Papers

We prove a dimension-free Poincaré inequality for isotropic unconditional log-concave probability measures, establishing the Kannan-Lovász-Simonovits (KLS) conjecture in the unconditional setting. The proof combines a new application of Dunkl operator theory combined with an appropriate transformation of the log-concave measure. …

Complete original asymptotic target: the smallest singular value of a uniform column subset tends to zero in probability for every admissible deterministic fixed-sparsity matrix sequence. The complete proof builds locally on macOS and its transitive axiom audit passes. Fresh sandboxed …

We construct a finite connected simple cubic graph $F$ such that every signing of its edges yields a signed adjacency matrix with an eigenvalue outside $[-2\sqrt2,2\sqrt2]$. This disproves the Bilu--Linial signing conjecture for general regular graphs. The graph $F$ is …

We study the uniform null-controllability, in the vanishing viscosity limit, of the transport--diffusion equation $u_t + M u_x = \varepsilon u_{xx}$ on $[0,L]$ with a Dirichlet boundary control at $x=0$. Our results concern the uniform null-controllability time $T_{\mathrm{unif}}$, defined as …

Spencer's 1985 ``six standard deviations suffice'' theorem shows that every $A \in [-1,1]^{n \times n}$ has a sign vector $x \in \{-1,1\}^n$ with $\|Ax\|_\infty \le 6\sqrt{n}$. We show the upper bound $\sqrt{3\operatorname{arsinh}(10)}\sqrt{n}+4 < 2.9992 \sqrt{n} + 4$ by directly rounding …

Let $0<a_1\le\cdots\le a_N$ and $H>a_N$. On every compact connected orientable surface with $b\ge1$ boundary components and genus at least an explicit integer $\Gamma(N,b)$, we construct a smooth metric whose first $N$ positive Steklov eigenvalues are $a_1,\ldots,a_N$ and whose next eigenvalue …

This paper settles the Matroid Secretary Problem with an $e$-probability-competitive algorithm. The algorithm is ordinal and accesses arrived elements only through comparison and independence oracles, and has expected polynomial time and oracle complexity.

It is well known that the \Lovasz-Theta function of a random graph $G(n,\tfrac{1}{2})$ is $Θ(\sqrt{n})$. More precisely, it is tightly concentrated in the interval \( [\sqrt{n},\, 2\sqrt{n}], \) where the upper bound follows from an explicit dual witness for the …

Determining a noisy quantum channel's classical capacity and entanglement cost generally requires regularization over many channel uses. We remove both regularizations for every qubit-to-qubit channel admitting a pure output. For each such channel, Holevo information, channel entanglement of formation, and …

Complete negative answer to the original direct induced p-to-2 norm uniqueness conjecture, with two distinct global minimizers at the exact admissible p=4 witness.

We solve a problem posed by Audenaert and Kittaneh in 2012 by proving that the Aujla--Bourin subadditivity inequality can be strengthened to an optimal form. We also answer questions posed by Bourin and the second author: (1) we significantly extend …

For an integer $m\ge3$ let $\lambda_{n,1}\le\cdots\le\lambda_{n,n}$ be the eigenvalues of the Toeplitz matrix $T_n(g)$ generated by $g(\theta)=(2-2\cos\theta)^m$, and put $h=(n+2)^{-1}$. There are real $C^\infty$ functions $d_0=g,d_1,\ldots,d_{2m}$ on $[0,\pi]$, given by an explicit phase built from the stable roots of the …

Complete symbolic Lean proof of the original three-part target, continuous-family obstruction, smooth common coefficients, sharp finite-head and global errors, and exact Fourier-matrix correspondence. All five contracts passed actual non-root Linux Comparator and default-kernel acceptance at the recorded immutable proof revision.

Let $m,n\ge 2$. For $m\times n$ complex matrices $A$, $C$ and an $n\times m$ matrix $B$, Nobori (Linear Algebra Appl. 725 (2025) 135--144) proposed the following conjecture on the generalized Böttcher-Wenzel inequality: $$ \|ABC-CBA\|_F^2 \le 2\|B\|_2^2\|A\|_{(2),2}^2\|C\|_F^2, $$ where $\|\cdot\|_F$ is …

We show that unless P = NP, there cannot be a polynomial-time (or even pseudo-polynomial-time) algorithm for output feedback stabilization of a linear dynamical system with a linear controller. This settles one of the best-known open problems in control theory. …

Full original coefficient-one bound for every complex positive definite pair of trace sum one in every positive dimension. All twenty frozen contracts pass local Lean and actual source-matched non-root Linux Comparator/default-kernel verification; two independent nonauthor final reviews pass.

