Papers

Poisson algebras are the quasiclassical limits of associative algebra deformations of commutative associative algebras. This paper extends this process to the level of bialgebras. We derive Poisson bialgebras as the quasiclassical limits of antisymmetric infinitesimal bialgebra deformations of commutative and cocommutative antisymmetric infinitesimal bialgebras. It might be regarded as the “infinitesimal" version of the quantization process of Lie bialgebras in terms of Hopf algebras. Such deformations-quasiclassical limits process for a Poisson bialgebra is equivalently characterized in terms of the introduced notions of deformations of a matched pair of associative algebras as well as a standard Manin triple of associative algebras, whose corresponding quasiclassical limits are a matched pair of Poisson algebras and a standard Manin triple of Poisson algebras, respectively. We illustrate these equivalent deformations-quasiclassical limits processes via coherent derivations.

We study the eigen-structure of the penalty matrix arising from cubic smoothing splines on equally spaced knots. Using purely matrix-theoretic arguments, we show that its positive eigenvalues are simple, that the associated eigenvectors alternate between even and odd, and that the eigenvector for the $k$th largest eigenvalue has exactly $k+1$ sign changes. The approach provides a direct and transparent alternative to existing variational proofs of the oscillation property. These results show that equally spaced knots support a spline basis with both a parity structure and an oscillation pattern.

We develop the infinitesimal deformation theory of generalized parabolic Hitchin pairs (GPHs) on irreducible nodal curves. For any GPH $(E,\phi,F(E))$ we associate an explicit three-term deformation complex $\mathcal{C}_{(E_\bullet,\phi)}$ whose first hypercohomology $\mathbb{H}^1(\mathcal{C}_{(E_\bullet,\phi)})$ parameterizes first-order deformations. As the main application, we prove that the coarse moduli space $\mathcal{M}_{\mathrm{GPH}}$ of $1$-stable generalized parabolic Hitchin pairs of rank $n$ and degree $d$ with $\gcd(n,d)=1$ is a local complete intersection, smooth in codimension one, and hence a normal variety. We further show that this moduli space carries a natural Poisson structure.

Let $\mathcal{N}$ be the set of non-Brjuno real numbers. We determine the exact critical exponent of $\mathcal{N}$ for both logarithmic Hausdorff measure and logarithmic capacity. For the gauges $h_\delta(r)=(\log(1/r))^{-\delta}$, the critical exponent is $2$: the $h_\delta$-Hausdorff measure is infinite locally for $0<\delta\leq2$ and vanishes globally for $\delta>2$, while the logarithmic capacity is positive exactly for orders $0<s\leq2$. In particular, $\dim_{\mathrm{cap}}\mathcal{N}=2$.

Special relativistic hydrodynamics (SRHD) equations arise in the modeling of high-speed fluid flows encountered in astrophysical phenomena such as jets, supernova explosions, and gamma-ray bursts. Owing to their highly nonlinear hyperbolic nature, solutions often develop strong discontinuities, making the design of stable and accurate numerical schemes challenging. Although Weighted Essentially Non-Oscillatory (WENO) schemes are widely used for such problems, component-wise WENO reconstruction may produce spurious oscillations near discontinuities. On the other hand, characteristic-wise WENO reconstruction provides accurate non-oscillatory solutions for systems of conservation laws, but it involves the computation of eigenvectors in each cell, which leads to high computational cost. In this work, we intend to develop hybrid schemes which maintain the non-oscillatory feature of characteristic-wise WENO while being less costly. We propose three hybrid schemes, namely the H1-WENO, H2-WENO, and H3-WENO schemes, based on a new troubled-cell indicator constructed from the smoothness indicators of the WENO scheme. The proposed troubled-cell indicator effectively distinguishes smooth and discontinuous regions, allowing the hybrid schemes to employ inexpensive reconstructions in smooth regions and the characteristic-wise WENO reconstruction only near discontinuities. Numerical experiments demonstrate that the proposed schemes retain the accuracy and robustness of characteristic-wise WENO methods while significantly reducing the computational cost. In particular, the H1-WENO scheme achieves an approximately 30–40% improvement in computational efficiency compared to the standard WENO scheme in 2D test cases.

