Papers

For a given graph $F$, a graph $G$ is called $F$-free if it does not contain $F$ as a subgraph. A graph is color-critical if deleting one of its edges decreases its chromatic number. Let $F_1, F_2, \cdots, F_t$ be $t$ disjoint color-critical graphs with chromatic number $r+1$. For $2 \leq s \leq r$ and sufficiently large $n$, we determine the unique extremal graph with the maximum $s$-clique spectral radius among all $n$-vertex $\bigcup_{i=1}^t F_i$-free graphs.

Game semantics is an elegant approach to the formal semantics of reasoning and computation that grounds model-theoretic concepts of truth and validity in game-theoretic concepts that emphasize the dynamic and interactive aspects of logical reasoning. In Hyland-Ong games, plays are traces of interactions between a player and an environment and such games provide a naturally appealing semantics for computation that is derived from proof-search in logical systems. Such a semantics can be seen as providing an intensional theory of meaning for systems of logic in terms of (the computation of) proofs. In logic, an intensional theory of meaning for systems of logic is offered by proof-theoretic semantics; in particular, by base-extension semantics (B-eS), in which the model-theoretic interpretation of atomic propositions in a satisfaction relation is replaced by a validity relation which uses provability in `bases' of atomic rules. We establish a fully abstract correlation between B-eS and Hyland-Ong game semantics, employing techniques similar to those used by Sandqvist to give a sound and complete B-eS for intuitionistic propositional logic. We illustrate our semantics through the example of 4x4 Sudoku.

In this paper we study the problem of change point detection in functional time series where the observations are allowed to vary on both sparse and dense support. We address the problem of mean shift as well as the process volatility. Our methodology is based on the maximization of the conditional probability of change point given all the other parameters. Further, it has been proved that the proposed estimator is consistent.

We extend a previously introduced one-dimensional diffusion model on the space of probability measures, defined via the rearranged stochastic heat equation by, penalizing the dynamics with an additional entropy-driven gradient-descent term. By means of a splitting argument, we prove that despite the opposite effects of rearrangement and entropy minimization, the resulting penalized stochastic heat equation is well defined. We study several properties of the associated dynamics and show, in particular, that solutions admit a density satisfying a corrected version of the Dean–Kawasaki equation. The smoothing properties established for the stochastic heat equation are shown to persist.

The Pósa-Seymour conjecture establishes the minimum degree threshold required to guarantee the presence of the $k$th power of a Hamilton cycle in a graph. Following numerous partial results, Komlós, Sárközy, and Szemerédi confirmed the conjecture holds for all sufficiently large graphs. Treglown later conjectured the analogous minimum semi-degree threshold for forcing the $k$th power of a Hamilton cycle in a digraph. Subsequently, DeBiasio et al. proposed a conjecture on the minimum total degree threshold for the same problem. In this paper we settle the conjecture of DeBiasio et al. for $k=2$. Specifically, we prove that every sufficiently large $n$-vertex digraph with minimum total degree at least $8n/5-c$ contains the square of a Hamilton cycle, where $c=2$ if $n\equiv2,4\pmod 5$, and $c=1$ otherwise.

Let $D$ be a tournament and let $X\subseteq V(D)$. The inversion of $X$ reverses all arcs whose both endpoints lie in $X$ and leaves every other arc unchanged. A family of inversions is a decycling family if applying all of them produces an acyclic, equivalently transitive, tournament. The inversion number $\inv(D)$ is the minimum size of such a family. Let $Q_n$ be the tournament on $[n]$ obtained from the natural transitive tournament by reversing precisely the consecutive pairs $12,23,\ldots,(n-1)n$. Belkhechine, Bouaziz, Boudabbous, and Pouzet conjectured in their unpublished manuscript that a natural path-reversed family has inversion number exactly $\left\lfloor(n-1)/2\right\rfloor$. The same problem was later recorded by Bang-Jensen, da Silva, and Havet and by Alon, Powierski, Savery, Scott, and Wilmer. In this paper we resolve this conjecture.

