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MathDB 361178 的反例 一、反例 对图 $X$，记 $Z^+_{(1)}(X)$ 为它的 1-泄漏正半定强迫数。MathDB 361178 的猜想断言：对任意图 $X$ 和任意边 $f\in E(X)$， … 取顶点集 … 边集 … 取 $e=05$，并记 … 图 $G$ 的 graph6 编码为 HCRUVjY。下面将证明 … 因此 … 故…

A Counterexample to MathDB 337797 1. Counterexample MathDB 337797 states the following conjecture. If $A\in\{0,1\}^{n\times n}$ has the 2-consecutive-ones property, meaning that th…

A Counterexample to the Triangular Gröbner-Basis Conjecture for Chemical Reaction Networks Proposition The triangular Gröbner-basis conjecture of Ferrari et al. is false as stated.…

Parity observation. If $G$ is bipartite and $x,y$ lie in opposite bipartition classes, then

A quadratic estimate. For $|\zeta|=1$, the function $\lambda\mapsto H(\lambda\zeta,\lambda)$ has positive real part on $\mathbb D$ and constant coefficient $1$. The one-variable Ca…

Orthonormal Wavelet ⇒ Wavelet-Set Conjecture DR. ARIE-ARIADNE DEWATSON-LE’DETSAMBALI A rigorous research report: from the original statement to the current measurable-cocycle front…

A Counterexample to the Chemical Antiregular Graph Conjecture For a chemical graph $G$ of order $n$, define … Here a chemical graph is a finite connected simple graph with maximum…

The sharp universal Gaussian EOT coupling bound Statement Let A,B be positive-definite covariance matrices on R^d. Let piepsilon be their Gaussian entropic optimal-transport coupli…

Lemma. Let $\overline d$ be an upper density on $G$. Suppose $q$ is a finitely additive probability measure on all subsets of $G$ such that

MathDB 363496: a four-element counterexample Result Conjecture 3.13 of arXiv:2406.07166v2 is false. The counterexample uses three very simple pseudoultrametric-preserving maps. For…

Lemma. If a Borel set $E\subset\mathbb R^N$ has nonempty interior, then $\dim_F(E)=N$.

MathDB 365102: the logarithmic-growth barycenter conjecture is false Result The literal Conjecture 1 of arXiv:2410.02715v2 is false. A smooth counterexample is … This potential sat…

Orthonormal Wavelet ⇒ Wavelet-Set Conjecture A rigorous research report: from the original statement to the current measurable-cocycle frontier Prepared as a mathematical research…

FROM ORTHONORMAL WAVELETS TO WAVELET SETS Incidence Graphs, Measurable Markers, and a Z/2Z-Cocycle Formulation Expanded journal-style research manuscript Prepared as a rigorous pro…

A rational-inner witness gives a new exact upper bound An exact rational-inner construction gives … Here $R=P/10^{265}$, with the full integer $P$ recorded in the attached manuscri…

Signed edge coloring of two cliques joined by one edge Theorem Let $G_n$ be the graph obtained from two vertex-disjoint copies of $K_n$ by adding one edge between them. For every s…

A weighted-gauge counterexample to the residue-class decomposition Statement Let $m$ be normalized Lebesgue measure on the unit circle $\mathbb T$. For a continuous, normalized,…

MathDB 351036 -- correction, proof, and divisor formula Corrected statement For positive integers n,a,d, define the geometric sum … The source context requires k to divide phi(n),…

MathDB 347688: a five-dimensional representation of $X_{2,9}$ Result The answer is yes: … The representation is defined by integral affine inequalities and has integral witnesses f…

Exact deficit and the asymptotic size of the cone Statement For $k\ge 2$, put … Section 4 of Bovdi and Leung's construction gives a maximal commutative algebraic system … The autho…

Exact excess and the asymptotic size of $\mathcal D$ Statement For $k\ge 2$, put … Section 4 of Bovdi and Leung's construction gives a maximal commutative algebraic system…

Balanced complete bipartite graphs uniquely minimize the spread Statement and scope As in the source's global convention, all graphs below are finite, simple, connected, and undire…

A Bernstein-basis proof of the stopping-region monotonicity Source correction There are two typographical indexing errors in the source formulas. They can be resolved from the prob…

MathDB 332161: a two-element counterexample Result The stated Schur--Brauer conjecture for semimodules is false. A counterexample is the regular one-dimensional module over the fie…

A counterexample to the pointwise Barbashin criterion Result The conjecture in Section 7 of Popa--Ceaușu--Megan, On exponential stability for linear discrete-time systems in Banach…