Given any closed hyperbolic n-manifold M ($n\ge 3$), we show that for a fixed $\kappa>\frac{3}{n-2}$, a random closed geodesic of length close to r has the collar of width $r^{-\kappa}$ when r is large enough. In particular, most closed geodesics of length close to r are simple. This theorem positively answers Problem 3.16 from Kirby's list which asks whether every closed hyperbolic 3-manifold contains infinitely many simple closed geodesics.

We study the Schur-positive cone in $\Rspace_{n,\mathbb R}\coloneqq \operatorname{span}_{\mathbb R}\{p_d^{n/d}:d\mid n\}$ through its basis $Q_{n,d}\coloneqq\ell_{n/d}^{(1)}[p_d]$, where $\ell_m^{(1)}$ is the Frobenius characteristic of the representation induced to $S_m$ from a faithful linear character of the subgroup generated by an $m$-cycle; brackets denote plethysm. A Boolean $Q$-sum is a sum of distinct elements of this basis. We give a unified proof of four conjectures of Sundaram on Schur positivity by classifying all Schur-positive Boolean $Q$-sums; the case of sums over divisors up to a prescribed bound recovers Hou's theorem. Specifically, for a nonempty set $J$ of divisors of $n$, the sum $\sum_{d\in J}Q_{n,d}$ is Schur-positive exactly when $1\in J$ and, for even $n$, $n\in J$ implies $n/2\in J$. The same character estimates prove the Ramanujan-square conjecture of Shareshian and Sundaram: the function $\sum_{d\mid n}c_d(n/d)^2p_d^{n/d}$, where $c_d(r)$ is the Ramanujan sum, has a positive coefficient of $s_\lambda$ for every $n\ge1$ and $\lambda\vdash n$, except when $n\equiv2\pmod4$ and $\lambda=(1^n)$, in which case the coefficient is zero. We prove that an element of this space has integral Schur coefficients if and only if its $Q$-coordinates are integral. The Boolean classification also determines the convex hull of the Schur-positive Boolean points with $Q_{n,1}$-coordinate $1$. We compute its Ehrhart polynomial and volume, prove its integer decomposition property, and determine the Hilbert basis of its cone. For $n\ge18$, we prove that setting the coefficient of $s_{(n)}$ or $s_{(1^n)}$ equal to $0$ or $1$ defines a facet of the section of the Schur-positive cone with $Q_{n,1}$-coordinate $1$. The positivity results and coordinate formulas also yield inequalities for major-index residue multiplicities.

Harwit and Sloane conjectured that every nonsingular entrywise-nonnegative matrix $A\in\mathbb R^{n\times n}$ satisfies $\|A^{-1}\|_F\ge 2n(n+1)^{-1}\|A\|_{\max}^{-1}$, with equality precisely for positive multiples of $S$-matrices. Zhang has given a complete proof of this conjecture by a centered pseudoinverse and spectral variance method. We present an independently obtained, structurally different proof of the strict even-dimensional inequality. Starting from the structural identities of Frankel and Urschel, we derive an exact global defect budget and combine binary rounding, fixed intersections, and Gram projection. A ten-row obstruction handles every even $n\ge66$; a finite exact calculation handles $4\le n\le64$, $n\ne6$; and a multi-column energy argument treats $n=6$. The order-two case is elementary. The even-dimensional argument is formalized in Lean 4, conditional on Frankel–Urschel Lemma 2.1 as an explicit external mathematical input. The finite evaluations use Lean's native evaluator; their trust boundary and exact certificates are documented. Together with Cheng's odd-dimensional theorem, the argument recovers the full S-matrix theorem and its equality characterization.

We introduce a variant of reduced Khovanov-Rozansky homology defined in terms of the variables alpha_1, ..., alpha_{n-1} that correspond to the regions between the strands. We rigorously reestablish several folklore results in terms of our framework, including the equivalence between Khovanov's and Rasmussen's approaches. Furthermore, we show that Rasmussen's chain complex is a free resolution of the Rouquier chain complex.

