Papers

Recent research has demonstrated the possibility of exploiting the thermodynamics of coupled electrical oscillators to implement computational tasks such as matrix inversion. While physical implementations rely on thermal noise to drive equilibration, we show that the underlying dynamics reduce to a deterministic iterative algorithm. Building on the framework of Aifer et al., we analyze the moment evolution of the Ornstein-Uhlenbeck process governing thermodynamic symmetric positive definite (SPD) matrix inversion. We prove that to a first-order approximation, the covariance dynamics are mathematically identical to preconditioned gradient descent on the Frobenius norm of the residual $\tilde{A}^{-1}A-I$. This equivalence demonstrates that thermal fluctuations, while essential for physical thermodynamic hardware, are algorithmically redundant for convex problems with a single global minimum. We validate the resulting algorithm against Thermox (Duffield et al.), a stochastic thermodynamic simulator, achieving speedups exceeding 100,000-fold while remaining competitive with the Newton-Schulz iteration. We also demonstrate acceleration through Schur complement techniques. These results establish a rigorous link between analog thermodynamic computing, statistical physics, and deterministic optimization methods.

Observation tasking is critical to satellite operations, enabling observation of planetary phenomena, orbital debris monitoring, and space domain awareness. To improve the effectiveness of satellite operations, orbital maneuverability is introduced as a leading-edge concept of operations within constellation reconfigurability for response to dynamic events. Previous investigations regarding constellation reconfigurability consist of deterministic mission environments, in which the formulations consider a priori knowledge of the environment; however, observation objectives inherently involve uncertainty at any given time. As such, scheduling satellite operations must account for uncertainties to ensure adequate observation tasking. In response, we present a stochastic variant of the Multistage Constellation Reconfiguration Problem (MCRP) which is solved using Stochastic Dual Dynamic Integer Programming (SDDiP). We additionally explore other stochastic problem-solving techniques for solution quality comparison. To demonstrate each solution method, two computational experiments with stochastic target properties are conducted. The first concerns random orbital targets, and the second concerns simulated hurricanes. The results of the experiments demonstrate the effectiveness of the SDDiP solution approach over other stochastic problem-solving methods. Overall, the stochastic MCRP accounts for target stochasticity while obeying visible time windows and maneuver feasibility.

Let $(M,g)$ be a smooth orientable $2d$ Riemannian manifold of genus $\mathfrak{g}$ with Riemannian metric $g$ and connected boundary $\Gamma$. Let $\Lambda$ be the Dirichlet-to-Neumann map on $\Gamma$ and let ${\rm det}_\zeta(\Lambda)$ be its (modified, i. e. with zero mode excluded) $\zeta$-regularized determinant. It is well-known that the quantity ${\rm det}_\zeta(\Lambda)/|\Gamma|$ (where $|\Gamma|$ is the length of $\Gamma$) is a conformal invariant. It was shown by Edward and Wu (\cite{EV}) that this invariant equals one for $\mathfrak{g}=0$; in the case $\mathfrak{g}>0$ Guillarmou and Guillopé \cite{Guillarmou} found two explicit expressions for this invariant through the Rouelle and (respectively) the Selberg zeta-functions of the two surfaces of negative constant curvature from the conformal class of $(M,g)$: one is of infinite volume and complete whereas another has geodesic boundary. We present an elementary counterpart of the formulae of Guillarmou and Guillopé using the periods of holomorphic differentials on the double $2M$ of $M$ only. Our approach is based on the properties of the Hilbert transform of $M$ \cite{B,HilbKor} and the Kontsevich-Vishik-Friedlander-Guillemin regularization of the determinants of pseudodifferenial operators \cite{KV,Ww,F}. In particular, a connection between the length spectra of (uniformized) $M$, $2M$ and the periods of holomorphic differentials on $2M$ is established.

We study estimation of tail heterogeneity for non-identically distributed extreme observations subject to random right-censoring. In the uncensored setting, such heterogeneity is described by the event scedasis function, which measures the relative contribution of different design points to the upper tail. Under censoring, however, the observed tail heterogeneity is contaminated by the censoring scedasis functions, and applying uncensored techniques targets the wrong object. We propose a Beran-type estimator of the relative event scedasis, which is consistent under mild conditions. To obtain these results, survival analysis representations at an upper order statistics are extended to the non-identically distributed case; specifically, we develop conditional Nelson–Aalen and Beran theory on increasing intervals whose random endpoint is dominated, with probability tending to one, by a deterministic high local quantile. In particular, we derive a martingale array representation of the conditional Nelson–Aalen estimator with explicit error bounds depending only on the sample fraction and the bandwidth. Simulations demonstrate the finite-sample performance of the method, and an application to French property-casualty insurance claims illustrates how heterogeneous censoring can distort naive scedasis estimates.

