Neural network (NN) controllers achieve strong empirical performance on nonlinear dynamical systems, yet deploying them in safety-critical settings requires robustness to disturbances and uncertainty. We present a method for jointly synthesizing NN controllers and dissipativity certificates that formally guarantee robust closed-loop performance using adversarial training, in which we use counterexamples to the robust dissipativity condition to guide training. Verification is done post-training using alpha,beta-CROWN, a branch-and-bound-based method that enables direct analysis of the nonlinear dynamical system. The proposed method uses quadratic constraints (QCs) only for characterization of non-parametric uncertainties. The method is tested in numerical experiments on maximizing the volume of the set on which a system is certified to be robustly dissipative. Our method certifies regions up to 78 times larger than the region certified by a linear matrix inequality-based approach that we derive for comparison.

Search engines and information platforms are increasingly scrutinized for their role in spreading misinformation. Traditional responses often focus on detecting falsehoods or verifying the ultimate validity of claims. This paper argues that such a validity-centered framing is inadequate for the epistemic challenges of search environments.

For the partition graph $G_n$ on the set of partitions of $n$, we study the stratification induced by the local simplex dimension $\dim_{\mathrm{loc}}(λ)$, defined as the maximal dimension of a simplex of the clique complex $K_n=\mathrm{Cl}(G_n)$ containing $λ$. Using the previously established description of maximal cliques through a vertex in terms of star and top capacities, we define the simplex layers $L_r(n):=\{λ\vdash n:\dim_{\mathrm{loc}}(λ)=r\}$ and study their global structure. We formalize the resulting layer stratification, rewrite layer membership in terms of local capacities, and record its basic consequences, including conjugation invariance. We then investigate first occurrence of layers across $n$, introducing the indices $n_r^{\mathrm{first}}$ and the corresponding first-occurrence sets $\mathcal{F}_r$. For the initial layer values, we obtain explicit exact results; more generally, we record a finite first-occurrence table and several natural sequence questions. We also define the adjacent-layer edge boundary $\partial^E_{r,r+1}(n)$, consisting of edges joining $L_r(n)$ to $L_{r+1}(n)$, together with the associated one-sided and vertex-boundary variants. This provides an exact interface language for the layer stratification, distinct from the broader shell-type geometric language used elsewhere in the project.

For each positive integer $n$, let $G_n$ be the graph whose vertices are the partitions of $n$, with edges given by elementary transfers of one unit between parts, followed by reordering. We study the local simplex dimension in the clique complex $K_n=\Cl(G_n)$ as a geometric thickness invariant of $G_n$. For a partition $λ\vdash n$, let $τ_n(λ):=\dim_{\mathrm{loc}}(λ)$ be its simplicial thickness. This gives threshold thick zones $T_{\ge r}(n)=\{λ: τ_n(λ)\ge r\}$ and, relative to the boundary framework of $G_n$, a shell/core decomposition into outer shells $Sh_r(n)$ and inner cores $Core_r(n)$. Using local-morphology results established earlier in the series, we work with simplicial thickness as a local invariant. We prove that it is preserved by conjugation, that the induced thick zones, shells, and cores are conjugation-invariant, and that the antennas remain strictly one-dimensional in the simplicial sense and are excluded from all nontrivial thick zones. The first shell order at which a nontrivial shell can occur is therefore $2$, and the corresponding shell $Sh_2(n)$ is the triangular skin, while higher simplicial regimes form nested higher-order shells inside the triangular regime. We also develop a complete finite computational atlas for $1\le n\le 30$, giving first-occurrence tables for the regimes $T_{\ge r}(n)$ and supporting a finite-range rear-central thickening pattern.

