Draft notes collected lattice evidence suggesting a positive Yang--Mills spectral gap.
Surveyed foundational articles and outlined approaches for formal analysis.
Reviewed lattice arguments in arXiv:1507.04087 supporting a nonzero gap.
The area-law fit $W(A)=e^{-\sigma A}$ with $\sigma=0.2$\,GeV$^2$ reproduced Wilson loops $W(1)=0.819$, $W(2)=0.670$, and $W(3)=0.549$.
Strong-coupling estimates using $m=-\ln(\beta/6)/a$ at $\beta=5.7$ and $a=0.1$\,fm yielded $m=0.10$\,GeV; a propagator fit gave $0.48$\,GeV.
A $4\times4$ $SU(2)$ lattice produced an area law $W(C)\approx e^{-0.18A}$, indicating a string tension $\sigma\approx0.18$.
Integrating the $2+1$ dimensional beta function gave $g(\mu)=\frac{g_0}{\sqrt{1+\frac{11 g_0^2}{6\pi}\ln(\mu/\mu_0)}}$, showing a slow ultraviolet decrease.
The $O(3)$ sigma model generated a gap $m\approx0.0019\,\Lambda$ for $g=0.5$.
Polyakov loop data gave $P(5)\approx0.67$ for $\sigma=0.2$.
Two-loop running from $g(1)=1$ yielded $g(10)\approx0.80$.
The estimate $m_G\approx1.6\sqrt{\sigma}$ with $\sigma=0.18$ gave $m_G=0.678$.