We show that the numerical invariants $\nu$ and $|\varepsilon|$ coming from knot Floer homology are knot $n$-trace invariants for any integer $n$, resolving the remaining case in Hayden-Mark-Piccirillo. This extension allows us to construct new families of exotic pairs using Yasui patterns. Moreover, by studying the invariant $\widehat{\nu}(K)=|\varepsilon(K)|(2\nu(K)-1)$, we give a new topological obstruction to a $4$-manifold being a strong symplectic filling of any contact structure on the boundary.

We study the self-intersection time of the non-backtracking random walk on connected undirected graphs. For every fixed $\Delta \geq 3$ we show that the expected self-intersection time is $O(\sqrt{n} \log n)$ on $n$-vertex graphs with minimum degree at least $3$ and maximum degree at most $\Delta$. For regular graphs with a uniform spectral gap, we improve this to $O(\sqrt{n})$. We also show an $\Omega(\sqrt{n})$ lower bound on a class of regular expanders. Our upper bound on the expected self-intersection time implies an improved mixing time bound on Glauber dynamics for the Ising model on $\Delta$-regular graphs at the tree uniqueness threshold.

Alon introduced $d(s)$ as the largest integer $d$ such that every digraph on $2n$ vertices with minimum out-degree at least $s$ contains a subdigraph on $n$ vertices with minimum out-degree at least $d$. He proved that $s/2-d(s)=O(\sqrt{s\log s})$, and further asked whether this deficit can be bounded by an absolute constant. Steiner answered this question in the negative by constructing suitable tournaments, and showed that $s/2-d(s)=\Omega(\log s)$. Using a different construction, we show that the deficit grows at least on the square-root scale, rather than merely logarithmically, improving the best known lower bound due to Steiner from $\Omega(\log s)$ to $\Omega(\sqrt{s})$ and leaving only a factor of $\sqrt{\log s}$ between the lower and upper bounds. This also completely settles a question raised by Steiner for tournament hosts. More generally, in the broader setting considered by Alon, our construction applies whenever the prescribed subdigraphs contain any fixed positive proportion of the vertices of the host digraph rather than specifically one half.

In this article, we study two consequences of abstract six-functor formalisms. Firstly, we show that an abstract six-functor formalism can be extended to specific Ind- and Pro- categories of geometric setups. As an application, we can define the motivic stable homotopy theory for ind-pro-algebraic stacks such as the Hecke stack. Secondly, we also show that Cohomological Purity is functorial using the multisimplicial language developed by Liu-Zheng.

We formulate gene regulatory network reconstruction from RNA velocity as a sparse dynamical inverse problem. We show that control-only data can be structurally nonidentifying and characterize excitation conditions under which controlled perturbations restore identifiability by generating complementary regulator trajectories. To enable stable reconstruction from noisy data, we develop an integral sparse estimator that avoids numerical differentiation and derive recovery bounds separating stochastic error from systematic contributions due to latent-time uncertainty, kinetic-parameter error, numerical quadrature, and model misspecification. Synthetic experiments illustrate perturbation-assisted identifiability, improved conditioning, and the robustness of integral reconstruction. Applied to perturbation-resolved RPE1 RNA-velocity data, the framework yields an empirically full-rank design whose conditioning improves with perturbational diversity and a reconstructed network core stable under perturbation subsampling. Held-out evaluation further shows that identifiability and reconstruction stability do not imply uniform predictive improvement. These results connect perturbational excitation, identifiability, and stable sparse recovery in regulatory dynamical systems.

A path in an edge-coloured graph is called conflict-free if a colour is exclusively applied to one of its edges. A graph $G$ is considered conflict-free connected if every pair of vertices in $V(G)$ is connected by a conflict-free path. The minimum number of colours required to render a connected graph $G$ conflict-free connected is referred to as the conflict-free connection number. In this paper, we introduce several sharp conditions on the minimum degree sum of any $4$ independent vertices in $G$ to ensure that the conflict-free connection number of $G$ is bounded.

