We study the free-energy gradient flow associated with the depth-N deep linear network on the space of invertible real d× d matrices. The loss observes only the diagonal, E_d(W)=1/2∑ᵢ₌₁ᵈ(Wᵢᵢ-1)², and the regularizer is the Boltzmann entropy of the balanced factorization fiber computed by Menon and Yu. This is a natural higher-dimensional version of a matrix-completion problem posed by Menon as a test of whether entropy selects among a noncompact family of minimizers. For diagonal completion, every width d≥2, depth N>2, and inverse temperature β>0, we prove that the free energy is unbounded below and has exactly 2ᵈ full-rank critical points, one in each diagonal sign chamber. Every critical point is diagonal and hyperbolic. Its unstable dimension is binom d2 and its stable dimension is d(d+1)/2. We also prove that a full-rank trajectory cannot converge to a finite rank-deficient matrix. Consequently, almost every full-rank initial condition has an unbounded forward orbit. Thus finite-temperature fiber entropy does not provide an equilibrium selection principle for diagonal completion.