The path the estimate runs along

§4 slab transfer, §8 affine-path regularity — and the mechanism U_GJS is imported at

What is happening

To compare ω_g with ω_g′ the manuscript does not jump between them. It joins them by the straight line g_s = (1−s)g + s g′, s ∈ [0,1], differentiates the state along it (Lemma 10.1), and integrates the derivative. That is why the estimate is called a one-path estimate.

The path has to stay in the class for this to mean anything, and it does: 0 ≤ g_s ≤ 1 pointwise because the constraint is convex, and the support is contained in the union of the two supports, so it stays compact. That is Lemma 3.1.

This is also where U_GJS lives. The source proves its Theorem 1.1.1 along exactly this construction — the affine path h₁ + α(h₂ − h₁) between two arbitrary cutoffs of its class — and applies its Theorem 1.1.7 at every cutoff of that path with one h-independent constant. The cutoff-uniform quantifier the manuscript imports as A8(vi) is the one the printed proof uses here, which is what the scope amendment of 5 September 2026 records.

The passages are quoted verbatim in A8(vii′); see imports.