Papers

We classify all Rota-Baxter operators of weight $\lambda$ on $M_n(F)$ over a field $F$ of characteristic zero whose minimal polynomial has maximal degree and show that in both cases $\lambda = 0$ and $\lambda \neq 0$ such an operator is conjugate to exactly one operator.

We consider the problem of minimizing error in quantized matrix multiplication $C=AB$. Scalar quantization of the factors introduces rounding errors whose scale depends on the maximum absolute entries – the ranges – of their rows and columns. These ranges determine the quantization grid steps. To reduce the error, we optimize over product-preserving transformations that alter the factor ranges and grid steps without changing $C$. Specifically, we seek the smallest leading expected squared error over invertible inner changes of basis and orthogonal outer rotations. Under independent, zero-mean subtractive dither noise on an unbounded lattice, we prove the output-only bound $E_{\rm lead} \ge (c_A+c_B)/K \Vert AB\Vert_*^2$, where $K$ is the inner dimension, $c_A$ and $c_B$ are normalized noise variances, and $\Vert AB\Vert_*$ is the nuclear norm. The bound is tight: an SVD-aligned Hadamard construction attains the infimum whenever a Hadamard matrix of order $K$ exists, including every power of two, while an SVD-aligned DCT construction is within a factor of two for every $K$. Without outer rotations, Gram-matrix balancing minimizes factorization energy, and finite-set flattening achieves the bound within $C\log(K(m+n))$. For power-of-two $K$, conditional expectations deterministically select the Hadamard signs in $O((m+n)K^2)$ exact-real operations. Synthetic experiments verify both constructions and illustrate the tradeoff between regularization and conditioning. These results characterize the full-gauge optimum and quantify the cost of preserving row and column indices.

Let \(T_\Lambda=\mathbb R^d/\Lambda\), \(d\geq2\), be a flat torus with quotient metric \(\rho_\Lambda\), and consider the distance energies \(I_\alpha(\mu)=\iint \rho_\Lambda(x,y)^\alpha\,d\mu(x)\,d\mu(y)\) of Borel probability measures \(\mu\). We prove a quantitative Fourier signature of the cut locus: for every Voronoi facet and every \(\alpha>0\), there is a sequence of dual-lattice frequencies approaching the facet normal along which the Fourier coefficients of \(\rho_\Lambda^\alpha\) are positive, with an explicit leading asymptotic determined by the facet. As direct consequences, Haar measure is not a local maximizer for any positive distance power, even among smooth densities, and every flat torus of dimension at least two has supremal negative type and generalized roundness zero. We then solve the global maximization problem for two classes of flat tori. On an orthogonal rectangular torus, the maximizers undergo a transition at \(\alpha=2\): for \(1\leq\alpha<2\) they are the translated uniform measures on the two-torsion subgroup; at \(\alpha=2\) all balanced couplings on translates of that subgroup are extremal; and for \(\alpha>2\) only equally weighted diametral pairs remain. On the regular hexagonal torus, the maximizers are precisely the uniform measures on translates of a distinguished cyclic subgroup of order three for every \(\alpha\geq1\).

Estimating a low-dimensional subspace shared across noisy data matrices is a fundamental problem in multi-view matrix estimation. We study this problem under the two-view JIVE model, where each data matrix contains shared and view-specific low-rank components. We demonstrate that standard plug-in subspace intersection, including AJIVE, suffers from a second-order bias caused by direction-dependent leakage of the empirical singular vectors. We propose bias-corrected subspace intersection (BCSI), which removes this bias before estimating the shared subspace. We establish finite-sample risk bounds for BCSI that accommodate unequal view dimensions, signal strengths, and view-specific ranks and require no condition-number assumptions on the signal matrices. When the shared and view-specific ranks are comparable, these bounds match our minimax lower bounds up to universal constants. The resulting minimax rate contains a new second-order term, arising from quadratic leakage perturbations relative to the shrinking spectral gap when the view-specific subspaces are nearly aligned. This term is absent from previous JIVE minimax lower bounds. Numerical experiments demonstrate the advantage of BCSI over AJIVE when the leakage bias is pronounced. Along the way, we establish a nonasymptotic concentration result for the bias-corrected leakage Gram matrix of a rectangular spiked matrix, which may be of independent interest.