For the partition graph $G_n$, whose vertices are the partitions of $n$ and whose edges correspond to elementary unit transfers between parts, we develop a degree theory with three levels: exact value theory, exact profile theory, and fibre-level geometry. Writing $n=T_s+q$ with $T_s=s(s+1)/2$ and $0\le q\le s$, we prove that every degree-maximizing partition lies in the support-maximal stratum and obtain the exact formula \[ Δ_n=s(s-1)+\lfloor\sqrt{4q+1}\rfloor-1 \] for the maximal degree in $G_n$. For a support-maximal partition $λ$, let $A(λ)$ and $B(λ)$ denote the numbers of active gap bonuses and multiplicity bonuses. We prove that the set of realized maximizing profiles is \[ Π_n=\{(a,b)\in\mathbb Z_{\ge0}^2:a+b=ρ(q),\ T_a+T_b\le q\}, \qquad ρ(q)=\lfloor\sqrt{4q+1}\rfloor-1. \] Thus the exact global theory stops at the profile level. For each realized profile we then study the corresponding fibre of maximizers: we prove nonemptiness, construct canonical representatives, obtain lower bounds for mixed fibres, and show that conjugation induces a bijection between the fibres for $(a,b)$ and $(b,a)$. We also classify exactly the first near-triangular fibre windows and formulate localization and stability questions for the remaining fixed-$q$ regime.

We study the numerical topology of the clique complex $K_n=\mathrm{Cl}(G_n)$, where $G_n$ is the partition graph on the set of integer partitions of $n$. Building on the previously established homotopy equivalence $K_n \simeq \vee^{\,b_n} S^2$, we shift the focus from qualitative topology to its numerical content. Our main objects are the Euler characteristic $χ(K_n)$, the derived sequence $b_n=χ(K_n)-1$, the clique counts $c_r(n)$, and several related maximal-simplex counts. We develop two exact counting languages for the same invariant. The first is the direct clique-counting formula $χ(K_n)=\sum_{r\ge 1}(-1)^{r-1}c_r(n)$, which expresses Euler characteristic through clique counts in the partition graph. The second is a nerve-side formula arising from the canonical good cover by distinct full star- and full top-simplices, which yields $χ(K_n)=χ(N_n)$, where $N_n$ is the corresponding nerve. We further use the classification of maximal simplices into star-, top-, and edge-type pieces to formulate a local-to-global counting framework based on local admissibility data and global deduplication. The paper is primarily organizational and computational. It fixes a consistent counting dictionary, separates intrinsic global counts from auxiliary based counts, records exact data for the full main sequence package on $1\le n\le 25$, and extends the low-dimensional clique-count layer through $n=60$. We do not claim closed formulas for $χ(K_n)$ or for the full family of clique counts. Rather, the paper provides a framework in which such questions can be studied systematically.

We study the partition graphs $G_n$ as a growing family of discrete geometric objects and introduce a formal framework for comparing their structures across different levels. The main tool is a family of Ferrers-translation maps \[ T_τ:G_n\to G_{n+k},\qquad (T_τ(λ))'=λ'+τ', \] defined for fixed partitions $τ\vdash k$. We prove that these maps are induced graph embeddings, giving a rigorous notion of translation overlay: an induced copy of $G_n$ inside $G_{n+k}$. As a consequence, every finite rooted induced motif persists to all higher levels under translation overlays, and every overlay-monotone finitely witnessed property has a stable emergence threshold. We apply this framework to obtain monotonicity for the extremal local invariants $Δ_n$, $Ω_n$, and $S_n$, and to establish strict threshold statements for a canonical family of theorem-safe motifs drawn from boundary, axial, and rear morphology. This yields a conservative structural language for discussing growth across $n$ while keeping exact transport separate from stronger typed or visual interpretations. We also record a compact atlas framework for first appearances, repeated patterns, and comparative growth profiles. In this way the paper isolates a theorem-level core for persistence and thresholds, and complements it with a weaker notion of self-similarity based on recurring finite motifs and repeated local fragments.

