Papers

We present reflections on an experimental seminar studying the human and cultural roots of modern mathematics.

We prove a Michael-Simon inequality for a class of varifolds in Euclidean space that includes all integral varifolds with locally bounded first variation of arbitrary dimension and codimension. In codimension $2$, this inequality is sharp and implies the sharp isoperimetric inequality for varifolds, which attains equality precisely for the round disk. Our proof uses the optimal transportation approach of Brendle-Eichmair.

We study signal prediction and smoothing in hidden Markov models whose latent signal is a multi-type Wright-Fisher diffusion and whose discrete-time observations are categorical. We consider a repeated-sampling regime with finitely many inspection times and large within-time samples. Under neutrality and parent-independent mutation, the Bayesian predictive and smoothing laws admit exact representations as mixtures of Dirichlet distributions. We show that the one-step predictive and local smoothing laws converge in total variation to the corresponding Wright-Fisher transition and bridge laws determined by the neighboring population frequencies, thereby justifying plug-in approximations based on empirical proportions. For prediction, we derive explicit finite-sample bounds for this approximation and for bounded predictive functionals such as diversity, together with a finite-sample concentration bound for the marginal smoother. The joint posterior at the inspection times concentrates at the target frequencies and satisfies a Bernstein-von Mises limit with distinct interior and boundary regimes, with Gaussian and Gamma limits on different scales. The results allow arbitrary dependence across inspection times. A key technical ingredient is a fixed-interval tail bound for the coalescent-based dual death process.

In recent years, due to the growing reliance on large amounts of data that are communicated, analyzed, and utilized, which inherently contain sensitive and personal information, secrecy and privacy in communication have become increasingly important. We present nonasymptotic information-theoretic analyses of two fundamental secrecy problems under channel uncertainties: the information hiding problem and the compound wiretap channel. The former admits a game-theoretic formulation, where one party (an information hider and a decoder) seeks to embed secret messages into a host signal for later reconstruction, while the opposing party (an attacker) attempts to remove or degrade the embedded information. The latter generalizes Wyner's wiretap channel by allowing multiple potential channel states. The information hiding problem concerns active attacks during data transmission, while the compound wiretap channel addresses passive eavesdropping and information leakage. The two problems, both of which consider channels with uncertainties, can be studied under a unified framework by utilizing a covering argument and the Poisson matching lemma by Li and Anantharam. We derive novel one-shot achievability results for both problems that are applicable to any source distribution and any class of channels (not necessarily memoryless or ergodic), and that apply to both discrete and continuous cases. We also show that existing asymptotic results on both problems can be recovered by applying our results to discrete memoryless channels.

Given a complete, 3-dimensional, asymptotically flat initial data set for the Einstein-Maxwell equations with vanishing magnetic field, we show that there exists a small universal constant $\mathcal{C}$ such that $m\geq\mathcal{C}(\sqrt{\mathcal{A}/(16\pi)}+\mathcal{Q}^2\sqrt{\pi/\mathcal{A}}\,)$. An analogous statement holds for asymptotically hyperboloidal initial data sets with the ADM mass replaced by hyperbolic energy. Under the additional assumption of axisymmetry, the inequality may be further strengthened to include angular momentum. These results extend the Penrose-type inequalities recently obtained by Allen-Bryden-Kazaras-Khuri to the charged setting.

We show that if a sequence of expander graphs equi-coarsely embeds into a Banach space, then this Banach space fails Kasparov and Yu's Property (H). Consequently, no Banach space with Property (H) can be coarsely universal for all countable groups. We provide a Lean verification of our results.

We study the canonical formal solution space of a homogeneous $A$-hypergeometric system in a generic direction. We present as a finite-dimensional cokernel the quotient of that space by the span of the series obtained by intrinsic perturbation, taken over all exponents occurring in that space and all ordered negative support families, and we compute the codimension of that span. We show that the presentation and the codimension transfer to the holomorphic solutions on a common nonsingular domain. We apply these results, for every integer $q\geq 1$, to a homogeneous $A$-hypergeometric system of lattice rank 2 with $5q$ columns. We show that the canonical formal solution space of this system has dimension $12q^2$ and that the span of the series obtained by intrinsic perturbation has codimension exactly $\lceil 3q^2/4\rceil$ in that space.

