This paper introduces a new method of finding the sum and product of roots of quadratic equations.The approach adapts the half-life concept from radioactive decay in Physics and Chemistry. During the derivation of the quadratic formula by completing the square, the step where we add half the coefficient of x to both sides is replaced by the half-life relation 1/2 = ln 2/k, leading to k = 2 ln 2. This substitution produces an auxiliary formula that yields two values x1and x2whose sum equals the Vieta sum.A second relation involving ln 2 then recovers the exact Vieta product. The method, called the Logarithmic Root-Sum-Product Extractor (or Half-Life Quadratic Formula), gives results identical to the conventional method while creating a strong link between mathematics and science.

The mass gap of Yang-Mills theory is derived analytically by establishing a correspondence between the classical kinetic energy coefficient and the quantum half-life decay constant. The replacement 1/2→ln2/k transforms the energy equation into a form compatible with gluonic bound states. The dimensionless constant k is de- termined from the running coupling as k = 2π/αs≈ 18. The trace anomaly, with an explicit derivation from the beta function and gluon condensate scaling, yields m0++= 4 √σ. Utilizing the lattice-calibrated string tension √σ = 7/16 GeV, the mass gap is computed as 1.75 GeV. The kinetic-half-life correspondence then predicts the flux tube thickness L =4 ln2/k√σ≈ 0.352 GeV^-1, consistent with the lattice value 0.343 GeV^-1. Four independent confirmatory calculations—lattice error analysis, gluon condensate sum rule inversion, excited-state Regge trajectory matching, and decay width half-life consistency—converge to this value within standard uncertainties. A constructive outline for the continuum limit and the positivity of the mass gap is provided, demonstrating that the derivation is consistent with the rigorous requirements of the Yang-Mills Millennium Problem.

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In \cite {ChenFengLiXiXu2026}, a reverse projection inequality for general convex bodies was established in three dimensions. In this paper, we completely characterize the equality cases of this inequality. We further show that the reverse projection inequality fails in every dimension $n\geq9$. The counterexamples are constructed as full-dimensional polytopes with at most $n+2$ facets.

Let $K\subset\R^3$ be a convex body and let $\Pi K$ be its projection body. We prove \[ \frac{V_3(\Pi K)}{V_3(K)^2} \ge \frac{3\pi^2}{4}, \] with equality exactly for ellipsoids. The proof combines a section estimate of Saroglou with a sharp one-dimensional functional inequality. The latter is proved by passing to the distribution function and approximating it by step functions. We also prove the complementary sharp upper bound \( V_3(\Pi K)/V_3(K)^2\le18, \) which is attained by tetrahedra.