We develop a methodology for designing infinite products of rational blocks whose exponents are polynomials in the index $k$. Matching power sums of the slot constants through order $n$ forces the Type-$N$ product with binomial exponent $\binom{n+k-2}{n-1}$ to converge to a ratio of Vignéras multiple gamma values $\Gamma_n$; an analogue finder lifts any Type-1 evaluation to every higher type, and integer combinations of binomial exponents then realise arbitrary integer-valued polynomial exponents, yielding explicit products with exponents $k$, $k^2$, $k^3$, ... for constants such as $\pi/2$, $\sqrt{2}$, $e^{2K/\pi}$, and rational multiples of $\pi^{M!}$ (Part I). As the main application (Part II) we prove that for every positive integer $n$, a finite multiple-gamma template $\mathcal{S}_n$ with generalised Eulerian weights $T(n,k)$ (OEIS A225118) evaluates the Duke-Imamoğlu expression $\mathcal{D}_n = \beta'(-n) + (\log 4)\,\beta(-n)$. For odd $n$ this yields a convergent Wallis-Eulerian product for $e^{\beta'(-n)}$; for even $n$ the raw product diverges. The proof expands the template through the multiple-gamma functional equation, evaluates the quarter-integer coefficients in closed form, and identifies the resulting Eulerian-binomial sums with Duke's polynomials $P_{n+1,\ell}$.

We study two boundary cases of the stability condition for the $J$-equation. First, under the $J$-semistable condition, we show that the destabilizing prime divisors form an exceptional family in the sense of Boucksom, with a uniform numerical gap away from them. The related modified nef estimate can remove the $J$-big assumption in Liu's work~\cite{Liu2026Boundary}. Second, under the smooth boundary cone condition, we obtain a uniform global $C^0$ estimate for solutions of the approximating twisted $J$-equations. As a consequence, we construct a bounded-potential Bedford–Taylor solution of the $J$-equation, which is smooth outside the destabilizing prime divisors.

Motivated by the concept of polarity introduced by G. Birkhoff for a binary relation on a set, we introduce a concept of orthogonality in a poset with $0$. A pair of operators $f$, $g$ on a poset with $0$ is called orthogonally adjoint if $f(x)$ is orthogonal to $y$ if and only if $x$ is orthogonal to $g(y)$. We characterize the existence and the uniqueness of $g$ for given $f$ and describe basic properties of orthogonal adjointness. We present constructions of orthogonally adjoint pairs in pseudocomplemented posets. If a given operator $f$ is an order-isomorphism of a pseudocomplemented poset satisfying some natural properties then the corresponding adjoint $g$ can be described explicitly. Moreover, if $f$ and $f^{-1}$ are bijective $\perp$-morphisms then they are orthogonally adjoint, too. Finally we show that a given pair of orthogonally adjoint mappings on a poset $\mathbf P$ may not be extendable to the Dedekind-McNeille completion of $\mathbf P$ and we present sufficient conditions for the existence of such an extension. We also provide sufficient conditions for the existence of an extension of orthogonally adjoint mappings to the lattice of ideals. Our results are illustrated by numerous examples.

The interconnection of port-Hamiltonian systems is usually understood as structure-preserving due to the geometric fact that the composition of the underlying Dirac structures produces again a Dirac structure. Pointwise, this is doubtless true, but when comparing dynamical systems, their trajectories are usually compared, and trajectories usually obey some regularity assumptions. It is shown that both classical and weak solutions do not necessarily exhibit a one-to-one correspondence between solutions of the interconnected port-Hamiltonian system, and the dynamical system realising the interconnection. In the mostly studied cases of constant Dirac structures or port-Hamiltonian ODE systems, both systems are behaviourally equivalent; for non-constant Dirac structures, a sufficient condition is given.

We develop simplicial-set models for dagger $(\infty,1)$-categories. We first establish a model structure on dagger simplicial categories whose weak equivalences are local weak equivalences that are essentially surjective up to coherently unitary equivalence. We then construct the dagger Joyal model structure on dagger simplicial sets and prove that the dagger rigidification–nerve adjunction is a Quillen equivalence. We next introduce the unitary core and use it to characterize dagger Joyal equivalences intrinsically. Finally, we compare the dagger Joyal model structure with the DCH anti-involutive Joyal model structure.

We study the periodic fractional stochastic Burgers equation $(\partial_t+\Lambda^\gamma)u=\partial_x(u^2)+|\partial_x|^{1-\alpha}\xi$, where $1<\gamma\leq 2$ and $\xi$ is space-time white noise. Under the condition $\alpha>\max{(7-4\gamma)/2,(15-8\gamma)/6}$, we establish pathwise $L^1$, energy, and Besov estimates for a transformed remainder. The argument combines a response decomposition with a conservative Zvonkin transformation. For $1<\gamma<2$, this produces a compensated nonlocal diffusion operator whose coercivity is coupled to a cubic-increment estimate adapted from the modified Karman–Howarth–Monin argument.