Let $\Omega\subset\mathbb R^2$ be a smooth bounded domain containing the origin and invariant under reflection across the coordinate axes, and let $0<\lambda<\lambda_1(\Omega)$, where $\Lambda_1 (\Omega)$ is the first eigenvalue for $-\Delta$ on $\Omega$ under Dirichlet boundary conditions. For every fixed integer $k\ge1$ and all sufficiently small $\varepsilon>0$, we construct a positive solution of \[ -\Delta u=\lambda u e^{u^{2+\varepsilon}} \quad\hbox{in }\Omega, \qquad u=0\quad\hbox{on }\partial\Omega, \] which blows up at the origin as a tower of $k$ Liouville bubbles. The concentration scales are strongly separated. The proof is based on a Lyapunov–Schmidt reduction adapted to these different scales.

Weighted Hardy inequalities on a domain $\Omega\subset\R^n$ can be deduced from lower Hausdorff-content bounds on suitable visible parts of the boundary, and it is natural to look for visible sets described by straight segments, which are easier to verify than the curve-based visual boundary of Koskela and Lehrbäck. We show that this approach is subject to an intrinsic obstruction. If a boundary point $\xi$ is seen from $x\in\Omega$ along a segment satisfying a uniform thickness condition, then $\Omega$ contains a truncated cone with vertex $\xi$; consequently the quantitatively straight visible boundary $\partial^{\mathrm{str}}_{x,\alpha}\Omega\cap B(x,C_0d_\Omega(x))$ is a finite union of Lipschitz graphs, is $(n-1)$-rectifiable, and carries no $\lambda$-Hausdorff content for any $\lambda>n-1$. Since content conditions with $\lambda\le n-1$ already imply the $(p,\beta)$-Hardy inequality for $\beta<p-n+\lambda$ without any visibility assumption, segment-based criteria are either unsatisfiable or redundant: they never reach the regime $\beta\ge p-1$ in which visibility is genuinely needed. We show that the exponent $n-1$ is attained, so that the rectifiability theorem is sharp; we exhibit a boundary point lying in the visual boundary of Koskela and Lehrbäck but in no $\partial^{\mathrm{str}}_{x,\alpha}\Omega$; and we observe that raw straight visibility, without the thickness condition, escapes the obstruction, is sufficient for $\lambda\le n-1$, and remains open for $\lambda>n-1$.

Let $\Lambda=KQ/I$ be a finite-dimensional zero-relation algebra. We encode Crawley–Boevey tree modules over $\Lambda$ by finite rooted trees labelled by arrows of $Q$ and their formal inverses, and construct a deterministic finite bottom-up tree automaton recognizing exactly these encodings. We define an accepted tree to be an automata-induced tree brick when it has no non-trivial factor–image self-overlap, and use Crawley–Boevey's graph-map basis to prove that this is equivalent to brickness of the associated tree module. We also introduce local colourings of $Q_1$ and show that the arrow alphabet can be compressed without changing the tree data, graph maps, or brick property. The optimal number of colours for such a compression is the maximum of the in-degree and out-degree of $Q$. We conclude by asking whether the tree language consisting only of bricks is regular.

Choose uniformly among the combinatorial types of convex three-dimensional polyhedra with a fixed admissible number $e$ of edges, and let $V_e$ be the number of vertices. This resolves a fixed-edge limit-distribution question posed by Rüdinger: if $\beta_e=V_e/(e+2)$, then $\sqrt e(\beta_e-1/2)\Rightarrow N(0,1/32)$, equivalently $\operatorname{Var}(V_e)\sim e/32$. Using the classical rooted enumeration and asymmetry results of Bender and Wormald, we derive a relative lattice local limit theorem on every $o(e^{3/4})$ window, a quartic correction from the rate function on every $o(e^{5/6})$ window, precise moderate-tail constants, a quadratic moderate-deviation principle for every $a_e\to\infty$ with $a_e=o(\sqrt e)$, and a full speed-$e$ large-deviation principle with an explicit good rate function. All fixed standardised moments converge. When $e=3m$, the two extremal vertex counts have equal probability asymptotic to $\frac{6561}{32\sqrt2}(4/27)^m$. Rooted and unrooted fixed-edge laws are also uniformly exponentially close, with relative discrepancy $O(\rho^e)$.