We give a candidate proof of Conjecture 5 in the third arXiv version of Latala and Strzelecka Operator lp lq norms of Gaussian matrices.

Seven chronological products cover every complex polynomial through degree 44 in the full 129-coefficient closure model, refuting the original equality 42. The later exact maximum 47 is outside this formalization. Local serial Lean, two nonauthor final reviews and two published-source …

The real nine-point squared distance matrix has nonnegative rank seven, including arbitrary real factors and zero or degenerate factor columns. Local serial Lean, two nonauthor final reviews and two published-source Linux verification runs pass.

Complete Lean formalization of the negative answer to the original PF-03 universal question, with arbitrary positive finite factor widths and frontier in the real symmetric-matrix subtype. Local serial Lean, two independent nonauthor final reviews and two published-source Linux Comparator/kernel/sandbox runs …

We construct Laurent polynomials that are real-valued on a Jordan curve surrounding the origin but whose banded Toeplitz matrices have nonreal limiting spectrum. This refutes the weakest surviving form of a conjecture of Shapiro and Štampach, and consequently also the …

In this second paper on descent methods as proof mechanisms in harmonic analysis (the first treated Bourgain's $Λ(p)$ selection theorem), entropic mirror descent gives an improved restricted isometry bound, improving sampling and recovery bounds in the Fourier Ratio program. Let …

We prove that the Y-partition into three equal sectors is the minimal spectral three-partition both for the Dirichlet Laplacian on the unit disc and for the planar harmonic oscillator $-Δ+|x|^2$, with minimal energies $j_{3/2,1}^{2}$ and $5$; for the disc, this …

For every complex matrix dimension n≥1 under the original positivity, unique stabilizing family, nonsingular right-limit, regular-pencil and simple-unit-root assumptions, the Hermitian imaginary-part rank equals half the unit-circle root count.

For every fixed nonempty compact complex matrix family in every positive finite dimension, prove a one-sided linear lower perturbation bound for its joint spectral radius, uniformly over arbitrary nearby nonempty compact families. The proof includes the actual critical product-bounded exterior-power …

Complete Lean verification of the original Rowland-Wu coefficient identity for every strictly positive real rectangular matrix, including positive scaling existence and uniqueness, empty and singular minors, and all transition signs.

Complete Lean verification of the original local two-family Holder assertion for arbitrary nonempty compact complex matrix families. All 14 exact contracts passed local compilation, two independent final source reviews, and actual canonical non-root Linux verification at the immutable proof revision.

For every complex Jordan block in the original IE-02 range, ideal GMRES equals worst-case GMRES. All polynomial minima, sphere suprema and their attaining witnesses use the actual Euclidean matrix and vector norms.

The original finite-unitary-class assertion is false: for every M there are M+1 complex 9-by-9 matrices with every ordered singular value equal for every complex shift, and no unitarily similar pair.

Starting from $1,x$ and allowing arbitrary complex linear combinations for free, we determine the largest degree whose entire coefficient space lies in the Zariski closure of outputs computable with seven products. The answer is $47$. A $49$-parameter polynomial family has …

Starting from $1,x$ and allowing free complex linear combinations, we prove that the Zariski closure of the polynomials computable with seven products contains every polynomial of degree at most $44$. This refutes the equality to $42$ conjectured by Jarlebring and …

Full complex rectangular MI-13 inequality with coefficient two, actual Euclidean operator and Frobenius norms and the two largest actual singular values. All ranks, both dimension orderings and every m,n at least two are retained.

