Papers

Let $n$ be a positive integer. A translation-invariant total order (TITO) with period $n$ is a total order of the integers that is invariant under translations by multiples of $n$. These structures arise naturally in the study of Coxeter groups. In particular, real $n$-TITOs are in bijection with biclosed sets of positive roots of the affine symmetric group $\widetilde S_n$. Barkley and Defant recently introduced pattern avoidance for TITOs and used it to define the affine Tamari lattice. The enumeration of TITOs avoiding a single pattern of length $3$ is due to Crites and Barkley–Defant. We extend this work to TITOs avoiding two patterns. Our main results include a complete enumeration of TITOs that avoid a pair of patterns in $S_3\times S_3$, as well as of TITOs that avoid a pair $(p, q)$ with $p \in S_3 \setminus \{123, 321\}$ and $q \in S_4$. Furthermore, we provide an explicit construction of the inverse of the bijection between $312$-avoiding TITOs and noncrossing arc diagrams, thereby extending the combinatorial framework introduced by Barkley.

We study the semiclassical dynamics of solitary waves for the three-dimensional cubic–quintic nonlinear Schrödinger equation with an external potential. For initial data given by a modulated free ground state $P_\omega$, we prove that the wave-packet center follows the associated classical Hamiltonian flow. For $\omega\in\mathcal I$, we obtain qualitative persistence near the soliton orbit. Under the additional positive-slope condition $\omega\in\mathcal I_+$, we establish an $O(\eps)$ approximation in the scaled $H^1$ norm on every fixed time interval. We also derive $O(\eps^2)$ concentration estimates for the mass, momentum, and center of mass.

We introduce the chordal isometric circle of a Möbius map, and use this to give a factorization of any Möbius map as the composition of a chordal isometry and either a reflection, or a rotary reflection, across a circle. We then use this to find the chordal, and spherical, Lipschitz constants of a Möbius map, and compare this with related results in the literature.

We study Yang's higher numerical invariants for the quasi-homogeneous submodules \[ M_{k,\ell}=[z^k-w^\ell]\subset H^2(\mathbb D^2), \qquad k,\ell\in\mathbb N,\quad k\neq\ell. \] The Hilbert–Schmidt property and the low-order invariants $\Sigma_0$ and $\Sigma_1$ for this family follow from earlier results on $M_{\theta,\varphi}$-type submodules. Our aim is to determine the complete higher-order sequence. Using the weighted homogeneous decomposition associated with $z^k-w^\ell$, we construct explicit orthonormal bases for the two defect spaces and determine the exact support of the shifted defect inner products. For $j\geq1$, set \[ A_j=\left\lceil\frac{j}{\ell}\right\rceil, \qquad B_j=\left\lceil\frac{j}{k}\right\rceil. \] We obtain \[ \Sigma_0(M_{k,\ell})=\frac{\pi^2}{6}, \qquad \Sigma_j(M_{k,\ell})=F(A_j,B_j), \] where $F$ is an explicitly evaluated symmetric function on $\mathbb N^2$. The sequence is nonincreasing, tends to zero, and satisfies $\Sigma_j>\Sigma_{j+1} \Longleftrightarrow k\mid j\ \text{or}\ \ell\mid j$. Hence its strict descent set is $k\mathbb N\cup\ell\mathbb N$. This descent set determines the additive semigroup $\langle k,\ell\rangle$, while the complete numerical invariant sequence determines the unordered pair $\{k,\ell\}$. The order of the two exponents is not recoverable because of coordinate-swap symmetry.

We introduce a broad generalization of a recursive formula for the Möbius function $\mu(n) = 1-n-\sum_{d=2}^{n-1}\mu(d)\left[\frac{n}{d}\right]$ due to George Spencer-Brown. By replacing the greatest integer function $\left[\frac{n}{d}\right]$ in this classical recurrence with an arbitrary arithmetic function $f\left(\left[\frac{n}{d}\right]\right)$ with $f(1)=1$, we define a new generalized family of functions, denoted $\star(n)$. We prove that $\star(p) = -1$ if and only if $p$ is prime. This result yields a surprising algebraic characterization of primes and reveals a deep structural property underlying divisor sums and the greatest integer function.

