Papers

This report studies the Euclidean distance degree $D_m$ of the standard real symplectic group, for which Baaijens and Draisma suggested the formula $D_m=2^{m^2}+2^{2m-1}$. Building on an earlier regular $m^2$-variable reduction, we identify its denominator exactly and put the full generic …

For the classical three-point Krause pair $A=\operatorname{diag}(1,\tfrac{4+5i\sqrt3}{13},\tfrac{-1+2i\sqrt3}{13})$ and $B=-A$, an exact trace sum-of-squares certificate proves that the unitary-orbit distance of $A\otimes I_k$ and $B\otimes I_k$ is $\sqrt{27/13}$ for every $k\ge1$. The bottleneck eigenvalue-matching distance is $\sqrt{28/13}$, so the result lies …

The three-point pair $A_0=\operatorname{diag}(1,\tfrac{4+5i\sqrt3}{13},\tfrac{-1+2i\sqrt3}{13})$, $B_0=-A_0$, whose unitary-orbit distance is unchanged by every finite amplification, is extended to an open neighborhood in the space of six independently variable complex eigenvalues. Throughout this neighborhood, unitary-orbit distance is unchanged by every finite amplification, …

Consider an explicit pair of $3\times3$ normal matrices with three distinct eigenvalues each. Its unitary-orbit distance is strictly below the eigenvalue-matching distance, and earlier rounds of this work show it is unchanged by every finite amplification. In an arbitrary finite …

For the dimension-independent constant $C_p(m)=\sup_n\sup\|\sum_jA_j\|_p/\|\sum_j|A_j|\|_p$, we prove the computer-assisted bound $C_{4/3}(2)\le(861/800)^{1/4}$. An exact rational certificate excludes an entire parameter interval in a positive cubic system, using 184 coefficient identities and 44 positive-definite Gram multipliers; the eight-stage certificate chain behind the …

For convex polygons $P,Q$ with $\operatorname{xc}(P)=\operatorname{xc}(Q)=6$, the note proves $\operatorname{xc}(P\times Q)=12$ by a finite, certificate-based support argument. Every such polygon has between six and nine vertices. A hypothetical eleven-term nonnegative factorization of the complete product slack matrix must satisfy explicit …

Exact rational pairs of normal matrices are constructed whose ratio $d_\infty(A,B)/\|A-B\|_2$ exceeds the classical value $\sqrt{28/27}$ of Krause's example. The strongest has dimension seven and proves $C_{\mathrm{normal}}>103077/100000$. Its proof consists of an exact orthogonal-reflection construction, a deficient matching block, and …

This report proves the spread of the best rank-two block matrix as an upper bound for every endpoint matrix whose centered sign matrix has rank at most two, in every dimension and throughout the normalized interval $-1\le a<1$. An analytic …

For a continuous nonvanishing zero-index symbol, geometric decay of all sufficiently large top-right inverse corners of its Toeplitz matrices is equivalent to an inner-annulus extension. The proof of the difficult implication uses an exact bordering identity: summable corner increments control …

Let $S_p(k, \varepsilon)$ be the worst-case minimum support of a nonnegative strong coreset on original rows for Euclidean subspace costs of power $p$, with no restriction on input rank. For every fixed real $p > 2$ that is not an …

We consider relative trace estimation of an arbitrary real positive-semidefinite matrix of order $N=n^q$ when an oracle returns the full matrix-vector response only on real product inputs. Two new upper bounds are established. A fractional-moment analysis of product-spherical empirical averaging, …

An exact minimum-energy criterion characterizes when a given linear feature space contains a valid spectral rank-$k$ approximation. It includes singular tail features and unequal leading singular values. For Gaussian block-Krylov features on interpolation nodes, the criterion becomes a $k\times k$ …

The graph complement conjecture asks whether $\operatorname{mr}(G)+\operatorname{mr}(\overline G)\le n+2$ for every graph on $n$ vertices. Take a bipartite graph whose vertices on one side have degree at most $d$. Vectors $(1,t_i,\dots,t_i^d)$ at distinct positive nodes and the coefficient vectors of …

For the real isospectral Nyström minimax problem, an oversampled limiting-profile estimate is proved and used to obtain an upper bound for every positive finite spectrum: $R_{n,k}\le\min\{k+1,n-k,C\sqrt{k}\}$ for $2\le k\le n-2$, with the explicit universal constant $C=2^{34}e^{2560}$. The argument completes the …

Complete Lean formalization of the negative n = 7 counterexample to the prescribed real Boolean matrix nonnegative-rank equality, with ten Comparator-boundary exports and the full source correspondence retained.