We determine the finiteness boundary of the Novikov exponential moment for a one-dimensional constant-volatility mean-reverting diffusion. A localised change of measure cancels the squared drift and leaves a Brownian Feynman-Kac functional with potential $q/2$, where $q=-\lambda'$. If $q(\theta+y)\sim\kappa_\infty y^2$, the exact spectral boundary is $\kappa_\infty\sigma^2T^2=\pi^2$; for cubic drift it becomes $c\sigma^2T^2=\pi^2/3$, and equality is divergent. On the spectral equality surface, the first lower-order transition occurs at power $4/3$, where an explicit coefficient separates the two sides. For symmetric finite pure-power tails, exact tuning generates the recursion $\beta_n=1+3^{-(n+1)}$. We prove that this recursion gives a complete classification of the class. Each individual tail is decided after finitely many comparisons, but the required depth is unbounded: arbitrarily long common critical prefixes can lead to opposite outcomes. The drift-removing stochastic exponential nevertheless remains a true martingale; under the physical law it has no higher moments on the steep-drift class, while the reverse density is essentially bounded.

We categorify a lifting of the integral form of the Cartan subalgebra for quantum sl(2), via categories of equivariant coherent sheaves on projective spaces.

Given a tournament $T$, Aboulker, Aubian, Charbit, and Lopes (2023) defined its clique number $\vec{\omega}(T)$ as the minimum clique number of a backedge graph of $T$, and raised the question: Which classes of tournaments are polynomially $\vec{\chi}$-bounded? Aboulker, Duron, Jacob, Kimbrough, Thomassé, and this work's authors (2026) showed that this holds for classes of tournaments whose arc sets may be written as the union of a bounded number of comparability digraphs. What about classes of tournaments that do not admit such a decomposition? The crossing tournaments of Nguyen, Scott, and Seymour (2025) are an example of such a class, as shown in the aforementioned 2026 work; we show that nonetheless crossing tournaments are polynomially $\vec{\chi}$-bounded by adapting a method of Davies and McCarty (2021) and Davies (2022). We additionally show that we cannot extend this result for crossing tournaments to tournaments with chordal graphs as backedge graphs.

In this paper, we show that the tensor rank of the $5\times 5$ permanent tensor is exactly $16$ over fields of characteristic zero, by proving the lower bound matching the known upper bound. The $5\times 5$ permanent tensor is a symmetric tensor corresponding to the monomial $x_1x_2x_3x_4x_5$, whose Waring rank is known to be $16$. Previously, it was known, by the higher-order Koszul flattening and Fischer's formula, that its tensor rank is either $15$ or $16$.

Embedding trained neural networks as surrogates within optimisation problems is an established practice in operations research. The prevailing approach uses feedforward neural networks (FNNs) with ReLU activations, whose piecewise-linear structure admits an exact but computationally intensive mixed-integer programming (MIP) reformulation as the networks grow. We advocate input convex neural networks (ICNNs) as structurally superior surrogates when the underlying response is approximately convex or concave. The convex architecture offers two computational advantages. First, the ICNN-MIP formulation tends to yield a tighter linear programming (LP) relaxation than its FNN-MIP counterpart, with no integrality gap in favourable instances. Second, ICNNs uniquely admit an LP-based reformulation via epigraph representations of ReLU activations, though this embedding is not always exact. When it is not, we exploit the properties of ICNNs to construct the strongest continuous relaxation over box domains, namely, the convex hull of the ICNN's graph, bounded below by the epigraph and above by the concave envelope; this construction is tractable under input convexity but hard for general ReLU networks. On this basis, we develop a branch-and-bound algorithm that builds this relaxation at each node, branches directly on input variables rather than intermediate variables as in MIP reformulations, and terminates at the root node whenever the epigraph embedding is valid. Case studies on humanitarian food aid, oil well routing, and wine blending show that ICNN surrogates match FNN accuracy and deliver gains in solve time and scalability, supporting ICNN as the default surrogate when the underlying function is convex, concave, or well-approximated as such.