Let $(X_1,\ldots,X_n)$ be a random variable in $(\mathbb{R}\setminus\{0\})^n$ that is both exchangeable and sign-invariant. For every $k\in[n]$, let $S_k=\sum_{i=1}^kX_i$. Define the weak persistence probability as $\mathbb{P}(S_1,\ldots,S_n\geq0)$, and the strong persistence probability as $\mathbb{P}(S_1,\ldots,S_n>0)$. From previous results, the optimal lower bound for $\mathbb{P}(S_1,\ldots,S_n\geq0)$ and the optimal upper bound for $\mathbb{P}(S_1,\ldots,S_n>0)$ are known. We complete the picture by determining the optimal upper bound for $\mathbb{P}(S_1,\ldots,S_n\geq0)$ and the optimal lower bound for $\mathbb{P}(S_1,\ldots,S_n>0)$ as follows. \[\frac{1}{2^n}\binom{n-1}{\lfloor (n-1)/2\rfloor}\leq\mathbb{P}(S_1,\ldots,S_n>0)\leq\frac{1}{4^n}\binom{2n}{n}\leq\mathbb{P}(S_1,\ldots,S_n\geq0)\leq\frac{1}{2^n}\binom{n}{\lfloor n/2\rfloor}.\] In particular, this implies that the weak and strong persistence probabilities of every exchangeable and sign-invariant random variable $(X_1,\ldots,X_n)$ in $(\mathbb{R}\setminus\{0\})^n$ are of the order $n^{-1/2}$. We also obtain some related results, one in the deterministic setting, and one when the random variables are allowed to take the value 0.

Let $d=(n-1)^2+1$, the dimension of the real linear span of the $n\times n$ permutation matrices. We prove that $d$ independent uniformly random permutation matrices fail to form a basis with probability $(1+o(1))n^2(1-1/n)^d=(e^{3/2}+o(1))n^2e^{-n}$. The same asymptotic holds for a uniformly random $d$-element subset, confirming a conjecture of Kushwaha and Tripathi and identifying the leading obstruction: a matrix position avoided by every sample. More generally, for any fixed number of additional samples, we determine the first three exponential orders of the failure probability. To prove these results, we develop support estimates valid over arbitrary fields, derive Fourier bounds from permanental minors, and introduce a counting argument for concentrated assignment functionals over large prime fields. We also give a sparse lifting argument showing that real rank deficiency without an annihilating functional of small support has probability $o(e^{-Cn})$ for every fixed $C>0$.

In this paper we obtain all doubly-stochastic solutions to the Yang-Baxter-like matrix equation $AXA = XAX$ when $A$ is a permutation matrix. We characterize the permutation solutions, proper doubly stochastic solutions, and we accompany our results with suitable examples.

We introduce, motivated by noncommutative Carrollian geometry, the notion of a Carrollian–Poisson bundle understood as a Carrollian manifold with a compatible Poisson structure. Compatibility is expressed as the inclusion of the Carroll distribution in the image of the Poisson anchor. We explore the geometric consequences of this compatibility and present several simple examples of Carrollian–Poisson bundles, including the (outer) Kerr horizon, the BTZ horizon, and future null infinity. Furthermore, we establish that every Carrollian–Poisson bundle can be equipped with a Vaisman (contravariant) connection that is both metric-compatible and clock-compatible, and thus geometric kinematics can be defined and studied.

We prove that every strictly positive probability distribution on \(\{-1,1\}^n\) is represented exactly by a sigmoid belief network with finite parameters. This answers a question of Sutskever and Hinton. The proof upgrades their probability-sharing approximation to exact representation using Brouwer's fixed-point theorem.

The Erd\H os–Sós conjecture states that every $n$-vertex graph with more than $(k-2)n/2$ edges contains every $k$-vertex tree. We prove that for every $\gamma$ there is an $n_0$ such that for all $n\ge n_0$ and $k \ge \gamma n$ the conjecture holds. As a corollary of our result, we obtain a solution of a 51-year-old problem of Erd\H os and Graham on the multicolor Ramsey numbers of trees.

