumptions used in earlier tent space theories. This allows us to prove optimal weighted tent space extrapolation from local boundedness plus off-diagonal decay. The resulting framework yields new bounds for operators arising from non-autonomous elliptic and parabolic PDEs with non-smooth coefficients.

We study the Gödel–Löb provability logic $\mathbf{GL}$ for finite derived topologies on the ordinals. Assuming GCH and the existence of a measurable cardinal $\kappa$ which carries a coherent sequence of normal measures $\vec{U}$ with $o^{\vec{U}}(\kappa)\geq\kappa$, we obtain a cofinality and GCH preserving generic extension in which $\text{Log}_d(\kappa,\tau_n)=\mathbf{GL}$ for every $n<\omega$, where $\tau_n$ is the $n$-th derived topology. The result answers a well-know open question. Extending Beklemishev's theorem, we show that in the same model, for every $k<\omega$, $ \text{Log}_d(\kappa;\tau_{2k},\tau_{2k+1})=\mathbf{GLP}_2, $ where the two modalities of $\mathbf{GLP}_2$ are interpreted respectively by $d_{2k}$ and $d_{2k+1}$.

We study the phase boundary of the finite-lifespan frog model. For rate-one continuous-time simple random walk on an infinite, connected, locally finite transitive graph of superlinear growth, we prove that the critical-density curve is continuous and strictly decreasing, with $-\log\lambda_c$ locally bi-Lipschitz. The inverse critical-lifespan curve has the corresponding regularity wherever it is finite. Thus the critical-curve conjecture of Angel, de la Riva, Hermon, and Shi reduces to finiteness of the critical lifespan at every positive density. We identify the exact short-lifespan limit $t\lambda_c(t)\to\Delta\log(1/(1-p_c(G)))$, where $\Delta$ is the degree and $p_c(G)$ is the bond-percolation threshold, and extend critical-curve regularity and sharpness to a class of long-range frog models. The main tool is a local-to-global principle for activation processes generated by independent finite rooted ranges: local one-hit comparison implies comparison of global reachability and survival. The comparison allows disconnected ranges and source-dependent laws, and therefore also applies to activation models beyond random walks on a fixed graph.

We construct a compact, equiregular, ideal sub-Riemannian manifold of step $2$ such that, for every smooth positive measure, the $\mathrm{MCP}(K,N)$ fails for all $K\in\mathbb{R}$ and $N \in (1,\infty)$. This shows that the real-analyticity assumption in the measure contraction theorem of Badreddine and Rifford in arXiv:1712.09900v2 cannot be replaced by smoothness. Moreover, our structure is ideal, i.e. it admits no non-trivial abnormal minimizing geodesics. Although failures of the measure contraction property for ideal structures were recently obtained in the higher-step setting, our construction shows that the phenomenon can already occur on compact, equiregular, ideal structures of step $2$. The proof exploits a differential consequence of the measure contraction property, namely a uniform upper bound for the sub-Laplacian of the squared distance near its base point, and constructs a structure for which this quantity is unbounded.

In this paper we study the resultant of systems of homogeneous multivariate polynomials which are equivariant under the action of a direct product of symmetric groups. We first treat, in detail, the case of a product of two symmetric groups, and establish a decomposition formula for the resultant of such systems. We then show that this decomposition, together with the underlying combinatorics, extends to an arbitrary (finite) direct product of $r$ symmetric groups. Thanks to these decomposition formulas, we prove that the discriminant of a multivariate homogeneous polynomial invariant under a direct product of $r$ symmetric groups splits into a product of resultants of smaller size that are easier to compute.

For a $p$-adic field of any residual characteristic, we determine irreducible decompositions of the Weil representation restricted to the maximal open compact subgroups and the Iwahori subgroup by using the Schrödinger model. In particular, these decompositions show that the restrictions are multiplicity free.

This work studies modifications of the Gromov–Hausdorff distance in which the infimum is taken over pairs of morphisms of a given class $\FF$ — the class of morphisms of a subcategory of the category of metric spaces (the $\FF$-Gromov–Hausdorff distance). We prove that the $\FF$-distance depends Lipschitz-continuously on the class $\FF$ in the Hausdorff metric on morphisms. As a consequence, for second countable $C^k$-manifolds equipped with a metric compatible with the topology, the smooth distance coincides with the continuous one: $\dGH{C_k} = \dGH{C}$. We also introduce the partially continuous and partially locally constant distances (continuity is weakened to continuity on an open subset of the support of full measure) and prove that on the class of metric spaces they coincide with the classical Gromov–Hausdorff distance.

