Deterministic gaps between positive factorization cost and the entrywise norm

For a positive-semidefinite matrix, the sum of the squared \(\ell_1\) norms of its positive rank-one factors is at least the entrywise absolute sum of the matrix. We construct rational orthogonal projectors for which the minimum factorization cost exceeds this lower bound by a growing factor. Applying the deterministic almost Euclidean subspaces of Guruswami, Lee and Razborov gives an algorithm using polynomially many bit operations in the matrix dimension \(n\), with ratio at least \(\sqrt n\exp[-O(\log\log n\,\log\log\log n)]\). The estimate holds for both real and complex factorizations. The proof uses the fact that positivity confines every factor to the projector range, together with a trace estimate; complexification preserves the subspace inequality without loss. An elementary construction from additive characters of the cubic graph over \(\mathbb F_q\), where \(q=2^s\), gives rank-\(q\) projectors in dimension \(q^2\) with ratio at least \(q^{1/2}/3\). These elementary projectors also specify a rational positive operator with absolutely summable entries for which every countable positive rank-one factorization has infinite total squared \(\ell_1\) cost.