By Tutte's classic theorem of 1956 that every 4-connected planar graph is Hamiltonian, every planar graph of order at least three with toughness greater than $\frac{3}{2}$ is Hamiltonian. In 1999, Owens constructed a sequence of maximal planar graphs whose toughness approaches $\frac{3}{2}$ from below and which do not contain even a 2-factor, and he asked whether there exists a maximal planar graph with toughness exactly $\frac{3}{2}$ and with no 2-factor. In 2025, Shan constructed a $\frac{3}{2}$-tough plane triangulation with no 2-factor. In that construction, there are many pairs of vertices of degree $3$ that have a common neighbor. By imposing a distance condition on the vertices of degree $3$, Hao, Ma, Shan, and Yang recently proved that every $\frac{3}{2}$-tough plane triangulation of order at least three whose vertices of degree $3$ are pairwise at distance at least $3$ has a 2-factor, and they asked whether every such graph is Hamiltonian. We answer this question in the negative, and in fact prove the following stronger statement: for every positive integer $\ell$, there exists a $\frac{3}{2}$-tough non-Hamiltonian plane triangulation whose vertices of degree $3$ are pairwise at distance at least $\ell$. Thus, although the distance condition guarantees the existence of a 2-factor, it does not guarantee that the graph is Hamiltonian: the essential obstruction to a Hamiltonian cycle is a certain local configuration involving a vertex of degree $3$, rather than the proximity of such configurations in the graph.

The Erd\H os–Sós conjecture states that every $n$-vertex graph $G$ with more than $(k-2)n/2$ edges contains every $k$-vertex tree. We solve the extremal cases of this conjecture, showing that for some fixed $\mu>0$, the conjecture holds for each $G$ that minimally satisfies the assumptions of the conjecture and has a subgraph~$H$ of minimum degree $\delta(H)\ge (1-\mu)k$. In our proof, we mainly have to deal with $H$ taking two different shapes: either $H$ is close to the complete graph $K_k$ or $H$ is close to the complete bipartite graph $K_{k,k}$.

We prove a necessary and sufficient condition for an abstract polyhedron $\Pi$ to be realized as a right-angled, ideal polyhedron in hyperbolic space.

For $2 \times 2$ matrices $A_1,\dots,A_k$, write $[A_1,\dots,A_k]$ for the left-normed iterated commutator $[\dots[[A_1,A_2],A_3],\dots,A_k]$, and $I_k$ for the ideal, in the $3k$-variable reduced-coordinate polynomial ring $R_k$, cutting out its vanishing locus. We prove, for every $k \geq 2$ over any field of characteristic $\neq 2$, and as four independently-established results rather than one bundled claim: $I_k$ has codimension 2; $I_k$ has exactly 3 minimal generators; $R_k / I_k$ is Cohen-Macaulay; and $I_k$ is radical. The last of these, together with an explicit component count resting on a non-containment argument, assembles into the Primary Decomposition Theorem: $I_k = P_2 \cap \cdots \cap P_k$ is an irredundant primary decomposition into exactly $k-1$ primes, following an explicit recursive block-involvement pattern. The proof identifies $I_k$ as the ideal of $2 \times 2$ minors of an explicit $2 \times 3$ matrix (a determinantal ideal, not merely one that looks determinantal), and invokes classical determinantal-ideal theory (Bruns-Vetter) and an explicit rank-2 Jacobian witness on every component (Serre's criterion) for the algebraic and radicality halves respectively.

We determine, up to a factor of $2^{o(k)}$, the number of $k$-sets $A \subset \{1, \ldots, n\}$ such that $|A + A| \leq m$, where $k = \Theta(\log n)$ and $m \leq k^{1 + \alpha}$, for small $\alpha > 0$, answering a question of Green and Morris.

We prove a weak-type $(1,1)$ estimate for the centered Hardy–Littlewood maximal function with respect to Euclidean balls with dimensional dependence $O(\sqrt{n} \log n)$. This improves the order of growth in the classical $O(n)$ estimate of Stein and Strömberg. The proof goes through a pointwise bound of the Hardy–Littlewood maximal operator by the heat maximal operator with $\sqrt{n}$ loss. The key technical aspect of our result is an improvement of the weak-type bound for the heat maximal operator from $O(\sqrt{n})$ to $O(\log n)$.

