Resolving the Edge of a Quantum Pyramid

Standing on the shoulders of giants, we resolve the quantum pyramids conjecture, confirming the globally information-optimal measurement for an ensemble of equiangular equiprobable pure states, as conjectured by Englert and Řeháček (arXiv:0905.0510). We do so by proving the remaining entropy inequalities of Holevo and Utkin (arXiv:2506.06700), which certify optimality for obtuse and flat pyramids. For obtuse pyramids, our key contribution is a rigorous proof that local minimizers of the corresponding entropy inequality cannot have three distinct coordinate values. We show that eliminating this family can be reduced to a neat algebraic reciprocal inequality relating branches of the Lambert W function, which may be of independent interest. For flat pyramids, we prove a tight ell^p inequality for zero-sum vectors that was recently conjectured, proved analytically in dimension d=3, and computationally verified for dleq 200 by Holevo and Utkin (arXiv:2603.24017). We prove this bound for all dgeq 2 via a technique in symmetric inequalities known as the equal variables method.