We develop a directional formalism for the partition graph G_n based on several canonical reference sets: the main chain, the self-conjugate axis, the spine, and the boundary framework. For each such set S, the graph distance d_S induces a shell structure and a local trichotomy of edges into inward, outward, and level classes. Passing from edges to paths, we define directional corridors as monotone inward geodesics toward a chosen reference set and prove that every vertex admits at least one. We then prove a structural non-equivalence theorem: for connected G_n, two nonempty reference sets induce the same edgewise directional field if and only if the difference of their distance functions is constant; in particular, distinct reference sets induce distinct directional fields. This gives a first precise formalization of anisotropy in G_n. We also show that every bounded neighborhood of a reference set is accessible by a monotone inward corridor, which gives a directional interpretation to previously established controlled regions around the axis, the spine, and the framework. Finally, we complement the strict theory with a computational atlas illustrating edgewise directional statistics, directional mixing, local invariant drift, and corridor-based transport profiles.

We study the outer geometry of the partition graph $G_n$, focusing on its canonical front-and-side framework, the family of nontrivial rectangular partitions, and the rear structures suggested by the visible geometry of the graph. We formalize the boundary framework $\mathcal B_n=\mathcal M_n\cup\mathcal L_n\cup\mathcal R_n$, where $\mathcal M_n$ is the main chain and $\mathcal L_n,\mathcal R_n$ are the left and right side edges, and we isolate the nontrivial rectangular family $\mathrm{Rect}^*(n)=\{(a^b):ab=n,\ a,b\ge2\}$ as a canonical discrete family marking the rear part of $G_n$. We prove that every nontrivial rectangular vertex $ρ=(a^b)$ has degree $2$, has exactly two explicitly described neighbors, and lies in a unique triangle of $G_n$. This leads to the notions of a rectangular ear, its attachment pair, and its support edge. We also prove that $\mathrm{Rect}^*(n)$ is an independent set in $G_n$, so the weak rectangular contour is not a graph-theoretic chain but a discrete rear marker family. For every genuinely rear rectangular ear, namely for $a,b\ge3$, we show that its support edge lies in a tetrahedral configuration of the clique complex $K_n=\mathrm{Cl}(G_n)$. To organize the interaction between different ears, we introduce support zones, support distances, and support corridors between attachment pairs. The paper also records a natural divisor-theoretic indexing of the rectangular family, presents a computational atlas in small and large ranges, and concludes with open problems concerning support-zone connectivity, inter-ear corridors, and canonical rear contours in $G_n$.

We study the degree landscape of the partition graph $G_n$, whose vertices are the integer partitions of $n$ and whose edges correspond to elementary transfers of one unit between parts, followed by reordering. Using the previously established local degree formula, we introduce the degree layers $D_d(n)$, the degree spectrum $Spec_D(n)$, and the numerical invariants $Δ_n$, $m_Δ(n)$, and $s(n)$. The main theorem provides an exact formula for the maximal degree. If $$ ρ(n):=\max\{r:T_r\le n\},\qquad T_r=\frac{r(r+1)}{2}, $$ and $$ ν:=n-T_{ρ(n)}, $$ then $$ Δ_n=ρ(n)\bigl(ρ(n)-1\bigr)+β_{ρ(n)}(ν), $$ where $β_r$ is an explicit budget function governed by a square--pronic threshold rule. We also prove that every maximal-degree vertex lies on the maximal-support stratum, and we obtain exact extremal classifications at the levels $n=T_t$, $n=T_t+1$, and $n=T_t+2$. The paper also includes a finite computation on the range $1\le n\le 60$, recording extremal multiplicities, representative extremal shapes, spectrum sizes, selected degree histograms, and first data on contact between the extremal layer and the self-conjugate axis. This computational part is deliberately limited in scope. It is descriptive rather than exhaustive, and is included only as a first numerical profile of the degree landscape.

