Theorem: Refinement stability and class-relative independence

In Definition def:refinement, suppose the complete non-flow error totals are

Suppose the flow factors correspond to intervals of lengths $h_{j,\ell }$, with

and have half-diamond errors bounded by $A{h_{j,\ell }}^{1+\alpha }$, where $A<\infty$ and $\alpha >0$ are independent of $j,\ell$. Then

If ${\delta }_j\to 0$ and the four non-flow totals tend to zero, the complete channels converge to $\Phi$.

Any two admissible refinement families satisfying these conditions for the same frozen target converge to the same limit.

Proof. The total flow error satisfies

Apply Lemma lem:telescoping to the complete matched factor chain and group the remaining factor errors by their declared roles. This proves (eq:refinement-budget).

For two families with bounds $b_j\to 0$ and ${b^{\prime }}_j\to 0$,

Both families consequently have the same channel limit.