We consider a discrete coagulation-fragmentation-diffusion system on a bounded domain of $\mathbb{R}^d$. In dimension $d\leq 2$, we prove mass conservation (that is absence of gelation) when the coagulation coefficient satisfies the critical additive bound $a_{i, j} \leq C(i + j)$ for some constant $C > 0$. We thereby close the remaining gap between the spatially homogeneous setting and the spatially inhomogeneous diffusive setting. In dimension $d\leq 4$, we impose a higher decay rate on the coagulation and fragmentation coefficients, but we nonetheless prove regularity and absence of gelation under mild assumptions on the diffusion rates.

An APN mapping $F:\mathbb{F}_{2^n}\to \mathbb{F}_{2^n}$ is a polynomial characterized by the non-vanishing property on 2-flats. In this work, we analyze notions that are closely related to this property. To understand which $k$-flats of $\mathbb{F}_{2^n}$ remain flats under $F$, we study the $k$-breaking. The function $x^{-1}$ has been studied in the past in this context—we extend this study to general mappings and characterize the 2-breaking of APN functions. Recently, two generalizations of the APN property have been introduced: $k$-strongly non-normality and $k$-th-order sum-freedom. Sum-freedom generalizes the non-vanishing property of APN functions to higher dimensional flats. We provide in-depth observations of the relations between the breaking property, strongly non-normality and sum-freedom. We introduce a fourth concept called $k$-strongly breaking, which implies the breaking property. We derive several structural results for both notions and give a characterization of a subclass of APN functions in terms of the 2-strongly breaking property. We propose a different perspective of the non-vanishing property via a natural character transformation, which is closely related to the sum-of-square indicator of the components of $F$. We derive a precise value for the total sum of the sum-of-square indicators of $F$. With this approach, we provide a simple answer to Open Problem 4 in IEEE Trans. Inf. Theory 52(9): 4160-4170, 2006. Moreover, it allows us to explore balancedness properties of polynomials, one of which characterizes component-wise APNness, for odd $n$, and provides a natural extension to any dimension. We show that Dillon's APN permutation and the Gold functions satisfy a related property, termed $k$-balanced, which is presented under our framework.

Let $r,t\ge 3$ be integers and $L=\{\ell_1,\ell_2,\ldots,\ell_s\}\subseteq [0,r-1]$ a fixed set of integers with $|L|\neq r$ and $\ell_1<\ell_2<\cdots<\ell_s$. For each integer $n$, let $\Psi_{r}(n,L,t)$ be the maximum number of $r$-cliques in an $n$-vertex graph whose $r$-cliques, viewed as a family of $r$-subsets of the vertex set, form a $t$-wise $L$-intersecting family. In this paper, we prove that $\Psi_{r}(n,L,t)=o(n^{|L|})$ when the sequence $\ell_1,\ell_2,\ldots,\ell_s,r$ does not form an arithmetic progression, and we give an asymptotic formula for $\Psi_{r}(n,L,t)$ when this sequence does form an arithmetic progression. When $t=2$, our results are exactly Helliar and Liu's results.

We determine the capacity region of a mixed binary multiple-access channel whose receiver can choose how each observation is formed. Two ports reveal the sum modulo two of the transmitted symbols with retention probabilities $\pg>\pb$; their roles are interchanged by an unknown state that remains fixed during the block. With one observation per channel use, no transmitter feedback, and zero switching cost, the region is $R_1+R_2\le\pg$ for every fixed error threshold below one, and a strong converse holds. For error thresholds below one half, the largest open-loop sum rate is $(\pg+\pb)/2$, whereas a fixed port permits only $\pb$. A causal receiver attains the larger region by learning the state from erasure flags while using every channel use for data. A posterior port rule makes only finitely many incorrect selections almost surely, with a uniformly bounded expected number of mistakes. Exact random linear-code formulas and output-counting converses quantify the finite-block consequences. At blocklength 256 with $(\pg,\pb)=(0.9,0.4)$, independent numerical validation supports 214 message bits at a one-percent ensemble failure target, compared with 148 bits for the best open-loop allocation in the same code ensemble. The resulting 44.59% increase is achieved with the same transmission and observation budgets. The analysis exhibits an explicit capacity gain from controlling the observation axis and separates that gain from information lost after a record has already been formed.