Inspired by interaction of gravitational waves, dark matters and dark energy, we study the Einstein scalar field equations with the nonzero cosmological constant for the Bondi-Sachs metrics. We provide the asymptotic expansions under the outgoing radiation condition. Unlike the case of the zero cosmological constant, the asymptotic expansions depend on four additional $\Lambda$-independent functions $B(u, \theta, \phi)$, $\tilde{X}(u, \theta, \phi)$, $\tilde{Y}(u, \theta, \phi)$ and $\tilde{I}(u)$. We show that, under suitable coordinate transformation, we can make $B(u, \theta, \phi)=0$. We prove the peeling property for the Einstein scalar field equations if $\tilde{I}=0$, and derive a loss formula of the Bondi energy-momentum for the nonzero cosmological constant. We remark that, for certain real data from gravitational waves, the loss formula is likely to indicate the loss property of the Bondi energy-momentum.

This paper characterizes all the singular inner eigenfunctions of the composition operators $C_\phi$ that arise from discrete measures, when $\phi$ is an automorphism of unit disk. By establishing a connection between Beurling and model invariant subspaces, we classify all the inner functions so that the corresponding Beurling subspace is invariant under the composition operators induced by non-elliptic automorphisms. This classification involves solving the eigenfunction equation for the composition operator. Further, we present some applications of the above-mentioned connection.

We study the uniqueness of the vortex wave system, i.e.\ the 2D incompressible Euler equations with vorticity consisting of a point vortex and a bounded part. We show that if the initial vorticity and point vortex location $(w_0,X_0)$ satisfy a constraint of the type $|w_0(x)-w_0(X_0)|=O(|x-X_0|^\alpha)$ for $\alpha> 2$ and the vorticity is in $L^\infty$, then solutions are unique, globally in time, even though the velocity is not log-Lipschitz.

Persistent homology summarizes a filtration by recording when homological features appear and disappear, but the resulting barcode does not determine a preferred homology basis or preferred cycle representatives. We study the additional information carried by a basis selected through matrix reduction and introduce the persistence pairing graph, a reduction-dependent directed graph that records how selected homology classes are re-expressed when their associated persistence intervals end. After fixing a filtered finite CW complex, a coefficient field, a filtration-compatible cell order, and a reduced factorization of the boundary matrix, the reduction determines selected birth cycles. When an associated persistence interval ends, the image of its selected birth class has unique coordinates in the basis formed by selected classes whose intervals survive beyond the same filtration value, and the nonzero coordinates define the graph edges. These coordinates can also be recovered through the Kronecker pairing with a dual cohomology basis. We prove that the active selected birth cycles form a homology basis at every cellwise filtration stage, characterize the kernel and image of each homological transition, and show that the graph is directed and acyclic, with every edge pointing toward an interval containing the source interval. In dimension zero, under the standard reduction convention, the graph agrees with the elder rule merge tree. We also prove that its edges can change while the persistence pairs and barcode remain fixed. Finally, we define a bounded fused graph discrepancy and separate the contribution controlled by ordinary persistence stability from the reduction-dependent graph structure. The graph is therefore a descriptor relative to a fixed reduction convention, rather than an invariant of the persistence module alone.

Cartan geometries are curved analogues of homogeneous spaces, and the correspondence space construction produces, from a Cartan geometry of one type, another of a different type over a larger base manifold. The unit tangent bundle of a Riemannian manifold is, in this language, a Cartan geometry of type $(\operatorname{Euc}(n),O(n-1))$ obtained in this way; we study Cartan geometries of this type in general, whether or not they arise as correspondence spaces. We show that every such geometry induces on its base manifold an almost contact metric structure, together with an orthogonal splitting of the tangent bundle and a compatible linear connection; conversely, these data determine the geometry, so that the correspondence is a bijection. When $n\ge3$, we single out a distinguished Cartan connection, which we call normal, by a condition on its torsion formulated in terms of the Spencer differential. This normalization is dictated by the correspondence space construction: the Cartan connection induced on the unit tangent bundle of a Riemannian manifold is normal.

