The actual smoothing displacement

Write

For the true displacement $\delta (H_1,H_2)$,

In particular,

The sign must be converted when the second cone is written in terms of $b-y$.

The suggested core

is not a complete localization proof. It supplies no common scaling for $r$, no tail estimate, and no physical asymptotic coefficient by itself.

Nor is this weighted scaling automatically Dang's Euler scaling. An Euler field acts as the identity on the normal ideal modulo its square Dang (Section 1). On the $(u,v,\xi )$ slice the weighted generator

sends ${\xi }^A$ to ${\xi }^A/2$, not to ${\xi }^A$ modulo quadratic terms. Applying an ordinary Euler-scaling extension theorem therefore requires an additional transfer argument.