Let $(X,T)$ be a topological dynamical system and $(\mathcal M(X),T_*)$ be its induced system. For a non-empty compact subset $K\subset X$, we define $\mathcal M(K)$ as the set of Borel probability measures supported on $K$. In this paper, we systematically study the relationship between various entropies of $(T,K)$ and of $(T_*,\mathcal M(K))$. We show that: \begin{equation*} \begin{aligned} & h_{\mathrm{top}}^{\mathrm{UC}}(T,K)>0 \iff h_{\mathrm{top}}^{\mathrm{UC}}(T_*,\mathcal{M}(K))=\infty, \qquad &h_{\mathrm{top}}^{P}(T,K)>0\iff h_{\mathrm{top}}^{P}(T_*,\mathcal{M}(K))>0, \qquad &h_{\mathrm{top}}^{B}(T,K)>0 \implies h_{\mathrm{top}}^{B}(T_*,\mathcal{M}(K))>0 , \end{aligned} \end{equation*} where $h_{\mathrm{top}}^{\mathrm{UC}}(T,K)$, $h_{\mathrm{top}}^{P}(T,K)$, and $h_{\mathrm{top}}^{B}(T,K)$ denote the upper capacity topological entropy, the packing topological entropy, and the Bowen topological entropy of $K$, respectively. Additionally, we present a counterexample involving a non-invariant set, demonstrating that the converse of the third assertion is not valid in general.

We give a negative answer to a question of Avramov, Buchweitz and Şega by constructing a commutative local finite-dimensional non-Gorenstein algebra $R$ with $\operatorname{Ext}_R^1(D(R),R)=0$; this question is related to the first Tachikawa conjecture. We also give a counterexample to a conjecture of Huneke, Şega and Vraciu on Tor vanishing over commutative finite-dimensional algebras. Finally, we construct a finite-dimensional commutative local self-injective algebra $R$ over $\mathbb{F}_2$ and an indecomposable non-projective $R$-module $M$ such that $\operatorname{Ext}_R^1(M,M)=\operatorname{Ext}_R^2(M,M)=0$, related to the second Tachikawa conjecture and answering a question of Dao.

We provide a general ansatz for the paracontrolled approach introduced by Gubinelli, Imkeller, Perkowski for treating singular SPDEs. The ansatz proposed covers a large class of equations. It is described via decorated trees that encode its coefficients and its stochastic data. The main novelty is the recursive definition of the paracontrolled stochastic iterated integrals that involve a well-chosen combinatorial set of words. The paralinearisation is performed via a lift of iterated paraproducts to modelled distributions. The main idea is to integrate localised iterated paraproducts in order to get a global object with a good regularity. We also provide a fixed point argument and the renormalised equation with this ansatz when one uses the preparation map formalism that encompasses the BPHZ renormalisation. The last main result is the convergence of the renormalised stochastic data via the BPHZ renormalisation using an adaptation of the spectral gap approach. One obtains in the end a general solution theory for parabolic singular SPDEs via the paracontrolled approach.

We study the growth of powers of mean-L-stable operators on Banach spaces. We show that, for linear operators, mean-L-stability is equivalent to uniform boundedness in density; this yields $\|T^n\|=O(n)$ on every Banach space. On Hilbert spaces we prove that mean-L-stability is equivalent to absolute Cesàro boundedness and obtain $\|T^n\|=O(n^{1/2-c_T})$ for some $c_T>0$. For positive mean-L-stable operators on abstract $L^p$-spaces, $1\le p<\infty$, we similarly obtain $\|T^n\|=O(n^{1/p-c_T})$, where in both cases the positive constant $c_T$ cannot be chosen uniformly over all such operators. For positive mean-L-stable operators on $p$-convex Banach lattices, we prove the bound $O(n^{1/p})$ and construct positive topologically mixing operators $T_p$ for which $\|T_p^n\|\asymp n^{1/p}$, where $1\leq p<\infty$. These operators satisfy a uniform weak $(p,p)$ orbit estimate, while the averages of $\|T_p^nx\|^s$ are bounded for $s<p$, of order $\log N$ for $s=p$, and of order $N^{s/p-1}$ for $s>p$. The operator $T_1$ is uniformly Kreiss bounded and has linear power growth, answering a question of Montes-Rodríguez, Sánchez-Álvarez and Zemánek (2005). Moreover, $T_1$ is mean-L-stable and mean Li–Yorke chaotic, while it is not distributionally chaotic. This answers a question of Bernardes, Bonilla and Peris (2020).

