Abstract
We exhibit nontrivial transversal logical multi-controlled-Z gates on [[N,Θ(N),Θ(N)]] quantum low-density parity-check codes and [[N,Θ(N),Θ(N)]] quantum locally testable codes with soundness Θ(1), combining nearly optimal code parameters with fault-tolerant non-Clifford gates for the first time. Remarkably, our proofs are almost entirely algebraic-topological, showing that such presumably intricate logical gates naturally arise as a fundamental topological phenomenon. We develop a general framework for constructing a rich new family of homological invariant forms which we call ''cupcap gates'' that induce transversal logical multi-controlled-Z and, building on insights from [Li et al., arXiv:2603.25831], covering space methods to certify their nontriviality. The claimed almost-good code results follow immediately as examples.