An explicit finite cylinder cover of the regular tetrahedron saving $3.86\times10^{-4}$ below the half-area bound
We give an explicit cover of a regular tetrahedron by $4096$ cylinders with compact triangular perpendicular bases. The sum of their base areas, divided by the tetrahedron's minimum orthogonal projection area, lies in $[0.499613830783714148,\,0.499613830783714149)$. Thus the saving below $1/2$ exceeds $0.000386169216285851$. The construction uses a finite tilt in the angular-sector framework of OpenAI's September 2026 preprint and rationally specified radial cutoffs. Coverage is certified by exact linear-programming dual witnesses, and independently verified by exhaustive rational polyhedron…
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