We develop an efficient online algorithm to sample a sequence of discrete random variables using an entropy source of i.i.d. fair coin flips, in a standard model of real computation where real-valued probabilities are represented by rational approximations. For any sequence $F_1, F_2, \dots$ of probability distributions, our sampler generates $n$ outputs $X_1 \sim F_1, \dots, X_n \sim F_n$ using at most $\mathbb{E}\left[H(F_1) +\dots + H(F_n)\right] + O(\log n)$ coin flips in expectation while carrying $O(\log n)$ bits of persistent space, where $H$ is the Shannon entropy. Under standard assumptions, we prove that the space used by our sampler to achieve this information-theoretically optimal entropy rate is asymptotically optimal. The key idea is to replace the global arithmetic-decoding sampling scheme of Han and Hoshi (1997) with a local discrete uniform state, yielding an exponential reduction in space for a given entropy loss. Our approach applies to distributions with irrational probabilities and countably infinite supports, generalizing recent randomness-recycling methods beyond finite rational distributions with bounded denominator.

We study the Kac particle system associated with the spatially homogeneous Boltzmann equation with non-cutoff collision kernels in the moderately soft potential regime $-1<\gamma<0$. We prove uniqueness in law for the particle system and, in particular, establish an explicit convergence rate from the empirical measure of the Kac particle system to the weak solution of the Boltzmann equation in Wasserstein-2 distance, assuming that the initial datum $f_0$ has finite Fisher information and polynomial moments. To the best of our knowledge, this provides the first quantitative convergence rate for the Kac particle system in the soft potential setting.

The problem is to select k columns with maximal volume from a real mxn matrix X. We show that the logarithm of the volume is a set submodular function on columns of X, and for full column-rank matrices X with sufficiently large singular values, it is a non-negative non-decreasing function. As a consequence, traditional Businger-Golub QR with column pivoting is a greedy algorithm, with a relative error of at most 37 percent. In contrast, Gu-Eisenstat strong rank-revealing QR is a 1-interchange algorithm, with a relative error of at most 50 percent. The higher accuracy, under this metric, of the simple QR with column pivoting confirms its well known effectiveness in practice. For general, possibly rank-deficient matrices, we derive probabilistic bounds for the absolute error based on a smoothed analysis. The above analyses are extended to finding kxk submatrices of maximal volume in symmetric positive-definite matrices.

We show that a diagonal subfactor arising from a finite family of automorphisms of a $II_1$-factor $Q$ is regular precisely when the classes of the defining automorphisms occur with same cardinality and form a subgroup of $\mathrm{Out}(Q)$, a subgroup that happens to be isomorphic to the generalized Weyl group of the subfactor. Moreover, it turns out that the cleanest picture of regularity in diagonal subfactors is graph-theoretic, namely, a diagonal subfactor is regular precisely when its principal graph is a complete, regular, balanced bipartite multigraph, with the generalized Weyl group fixing its size and the common multiplicity of the defining automorphisms determining its regular edge multiplicity. Prior to the characterization of regularity, revisiting Bisch and Popa's observations on the standard invariant and depth of diagonal subfactors, we give an exact criterion for a diagonal subfactor to have any prescribed depth, in terms of a stabilizing sequence of subsets of $\mathrm{Out}(Q)$ consisting of non-reduced alternating words in the classes of the defining automorphisms, which proves useful in the characterization of regularity.

We study commuting isometric pairs $(V_1,V_2)$ that admit a factorization of the form $(V_2V_3,\,V_2)$ for some isometry $V_3$. To investigate this class, we introduce the notion of core operator associated with a commuting isometric pair $(V_1, V_2):$ \begin{equation*} H(V_1, V_2):=V_2V_2^*-V_1V_1^*-V_2^2V_2^{*2}+V_1V_2V_1^*V_2^*. \end{equation*} This core operator is closely associated with the domain Hartogs triangle. In particular, we establish a model for pure commuting isometric pairs satisfying $H(V_1,V_2)\geq 0$ via the coordinate multiplication operators on a vector-valued Hardy space over the Hartogs triangle. We also investigate the properties of the core operator and its applications to commuting isometric pairs.

We construct a short-time parametrix for the Fokker–Planck semigroup in Euclidean space. Among possible applications, we obtain smoothing and localization properties of the semigroup, the derivation of an approximate short-time polar decomposition for the semigroup, the construction of a parametrix of the resolvent, pseudospectral estimates, and estimates of the asymptotics of the number of eigenvalues of the Fokker–Planck operator. As a step in the proofs, we introduce a class of operators, which we call subsectorial operators, to which the Fokker–Planck operator belongs, and describe some of their functional analytic and spectral properties.