Arithmetic terms are fixed finite compositions of addition, truncated subtraction, multiplication, integer division and exponentiation on natural numbers. We construct such terms for the number of affine solutions of $ y^2 = x^3 + Ax + B $. For arbitrary moduli $ n \geq 1 $, a specialization of Prunescu's general construction reduces the number of monomial contributions from fourty-nine to fifteen and the size of the packed integer from approximately $ 2n^{11} $ to approximately $ 2n^5 $ binary digits. For prime moduli $ p \geq 17 $, the Hasse invariant and the trace of Frobenius give a term of about thirty operations. For curves defined over $ \mathbb F_p $, a further term counts the points over $ \mathbb F_{p^k} $ with $ k $ as a variable. For primes $ p \geq 5 $, we also obtain arithmetic terms for the counts over $ \mathbb Z / p^k \mathbb Z $ with $ k $ variable, covering good reduction, nodes and cusps, including non-minimal equations and zero discriminant.

In the uniform sparsest cut problem we are asked to find a vertex set that cuts few edges relative to the number of vertex pairs it separates. The Goemans-Linial SDP coupled with the Arora-Rao-Vazirani rounding gives an $\mathcal{O}(\sqrt{\log n})$ approximation on arbitrary graphs on $n$ vertices. We study this relaxation on finite Abelian Cayley graphs. First we show that when the second normalized Laplacian eigenvalue of $G= \mathrm{Cayley}(\Gamma, S)$ is realized by a Fourier character with image size at most four then $\lambda_2(G)=\mathrm{SDP}_{\mathrm{GL}}(G)=\psi(G)$. Geometrically, a character maps the vertices onto a regular polygon where the squared chord distance satisfies the triangle inequalities exactly when the polygon has at most four vertices. Grouping equal character fibers gives a cyclic quotient where the optimal cut can be found exactly and so the relaxation is exact on finite Abelian Cayley graphs on groups of exponent at most four. Second, we replace each generator $s$ of $S$ by a uniformly random element of its cyclic subgroup (including identity). If $r_s$ is the order of $s$, we let $\alpha(r_s)$ to be the average number of $\pm s$ steps needed to simulate such a move, and let $\rho(S)=\max_{s\in S}\alpha(r_s)$ be its worst case. Full cyclic averaging eliminates character phases and choosing a nontrivial character $\chi^*$ minimizing the auxiliary eigenvalue and taking $K=\mathrm{ker}\chi^*$ gives \[ \psi(G)\leq\psi_G(K)\leq\frac{q^*}{q^*-1} \cdot\rho(S)\cdot\mathrm{SDP}_{\mathrm{GL}}(G)\leq 2\rho(S)\cdot\mathrm{SDP}_{\mathrm{GL}}(G), \] where $q^*=|\chi^*(\Gamma)|$. If all generator orders are at most $R$, this is an $R/2$ approximation. Finally, we construct an infinite family of finite Abelian Cayley graphs with Goemans-Linial integrality gap exactly $16/15$.

McDuff and Segal proved homological stability for unordered configuration spaces of connected manifolds with non-empty boundary. Later, Church proved representation stability for ordered configuration spaces on compact manifolds. These stability patterns extend to surface braid groups and pure surface braid groups, respectively, whenever the associated configuration space is an Eilenberg–Maclane space. However, these results do not apply to braid groups and pure braid groups on $S^2$ and $\RP^2$ since ordered and unordered configuration spaces on these surfaces are not Eilenberg–Maclane spaces. This paper will use results by Cohen–Pakianathan and the theory of $\FI$-homology to extend known stability patterns to these unknown cases. This is a special case of a more general result proven in this paper: the equivariant homology of ordered and unordered configuration spaces exhibit representation stability and stable periodicity respectively.

Klaja and Ransford exhibited a complex unital Banach algebra in which the exponential spectra of \(ab\) and \(ba\) differ away from zero, and asked whether this can occur in an algebra of bounded operators. We answer their question affirmatively. For a Bourgain–Delbaen space \(X\) obtained from Motakis' construction with Calkin algebra isomorphic as a Banach algebra to \(C(S^4)\), we construct \(S,T\in\B(X\oplus X)\) such that \[ \frac12\in\varepsilon_{\B(X\oplus X)}(ST) \quad\text{and}\quad \frac12\notin\varepsilon_{\B(X\oplus X)}(TS). \] Moreover, \(\ltsr\B(X\oplus X)=\rtsr\B(X\oplus X)=2\), which is the least possible stable rank for such an example. More generally, if \(K\) is compact metrisable and \(X\) is any space arising from Motakis' construction for \(C(K)\), then, for every \(n\geqslant1\), \[ \ltsr\B(X^n)=\rtsr\B(X^n)= \begin{cases} \left\lceil \lfloor\dim K/2\rfloor/n\right\rceil+1,&\dim K<\infty, \infty,&\dim K=\infty. \end{cases} \] This also gives counterexamples of every finite stable rank at least two and of infinite stable rank. Finally, we relate every failure of exponential-spectral commutativity to the kernel of the first matrix stabilisation map on the index group.