In this paper we consider a two-dimensional generalized continuous-state branching process with mutually enhancing two-way interactions characterized by two stochastic differential equations driven by Brownian motions and spectrally positive $\alpha$-stable random measures. Such continuous-state population dynamics can also be identified as a stochastic Lotka-Volterra type population model. Infinite behavior such as explosion/nonexplosion for this population and staying infinite/coming down from infinity are derived under various conditions on the coefficients involved in the model and the conclusions are rather sharp

For even k, we derive a reflection identity for the basic count distributions in the problem of maximizing the probability that all n k-wise independent Bernoulli variables equal one. After separating the distinguished node n, the other support nodes reflect by s -> n-1-s while the Bernoulli parameter changes from p to 1-p. Corresponding basic weights differ by an explicit positive factor. Thus interior feasibility sets, and local changes of optimal support with their multiplicities, are reflected. This gives a proof of the exceptional-point reflection in Conjecture 5.11 of Berend, Ernst, Kontorovich and Kumar. The argument uses Lagrange interpolation and an elementary binomial reweighting identity; it does not require a conjectured ordering or connectedness of the feasibility sets.

Integrated Sensing and Communication (ISAC) enables the joint design of communication and sensing functionalities using a shared waveform, but its performance critically depends on the accuracy of the transmitted signal available at the sensing receiver. In practical systems, this signal is obtained by finite blocklength decoding . This paper develops an analytical framework for quantifying the impact of the associated decoding uncertainty on sensing performance. We model decoding errors via an equivalent noise formulation that captures their second-order effect on the sensing receiver, and derive the corresponding Fisher Information and Cramér–Rao Bound (CRB). The resultant characterization reveals a fundamental sensing–communication tradeoff governed by the communication rate and decoding reliability. This shows that the classical finite blocklength reliability results in a fundamentally new insight concerning ISAC systems: when the decoded communication waveforms are reused as sensing references in a data-aided fashion, the communication reliability boundary also acts as a sensing-information bound. {We identify and characterize a sharp transition in sensing performance – referred to as the Tradeoff Cliff – arising from finite blocklength reliability effects.} This transition separates the regimes of near-ideal sensing performance from those of excessive estimation error as the communication rate approaches capacity. Furthermore, we provide an explicit expression for the critical rate at which this transition occurs, showing its dependence on blocklength, channel dispersion, and target error probability. Monte Carlo simulations under both AWGN and block fading channels validate the theoretical analysis and confirm the trend observed. The results provide design insights for operating ISAC systems with an appropriate reliability margin to avoid severe sensing degradation.

The aim of this article is to leverage Enomoto's classification of exact structures to the dg level. Let $\mathscr{A}$ be a connective additive idempotent complete dg category. We establish an intrinsic Enomoto-type classification of exact dg structures on $\mathscr{A}$ in terms of bi-Serre subcategories in $\mathsf{tr}(\mathscr{A})$. Our proof is a direct dg argument based on dg totalizations of $3$-term complexes and dg duality, rather than a reduction to the greatest exact dg structure and the classification of its substructures. As a consequence, we establish a bijection between exact dg structures on $\mathscr{A}$ and bi-Serre subcategories in $\mathsf{tr}(\mathscr{A})$. Moreover, this bijection is an isomorphism of posets and yields a complete lattice structure on the class of exact dg structures on $\mathscr{A}$. In particular, we provide another construction of Rump–Chen's greatest exact dg structure on $\mathscr{A}$.

We prove the following: If $X$ is a plane separating continuum, then every embedding of $X$ into $\mathbb{R}^2$ contains the vertices of a Euclidean rectangle. We arrive to this result by extending a known result by H. Vaughan for Jordan curves to a wider class of topological objects via shape theory and Steenrod homology.

