Explain monotonicity of the invariant obstruction dimension along chains

Explain why the dimension of the O(D(L),q)-invariant subspace S⁵ᐟ²(ρ_L) is non-decreasing along every chain in the census and why it passes through the strata S^O = 0, S^O = 1, and S^O ≥ 2 in that order.

Background

The census organizes rank-three root bases into chains determined by their Cartan edge data. Computations show that dim S⁵ᐟ²(ρ_L)^O is non-decreasing along every chain and that the invariant dimension stratifies each chain into consecutive vacuous, selection-possible, and potentially split-shadow regions.

The paper establishes this behavior computationally but does not identify a theoretical mechanism explaining it.