This paper gives a survey of propositional modal logic in an intuitionistic setting. Mainly intended for readers already familiar with classical modal logic, we discuss why there are two prominent families of such logics, intuitionistic modal logics (IMLs) and constructive modal logics (CMLs), what motivates their study, and how they compare with each other. We then take a look at the logics lying between the CML and IML traditions, and survey various extensions and $\Diamond$/$\Box$-free fragments. We further record several observations which seem to be implicit or absent in the literature; we point out a small defect in the original semantics for $\textbf{CK}$ and prove Lyndon interpolation theorems for $\textbf{CK}$ and $\textbf{WK}$.

In this paper, we will construct periodic solutions to the viscous steady Burgers equation with an external force $f(x)$, based on the following formal expansion $u^\varepsilon(x)=u_0(x)+\varepsilon u_1(x)+\cdots+\varepsilon^n u_n(x)+\cdots$, where $\varepsilon \ge 0$ represents the viscosity and $u_0(x)$ is a solution to non-viscous steady Burgers equation with the external force $f(x)$. We will focus on the solutions which are uniformly bounded with respect to the viscosity. In our previous work, starting from $u_0=-(2+\cos x)$, the authors constructed the periodic solutions to the viscous steady Burgers equation with the external force $f=u_0(x)u_{0x}=-2\sin x-\sin x\cos x$. In this paper, we will extend the main result obtained in our previous work and construct the solutions starting from general $u_0$ and $f$ satisfying the non-viscous steady Burgers equation $u_0(x)u_{0x}=f$. It will be shown that there exists a $\varepsilon_0>0$, which depends on $n$, such that for any $0<\varepsilon<\varepsilon_0$, the viscous steady Burgers equation with the external force $f$ has a periodic solution $u^\varepsilon(x) \in C^2([0,2\pi])$, satisfying $|u^\varepsilon(x)-u_0(x)-\varepsilon u_1(x)-\cdots-\varepsilon^n u_n(x)| \leq C\varepsilon^{n+1}$, where $C>0$ is a constant which depends on $n$, but is independent of $\varepsilon$. Compared with Jauslin-Kreiss-Moser's result, we present a new approach to construct periodic solutions to the viscous steady Burgers with a general external force. The constructed solutions will tend to the ones of the non-viscous Burgers equation with a sharper convergence rate (up to higher order) when the viscosity vanishes.

We prove a refined quadratic normal approximation for the first absolute moment of normalized weighted Rademacher sums with bounded maximal coefficients. For any weight vector $w\in\mathbb{R}^{n}$ satisfying $\|w\|_{2}=1$ and $\|w\|_{\infty}\leq\beta$ with sufficiently small $\beta>0$, we establish the uniform error bound $|\mathbb{E}|\sum_{i=1}^{n}w_{i}X_{i}|-\sqrt{2/\pi}|=O(\beta^{2})$ over all admissible weight configurations. Our proof combines zero-bias Stein's method and refined small-ball probability estimates to exploit symmetry cancellation and control the non-smooth residual of the absolute-value test function. An explicit extremal construction further verifies the optimality of this quadratic convergence rate. As an application, we establish an asymptotically sharp refinement of the Friedgut–Kalai–Naor (FKN) theorem for Boolean functions, also known as the sharp $\pi/2$ theorem, characterizing the level-1 Fourier energy for functions deviating far from dictatorships.

We completely characterize the $n$-fold connected sums of $S^3\times S^4$ that admit Sasaki-Einstein structures by proving that such a structure exists if and only if $n$ is even.

We answer two questions of Kozlov about polynomial-in-momentum first integrals of billiard systems. For billiards in bounded polytopes, such integrals are independent of position. For hard-ball gases in a rectangular box, every polynomial-in-momentum first integral is a polynomial in the total kinetic energy whenever the collision graph is connected.

The paper introduces a successive backstepping observer design for strictly feedforward parabolic PDE-ODE systems, in which the coupling structure determines the order of error stabilization and the corresponding transformations. First, a transformation based on a virtual measurement stabilizes the ODE observer error subsystem, which is most distal from the measurement, while decoupling it from the PDE error state. Second, a Volterra integral transformation is employed to stabilize the PDE error subsystem and to map the overall error dynamics into a cascade of exponentially stable ODE and PDE subsystems. The design is shown to be dual to a recently proposed multi-step state feedback design for parabolic PDE-ODE systems in strict feedback form, thus explaining the structure of the presented observer design.