The article focuses on the inviscid and non-resistive limit of the incompressible magnetohydrodynamics (MHD) equations in a bounded domain. We derive sufficient conditions for the limit of Leray-Hopf solutions of the MHD system, which converges to the weak solutions of the ideal MHD system when $\mu,\nu\rightarrow0$, under the assumptions of uniform interior regularity and uniform equi-continuity at the boundary. Therefore, we establish an analogue of the Kato-type criterion to show that the energy dissipation rate is independent of the coefficients of viscosity and resistivity. Moreover, to analyze the different decay rates for the viscosity and resistivity, we obtain the limits of infinite and vanishing magnetic Prandtl number ($Pm=\frac{\mu}{\nu}$) for the incompressible MHD system. To the best of the authors' knowledge, this is the first rigorous result from a mathematical viewpoint on the limit of the magnetic Prandtl number.

We construct an explicit strict Lyapunov function for the one-dimensional linear wave equation with a locally distributed damping and Dirichlet boundary conditions.

A decision-maker's risk preference is inherently unstable and may adjust in response to external information, shaping subsequent choices and their outcomes. This paper develops a persuasion framework to study how information can be designed to steer risk preferences and decision results. In our framework, a receiver starts with an initial risk preference represented by a coherent risk measure and revises it after observing a system state generated by an information rule claimed by a sender. The revision must preserve time consistency of risk evaluations before and after the state realization. We investigate two models under this framework: one focuses purely on how receiver's risk preference is influenced by information, the other also incorporates receiver's decision. For both models, we investigate the sender's optimal information design problem by analyzing the induced distribution of posterior beliefs over states. Each belief leads to specific preference revisions and corresponding conditional risk assessments. We identify conditions under which information design benefits the sender across several settings and illustrate the framework's potential in risk management through an application to reinsurance design.

So far the global well-posedness of strong solutions for the 3D non-diffusive Boussinesq system with large initial data remains a remarkable open problem. In this paper, we solve this problem in the regime of large Prandtl number. More precisely, we prove the global existence and uniqueness of strong solution for this 3D Boussinesq system associated with initial data $(u_0,\theta_0)\in H^{\frac{1}{2}}(\mathbb{R}^3) \times (L^1\cap L^s(\mathbb{R}^3))$ with $s>3$, provided that the Prandtl number is sufficiently large (the threshold depends only on a scale-invariant norm of $(u_0,\theta_0)$); moreover, for the non-constant temperature patch initial data, we establish the global persistence of $C^{1,\gamma}$, $W^{2,\infty}$, and $C^{2,\gamma}$ ($0<\gamma<1$) boundary regularity of the evolved temperature patch, with corresponding estimates uniform in the large Prandtl number regime. Furthermore, we rigorously justify the limit as the Prandtl number tends to infinity and show that the patch solution of the 3D Boussinesq system converges to the unique patch solution of the 3D Stokes-transport system, and that the patch boundary regularity in $C^{1,\gamma}$, $W^{2,\infty}$, and $C^{2,\gamma}$ is preserved globally in time. In particular, our result for the 3D Stokes-transport system can be viewed as the 3D analogue of the main result in Grayer II [ARMA 2023] concerning 2D Stokes-transport system.

This paper establishes a direct analogue of the classical Coding Theorem in the setting of symmetry groups. We consider computable bijections on the set of binary strings and define the symmetry prior of a string x as the probability that a randomly chosen symmetry from a given group G has x as its unique fixed point. We show that for any fix-retractable symmetry group G, a group admitting a computable section that selects an isolating symmetry for every string, the symmetry prior is a universal lower semi-computable semi-measure. In this case, the Geometric Coding Theorem holds. This result is a coding-theoretic restatement of a theorem of Trejo, Kreinovich and Longpré, who showed that the complexity of describing a string by a symmetry with that string as its unique fixed point equals its Kolmogorov complexity. Our contribution is to recast it in terms of algorithmic probability and to treat the symmetry group as a parameter, identifying fix-retractability as exactly the condition under which symmetry complexity collapses onto Kolmogorov complexity.

Let $T$ be a $\rho$-contraction on a Hilbert space. We establish sharp bounds for the functional calculus of $T$ for matrix-valued analytic functions on the unit disc. These are complete versions of bounds established by Drury and by Schwenninger and the second author.