We study the partition graph $G_n$, whose vertices are the integer partitions of $n$ and whose edges correspond to elementary transfers of one unit between parts. We introduce the simplex stratification of $G_n$: for each vertex $λ$, let $\dim_{\mathrm{loc}}(λ)$ denote the largest dimension of a simplex of the clique complex $K_n = \mathrm{Cl}(G_n)$ containing $λ$. This defines a decomposition of $V(G_n)$ into layers $L_r(n)=\{λ\in V(G_n): \dim_{\mathrm{loc}}(λ)=r\}$. We formalize the graph-theoretic interfaces between consecutive layers, called phase boundaries, and study the associated interface graphs and boundary thresholds. Using the previously established star/top description of cliques through a fixed vertex, we show that $\dim_{\mathrm{loc}}(λ)$ is determined exactly by the maximal star and top capacities through $λ$. This yields explicit local criteria for membership in higher simplex layers and reformulates their first appearance in terms of local star/top capacity thresholds. We also present an exhaustive computational study for $n\le 30$, including exact-layer thresholds, boundary thresholds, selected layer profiles, and the behaviour of the boundary framework. The computations suggest a rigid threshold pattern related to staircase partitions and their one-cell extensions, while the corresponding global statements are left as conjectures and open problems.

We study the partition graph $G_n$, whose vertices are the partitions of $n$ and whose edges correspond to elementary unit transfers between parts. We define the self-conjugate axis, its distance neighborhoods, and the thin spine, a first off-axis layer built from common neighbors of distinct axial vertices. We prove that distinct self-conjugate vertices are never adjacent, that the thin spine is a conjugation-invariant induced subgraph, and that axial and spinal concentration radii differ by at most one. Computations for $1 \le n \le 30$ show that the main local invariants are maximized near the axis and the spine.

For each positive integer $n$, let $G_n$ be the graph of integer partitions of $n$, where two partitions are adjacent if one is obtained from the other by an elementary transfer of a cell in the Ferrers diagram, followed by reordering. Previous work has studied the global homotopy type of the clique complex $Cl(G_n)$ and the local combinatorics of $G_n$ at a fixed vertex. This paper initiates the study of $G_n$ itself as a growing discrete geometric object. It introduces a structural language for the large-scale morphology of partition graphs, centered on the antenna vertices, main chain, boundary framework, self-conjugate axis, simplex layers, degree landscape, central region, and spine. Using local invariants from the companion local theory, it also defines canonical vertex layerings of $G_n$. A small computational atlas for $1 \le n \le 12$ is included to illustrate how these structures emerge and interact. The paper is intended as a foundational and exploratory contribution, providing a vocabulary, a first structural picture, and a set of open directions for future quantitative and asymptotic work.

For a fixed integer $n$, let $G_n$ be the graph whose vertices are the partitions of $n$, with adjacency defined by a single elementary transfer of a cell in the Ferrers diagram. In a previous paper, the clique complex $K_n = \mathrm{Cl}(G_n)$ was studied from a global homotopy-theoretic point of view. This paper studies instead the local combinatorics of the graph $G_n$ itself. For a partition $λ=(s_1^{m_1},\dots,s_t^{m_t})$, where $s_1>\dots>s_t>0$, we describe the admissible transfers from $λ$ in terms of its block structure. This yields a bipartite graph $B(λ)$ obtained from $K_{t,t+1}$ by deleting two explicitly determined families of edges, corresponding to singleton support blocks and unit support gaps. We prove that the graph induced on the neighborhood of $λ$ in $G_n$ is isomorphic to the line graph $L(B(λ))$. As consequences, we obtain an explicit formula for the degree of $λ$, a classification of all cliques through $λ$, and a formula for the maximal dimension of a simplex of $K_n$ containing $λ$. These local invariants are shown to depend only on an ordered binary datum associated with the support of $λ$. The results provide a local structural description of the partition graph and a combinatorial language for the study of larger-scale features of $G_n$.