In Weinstein's symplectic "category", objects are symplectic manifolds and morphisms are canonical relations, which may not be composable; it is therefore only morally a category. We construct a true differential graded category which is morally dual to Weinstein's "category". The objects in this category are prequantum systems, associated to integral symplectic manifolds. The morphisms are a complex of differential forms twisted by a prequantum line bundle. We also consider the cohomology category, which turns out to be an (ordinary) linear category. In each case, Weinstein's morphisms are associated with currents dual to the forms. For the differential graded category, these are isotropic, or if preferred Lagrangian, currents, twisted by a section of the prequantum line bundle; in the case of cohomology, these currents are supported on integral Lagrangian submanifolds, equipped with a global covariant constant section of the prequantum line bundle. We then apply our methods to quantization, and show that for Kahler manifolds, a quantization in this category is related to holomorphic quantization. Along the way we show that in the case of prequantum systems, the Lefschetz action of ${\mathfrak {sl}}(2,\R)$ on differential forms, studied by Brylinski, Mathieu, Guillemin, and Tseng-Yau in the symplectic case, is promoted to an action of the superalgebra ${\mathfrak {osp}}(1|2)$ on twisted differential forms, with ${\mathfrak {sl}}(2,\R)$ as the even subalgebra. A first application of this superalgebra action is the vanishing theorem which shows the cohomology category is supported in middle dimension.

In 1970, Klingenberg proved a fundamental result on closed Riemannian manifolds with Anosov geodesic flows, asserting that such manifolds are without conjugate points. The proof, based on the Morse theory of the energy functional, had crucial gaps pointed out by Anosov, who saved the theorem by providing a completely independent proof. In this article, we fill the gaps in the original argument, and thus provide a full Morse theoretic proof of Klingenberg's theorem.

We study an inverse initial-data problem for a quasilinear transport equation with nonlinear and memory effects. The unknown initial state is reconstructed from time-dependent measurements on the outflow boundary, with prescribed inflow data. We first apply a Legendre–exponential time-dimensional reduction to transform the governing equation into a finite nonlinear system in space. We then develop a Carleman-weighted and Tikhonov-regularized Picard method. At each iteration, the nonlinear terms are evaluated using the previous iterate, leading to a linear minimization problem with a unique solution. A Carleman estimate for the principal transport operator is used to prove that, for a sufficiently large Carleman parameter, the resulting Picard map is contractive on a prescribed admissible set. Consequently, the method converges from an arbitrary initial guess in that set. We also establish stability with respect to noisy outflow data. These analytical results concern the truncated and regularized reduced problem. Two-dimensional numerical experiments demonstrate accurate reconstruction of single and multiple inclusions, robustness with respect to noise, and rapid convergence of the Picard iteration.

For a field $F$ of characteristic zero with a derivation $\delta$, we provide a necessary and sufficient condition for a system of differential equations \begin{equation*} \delta y_1=f_0, \quad\delta y_2=f_1,\quad\dots\quad,\delta y_n=f_{n-1}, \end{equation*} where $f_0\in F$, $f_1\in F[y_1]$, $\dots$, $f_{n-1}\in F[y_1,\dots,y_{n-1}]$, to have elementary first integrals.

We study minimal models and flops for foliations. We show that if $\mathcal F$ is a rank one foliation with canonical singularities on a normal projective $\mathbb Q$-factorial variety and $K_{\mathcal F}$ is pseudo-effective, then any two outputs of the $K_{\mathcal F}$-MMP are isomorphic. For co-rank one foliations on threefolds, we prove existence results for $D$-flops in the klt setting and, under additional hypotheses, in the F-dlt setting. By contrast, we construct examples showing that rank one foliations display pathologies absent from the classical MMP: flopping contractions need not admit $D$-flops, and nef and big canonical divisors need not give rise to canonical models, even in the category of algebraic spaces.

Next-generation data-channel applications of polar codes demand decoding algorithms with high throughput and low latency. The recently proposed speculative successive cancellation (Spec-SC) decoding reduces average decoding latency by speculatively executing the right branch of successive cancellation (SC) decoding in parallel with the left branch, verifying the result once the g-function is computed. In this paper, we propose new acceptance criteria based on Hamming distance and ellipsoidal distance that improve acceptance rates, enabling greater speed-ups for lower-rate nodes. We further show that combining code membership testing with speculative decoding accelerates high-rate nodes as well. Numerical results confirm that both approaches outperform fast simplified SC (FSSC) decoding and Spec-SC with the original acceptance condition in terms latency by a wide margin, with virtually identical error-rate performance.