We determine the capacity region of a two-user binary interference channel with receiver-controlled observation and prove that causal port selection attains the interference-free limit. The physical–observation dual-axis formulation makes the observation kernel a design variable subject to explicit causal and resource constraints. For two equiprobable, initially unknown block states, matching converses and coding constructions give exact sum capacities of 1, $4/3$, and 2 bits per channel use for fixed-port, open-loop, and causal policies, respectively. These results hold for every fixed joint error threshold below $1/2$, under both average and maximal message error, with error averaged over the block state. Every receiver makes one observation per slot and switches ports at most once; the transmitters receive no feedback and the receivers do not cooperate. Causal observation thus doubles the optimal fixed-port sum capacity and exceeds the optimal open-loop sum capacity by 50%. With independent erasures of retention probability $p$, the exact causal region is $[0,p]^2$. A finite-blocklength bound accounts jointly for state-identification pilots and coding redundancy. We also optimize observation policies for fixed codes: an exact finite-horizon solution reduces message error from 10.76% to 7.52%, and a second-order approximation bound controls optimization over continuous observation directions. Linear-array designs and coding experiments examine the corresponding training, reliability, and control costs. The capacity result identifies a setting in which observation design removes the entire interference penalty, with the gain established by a converse as well as an achievable construction.

In this article, on Descartes' famous "circle theorem," we first attempt to explain how the philosopher could have arrived at the equation that he presents without demonstration in a letter to Princess Palatine Elizabeth of Bohemia in November 1643 (a previously unpublished proof, to our knowledge). Then we report some stages of the subsequent generalization of this theorem, whose Cliffordian flavor, via the equivalence of a quadratic form and the square of a linear form, still mobilizes mathematicians today. This statement, which, over time, has undergone different extensions, Euclidean and non-Euclidean, has found, in fine, a spinor formalization. We see there the proof of what we may call, in a Bachelardian style, an "inductive value" of the truth, which extends by successive generalizations. We conclude, more briefly, with the contacts between Elizabeth and Descartes, and the perhaps symbolic meaning of these considerations on "kissing circles," as they are called in Anglo-Saxon countries, in the context of their correspondence on the soul and the body, and the question of passions.

Two immiscible viscous fluids in pressure-driven channel flow can be unstable through their interface when their viscosities differ, at arbitrarily small Reynolds number — the instability Yih found in 1967. The unstable state is the interface itself: a curve inside the domain, separating the two fluids, whose displacement is governed by a partial differential equation of its own and which moves the fluid domain with it. Actuation is at one wall only, so the state to be controlled is reached across a fluid. We stabilize this interface at any rate below the limit set by the streamwise mean flow, by static feedback of the two velocity components of that wall, through a Fredholm backstepping transformation acting jointly on the two layers. The target is the two-fluid Stokes problem, shifted, with the transmission of stress between the fluids kept and the coupling of the interface into them removed. Two findings about the two-fluid spectrum carry the design: joined at the interface, the eigenvalue sequences of the two layers, which collide for a dense set of layer ratios when the layers are uncoupled, avoid each other uniformly, so no degeneracy has to be assigned; and the slowest mode left to viscosity is not a bulk mode but the capillary relaxation of the interface, whose rate is linear rather than quadratic in the wavenumber, which sets the band of actuated wavenumbers. The controller is tested on the nonlinear two-fluid channel, switched on against a saturated interfacial wave.

Let $z(n)$ be the maximum number of edges in a tripartite $3$-uniform hypergraph with $n$ vertices in each part and no copy of $K_{2,2,2}^{(3)}$ (a “box”). Erdős (1964) proved that $z(n) = O(n^{11/4})$, whereas the best previous lower bound, due to Katz, Krop, and Maggioni (2002), was $\Omega(n^{8/3})$. For each $q = 2^m$, we construct a box-free hypergraph with $q^4$ vertices in each part and $q^{11}$ edges, showing that $z(n) = \Theta(n^{11/4})$. The construction uses the power map $\tau(s) = s^{q^2-q+1}$ on $\F_{q^3}$, which sends the fibers of $s \mapsto \tau(s+1) + \tau(s)$ to pairwise skew affine lines over $\F_q$.

