Stage 3: PEPS existence, algorithms, and approximation budgets

On an open $L\times L$ square, for the stated unique uniformly gapped nearest-neighbour class, the external companion asserts existence of a nonzero PEPS ${\Psi }_L$ such that

with uniform constants for fixed local parameters. The tensors may depend on $H$ and $\Omega$. No efficient tensor-production or contraction algorithm is asserted in the frozen record OpenAI265B. This statement is not assumed in the proofs below.