A graph is $k$-degenerate if every subgraph has a vertex of degree at most $k$. We extend this notion to matroids, defining a loopless matroid $M$ to be $k$-degenerate if every restriction of $M$ contains a cocircuit of size at most $k$; $M$ is minimally $k$-degenerate if it has cogirth $k$ and every proper restriction of $M$ has cogirth at most $k-1$. Our main result characterizes extremal minimally $k$-degenerate matroids. We also extend the known arboricity bound for matroids, showing that $k$-degenerate matroids have arboricity at most $k$ and providing sharper bounds.

A finite subset $X$ of ${\mathbb R}^d$ is called a Ramsey set if for any number of colours $k$ there exists a dimension $n$ such that whenever ${\mathbb R}^n$ is $k$-coloured there exists a monochromatic congruent copy of $X$. The classification of Ramsey sets is one of the major unsolved problems in the field of Euclidean Ramsey theory. Towards this, Ivan, Leader and Walters recently asked whether adding a point to a Ramsey set outside of its affine hull necessarily produces another Ramsey set. In this note, we answer their question in the affirmative.

We prove exact boundary controllability for a class of one-dimensional singular/degenerate wave equations of Sturm–Liouville type, \[ u_{tt}-(x^\alpha u_x)_x-\beta x^{\alpha-1}u_x-\mu x^{\alpha-2}u=0, \qquad x\in(0,1), \] with a Dirichlet condition at the regular endpoint and a boundary control acting at the singular endpoint \(x=0\). The analysis is carried out in the fractional energy space % \[ X=\mathcal H^{\nu+1/2}\times \mathcal H^{\nu-1/2}, \] associated with the corresponding singular Sturm–Liouville operator. Using the spectral decomposition induced by Bessel functions, we establish admissibility of the boundary observation operator and derive precise lower estimates for the observation coefficients. Exact observability is obtained through Ingham-type inequalities together with an abstract observability result for unitary groups. The proof treats simultaneously the subcritical, critical logarithmic, and limit-point regimes by means of singular Sturm–Liouville theory and boundary traces identified through the Lagrange bracket of the singular Sturm–Liouville expression.

We prove that, for $n\geq 3$, there is no constant $C_n>0$ depending only on $n$ such that the Steklov eigenvalues $\sigma_k(\Omega)$ satisfy $|\partial\Omega|^{\frac 1{n-1}}\sigma_k(\Omega)\leq C_nk^{\frac1{n-1}}$ for every smooth bounded domain $\Omega\subset\mathbb R^n$ and every $k\geq1 $, providing a negative answer to an open problem posed by Girouard and Polterovich \cite{GiPo}. On the other hand, we prove that there exists a constant $C_n>0$ depending only on $n$ such that $|\Omega|^{\frac 1n}\sigma_k(\Omega)\leq C_nk^{\frac1{n-1}}$ for every smooth bounded domain $\Omega\subset\mathbb R^n$ and every $k\geq 1$. This estimate, combined with the isoperimetric bound of Colbois, El Soufi and Girouard \cite{colboisgirouard_steklov}, implies the bound $|\partial\Omega|^{\frac 1{n-1}}\sigma_k(\Omega)\leq C_nk^{\frac1{n-1}+\frac{n-2}{n(n-1)^2}}$, where the exponent of $k$ turns out to be sharp.

Diffusion-based quantum state tomography (QST) has shown promising results, but all existing methods implicitly adopt a single parameterization (typically Cholesky) without systematic evaluation. We present the first design space study of density matrix parameterizations for diffusion QST, introducing a geometric framework based on the Jacobian Gram matrix $\mathbf{J}^\top\mathbf{J}$. Our calibration of seven parameterizations at 2- and 3-qubit scales, validated by end-to-end training, reveals that geometric conditioning alone does not predict end-to-end performance: at 3-qubit scale, Hermitian direct ($\kappa = 2.0\times$) performs worse than Cholesky ($\kappa = 27\times$) at all shot levels—a $13.5\times$ isotropy advantage that translates into a fidelity disadvantage of up to $+0.51$. The 2-qubit ranking (Hermitian $>$ Bloch) reverses at 3 qubits (Bloch 0.907 vs.\ Hermitian 0.394). We provide a geometric explanation: unbounded parameterizations suffer projection-induced information loss because the PSD constraint couples diagonal and off-diagonal coordinates in ways the unconstrained model cannot respect, whereas the Bloch representation places the maximally mixed state at the center of the valid region, minimizing projection loss.