Minimal recurrence operators for the moment ladders of the root lattice family, with identification theorems

The even moments of the Voronoi cells of the root lattices are classical quantities; the second moments are the quantizer constants tabulated by Conway and Sloane, and no closed recurrences for the full moment sequences of these cells appear to have been available. This record supplies them: for each of A2, A3, D4, E6, E7, and E8, a linear recurrence with integer polynomial coefficients for the scaled moment ladder, of orders 2, 4, 4, 7, 9, and 9, together with a machine certified identification theorem proving that the recurrence is the one produced by elimination from the cell geometry, that its order is minimal over the rational function field at every degree, and that it holds at every index from order plus one onward. The leading coefficients split along the squared circumradii of the facet classes, tying each operator to the face geometry of its cell. The record further documents the transposed differential equations with their weight mechanism, a previously unrecorded twin class split at E7, closed tail amplitudes for five lattices including 768 for E6 and 45927/16 for E7, and the two adic structure of the ladders. Every claim reproduces from the Gram matrices alone with the included standard library instruments, the deposit is checked by an expectation free validation engine whose code and report are included, and a text audit instrument, also included, re derives every number stated in the announcement from the deposit files. Companion to doi 10.5281/zenodo.21972708.