This paper introduces the Dynamical Vehicle Orienteering Problem (DVOP), a generalization of the Orienteering Problem (OP). The OP maximizes the reward collected from spatial targets under a limited travel budget; the DVOP extends it by accounting for both external and vehicle-actuated forces. We study the DVOP in the context of multi-rotor Unmanned Aerial Vehicle (UAV) flight planning, using a three-dimensional Point-Mass Model (PMM) constrained by maximum velocity and acceleration magnitudes and subject to gravitational acceleration, with the travel budget expressed as a maximum flight time. Because the DVOP couples reward maximization with time-optimal trajectory planning, it cannot be formulated as a simple graph problem and solved exactly without relaxing or under-actuating the vehicle dynamics. We therefore propose two solution approaches: a Branch-and-Bound (BnB) procedure that combines Non-Linear Programming (NLP) and Mixed-Integer Linear Programming (MILP) to provide high-quality solutions, and a Large Neighborhood Search (LNS) metaheuristic that supplies an initial reward bound and scales to instances intractable for the BnB. The BnB relies on a novel MILP formulation of travel costs based on minimum-time trajectory primitives through target triplets, yielding a tight reward upper bound, while the LNS uses limited thrust decomposition to compute fast, high-quality PMM trajectories. Experiments on benchmark instances show improvements of up to 37 % over state-of-the-art solutions for the Kinematic Orienteering Problem, and a real-world deployment on a multi-rotor UAV verifies the proposed PMM solution trajectories.

We study uniform versions of the diameter two properties, that is, the uniform slice-D2P, the uniform D2P and the uniform SD2P, as the property that every ultrapower over any free ultrafilter over $\mathbb N$ has the respective diameter two property. The aim of this note is to prove that all the uniform diameter two properties are different eachother.

We give an elementary proof that the coordinate ring of the variety of nilpotent matrices is $F$-regular; over an infinite field $K$, this ring also arises as the nullcone for the conjugation action of the general linear group $\textrm{GL}_n(K)$ on the polynomial ring $K[X]$, where $X$ is an $n\times n$ matrix of indeterminates. We prove that the divisor class group of the coordinate ring is the cyclic group $\mathbb{Z}/n\mathbb{Z}$. We then study the case of symmetric nilpotent matrices, where the picture is completely different: the coordinate ring is not normal for $n\geqslant 2$; for $K$ algebraically closed of characteristic other than two, we prove that the coordinate ring is an integral domain precisely when $n$ is odd.

We prove local uniform finite cyclicity of the labelled $H_{14}^{3}$ hemicycle in a full neighborhood of its base field in the twelve-dimensional space of planar quadratic vector fields. Thus there is one fixed two-sided annular neighborhood of the compactified graphic in which the number of isolated limit cycles is bounded uniformly for every sufficiently close quadratic field. A local analytic slice theorem is part of the result: the displayed five-parameter source-normalized family is transverse at the $B=0$ field to the seven-dimensional action of affine phase changes and positive constant time rescalings. This removes the normalization and transfers the same cyclicity bound to the full quadratic coefficient space. The normalized problem combines a noncompact period annulus, two semihyperbolic endpoints at infinity, and a degeneration at the upper equatorial point. We replace the unavailable global Poincare map by finitely many stopped transition maps and prove an exact multiplicity-preserving correspondence between collar cycles and displacement zeros. At the noncompact source a matched nonlinear return on a common physical domain, controlled through six derivatives, gives a two-zero bound by Rolle's theorem. At the two saddle-nodes an exhaustive physical incidence and scale decomposition reduces all limits to source, mixed, hyperbolic, central, and lips estimates. A finite specialization argument then extends these bounds across coefficient and identity strata. The resulting bound is existential and is not claimed to be sharp. In the terminology of the quadratic finite-cyclicity program, this completes the labelled $H_{14}^{3}$ open case.

Motivated by algebraic correspondences and linear operators associated with volume and Lorentzian polynomials, we introduce matroid correspondences and their polymatroid analogues. A matroid correspondence defines a functor between poset categories of matroids whose morphisms are matroid quotients, and various standard functors, including deletion, contraction, free extension, truncation, intersection, union, and pullback, arise in this way. We show that these correspondences preserve representability and algebraicity under natural hypotheses. In the polymatroid setting, we establish compatibility with multisymmetric lifts. Finally, we relate this construction to the supports of linear operators with Lorentzian symbols.

