Sharp Neumann eigenvalue estimates and $C^2$ elliptic regularity in non-obtuse polyhedral domains

For integers $n\ge 1$, consider assertions: $\mathbf{(P_n)}$: Let $Ω\subset S^n$ be a spherical domain enclosed by totally geodesic $S^{n-1}$'s with non-obtuse dihedral angles. Then in $[0,2(n+1)]$, its Neumann spectrum can only take values among $\{0,n,2(n+1)\}$. Moreover, $n$ is a Neumann eigenvalue if and only if the corresponding eigenfunction is the restriction of a linear function in $\mathbb{R}^{n+1}$, while $2(n+2)$ is a Neumann eigenvalue if and only if the corresponding eigenfunction is restriction of a quadratic polynomial in $\mathbb{R}^{n+1}$. $\mathbf{(Q_n)}$: A weak solution $u$ to $Δu = f$ with the Neumann boundary condition, with $f$ Hölder continuous, in a conical polyhedral domain $Ω$ in $\mathbb{R}^n$ with non-obtuse dihedral angles, is in $C^{2,α}_{loc}(\overlineΩ)$. We prove the implications \[\mathbf{(Q_n)} \Rightarrow \mathbf{(P_n)},\qquad \mathbf{(P_n)}\Rightarrow \mathbf{(Q_{n+1})}.\] Consequently, both assertions hold in all dimensions. These give the optimal Neumann eigenvalue lower bound and $C^2$ elliptic regularity in non-obtuse Riemannian polyhedral domains.