Complete negative answer to the original RPCholesky polynomial trace-factor question. For every real C>0 and p>=0, a complex positive-definite matrix of order r+1 strictly violates the proposed bound after exactly r pivots. All normalized process and genuine ordered eigenvalue-tail semantics …

Complete original MF-07 result for arbitrary nonempty compact complex matrix families at joint spectral radius one, all positive dimensions and every switching word. All 18 exports passed actual canonical Linux default-kernel, Comparator, axiom and control checks at the immutable proof …

Complete Lean verification of the full original MI-04 necessity implication. All21 exports passed actual canonical Linux default-kernel, Comparator, axiom and control checks at the immutable proof revision.

For $X\in\mathbb{C}^{n\times n}$ the following are equivalent: (1) $\|P\|\le\|A+B\|$ in the operator norm for every positive semidefinite completion $P=\begin{pmatrix}A&X\\X^*&B\end{pmatrix}$; (2) the same holds whenever $A+B$ is a scalar multiple of $I$; (3) for every Hermitian $T$ the largest and smallest …

Complete all-dimensional inverse-M interval characterization by n-squared negative-sign vertices, implying the full original two-sign criterion without a regularity premise.

Complete formally checked proof of the full original MF-22 eventual polynomial-conditioning target, with sufficient exponent 2 and every fixed positive real parameter. All 22 targets passed canonical non-root Linux run 35053254275 at proof commit c701bfeea660473fc31ad9d0c74b76309be3b49f.

Complete formally checked proof for every nonnegative real polynomial growth exponent using a fixed positive-dimensional real matrix pair, all words at every positive length, genuine Euclidean operator norm and the actual nth-root limit one.

Complete formally checked negative resolution of the uniform exponential Gaussian-smoothed GEPP tail, with all 21 frozen obligations, genuine product Gaussian measures, the full nonsingular perturbation box and arbitrary positive real proposed constants.

The complete original symmetric/skew Rayleigh-minimum inequality for every real symmetric positive-definite pair and every n≥2, with actual attained sector minima. The stronger source result gives a real nonzero PSD eigenmatrix at the global Jordan minimum for every n≥1.

Complete original generic selection rule refuted: a nonempty open family of real 3×3 data matrices has invertibility, distinct squared singular values, a finite full stationary set, a unique least-absolute-multiplier pair and a strictly closer feasible matrix in the actual Frobenius …

Complete exact counterexample to the original universal connectedness question: a positive integer matrix of actual rank six and attained PSD rank three has a disconnected full real congruence quotient with its Euclidean quotient topology.

Complete exact counterexample to both canonical monotonicity claims: attained matrix-only spectral minimum and the literal Moore–Penrose approximation along actual exact LSMR iterates.

Derives the actual two residual-map executions and the maximum over the genuine Hermitian eigenvalue list, then refutes the complete original universal maximum identity by an admissible strict counterexample.

Exact diagonal examples refute a conjectured pairwise formula for four-step residual amplification in restarted Anderson acceleration. With $A=I-M$, $\alpha(v)=v^TAv/v^TA^2v$ and $R(v)=M[I-\alpha(v)A]v$, the residual after four steps is $R(R(v))$. For $M=\mathrm{diag}(1/10,1/2,3/5)$, where both $M$ and $I-M$ are positive definite, and $v=(1,1,1)^T$, …

We prove that every smooth properly embedded free-boundary minimal annulus in the unit ball is congruent to the critical catenoid. We also classify positive solutions of $(Δ_{\mathbb{S}^2}+2)u=0$ on smooth spherical annuli with $u=0$ and $|\nabla u|=1$ on the boundary; their …

We prove that unrestricted static output-feedback stabilization is NP-hard, already in the single-input multiple-output case. More precisely, we give a polynomial-time many-one reduction from the NP-complete Betweenness problem. Given an instance with $n$ elements and $m$ betweenness constraints, the reduction …

We give explicit nonnegative nilpotent weighted shifts of order $(m+1)^2$ whose singular values agree after every complex scalar shift, but whose norms under a common polynomial have ratio at least $(4/5)\sqrt m$. Thus no dimension-independent comparison constant exists. A two-by-two …

The asymptotic rank of the small Coppersmith--Winograd tensor $P=\mathrm{cw}_2$ over $\mathbb C$ satisfies $\widetilde R(P)<3.876919161$. The new ingredient is a three-dimensional isotropic space of matrices that is not obtained by selecting coordinates. It lets a literal copy of $P$ survive …