Given a real square matrix $A$ and a nonempty closed target set of complex matrices $\mathcal E$, invariant by complex conjugation and containing real matrices, does the subset of $\mathcal{E}$ consisting of the matrices nearest to $A$ in the Frobenius …

We study the fixed CG recurrence and adversarial scalar relative-error envelope of the problem. The scalar threshold is determined exactly by a degree-five rational inequality. In general dimension, set $H=\min\{K,64/\varepsilon^2\}$ and $D=1024H^2$, and let $m$ be the minimum of the …

For the $q$-stage Radau IIA matrix with $A_q^{-1}=L_qU_q$, $B_q=L_q^{-1}$ and $N_q=U_q-I_q$, the spectral disk for the stage preconditioner reduces to $\rho((I+\mu B_q)^{-1}N_q)\le\|N_q\|_2$ for every real $\mu>0$. For every stage number this report proves the inequality when $0<\mu\le1/(q\|B_q\|_2)$ or $\mu\ge2\|L_q\|_2$, gives …

For the $q$-stage Radau IIA collocation matrix we give a self-contained differentiation formula for its inverse, prove that every pivot of the no-pivot factorization $A_q^{-1}=L_qU_q$ is positive, reduce the spatial pencil exactly to the stage matrices $I_q+(I_q+\mu L_q^{-1})^{-1}(U_q-I_q)$ with $\mu>0$, …

For the collocation integration matrix $A$ on distinct positive nodes, the target is one positive diagonal $D$ for which $I-D^{-1}A$ is nilpotent. We give a self-contained all-stage existence theorem for nodes sufficiently close to a common positive point. The proof …

Complete negative resolution of the original IE-16 universal inequality from Sidney Holden’s exact nine-point, degree-four counterexample. All fifteen independently reviewed contracts passed actual non-root Linux Comparator, standard axiom checks and default-kernel replay.

For $d_1,d_2\ge 2$ let $W_d=x^{d-1}y\in\mathrm{Sym}^d(\mathbb C^2)$ (up to scale) be the W-state. We prove that the partially symmetric rank of $W_{d_1}\otimes W_{d_2}$ equals $2(d_1+d_2-2)$, so the known upper bound of Gałązka and of Canino–Casarotti–Santarsiero is sharp for every pair. This …

Two concrete pieces of the finite-accuracy transition for unrestricted adaptive matrix–vector queries are isolated, without closing the transition itself. First, a weighted Chebyshev interpolation argument proves matching bounds, for all finite parameters, for a precisely defined deterministic-width, full-block span-query class, …

After a common unitary change of basis, write the normal pair as $D_a$ and $UD_bU^*$ with $U$ unitary. A direct fourteen-term identity is established for every complex unitary of order four and arbitrary complex spectra. It writes $\det(D_a+UD_bU^*)$ as a …

Complete exact order-four counterexample to the original spectral-uniqueness locus condition, independently reviewed and verified by the actual non-root Linux sandbox, Comparator, permitted-axiom audit and default kernel.

Complete negative resolution of the original finite-Toeplitz-spectrum implication, independently reviewed and verified by the actual Linux sandbox, Comparator, permitted-axiom audit and default kernel.

Weighted orthogonal-complement duality proves the real square-root bound of the Goreinov–Tyrtyshnikov–Zamarashkin conjecture for Parseval frames (matrices with orthonormal columns) having at most $r+2$ nonzero row directions, using the established real two-column theorem. The unrestricted real conjecture is not resolved here. …

We establish the expected relative-error guarantee of randomly pivoted Cholesky at $\lceil Cr/\varepsilon\rceil$ pivots with the explicit constant $C=3$ when the eigenvalues after the target rank lie below a half-geometric envelope, and with $C=4$ under an inverse-square tail envelope. The …

We study relative trace estimation of arbitrary real positive-semidefinite matrices from adaptive, exact, real rank-one tensor matrix-vector queries. A degree-$q$ interpolation identity simulates one complex product query with $q+1$ real product queries. It also yields a counterexample, already for local …

Complete negative resolution of the original universal orthogonal GEPP QR-extremizer equality, by an exact order-eight counterexample and a strict gap to the genuine real supremum.

Complete formalization of the original block-Lanczos interval-occupancy conjecture, via the stronger full-prefix theorem and exact finite-dimensional real linear algebra.

Complete negative resolution of the original joint generic critical-count conjecture: the actual count at n=s=3 is three, whereas its formula predicts four.