We study finite-sample valid tests of the one-sided mean hypothesis $H_0:\mu\leq 1$ against $H_1:\mu>1$ for nonnegative random variables. To do so, we develop a leave-one-out dual certificate framework, where certain pointwise inequalities imply p-value validity under the conditional mean null $\mathbb{E}[X_i\mid\mathbf{X}_{-i}]\leq 1$, and which also gives conditions that allow combining dual certificates for p-values to show that their pointwise minimum is also a valid p-value. The framework proves finite-sample validity of Wang and Zhao's nonparametric likelihood-ratio statistic $T_{\mathrm{nplr}}$, yields a new p-value $T_{\mathrm{bin}+}$ extending the Clopper–Pearson binomial test to general nonnegative random variables, and shows that the pointwise minimum $\min\{T_{\mathrm{nplr}},T_{\mathrm{bin}+}\}$ is itself a valid and more powerful p-value. We establish sharp optimality results for such testing problems in two regimes: both $T_{\mathrm{nplr}}$ and $T_{\mathrm{bin}+}$ attain a universal detectability boundary for the null $H_0$ without moment or tail assumptions, and $T_{\mathrm{bin}+}$ attains a nonparametric power lower bound under $n^{-1/2}$-local alternatives to $H_0$. Efficient algorithms and numerical experiments demonstrate substantial finite-sample power gains over existing valid methods.

We determine the full Sprague-Grundy sequence and its least period for a class of three-move subtraction games whose sequences are periodic from the start. Write the move set as $S=\{a,b,c\}$, with $0<a<b<c$ and $\gcd(a,b,c)=1$. The case $a=1$ is known and is summarized separately. For $a\ge2$ and $c\ne a+b$, we give three explicit sufficient tests for pure periodicity, associated with the candidate periods $a+b$, $c+a$, and $c+b$. For fixed $a,b$, the tests depend only on $c\bmod(a+b)$ and are computed from the losing positions of the two-move game $\{a,b\}$. Whenever a test succeeds, we give the losing positions in closed form, reconstruct the remaining values, and prove that the least period is the smallest candidate whose test succeeds. The construction shows how the added move $c$ modifies the two-move pattern to form a repeating block. We also give a uniform formulation of the known additive case $c=a+b$. For $a\ge2$, we conjecture that the tests cover every purely periodic non-additive game. In this range, when $c\ge2(a+b)$, we prove necessity for purely periodic games with least period at most $a+b$.

In this note we study a two-regime representation of a loss random variable under quadratic error. For a finite law we compute exactly the change of the optimal risk when one atom crosses the cut. This turns the problem into a convexity question in cumulative-mass coordinates. On an equally spaced support, log-concavity gives this convexity, while weak symmetry locates the optimal cut, with an additional correction when the mean lies between two atoms. We also discuss the continuous analogue and extend the main identities to finitely supported random vectors, where a global optimal partition may be chosen as a halfspace.

Data-Driven Computational Mechanics (DDCM) replaces traditional phenomenological constitutive models by directly reformulating boundary-value problems in terms of local material state data obtained from experiments or fine-scale simulations. Standard DDCM formulations, only enforcing equilibrium and compatibility, do not inherently guarantee compliance with the second law of thermodynamics. This breakdown occurs particularly when input material data sets are subject to noise or local physical non-admissibility. In this work, we present a variational DDCM framework specifically tailored to diffusion–reaction problems. Taking advantage of the simplicity of the thermodynamic constraint in gradient-flux systems, we propose an augmented formulation that explicitly enforces the second law of thermodynamics as a hard constraint within the energy-minimization problem. Although the set of thermodynamically admissible states is non-convex and fails to be weakly closed in the ambient phase space, we establish existence of minimizers by proving that the intersection of the admissible set with the subspace of fields that are compatible and in equilibrium is weakly sequentially closed via a compensated compactness argument. To enable practical computations, we analyze both a Lagrange multiplier formulation and a penalization scheme. We prove the $\Gamma$-convergence of the penalized functionals to the exact constrained problem and establish a fully discrete convergence framework incorporating spatial finite-element discretization and empirical data-set approximations. Numerical experiments confirm that the proposed penalty scheme effectively restores thermodynamic consistency even in the presence of severely corrupted material data.