We introduce a transformation on partially ordered sets, termed the stellahedral transform, with notable features. It preserves the properties of being Eulerian, Cohen–Macaulay, and of being the face poset of a polytope. Furthermore, it admits an explicit geometric realization for convex polytopes and specializes to the construction that takes a simplex to the stellahedron. One motivation for this definition comes from the theory of toric $h$-polynomials and (augmented) Chow polynomials of Eulerian posets. We show that the right augmented Chow polynomial of an Eulerian poset $P$ agrees with the toric $h$-polynomial of the stellahedral transform of $P$. We use this perspective, together with $\mathbf{cd}$-index results due to Ehrenborg (2005) and Karu (2006), to prove two positivity results for augmented Chow polynomials: for Gorenstein* posets they are unimodal, and for face posets of polytopes they are $\gamma$-positive. Along the way we provide negative answers to two open questions concerning Eulerian and Gorenstein* posets. First, the question on the nonnegativity of Eulerian Chow polynomials, posed by Ferroni, Matherne, and Vecchi (2024). Second, the question posed by Athanasiadis and Kalampogia-Evangelinou (2023) on the real-rootedness of chain and Chow polynomials of Gorenstein* posets: these examples provide a novel application of a technique introduced by Murai and Nevo (2014).

We study the energy minimization problem for eight points on the unit sphere. For the logarithmic and Coulomb energies, we show that the unique global minimizer up to congruence is a square antiprism with height characterized by a unique stationarity equation. The proof is computer-assisted and fully verified in Lean. After this, we consider generalizations of the result to other important energies. For the Riesz $s$-energies, we provide a Lean-verified, non-computer-assisted proof that the square antiprism with height depending on $s$ is the unique global minimizer for all sufficiently large $s$, and a computer-assisted proof that this in fact holds for all $s\geq 0$. We also give examples of energies arising from completely monotonic potentials for which the square antiprism is not a global minimizer, answering in the negative a universality question of Cohn and Woo.

In this work, we develop a generalized Hamiltonian formalism for spatially nonlocal field theories whose Lagrangian densities depend explicitly on both the local field and its spatially reflected counterpart. Starting from a generalized variational principle, we derive generalized Euler-Lagrange equations and introduce a generalized functional derivative that consistently accounts for reflected-field contributions. The proposed formalism is applied to three spatially nonlocal nonlinear Schrödinger equations. For the Ablowitz-Musslimani equation, we construct, to the best of our knowledge, the first standard Lagrangian density formulated directly in terms of the complex fields and derive its complete Hamiltonian formulation. The same framework is subsequently applied to the two Lagrangian nonlocal nonlinear Schrödinger equations introduced by Velasco-Juan and Fujioka, yielding consistent Hamiltonian formulations that reproduce the corresponding generalized Euler-Lagrange equations. These results establish a unified Hamiltonian framework for a broad class of spatially nonlocal nonlinear field theories.

We derive explicit lower bounds for relative Fisher information by combining a variational principle with suitably orthogonalized Hermite-polynomial test functions. The resulting cumulant bounds are asymptotically sharp and yield lower bounds for Gaussian entropy deficits. We also establish quantitative versions of sharp min-entropy power inequalities in all dimensions. En route, we develop a stability result for Brzezinski's sharp bound for block sections of products of Euclidean balls, which may be of independent interest.

Cellular embeddings of graphs in surfaces have well-defined duality and minor (edge contraction and deletion) operations that interact in a natural way. A pseudosurface is obtained from a surface (compact 2-manifold) by a finite number of identifications of finite sets of points. Points that are created by the identifications do not have a neighborhood homeomorphic to an open disk and are known as pinchpoints. Embeddings of graphs in pseudosurfaces have been considered, both implicitly and explicitly, since the 1960s. Usually the condition that all pinchpoints correspond to vertices of the graph is imposed. However, this makes it difficult to define duality and minors for pseudosurface embeddings and have these operations interact in the expected way. We define the class of pseudocellular embeddings of graphs in pseudosurfaces, which allow pinchpoints at places other than vertices, in particular in the middle of faces or edges. A subclass known as quasicellular embeddings corresponds to previous embedding models due to Deneen, Shute, and Thomborson and to Huggett and Moffatt. Pseudocellular embeddings also generalize other structures, including the edge-point ribbon graphs of Ellis-Monaghan, Kauffman, and Moffatt, and cyclically ordered graphs or cogs (also known as rigid-vertex graphs). Duality and minor operations for pseudocellular embeddings have very simple and straightforward definitions using topological quotient operations. Pseudocellular embeddings of edgeless graphs have nontrivial structure, and we define some minor operations for those that are related to `t-minor' operations on bipartite graphs. We develop a family of polynomial invariants for pseudocellular embeddings and discuss connections to other polynomial invariants.

For every integer $d\ge4$, we prove strong stability for a subfamily of classical rotational hypersurfaces in $\mathbb H^d$ with normalized mean curvature one. The examples are complete, two-sided, properly embedded, and nowhere umbilic, with topology $\mathbb {R}\times\mathbb{S}^{d-2}$. An explicit positive supersolution yields a quantitative stability inequality for all compactly supported test functions. Consequently, endpoint horospherical rigidity fails in every ambient dimension at least four.