Let $\mathfrak{c}=2^{\aleph_0}$. We prove that $\ell_\infty$ contains $2^{\mathfrak{c}}$ pairwise non-isomorphic non-reflexive Grothendieck subspaces. They may all be chosen to contain the canonical copy of $c_0$ and to have infinite-dimensional reflexive quotient. This is optimal and answers by the authors in 2021. We also show that the relative $\ell_\infty$-Grothendieck property depends on the specified copy of $c_0$ and is not an isomorphic invariant of the subspace. More generally, for every closed $E\subseteq\ell_\infty$ there is an $\ell_\infty$-Grothendieck subspace isomorphic to $\ell_\infty\oplus_\infty E$; taking $E=c_0$ answers a problem by González et al.

We prove boundary partial regularity for the three-dimensional nonstationary Navier–Stokes equations in bounded domains with fractional Sobolev boundary graphs whose integrability exponent is greater than three. Within the fractional Sobolev graph setting, this lowers the boundary threshold from fifteen quarters to three. We work in Breit's boundary suitable weak-solution framework, with the mixed maximal-regularity pair $(5/3,15/14)$ explicitly included in the solution assumptions. The main difficulty is to recover pressure compactness without imposing higher regularity on the rough boundary coefficients. After a rigid rotation to the tangent plane and parabolic rescaling, we show that each boundary graph becomes small in both Sobolev multiplier classes needed for the local energy inequality and the critical Stokes estimate. Geometry and fluid excess enter a contradiction argument as independent small parameters, so the limiting problem is the standard flat Stokes system. A localized pressure decomposition separates the vanishing nonlinear and coefficient errors from the forced and homogeneous flat pressures. The error pressure converges strongly at the critical mixed integrability pair, whereas higher spatial integrability is used only for the flat problems. These estimates yield a one-step excess decay bound with a constant independent of the contraction radius. A physical-scale iteration, followed by interior–boundary Campanato patching, gives local parabolic Hölder continuity of the velocity at every regular boundary point when the force has integrability exponent greater than five halves. The boundary singular set is relatively closed and has zero parabolic Hausdorff measure in dimension five thirds. The geometric decay determines a uniform starting scale and does not restrict the final regularity exponent.

We study a two-parameter first-order hyperbolic relaxation approximation of the incompressible Navier–Stokes equations on the two-dimensional torus, under the derivative-level large-perturbation assumptions of Huang, Rohde and Zhang. Building on the auxiliary-system approach used in earlier pressure-convergence results, we introduce a non-autonomous linear corrector that carries the complete initial discrepancy, the linearized convection and the smooth consistency forcing. This corrector describes the full linear response; it is not asserted to be a purely acoustic component. An exact compensated energy identity, including a time integration by parts in the linearized convection term, yields an integrated gradient estimate for the velocity corrector. The two-dimensional Ladyzhenskaya inequality then bounds its quadratic self-interaction by $O(\delta^2)$ in squared space-time $L^2$ norm. Consequently, the nonlinear remainder has weighted energy $O(\delta^2)$, and its pressure and velocity components satisfy the $L^\infty(0,T;L^2)$ bounds $C_T\delta/\sqrt{\epsilon}$ and $C_T\delta$, respectively. In the sufficient parameter window $\delta^2\ll\epsilon\leq\mu_*\delta$, the pressure after subtraction of the linear corrector converges strongly to the incompressible pressure. A mean-zero Fourier-mode construction, transferred to the nonlinear system by the remainder estimate, shows that the uncorrected pressure need not converge even on time intervals bounded away from zero.

Let $\lambda_1(\Omega)$ and $T(\Omega)$ denote the first Dirichlet eigenvalue and torsional rigidity of a bounded convex domain $\Omega\subset\mathbb{R}^2$, and let $M=\max_\Omega u$, where $u$ is the torsion function. We prove the sharp inequalities $\pi^2/24<\lambda_1(\Omega)T(\Omega)/|\Omega|<\pi^2/12$. This proves the two-dimensional case of Conjecture 4.2 proposed by van den Berg, Buttazzo, and Pratelli. Its planar formulation was later restated as Conjecture 1.1 by Bañuelos and Mariano, who proved it for triangles and rectangles. The lower bound follows by combining Payne's strict estimate for $\lambda_1M$ with the sharp torsion-efficiency inequality $T(\Omega)\geq |\Omega|M/3$. For smooth strictly convex domains, we use an Airy stress potential to construct a convex body whose surface-area measure is the torsional first-variation measure. A sharp one-dimensional inequality for directional maximum profiles then yields a stronger geometric containment. For the upper bound, we factor the Pólya functional through the second torsion moment. A level-set torsion–perimeter inequality gives the factor $5/6$, while a sharp weighted one-dimensional estimate gives the factor $\pi^2/10$. Collapsing triangles and elongating rectangles show that both endpoint constants, as well as the efficiency constant $1/3$, are sharp.