We present the deepest systematic search for planets around mid-to-late M dwarfs to date. We have surveyed 8134 mid-to-late M dwarfs observed by TESS with a custom built pipeline and recover 77 vetted transiting planet candidates. We characterize the sensitivity of our survey via injection-recovery and measure the occurrence rate of planets as a function of orbital period, instellation, and planet radius. We measure a cumulative occurrence rate of $1.10\pm0.16$ planets per star with radii $>1\, R_\oplus$ orbiting within 30 days. This value is consistent with the cumulative occurrence rate around early M dwarfs, making M dwarfs collectively the most prolific hosts of small close-in planets. Unlike the bimodal Radius Valley exhibited by close-in planet population around FGK and early M dwarfs, we recover a unimodal planet radius distribution peaking at $1.25\pm0.05 \, R_\oplus$. We additionally find $0.954\pm0.147$ super-Earths and $0.148\pm0.045$ sub-Neptunes per star, with super-Earths outnumbering sub-Neptunes 5.5:1, firmly demonstrating that the Radius Valley disappears around the lowest mass stars. The dearth of sub-Neptunes around mid-to-late M dwarfs is consistent with predictions from water-rich pebble accretion models that predict a fading Radius Valley with decreasing stellar mass. Our results support the emerging idea that the sub-Neptune population around M dwarfs is composed of water-rich worlds. We find no hot Jupiters in our survey and set an upper limit of 0.012 hot Jupiters per mid-to-late M dwarf within 10 days.

Weighted graph states extend standard graph states by associating phases with entangling edges, and may serve as resources for measurement-based quantum computation (MBQC). We analyze how the two main fusion operations, Type-I and Type-II, act on weighted graph states. Type-I fusion operates identically to the unweighted case, merging two one-dimensional weighted graphs, while preserving edge weights and success probabilities. In addition, the pool of 2-qubit weighted graph states can be generated easily by GHZ states or Bell pairs. In contrast, Type-II fusion requires a logical qubit, which can be formed only for specific weight configurations, and with success probability below one-half, which is an obstacle one can avoid. When successful, it fuses the states correctly, but its failure outcomes destroy the structure of the graphs, removing the good-failure feature, known from ordinary graph states. We compute the change in the entanglement entropy of the resulting link due to the fused states being weighted graph states (for generalized fusion), and classify the resulting states of a general non-Bell projection. These results define the practical limits of the fusion-based construction of weighted graph states for MBQC.

We construct in detail the irreducible representations of the BMS group with super rotations in three and four dimensions that have the same rest frame momenta as the massive and massless Poincare point particles. We compare these representations to those of the Poincare group and also to the analogous representations of global BMS. We argue that these extended BMS representations are carried by a string rather than a point particle. The super rotations play a crucial role in our discussions.

We demonstrate 4.65-THz WDM/SDM transmission of 140-Gbaud PS-QAM signals over field-installed 12-coupled-core fiber cable with standard cladding diameter, achieving a record 0.455 Pb/s coupled-core capacity in a field environment. We also demonstrate 0.389 Pb/s over-1000-km transmission of spatial MIMO channels with >12 Tb/s/wavelength net bitrate.

We introduce the concept of quantum polymorphisms to the complexity theory of quantum constraint satisfaction. Via this notion, we build an algebraic framework of reductions between quantum CSPs, and we establish a Galois connection between quantum polymorphism minions and quantum relational constructions. By leveraging a contextuality property of quantum polymorphisms, we fully characterise the existence of commutativity gadgets for relational structures, introduced by Ji as a method for achieving quantum soundness of classical CSP reductions. Prior to our work, only a partial classification was known for a subclass of Boolean languages and for non-Boolean languages meeting specific structural conditions [Culf--Mastel, FOCS'25]. As an application of our framework, we prove that the quantum CSPs parameterised by odd cycles and the quantum CSP expressing quantum satisfiability of Siggers clauses are undecidable.

Can machine learning algorithms be implemented using chemistry? We demonstrate that this is possible in the case of support vector machines (SVMs). SVMs are powerful tools for data classification, leveraging Vapnik-Chervonenkis theory to handle high-dimensional data and small datasets effectively. In this work, we propose a chemical reaction network scheme for implementing SVMs, utilizing the steady-state behavior of reaction network dynamics to model key computational aspects of SVMs. This approach introduces a novel biochemical framework for implementing machine learning algorithms in non-traditional computational environments.