We construct an explicit quaternary Legendre pair of length $64$. To the best of our knowledge, this is the first reported construction at this length.

Let $0<\kappa<1$ and set $p_+=2/(1-\kappa)$ and $p_-=2/(1+\kappa)$. We construct a coefficient $\kappa\leq a\leq\kappa^{-1}$, smooth outside a compact set of Lebesgue measure zero, for which the $W^{1,2}_{p_+}$ estimate fails for the one-space dimensional nondivergence form parabolic equation. The corresponding solution has second spatial derivative in the weak $L_{p_+}$ space, but not in $L_{p_+}$. By duality, the $W^{1,2}_{p_-}$ a priori estimate also fails. Conversely, we demonstrate that the $W^{1,2}_p$ estimate holds and the equation is uniquely solvable for $|p-2|<c\kappa$, showing that the size of the solvability interval around $2$ has optimal order $\kappa$ as $\kappa\downarrow0$, which addresses a question raised in [17]. These results imply that the Alexandrov maximum principle holds for $p>2-c\kappa$ and this order is sharp as $\kappa\to 0$. The corresponding results for divergence form equations are also obtained.

We construct a sequence of finite Rodriguez-Villegas transforms converging locally uniformly to the Riemann $\xi$-function in the critical strip. The input is a positive symmetric profile on the unit interval obtained from the Riemann theta kernel through convolution with the hyperbolic-secant kernel and the logistic coordinate. The profile is a Stieltjes function of $x(1-x)$. Its Bernstein polynomials produce reciprocal numerators and exact finite functional equations. The same numerators admit an exact realization as fermionic supertraces, while the Bernstein polynomials are normalized Gibbs traces.

We investigate and characterize order automorphisms on the set of partial isometries in the finite-dimensional matrix algebra $M_n(\mathbb{C})$. Different from the classical order automorphisms of subspace lattices, which can be implemented by standard invertible or unitary transformations, the order automorphisms considered herein admit no such conventional matrix representations. Instead, they are essentially governed by matrices such that $I-(A+A^*)$ is either positive or negative invertible. The results reveal that the structural features of order automorphisms for partial isometries are substantially more intricate than those of classical subspace automorphisms.

We define a signed GLMY chain complex over $\mathbb{R}$ for signed digraphs using sheet-labelled regular paths. The complex is naturally isomorphic to the deck anti-invariant subcomplex of the ordinary GLMY complex on the signed double cover. The double-cover realization yields switching invariance and recovers ordinary GLMY homology for switching-balanced signings. Bidirected completion gives an orientation-independent homology theory for signed graphs. For a signed graph, the zero-dimensional homology identifies with the kernel of the signed Laplacian and has dimension equal to the number of balanced connected components. Signed GLMY homology is functorial under signed weak morphisms, which combine vertex maps with switching functions and allow compatible arrow contractions. For signed digraphs, the all-positive reduction retains the orientation sensitivity of ordinary GLMY homology, while explicit computations show additional sensitivity to the arrow signs. For a fixed digraph with five vertices and nine arrows, we classify all 512 arrow signings and obtain exactly four signed Betti vectors. Precisely 16 signings have nonzero second signed GLMY homology.

In their studies of pathologies in recursion categories, Montagna (1989) and Di Paola–Montagna (1991) introduce the algebraic systems $S'$ and $S'_T$, respectively, and claim that they are categories. We show that the proposed composition is not independent of the choice of representatives. For every consistent recursively enumerable extension $T$ of Peano arithmetic ($\mathrm{PA}$), we exhibit two unary programs whose partial functions are provably equal in $T$, separately at each standard input. Composing each after a program that searches for a $T$-proof of contradiction and returns its code yields programs that are not equivalent in this sense. An alternative proof uses the productivity of the complement of the diagonal halting set. Montagna's $S'$ is the case $T=\mathrm{PA}$. More generally, for consistent $T\supseteq\mathrm{PA}$, pointwise provable equality is a composition congruence exactly when $T$ proves every true $\Pi^0_1$ sentence, in which case it is extensional equality. This completeness condition fails for every consistent recursively enumerable $T\supseteq\mathrm{PA}$ by Gödel's second incompleteness theorem. For every extension $T\supseteq\mathrm{PA}$, the least composition congruence containing pointwise provable equality is extensional equality if $T$ is $\Sigma^0_1$-sound and the universal relation otherwise.