This article is devoted to the inverse problem of determining the zeroth- and first-order coefficients, depending on both the time and space variables, in a parabolic equation from partial boundary measurements of the flux generated by Dirichlet excitations. More precisely, we establish the unique determination of a time-dependent convection term and potential from the partial Dirichlet-to-Neumann map associated with the corresponding parabolic equation, where the Neumann measurements are restricted to the portion of the boundary illuminated from a point located outside the closure of the domain. Our main objective is to extend to parabolic equations several observations that have thus far been confined to the elliptic setting and to exploit these properties to recover a general class of coefficients depending on both time and space variables. To achieve this, we develop a suitable class of special solutions to the parabolic equation, introduce a new family of Carleman estimates with nonlinear weights, and combine these tools with partial data results for the X-ray transform. The principal difficulty of the present work stems from the fact that, unlike the existing literature on partial data inverse problems for parabolic equations, both the phase of the special solutions and the weight of the corresponding Carleman estimates are nonlinear. As a byproduct of our analysis, we establish a new class of Carleman estimates for parabolic equations with nonlinear weights, which may be viewed as the parabolic analogue of the notion of limiting Carleman weights in the elliptic setting.

Commensurating actions govern how a group can act on non-positively curved cube complexes. We obtain a complete picture of them for a class of finitely generated groups acting on rooted trees: contracting self-similar branch groups. The main application is a proof of Property FW for the iterated monodromy group of the subdivision rule generating the classical square Sierpiński carpet. This is the first example of an infinite finitely generated amenable group with Property FW, answering a question of Cornulier. As another application, we show that a contracting self-similar regular branch group does not have Property PW. In particular, the Grigorchuk group does not have Property PW, answering another question in the literature.

We study a nonlinear tensorial wave-Dirac system on $(1+3)$-dimensional asymptotically flat spacetimes as a semilinear model motivated by the Maxwell-Dirac and Einstein-Dirac systems. The purpose of this model is to isolate the interaction between the null geometry of antisymmetric tensor fields and the intrinsic first-order geometry of the Dirac equation while avoiding the derivative loss mechanism of the Einstein equations and the gauge structure of the Maxwell equations. Our analysis preserves the first-order nature of the Dirac equation throughout the nonlinear argument. The Dirac current provides the fundamental energy identity, while quantitative spacetime estimates are obtained from the wave equation arising from the squared Dirac operator. Combining these ingredients with integrated local energy decay estimates and $r^p$-weighted energy hierarchies, we establish a coupled energy method for the tensorial and spinorial components. A key observation is that the Clifford algebra is compatible with the null decomposition of antisymmetric tensor fields and excludes the most singular nonlinear interactions. As a consequence, we establish the global existence of small-data solutions together with quantitative weighted energy and decay estimates. Remarkably, combining the null structure with dyadic argument, weak decay such as $t^{-\frac12-\delta}$ is sufficient to obtain nonlinear stability of the system, in the spirit of \cite{DHRT}. We expect that the geometric ideas developed in this paper provide a useful starting point for the study of more general nonlinear Dirac systems, including the Maxwell-Dirac and Einstein-Dirac equations.

Gigena et al. [npj Quantum Inf. 11, 82 (2025)] proved the exact quantum maximum of the doubly-tilted CHSH functional $B_{\alpha\beta}=\alpha\langle A_0\rangle+\beta\langle B_0\rangle+\mathrm{CHSH}$ and observed that the NPA level required to reach it grows without evident bound toward the critical line $\alpha+\beta=2$. We quantify the mechanism on the symmetric slice $s=2-\alpha-\beta$: (i) the quantum value leaves the local bound cubically, $c_Q=4-s+s^3/6-s^4/36+O(s^5)$; (ii) each NPA level overshoots quadratically, $c_k(s)=4-s+a_k s^2+O(s^3)$, with the almost-quantum coefficient computed exactly, $a_{1+AB}=3/64$; (iii) the divergence of the required exact level is equivalent to positivity of the single sequence $(a_k)$ - proven for every $k$ in the companion paper. We prove the supercritical side completely: for all $\alpha,\beta\ge 1$ and every level the hierarchy is exact, via three explicit rational certificates realizing an affine identity. The hierarchy's exactness thus undergoes a phase transition at the critical line. On the subcritical side we certify the first four levels in exact arithmetic (rational pseudo-moments beating $c_Q$, confirmed by Sturm's theorem). We identify the exact mechanism: rescaled to the critical corner, the limiting obstruction is the Motzkin polynomial, the classical nonnegative-but-not-sum-of-squares form, so the finite-level failure sits in the restricted-certificate regime. The phase boundary has a precise geometric reading via Nie's finite-convergence theorem and Marshall's boundary Hessian condition: a self-tested optimum is finitely NPA-certifiable whenever its boundary Hessian is nondegenerate (contact order two), which holds for the single tilt and fails exactly at the doubly-tilted cubic touch. Three verified errata in the published polynomial system of Gigena et al. are documented.

