Beat or prove optimal the densest known packing of regular tetrahedra ($\phi=4000/4671$)

Consider packings of congruent regular tetrahedra in $\mathbb{R}^3$ (translations and rotations allowed, no overlaps). Let $\phi^*$ be the supremum of the packing fraction (fraction of space covered) over all such packings. The densest packing currently known achieves $\phi = 4000/4671 = 0.856347\ldots$. It is unknown whether $\phi^* = 4000/4671$ or whether a denser packing exists. Exhibit a periodic packing of regular tetrahedra with packing fraction $> 4000/4671$, or establish a nontrivial upper bound $\phi^* < 1$ with a verifiable certificate.

Acceptance. FULLY RESOLVES: a proof that $\phi^* = 4000/4671$ (optimality), or determination of $\phi^*$. PARTIAL (improve record): a periodic packing given as a fundamental cell (lattice basis) plus the vertex coordinates of the tetrahedra inside it, with packing fraction $> 4000/4671$; verifier -- confirm all tetrahedra (and their periodic images) are pairwise non-overlapping (separating-hyperplane / LP feasibility test between convex polytopes) and compute the density as (cell volume covered)/(cell volume). PARTIAL (upper bound): any rigorous $\phi^* \le c < 1$ with a checkable argument or certificate.