We study the supercritical LYZ equation on compact Kähler manifolds at the boundary of stability. Under numerical semistability, we prove a quantitative positivity estimate on the modified nef cone. It implies that the destabilizing prime divisors form a finite exceptional family. Assuming the existence of a smooth semisubsolution, we prove that the associated intersection form is negative definite on the span of classes of these divisors. Using this rigidity, we then construct a logarithmic singular subsolution along their union and obtain a bounded Bedford–Taylor solution which is smooth on its complement.

In connection with eigenvalue perturbation bounds, Li [Linear Algebra Appl., 278 (1998), pp. 317-326] formulated three minimization problems involving Hadamard products. The third asks how to relate $$ \min_{W\ \mathrm{nonsingular}}\|W\circ G\|_\mathrm{ui}\,\|W^{-1}\circ G^\mathrm{T}\|_\mathrm{ui} \quad\mbox{and}\quad \min_{P\ \mathrm{permutation}}\|P\circ G\|_\mathrm{ui}^2, $$ where $\|\cdot\|_\mathrm{ui}$ denotes a unitarily invariant norm and $\circ$ denotes the Hadamard product, i.e., entrywise multiplication. The first quantity never exceeds the second, and a reverse inequality with a constant independent of the matrix size was conjectured. Romeo and Tilli [Linear Algebra Appl., 326 (2001), pp. 161-172] showed that if such a bound holds for every $G$, then $\|\cdot\|_\mathrm{ui}$ must be uniformly equivalent to the Frobenius norm $\|\cdot\|_\mathrm{F}$. We prove that the two quantities coincide when $\|\cdot\|_\mathrm{ui}=\|\cdot\|_\mathrm{F}$: for every complex matrix $G$, $$ \min_{W\ \mathrm{nonsingular}} \|W\circ G\|_\mathrm{F}\,\|W^{-1}\circ G^\mathrm{T}\|_\mathrm{F} =\min_{P\ \mathrm{permutation}}\|P\circ G\|_\mathrm{F}^2. $$ The associated conjecture therefore holds with the optimal constant $c=1$.

We study Keller–Segel systems on $ \mathbb{T}\times\mathbb{R}$ subject to a stochastic Couette transport flow. We prove that sufficiently strong mixing suppresses chemotactic finite-time blow-up with high probability; thus yielding a unique global-in-time mild solution for initial data of arbitrary mass. Moreover, we obtain a quantitative estimate for the probability of blow-up, showing that it decays exponentially with increasing strength of the stochastic shear. Our analysis is based on the enhanced dissipation generated by the stochastic shear flow. To prove our main result, we first construct a maximal local mild solution through pathwise estimates for the random solution operator of passive scalar transport equations and a fixed-point argument. To obtain global control, we decompose the evolution into time blocks and exploit mixing-induced decay of the nonzero spatial Fourier modes. A nonlinear bootstrap argument then yields global existence above an almost-sure finite random threshold for the shear strength which is obtained by an application of the Borel-Cantelli theorem.

We prove a Positive Mass Theorem for $C^0$-asymptotically flat Riemannian metrics with nonnegative scalar curvature in a weak sense that are sufficiently uniformly close to Euclidean space. More precisely, we show that a $C^0$-asymptotically flat Riemannian metric that is a $C^0$ perturbation of Euclidean space with nonnegative scalar curvature in the sense of Ricci flow has nonnegative mass, where the mass is given by a $C^0$ analog of the classical ADM mass previously introduced by the author.