We formulate quantisation of gauge field theories on globally hyperbolic Lorentzian manifolds with marked hypersurfaces within the framework of perturbative algebraic quantum field theory (pAQFT) enriched by the Batalin, Fradkin, Vilkovisky formalism (BV/BFV). This allows one to incorporate some crucial aspects of the local functorial approach to gauge field theory on manifolds with boundary within the algebraic setting. In particular, we provide a pAQFT-formulation of the modified classical and quantum master equations (after Cattaneo, Mnev and Reshetikhin CMR), as well as a constructive way to build a renormalised quantum BFV operator correcting the failure of the Quantum Master Equation by means of boundary terms (in the appropriate sense). We find that the renormalised quantum homotopy dg Lie algebra arising from the Anomalous Master Ward Identity becomes curved when boundaries are considered. The failure of the quantum master equation is thus encoded by a nontrivial curvature term in an $L_\infty$ algebra. As a byproduct, we recover previous results of Hollands' on the relation between the (boundary) BRST charge and the BV operator, and we recover CMR's ansatz for the quantum BFV operator at leading perturbative order on causal cylinders in Abelian Yang–Mills theory.

Gigena, Panwar, Scala, Araujo, Farkas and Chaturvedi [npj Quantum Inf. 11, 82 (2025)] determined the quantum maximum of the doubly-tilted CHSH functionals, observed that the Navascues-Pironio-Acin level needed for exactness grows without evident bound toward the critical tilt, and asked whether any finite level suffices. We answer this in the negative. For the symmetric critical family $B_s=(1-s/2)(\langle A_0\rangle+\langle B_0\rangle)+\mathrm{CHSH}$ we prove: for every NPA level $k\ge 2$ there are an explicit rational $g_k>0$ and an $s^*_k>0$ with $c_k(s)\ge 4-s+g_k s^2$ on $(0,s^*_k]$; since the quantum value leaves the local bound only cubically, every finite level strictly overshoots on an interval: no finite level is exact on any neighbourhood of the critical point. Unconditionally $a_2>1/39$, $a_3>1/188$, $a_4>1/641$. The proof is a primal construction: an exactly feasible moment curve at each level, built from level-uniform structural laws and one level-independent signed witness – a closed-form class function $y^*$ with $N_k^T\Gamma(y^*)N_k=u_k u_k^T$ at every level. The mechanism forces the sign: for $k\ge 3$ no quantum state and no smooth curve of quantum models can realize the gain direction, so the overshoot lives strictly in the non-quantum part of the NPA tangent cone. The proof is computer-assisted in the strict sense: finite exact-integer verifications with proven degree bounds are constituent parts of the argument; the chain has been re-verified against independent implementations, including a symbolic per-regime proof of the witness identity and a clean-room implementation written from the paper text alone. The one external input is the published quantum value of Gigena et al., cross-checked to twelve digits.

We study likelihood-ratio tests for the hypothesis that a positive-semidefinite matrix has rank at most a prescribed value. The null hypothesis is stratified: points of maximal allowed rank lie on a regular boundary stratum, whereas lower-rank points are singular. Consequently, the usual chi-bar-square calibration on the top stratum does not by itself describe the whole composite null, especially along sequences whose rank changes at the local $n^{-1/2}$ scale. After profiling regular nuisance parameters, we derive a common reduced Gaussian experiment for every fixed null rank and for all admissible local rank transitions. On the top stratum, the classical chi-bar-square law is recovered. At lower ranks, the limit generally involves projection onto a nonconvex rank-constrained semidefinite set. Our main calibration result shows that, under isotropy, the top-stratum law is least favourable over all fixed null strata and all local null rank transitions. We also prove the corresponding transition dominance under arbitrary anisotropy when the active corank is one. Finally, on the top stratum, we obtain a conditional shape derivative for the limiting distribution and its critical value. Gaussian covariance models and finite-sample experiments illustrate nuisance profiling, rank transitions, anisotropy, and orientation sensitivity.