A pseudorandom code (PRC) is a keyed error-correcting code whose codewords are computationally indistinguishable from uniform strings. We study public-key PRCs over fixed alphabets against adversarial deletions, where the deletion channel may both depend on the public encoding key and the transmitted codeword. Let $\gamma_q^{\mathrm{LCS}}$ denote the asymptotic normalised longest-common-subsequence length of two independent uniform $q$-ary strings. We prove that for every fixed $q\ge2$ no multi-message public-key PRC with a single-output decoder is robust against all such $\delta$-deletion channels for any $\delta>1-\gamma_q^{\mathrm{LCS}}$. For $q=2$, the current rigorous bound $\gamma_2\ge0.792665992$ rules out every constant $\delta>0.207334008$. The proof uses pseudorandomness only to transfer an LCS event from uniform strings to independently sampled codewords and therefore also the resulting collision argument is information-theoretic and requires no secret key. We also extend the argument to list decoding: for every fixed constant $L$, provided the message space contains at least $L+1$ messages, no such PRC with output lists of size at most $L$ is robust for $\delta>1-\gamma_2^{(L+1)}$. Since $\gamma_2^{(m)}=1/2+\Theta(1/\sqrt m)$, these thresholds approach $1/2$. Our bounds are specific to public-key-dependent adversarial channels and do not apply to oblivious edit channels.

Call a permutation of a set of positive integers admissible if it contains no increasing or decreasing three-term arithmetic progression in position order. Davis, Entringer, Graham, and Simmons proved that the positive integers are not admissibly permutable, partitioned them into three admissibly permutable sets, and asked whether two sets suffice. We study the canonical dyadic candidate S_A, the union of the blocks (2^{k-1}, 2^k] over even k >= 2. We establish several structural restrictions on admissible permutations, including an orbit obstruction, an asymmetric balance law, and the impossibility of placing all sufficiently large dyadic blocks as contiguous runs. For the main result, we reduce admissibility of S_A to feasibility of an order gadget on (M, 2M]: the order must avoid monotone three-term progressions and satisfy fifteen precedence constraints induced by the values 15 and 16. We then prove that a three-constraint core of this gadget is inconsistent for every M congruent to 0 mod 8, using zigzag propagation on arithmetic-progression ladders, a transfer lock, and mirror-flood induction. Since every relevant dyadic scale lies in this residue class, S_A admits no admissible permutation. Thus the canonical dyadic partition does not solve the two-set problem, which remains open. Independent SAT encodings and certificate checks provide auxiliary verification of the human-readable proof.

We introduce folded-algebraic matroids. In such a representation, every matroid element is replaced by a finite tuple of algebraic quantities, and transcendence degree agrees with matroid rank after one uniform scaling. The resulting class contains both algebraic and folded-linear matroids and is contained in the class of almost-entropic matroids, whose rank functions are limits of scaled entropy functions. We prove that the latter containment is proper. Our main result concerns the classical rank-three matroids $M(p)$ of Gordon. For every prime $p$, we show that $M(p)$ has a folded-algebraic representation over a field $K$ if and only if $K$ has characteristic $p$. We then use a point-identification construction that preserves almost-entropicity to obtain a $13$-element rank-three $3$-connected matroid $C_{2,3}$ that is almost entropic but not folded algebraic. Choosing a common element as dealer also yields a connected $12$-participant port with incompatible characteristic requirements. Finally, we record compact explicit witnesses and size bounds for several other separating regions among the representation classes.

Bayesian synthesis, which generates synthetic data by sampling from the posterior predictive distribution, is a popular approach for privatizing sensitive personal data. However, how attribute disclosure risk is affected by feature dimensionality, the number of individuals in the original dataset, and the amount of released synthetic information remains poorly understood. We propose a mathematically tractable Bayesian synthesizer for multivariate normal data based on a conjugate normal-inverse-Wishart prior for the mean vector and covariance matrix. The conjugate structure yields closed-form posteriors and enables direct investigation of an adversary's ability to infer records under different data and release settings. We then demonstrate several intuitive properties of synthetic data generation through several simulations. Specifically, we show that disclosure risk decreases with the size of the original dataset, but increases with the dimensionality of the feature space and the amount of synthetic information released, whether through the release of larger synthetic datasets or multiple generator realizations. Finally, the proposed synthesizer was used to generate synthetic versions of a polygenic risk score dataset, with the synthetic data exhibiting distributional properties comparable to those of the original data.