We construct a framework for higher order unbounded rough drivers defined by rough paths over vector fields. This allows us to consider a purely Eulerian perspective for rough transport equations where the driving rough path vector field is allowed to have temporal $\mathfrak{p}$-variation for any $\mathfrak{p} \in (1,\infty)$. We use this framework to show the well-posedness of an equation with two transport terms; one driven by a rough path vector field with aforementioned $\mathfrak{p}$-variation in time, and the other driven by a DiPerna–Lions-type drift vector field.

The Koebe-Andreev-Thurston theorem assigns a 3-dimensional hyperbolic reflection group to each combinatorial polyhedron. A natural question is: which of them are arithmetic? In 2016, Kontorovich-Nakamura conjectured that all arithmetic reflection groups obtained in this way are commensurable to those obtained from the tetrahedron, square pyramid, or cuboctahedron. In this paper, we prove the conjecture. It is a consequence of the following result of independent interest: all arithmetic ideal, right-angled hyperbolic polyhedra are obtained by gluing together copies of one of three “seed” polyhedra.

We prove that if a real-valued function $f\in L^1$ on the complex unit circle has a gap of width at least $\pi$ in its essential range, then $\exp(\tilde f)$ is not integrable, where $\tilde f$ is the conjugate function. More generally, the exponential function can be replaced by any nonnegative convex function $Y$ satisfying $$ \int_0^\infty Y(x) e^{-x}\, dx = \infty. $$ We also establish a local version on arcs of the unit circle: Under a natural condition on the inverse images of the two sides of the gap, $Y(\tilde f)$ fails to be integrable on the arc; in particular, it suffices that one of these inverse images is not an interval modulo null sets. Furthermore, we obtain corresponding results for complex-valued functions and a $Y$-integrability version of Zygmund's result.

We consider collisions of simple random walks on comb graphs $\mathrm{Comb}(\mathbb{Z},H)$, which are obtained by attaching vertical segments of the form $[0,H_x] \cap \mathbb{Z}$ to any point $x$ of the integer axis. For $\mathrm{Comb}(\mathbb{Z},H)$ with profile $H_x(x) = |x| \log^\gamma(|x| \vee 1)$, we show that two independent simple random walks starting from the same site collide infinitely often almost surely if $\gamma \leq 2$. If the tooth profile is taken as a typical realization of i.i.d. heavy-tailed random variables with $\textbf{P}(H_x > z) \sim Cz^{-\gamma}$ (with some $C > 0$) as $z$ tends to infinity, we show that infinitely many collisions occur almost surely for two independent random walks if $\gamma > 1/3$, whereas finitely many collisions occur almost surely if $\gamma \in (0,1/3)$, and for any $\gamma \in (0,1]$, three independent random walks only collide finitely many times, almost surely.

We study multiplicative inequalities among the coefficients of Lorentzian polynomials through the notion of bounded ratios. Our main structural result completely characterizes the cone of bounded ratios for Lorentzian polynomials of degree $n$ in $k$ variables. We show that the dual of the cone of bounded ratios is generated by equivalence classes of M-convex functions modulo affine functions. For ternary Lorentzian forms of arbitrary degree $n\ge3$, we show that the cone of bounded ratios is generated by triangular ratios and determine the optimal bounding constant of every bounded ratio. Furthermore, we characterize the pairs $(n,k)$ for which the cone of bounded ratios can be computed by tropicalizing products of $n$ linear forms in $k$ variables with nonnegative coefficients.

For a finite set $S$ of positive integers, put $K(S):=-\min_{x\in\mathbb T}\sum_{s\in S}\cos(2\pi sx)$. Bedert recently proved the uniform lower bound $K(S)\geq |S|^{1/5-o(1)}$. We remove the subpolynomial loss and prove that $K(S)\geq c|S|^{1/5}$ for an absolute constant $c>0$. The proof combines two estimates from Bedert's argument with an exact averaging identity for the asymmetric boundaries of additive intersections. This identity replaces the multiplicative-amplification step responsible for the logarithmic loss.