Future e-mobility calls for efficient electrical machines. For different areas of operation, these machines have to satisfy certain desired properties that often depend on their design. Here we investigate the use of multipatch Isogeometric Analysis (IgA) for the simulation and shape optimization of the electrical machines. In order to get fast simulation and optimization results, we use non-overlapping domain decomposition (DD) methods to solve the large systems of algebraic equations arising from the IgA discretization of underlying partial differential equations. The DD is naturally related to the multipatch representation of the computational domain, and provides the framework for the parallelization of the DD solvers.

We present and analyze a new finite element method for solving interface problems on a triangular grid. The method locally modifies a given triangulation such that the interfaces are accurately resolved and the maximal angle condition holds. Therefore, optimal order of convergence can be shown. Moreover, an appropriate scaling of the basis functions yields an optimal condition number of the stiffness matrix. The method is applied to an optimal design problem for an electric motor where the interface between different materials is evolving in the course of the optimization procedure.

We investigate whether spontaneous symmetry breaking can persist up to arbitrarily high temperature in ultraviolet-complete quantum field theories in four spacetime dimensions. We focus on completely asymptotically free models with gauge group $\mathrm{SU}(N_{c1})\times \mathrm{SU}(N_{c2})$ and two complex scalar fields, each transforming in the fundamental representation of one gauge factor and singlet under the other. The scalar potential contains quartic self-interactions together with a negative portal coupling between the two sectors. In the Veneziano limit, this class of theories was previously shown to admit fixed-flow trajectories for which one scalar acquires a negative thermal mass at asymptotically large temperature, leading to symmetry non-restoration. Here we extend that analysis to finite numbers of colours and flavours. We derive the finite-$N$ fixed-flow equations, compute the leading $1/N$ corrections to the large-$N$ solutions, and solve the full finite-$N$ system numerically. We find explicit finite-$N$ benchmark theories for which the scalar potential remains bounded from below, the gauge sector is asymptotically free, and one scalar thermal mass stays negative at arbitrarily high temperature. This provides an explicit perturbative example of infinite heat order in a four-dimensional ultraviolet-complete quantum field theory with a finite field content.

Using learning objectives to define designer intents for communicative visualizations can be a powerful design tool. Cognitive and affective objectives are concrete and specific, which can be translated to assessments when creating, evaluating, or comparing visualization ideas. However, while there are many well-validated assessments for cognitive objectives, affective objectives are uniquely challenging. It is easy to see if a visualization helps someone remember the number of patients in a clinic, but harder to observe the change in their attitudes around donations to a crisis. In this work, we define a set of criteria for selecting assessments--from education, advocacy, economics, health, and psychology--that align with affective objectives. We illustrate the use of the framework in a complex affective design task that combines personal narratives and visualizations. Our chosen assessments allow us to evaluate different designs in the context of our objectives and competing psychological theories.

Optical biosensors are indispensable in medical and environmental diagnostics, yet existing approaches are fundamentally limited in their sensitivity due to ensemble-averaged measurements. Digital biosensing has emerged as a promising solution for resolving individual binding events, thereby providing signals at very low analyte concentrations down to the single-molecule level. Here, we present a novel concept for digital optical biosensing empowered by dielectric Mie voids, combining nanoparticle-based contrast enhancement and deep learning for ultrasensitive biomarker detection. The resonantly trapped light in the air cavities of the periodic Mie void arrays ensures strong overlap between the near-fields and the single gold nanoparticles that are captured on the surface in the presence of the protein biomarker. Remarkably, this strong interaction creates high-contrast digital signals for the precise counting of single nanoparticles located both within and outside the voids, yielding efficient use of the entire sensor area for high sensitivity. We employ deep-ultraviolet (DUV) lithography for the scalable and low-cost production of Mie voids in silicon wafers and automated image analysis with a convolutional neural network for robust nanoparticle counting. As a proof of our concept, we demonstrate the detection of an important disease biomarker, interleukin-6 (IL-6), from small sample volumes at concentrations as low as 1.84 pg/ml, within the physiological range of healthy individuals. Owing to its scalability, precision, and adaptability, our digital nanophotonic biosensing approach based on silicon Mie voids establishes a versatile route for applications ranging from bioanalytics to health and environmental monitoring.