Advances in machine learning (ML) have created new opportunities to complement traditional operations research (OR) methods. In particular, transformer models can capture complex interactions in token sequences by mapping tokens into a high-dimensional embedding space and propagating contextual information via attention. This makes them a candidate to model non-permutation flow shop scheduling with secondary resources as a next-token prediction task, where tokens represent job-machine-secondary resource tuples. For training, mixed-integer linear programming (MILP)-generated schedules are tokenized and used as next-token prediction data. During inference, partial token sequences (prefixes) are randomly generated and completed by the trained transformer through constrained decoding. A computational study is conducted on a flow shop with 8 jobs, 4 machines, and 3 secondary resources, where jobs are selected from a fixed pool of 20 jobs that is sampled during training and provides the candidates during prefix completion. The transformer achieves better solution quality (smaller makespans) compared to a genetic algorithm (GA), the NEH heuristic, and random search. It is outperformed only by the MILP model and the iterated greedy (IG) heuristic. The study concludes that transformer models can, to some extent, learn patterns from MILP-optimized non-permutation flow shop schedules and that transformer-based scheduling represents an interesting direction for future research, particularly in settings with a fixed, recurring job set.

We introduce latent geometry in BFSS/BMN models. Before an emergent geometry evaporates at the geometric transition, large-$d$ Gaussianization encodes its information in the exceptional $N=2$ sector selected by the baby fuzzy sphere, a single active spin-$1/2$ Pauli block. Dimensional reduction, finite-$N$ Gaussianization and exact Molien–Weyl projection transport this encoding to mass-deformed BFSS$_2$. At $N=2$, the same stable BFSS$_2$ oscillator admits inequivalent unitary $\mathbf{H}^2_\theta=\mathbf{EAdS}^2_\theta$ and $\mathbf{dS}^2_\theta$ decodings, realizing spacetime transmutation and the emergence of Lorentzian time.

The bosonic Kitaev chain under open boundary conditions has attracted recent attention due to its realization in driven-dissipative systems and its intriguing non-Hermitian boundary physics. The model is known to be solvable via local squeezing transformations in the position-momentum representation. In this paper, we present an alternative, purely algebraic solution that relies entirely on the standard bosonic Bogoliubov transformation. For an $N$-site chain, we propose an asymmetric plane-wave ansatz with unequal left- and right-moving momenta to analytically solve the associated $2N\times 2N$ non-Hermitian “associated matrix". The left eigenvalue problem yields $N$ distinct eigenvalues, each of which is twofold degenerate. By carefully resolving these degeneracies using the bosonic commutation relations, we construct the $N$ physical Bogoliubov quasiparticle operators. The construction reveals non-uniquenesses that in special cases exactly mirror the freedom in the local squeezing transformations of the original approach. The diagonal form of the Hamiltonian is obtained explicitly and is shown to be equivalent to the original Hamiltonian. The proposed asymmetric plane-wave ansatz and degeneracy-resolution technique are not limited to the present model and can be generalized to other bosonic pairing systems, including those with inhomogeneous pairing or hopping terms.

Planar cell polarity is preceded by the clustering of polarity proteins into discrete, low-turnover membrane subdomains (puncta), yet the minimal interactions that nucleate puncta, set their number, and segregate opposite orientations remain unclear. We address these questions with a mass-conserving reaction-diffusion model in which two diffusible monomers bind reversibly across a cell-cell junction into trans-complexes of two orientations, with feedback entering only through concentration-dependent rates. Above a critical density, the uniform state undergoes a long-wavelength mass-redistribution instability rather than a finite-wavelength Turing bifurcation. The form of the feedback then selects between two morphologies: broad mesas fixed by a Maxwell construction under saturating feedback, and narrow mass-limited spikes under unbounded feedback. For spikes we obtain closed-form expressions exhibiting a clean separation of amplitude (mass and feedback), width (the complex diffusion length), and spacing (the monomer screening length). Within a fast-monomer reduction we prove, for any number and arrangement of puncta, that like-oriented arrays coarsen, so multiplicity is metastable and kinetically determined. The second monomer reservoir introduces a second screening length that rate-limits competition by the harmonic mean of the monomer diffusivities, and the spectrum of a punctum remains free of oscillatory ("blinking") instabilities throughout. Finally, sign-definite cross-modulation of turnover converts clustering into orientation sorting - the mutual exclusion of orientations along a single contact: mass redistribution sets puncta number, and the sign of the cross-coupling determines whether orientations segregate. Puncta number and orientation sorting are thus governed by mathematically separable ingredients.