An unusual object in the sky raises two related questions: what kind of object it is and what physical process drives its motion. Departures from a gravity-only trajectory can provide evidence, but the same apparent acceleration may be generated by distinct dynamical mechanisms. We formulate this as an identification problem and establish conditions for identifiable uniqueness among three reference classes: inert dynamics, impulsive forcing, and active stabilization such as a controlled spacecraft. The distinction rests on how each class responds to disturbances. Although different mechanisms can produce similar average trajectories, the specified reference models imply different stochastic responses around those trajectories. We show when these differences uniquely determine the dynamical class from the observed trajectory and when several classes remain observationally compatible.

In this paper, we study local minimizers of the energy functional$J(\mathbf{u}) = \int_D (|\nabla \mathbf{u}|^2 + Q^2(x)\chi_{\Omega_{\mathbf{u}}})\,dx$ which give rise to a singular cooperative system. Here \(\mathbf{u}=(u_1,\dots,u_m): D\to\R_+^m\) is a vector-valued unknown function, \(\Omega_{\mathbf{u}}:=\{|\mathbf{u}|>0\}\) is the positive set, \(\chi_{\Omega_{\mathbf{u}}}\) is the characteristic function of the positive set \(\Omega_{\mathbf{u}}\), and \(Q(x)\) is the Bernoulli weight function. This problem was first introduced by Caffarelli, Shahgholian, and Yeressian in the poineer work (Duke Math. J. 167(10), 2018), where the regularity theory for minimizers and for the free boundary \(\partial\Omega_{\mathbf{u}}\) was established for nondegenerate Bernoulli weights, i.e., \(Q(x)\ge Q_{\rm min}>0\). The present paper studies the same free boundary problem but for a degenerate Bernoulli weight. The main difficulty lies in the coupling between the degeneracy of \(Q(x)\) and the vector-valued nature of the problem. We show that the the free boundary set \(\partial\Omega_{\mathbf{u}}\cap\{Q(x)=0\}\) decomposes into a nondegenerate part and a degenerate part. Nondegenerate points admit nontrivial blow-up limits; degenerate points have zero weighted density. We classify nondegenerate points into single-phase, multi-phase non-branching, and multi-phase branching points, analyzing their geometric structure. The branching-point singular structure for multi-phase points is purely vectorial and new in this setting. Finally, following Naber–Valtorta (Ann. Math. 185, 2017) and Edelen–Engelstein (Trans. Amer. Math. Soc. 371, 2019), we prove the nondegenerate set has locally finite $\mathcal{H}^{n-2}$-measure and is countably $(n-2)$-rectifiable.

Motivated by Eliahou's 64-modular Hadamard construction at the smallest open Hadamard order n=668, we introduce an explicit half-flip family of 32-modular Hadamard matrices at orders n=4L for L = 3 (mod 8). A master identity reduces the four-sequence Golay-quadruple condition under the half-flip ansatz (s, s*, sq, (sq)*) to a single-sequence type-restricted autocorrelation c_k^tau(s). The construction yields a closed-form expression for c_k^tau, a true Hadamard matrix at L=11, and 32-modular matrices at every L = 3 (mod 8), including the open orders n=716 and n=1132. Existence of 32-modular at L = 3 (mod 4) is due to Eliahou-Kervaire (2001); Eliahou's 2026 follow-up in J. Algebraic Combin. gives 64-modular matrices at L = 3 (mod 16) and L = 7 (mod 32) via the same ansatz and correlation identity we call the master identity, subsuming our construction on that residue subclass. Our contribution is therefore primarily structural. Applying Barrera Acevedo-O Cathain-Dietrich (2019) and Alvarez et al. (2020), the family is non-cocyclic over any group at every prime L in {11, 19, 59} in the YES set, yet pseudococyclic over the Goethals-Seidel Moufang loop GS_{4L} at every L. A half-flip H-set decomposition theorem parameterizes the symmetric difference of any two family elements by a single sequence flip set, giving the family the structure of a length-L Hamming cube. A symbolic mod-tower verifier (mod-8 is F_2-linear) classifies true Hadamards in the family through k=14 (L <= 115): the YES set is empirically bounded by k=7, refuting four H4 predictions and excluding L in {179, 283} within the ansatz. A Grobner basis at L=11 exhibits a previously unrecorded even-T0-block linear identity. All code and JSON certificates: github.com/michelkulhandjian/hadamard-halfflip-structural