We show that there are integers $N$ and odd $D$ such that no algorithm can decide, from the rational coefficients of a homogeneous polynomial vector field $F$ in dimension $N$ of degree $D$, whether the origin is Lyapunov stable for $\dot Y=F(Y)$. This proves a conjecture of V. I. Arnold. We provide a Lean formalization of the proof. The smallest dimension $N$ for which we were able to prove undecidability is $N = 5$. An analogous undecidability result holds for global asymptotic stability and global exponential stability for non-homogenous vector fields. For dimension $N=2$ and homogenous vector fields, we establish decidability for all commonly used stability notions, building heavily on existing results.

We develop a representation-theoretic framework for computing projection constants of finite-dimensional subspaces of $C(K)$, where $K$ is a compact homogeneous $G$-space. Within this framework, we characterize the finite-dimensional $G$-invariant subspaces for which the $G$-equivariant projection is unique. These are precisely the finite orthogonal sums of full isotypic components in the Peter–Weyl decomposition of $L_2(K)$. For such subspaces, averaging shows that this unique $G$-equivariant projection is minimal; it is the restriction to $C(K)$ of the $L_2(K)$-orthogonal projection. Consequently, their projection constants are given by the $L_1$-norm of an explicit reproducing-kernel slice. We apply this method to spaces of $d$-homogeneous polynomials in high dimension. The examples range from Fourier analysis on the torus, through spherical harmonic analysis on real and complex Euclidean spheres, to genuinely noncommutative harmonic analysis on the unitary group, where representation theory becomes indispensable. For all these families, we obtain precise high-dimensional asymptotics at the square-root-of-dimension scale.

We study incidence problems for sticky sets of tubes. In the first part, we prove a sharp Furstenberg set estimate for sticky sets of tubes in $\mathbb R^3$ satisfying convex Wolff axioms. In the second part, we prove the Kakeya set conjecture in $\mathbb R^4$ for sticky sets of tubes satisfying convex Wolff axioms.

Spatial room impulse responses (SRIRs) capture directional distributions of acoustic sound-sources and their reflections. However, collecting SRIRs of moving sound-sources remains a challenge, requiring complex interpolations across measurements that account for multi-path spatial-temporal dynamics. This paper investigates the Wasserstein metric and displacement for evaluating interpolated SRIR echo densities in the spherical harmonic domain. We present novel sum-of-magnitude square expansions for efficiently fitting probability density functions, maximizing likelihood, inverse sampling, and computing spherical sliced Wasserstein interpolations. Experiments compare the Wasserstein displacements and metric to linear and geometric interpolations of SRIR image-source densities on a line-path, and demonstrate model-order reduction.

An Askey-type confluence scheme is developed for Jacobi multiple orthogonal systems of mixed type, with $q$ row weights and $p$ column weights. Rescaling the Jacobi variable near $0$ and near $1$ produces Laguerre systems of the first and second kinds. A Hermite system with one Gaussian–gamma row and $q-1$ exponential convolutions is obtained from Laguerre of the first kind, directly from the second kind, and directly from Jacobi. Explicit limiting weights and orthogonal forms are given, together with convergence of their mixed moments. For each fixed index, the recurrence coefficients and the entries of the bidiagonal factors also converge in the limit to Laguerre of the first kind, under nonvanishing conditions. The Laguerre family of the second kind is described by additive gamma convolutions. A closed product formula for its moment determinants holds for arbitrary positive real shape parameters and near-diagonal row multi-indices, yielding explicit normality conditions. Its forms admit contour and Rodrigues representations with terminating hypergeometric components: a multiple Kampé de Fériet series for $A$, and at most $q+1$ Srivastava–Daoust series for each $B$-component when $q\ge2$. For the Hermite family, normality holds at every Laguerre-admissible index under the stated separation conditions, and these indices form an infinite set. Its components have finite Hermite expansions and terminating Srivastava–Daoust representations. The three routes give the same limiting moments and normalized components, the Jacobi one being a single limit and the other two iterated. An appendix treats several derivative rows, where the connection determinant vanishes beyond a finite index range, with exact rank loss and diverging components. The one-row reductions recover the classical multiple second-kind Laguerre and Hermite systems.