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

For integers $n\ge 1$, consider assertions: $({\mathbf{P}}_{\mathbf{n}})$: Let $\mathrm{\Omega }\subset S^n$ 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 ${\mathbb{R}}^{n+1}$. $({\mathbf{Q}}_{\mathbf{n}})$: A weak solution $u$ to $\mathrm{\Delta }u=f$ with the Neumann boundary condition, with $f$ H"older continuous, in a conical polyhedral domain $\mathrm{\Omega }$ in ${\mathbb{R}}^n$ with non-obtuse dihedral angles, is in $C_{loc}^{2,\alpha }(\bar{\mathrm{\Omega }})$. We prove the implications $$({\mathbf{Q}}_{\mathbf{n}})\Rightarrow ({\mathbf{P}}_{\mathbf{n}}),({\mathbf{P}}_{\mathbf{n}})\Rightarrow ({\mathbf{Q}}_{\mathbf{n}+\mathbf{1}})\mathrm{.}$$ 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.