We study $n$-torsionfree objects in abelian categories with enough projectives. Frobenius functors preserve $n$-torsionfreeness, and faithful ones reflect it. We prove that stabilization of the torsionfree filtration implies weak Gorensteinness. For Frobenius extensions satisfying a generator condition, we compare the terms of minimal injective resolutions and obtain transfer of Auslander-type conditions and of the Auslander–Gorenstein conjecture. We also compute a family of non-Gorenstein algebras whose torsionfree filtrations stabilize at level two and contain explicit nonprojective Gorenstein projective modules.

Let $G$ be a connected graph. The $k$-token graph of $G$ is the graph $F_k(G)$ whose vertex set consists of all subsets of $k$ vertices of $G$, where two of them are adjacent whenever their symmetric difference is an edge of $G$. Every automorphism of $G$ induces one of $F_k(G)$, as does complementation when $k=|G|/2$; automorphisms of this form are called induced. Fabila-Monroy et al.\ (Graphs and Combinatorics 42, 2026) show that token graphs can have many non-induced automorphisms, arising from twin cuts. These are cut sets $\{x,y\}$ whose two vertices have the same neighbours (other than themselves) in $G$. These non-induced automorphisms send configurations with a prescribed number of tokens on each component of $G\setminus \{x,y\}$, totalling $k-1$, and exactly one token on one vertex of $\{x,y\}$, to the configuration obtained by moving (flipping) the token at $\{x,y\}$ to the other vertex of $\{x,y\}$. These automorphisms send an induced $4$-cycle generated by moving two tokens on two disjoint edges to one generated by moving two tokens on a $4$-cycle; thus introducing what we call a twist. We prove a partial converse: if an isomorphism $\varphi\colon F_k(G)\to F_{k'}(G')$ sends some $4$-cycle generated by moving two tokens on two disjoint edges to one generated by moving two tokens on a $4$-cycle, then $G$ and $G'$ have twin cuts $\{x,y\}$ and $\{x',y'\}$, respectively. We also show that any twist can be undone by composing $\varphi$ with twin-cut flips.

We prove that every Deligne–Mumford superstack over $\mathbb{Z}[1/2]$ with finite inertia admits a coarse moduli superspace. This generalizes the Keel–Mori theorem to the super setting. As part of our argument, we introduce a notion of integral morphisms in algebraic supergeometry and use it to establish an effective descent result for étale morphisms of algebraic superspaces. We also include an example of a non-Deligne–Mumford superstack with finite inertia that does not admit a coarse moduli superspace.

A controlled comparative study of a mesh-free Kolmogorov–Arnold Physics-Informed Neural Network (KAN-PINN) applied to nonlinear strain-limiting partial differential equations is presented. Three distinct training runs across varying material-parameter pairs $(\alpha, \beta)$ are evaluated within a fixed computational pipeline comprising an exact hard boundary ansatz, a robust pseudo-Huber residual loss, and a multi-stage Adam-to-L-BFGS optimizer schedule. A fundamental metric inversion is revealed by the analysis: while training loss reflects operator-specific residual scaling and constitutive stiffness, the operator-independent relative $L_2$ error demonstrates that intermediate parameter regimes achieve superior solution accuracy and enhanced generalization.

