Minimal retained dimension for the observed Krylov reduction

Determine whether a retained interaction dimension smaller than eight can attain the five-vector complete-equation Krylov basis observed for the $325{,}260$-coordinate frozen Tet10/P0--IPC system.

Background

The retained-dimension sweep tested dimensions 6 and 8 around the transition to a five-vector Krylov basis. Dimension 8 was the first sampled dimension achieving that result, but dimension 7 was not tested.

Consequently, the data do not establish that eight is the minimal retained dimension; a smaller dimension could produce the same convergence behavior on the reported frozen system.

References

The sweep samples dimensions six and eight on either side of the observed transition; dimension seven was untested. Eight is therefore the first sampled dimension attaining five Krylov vectors, with minimality left open.

— TOGEARI: Interaction-Space Preconditioning for Condensed Finite-Element Systems with IPC Contact  (2608.14162 - Liu et al., 14 Aug 2026) in Section 5.1, paragraph following Equation (rank-8 certificate)