We analyze the behavior of the Euler method for delay differential equations under nonstandard assumptions on the right-hand-side function f, when evaluations of f are corrupted by informational noise. We provide theoretical upper bounds on the Euler discretization error and present results from the numerical experiments.

We study the impact of a minimal length, implied by generalized uncertainty principles and quantum gravity models, on unbounded (scattering) trajectories in the Kepler problem. The analysis is based on the precession of the Hamilton vector, which serves as a sensitive probe of orbital perturbations. Within the framework of the deformed Heisenberg algebra, we derive the correction to the trajectory arising from minimal length effects. It is shown that these quantum-gravitational corrections lead to a reduction in the scattering angle. In particular, for massless particles such as photons, the quantization of space results in a weakening of the gravitational lensing effect. Using available experimental data from the observation of the Einstein ring, we estimate the deformation parameter and the corresponding minimal length for the electron and Mercury. These findings highlight potential observational signatures of minimal length scenarios in high-energy astrophysics and gravitational optics.

The strategic demand for quantum hardware currently outpaces the availability of near-term devices, necessitating high-performance software emulators to validate novel protocols. We introduce Hyperion, a massively parallel, GPU-accelerated quantum emulator architected to bypass the classical memory walls inherent in strongly correlated quantum chemistry simulations. Hyperion leverages custom-optimized Sparse Matrix-Sparse Vector (SpMspV) kernels to natively accelerate exact matrix-vector multiplications, enabling strictly accurate State-Vector (SV) ADAPT-VQE simulations for up to 32 qubits on multi-node platforms. To scale beyond this hardware limit, we address the trade-off in pure Matrix Product State (MPS) emulators, where standard compression yields severe truncation errors and strict compression triggers intractable tensor rank explosions. We propose a novel partitioned emulation, namely the SV-MPS strategy: by routing non-interacting terms into an exact sparse SV core and delegating interacting terms to the MPS engine, this approach achieves emulation of 36 to 40 qubits with controlled approximations. This partitioning significantly reduces GPU resource requirements while maintaining robust accuracy across ADAPT-VQE iterations. Ultimately, Hyperion offers a high-fidelity platform dedicated to the development of new quantum algorithms for chemistry, enabling the modeling of realistic chemical systems at accuracies approaching the exact Full Configuration Interaction (FCI) / Complete Basis Set (CBS) limit.

We consider eight natural planar corridors, including the standard $\mathrm{L}$-shaped one, and characterize the rectangles that can move around their corners. As a bi-product we describe completely the corresponding rectangles with maximum area, as well as the rectangular parallelepipeds with maximum volume that can move around the corners of the spatial analogues of the considered eight planar corridors.

The Gaussian Process (GP) assumption is often used in functional data analysis. We propose a method to assess departures from the GP assumption, both in terms of the shape of the distribution and its potential dependence on covariates, using a sequence of functional moment regressions. Our methods are inspired by and applied to objectively measured minute-level physical activity data from the National Health and Nutrition Examination Survey (NHANES) 2011-2014 study. In this setting, we find that the GP assumption is not satisfied, quantify the associations between functional moments and covariates, and show that standard data transformations, such as the log transformation, do not resolve the discrepancy between assumptions and reality. We further show that when the effect sizes are moderate, inference on the functional fixed effects is largely unaffected by departures from the GP assumption. However, when effect sizes are small, both inference and prediction of subject-level data can be strongly affected. Extensive simulations support these findings. This pragmatic paper presents new methods for real data analysis, with implications for statistical methodology and for understanding human activity and health.