Let $G=(G_1,...,G_n)$ be a graph system on a common $n$-vertex set $V$. For $v\in V$, set \[ \delta(v)=\min_{i\in[n]}d_{G_i}(v), \qquad E_{\cap}=\bigcap_{i=1}^n E(G_i), \] and let $d^c(u,v)$ denote the length of a shortest rainbow $u$–$v$ path. We study Hamiltonicity conditions imposed only on pairs at small rainbow distance. Suppose that every $G_i$ is $2$-connected and that \[ \delta(u),\delta(v)<\frac n2, \qquad d^c(u,v)\le2 \quad\Longrightarrow\quad uv\in E_{\cap}. \] Our main result shows that any system satisfying this local Fan-type condition but containing no rainbow Hamilton cycle must satisfy \[ E_{\cap}=\varnothing, \qquad \sum_{v\in V}\delta(v)\le\frac{n^2-1}{2}. \] Consequently, either $E_{\cap}\ne\varnothing$ or $\sum_{v\in V}\delta(v)\ge\lceil n^2/2\rceil$ guarantees a rainbow Hamilton cycle. We further prove that the local Ore-type condition \[ \delta(u)+\delta(v)<n, \qquad d^c(u,v)\le2 \quad\Longrightarrow\quad uv\in E_{\cap} \] already forces a rainbow Hamilton cycle. Along the way, the Fan-type hypothesis yields a rainbow Hamilton path and a rainbow $(n-1)$-cycle. For diagonal systems, our conditions reduce to the classical Fan and local Ore conditions. The proofs combine rainbow rotation–extension with a double-counting argument in an auxiliary digraph. Keywords: graph system; rainbow Hamilton cycle; rainbow distance; Fan-type condition; Ore-type condition; rotation–extension.

Let $\mathbb{R}\langle X,Y\rangle$ be the free algebra on two symmetric letters. A polynomial is trace-positive if it has non-negative trace on every pair of real symmetric matrices, and it lies in the cone $\Theta$ if it is cyclically equivalent to a sum of Hermitian squares. Burgdorf and Klep proved that the two notions coincide in degree at most four; from degree six onwards they differ. We give a complete, explicit description of both cones on the six-dimensional space $Z$ of symmetric homogeneous sextics in two symmetric letters whose abelianization vanishes identically, and show that on $Z$ they coincide: for $f\in Z$, trace-positivity is equivalent to membership in $\Theta_3$, to non-negativity of the trace on pairs of $2\times2$ symmetric matrices, and to being cyclically equivalent to a sum of four Hermitian squares of cubics with vanishing abelianization. Membership in $\Theta$ reduces to a $4\times4$ positive semidefinite Gram matrix whose diagonal blocks are fixed by the content-$(4,2)$ and $(2,4)$ parts of $f$; each of those parts ranges over a two-dimensional cone with two linear facets, and the content-$(3,3)$ part is then confined to a rectangle whose sides are the geometric means $\sqrt{p_+q_+}$ and $\sqrt{p_-q_-}$ of the eigenvalues of those blocks. Each facet is certified by one of two families of pairs of $2\times2$ symmetric matrices. The abelianization hypothesis is natural: every bivariate degree-six example we are aware of – a trace-positive polynomial outside $\Theta$ – arises by lifting a non-negative, non-SOS polynomial in two variables to a cyclically sorted polynomial; and that lifting cannot operate when the abelianization vanishes. Symmetry of $f$ is essential: dropping it adds one antisymmetric direction, on which the trace vanishes identically, yet whose non-zero elements all lie outside $\Theta$.

Burde proved that the Lie algebra $\mathfrak{sl}_2(\mathbb{K})$, over a field $\mathbb{K}$ of characteristic $p > 2$, admits compatible left-symmetric algebra structures if and only if $p = 3$. Since the even part of the Lie superalgebra $\mathfrak{osp}(1|2)$ is isomorphic to $\mathfrak{sl}_2(\mathbb{K})$, we ask whether Burde's structures can be extended to compatible left-symmetric superalgebra structures on $\mathfrak{osp}(1|2)$. We verify Burde's left-symmetric structures on $\mathfrak{sl}_2(\mathbb{K})$, and then test their possible extensions to $\mathfrak{osp}(1|2)$. We show that no such extension exists in characteristic 3. Together with the obstruction coming from the even part for characteristic $0$ and $p > 3$, this gives non-existence for a field $\mathbb{K}$ of characteristic different from $2$.