Recursive rational reconstruction from predecessor matrix elements

Prove that every to-be-reconstructed matrix element H_{M+i,M+j}^(M+K) of the partially tridiagonalized Hamiltonian can be defined recursively as a comparatively compact rational function of predecessor matrix elements with smaller row or column indices.

Background

The paper observes that explicit Gröbner-elimination formulas become prohibitively long as K increases, while hand-rearranged recursive formulas for the first several reconstructed entries are much more compact. The proposed recurrence proceeds through the diagonal entries a_i and off-diagonal products rho_i of the out-of-model-space tridiagonal block.

The conjecture extends the observed recurrence pattern to every reconstructed entry and every finite K. It is not proved in the paper; the displayed formulas and computations are presented only as constructive support.