For the fixed-partition HODLR approximation problem with $n=2^Lk$, this note develops the bounds $c\min\{n,kL/\varepsilon+k/\varepsilon^2\}\le q_*(n,k,\varepsilon)\le C\min\{n,kL^2/\varepsilon+kL/\varepsilon^2\}$. The upper bound is obtained by a row-isotropic perturbation inequality for truncated SVD, a noisy Gaussian Nyström calculation, a constant-accuracy perforated pilot, and a …

This manuscript studies the minimum number $\mu_{\mathbb R}(d,r)$ of real linear measurements that are injective on real $d\times d$ matrices of rank at most $r$. Its main result is an exact solution of a topological relaxation: one measurement below the …

For every admissible finite dimension, rank and accuracy, this note proves $q_{\mathrm{sp}}(n,k,\varepsilon)\ge c\,\frac{k}{\sqrt\varepsilon}\log\bigl(1+\frac{n\sqrt\varepsilon}{k}\bigr)$ with a universal positive constant $c$. The argument uses a determinant tilt of a singular Wishart law, its exact posterior under adaptive matrix–vector queries, and an integration-by-parts …

A computer-assisted exclusion of four-dimensional, eight-facet lifts proves that the regular $n$-gon slack matrix has real nonnegative rank nine for $17\le n\le20$; explicit nine-term factorizations give the matching upper bound. The reduction uses both factors of a nonnegative factorization, a …

For subsets $a,b\subseteq[n]$, let $C_n(a,b)=(1-|a\cap b|)^2$, with every entry prescribed. We construct nonnegative rational factors proving $$\operatorname{rank}_+(C_n)\le 2^{n-1}+n+\binom n2+\binom n4.$$ At $n=7$ this gives $\operatorname{rank}_+(C_7)\le127<128$, disproving the universal full-rank conjecture. In fact the displayed bound is strictly below $2^n$ for …

An explicit integer matrix of order $444$ and rank seven is constructed that is completely positive over the reals but has no nonnegative rational Gram factor of any finite width. All its entries are strictly positive, and its real cp-rank …

Graph matrices encode dependencies in random matrices built from shared random variables and arise in spectral algorithms, sum-of-squares (SoS), and high-dimensional statistics. We determine how finite graph structure controls their sharp spectral growth. For every fixed simple graph shape in …

The paper proves that every real completely positive matrix of order six has an entrywise nonnegative factor with at most nine columns, and exhibits a positive definite completely positive matrix of order six with cp-rank nine, so that the maximum …

For randomized block Krylov iteration with block size $b$, a Gaussian starting block and output $Z[Z^{\mathsf T}A]_k$, the manuscript proves that $q = \lceil C\varepsilon^{-1/2}(t\log(2/\Delta) + \log(n/(\delta\varepsilon)))\rceil$ iterations, with $t = \lceil k/b \rceil$, an absolute constant $C$ and $\Delta$ …

Let $A_k$ have $k$ rows and $k/100$ independent columns. Each column has exactly $s_k$ uniformly located nonzero entries, with independent signs and magnitude $s_k^{-1/2}$. We prove the sequence criterion $$\exists a > 0:\ \Pr\{\sigma_{\min}(A_k) \ge a\} \longrightarrow 1 \quad\Longleftrightarrow\quad \liminf_{k …

For an explicitly given finite family $\mathcal F \subset \mathbb{R}^{n \times n}$ of size $M \ge 2$ and a fixed unknown matrix $A$ accessible through products $Av$ and $A^{\mathsf T}v$, the manuscript gives a fully nonadaptive algorithm returning $\widehat B …

Complete negative answer to Johnson's original universal normalized-characteristic-polynomial derivative realizability conjecture, with genuine arbitrary-matrix trace certificates and the unchanged order-seven source witness.

For every order $n$ and every nonnegative spectrum $\lambda = (\lambda_1, \dots, \lambda_n)$, the dossier gives an exact finite algebraic prescription for the maximum $M_n(\lambda)$ of $\operatorname{per}(U^* \operatorname{diag}(\lambda) U)$ over unitary $U$. A maximizer is a critical point of the …

This manuscript studies the minimum number $\mu_{\mathbb R}(d,r)$ of real linear measurements that are injective on all real $d\times d$ matrices of rank at most $r$. The main exact result established here is $\mu_{\mathbb R}(4,1)=11$: a Grassmannian characteristic-class obstruction excludes …

For every pair of positive definite complex matrices $A,B$ with $\operatorname{tr}(A+B)=1$, we prove $$\|[B,\log(A+B)]\|_1\le-\operatorname{tr}(A)\log\operatorname{tr}(A)-\operatorname{tr}(B)\log\operatorname{tr}(B).$$ The proof uses Frenkel's integral representation of quantum relative entropy, in its positive-part form given by Hirche and Tomamichel. The key estimate is that $\operatorname{tr}(\rho-\gamma e^{itH}\sigma …

Complete negative answer to the original universal at-most-dimension bound for connected components of the real eigenvalue union of an independent-entry interval matrix.

Let $H_{i_1 \cdots i_m} = h_{i_1 + \cdots + i_m}$ for $0 \le i_j \le n-1$, over $\mathbb{C}$. Write $D = m(n-1)$, and let $r$ be the rank of the middle Hankel catalecticant. For $H \ne 0$, choose a homogeneous …

Complete proof of the canonical odd-summand conjecture, via the stronger exact least admissible constant k/4 for every k≥2 in every positive dimension.