Proposition: Homogeneous-origin route

Suppose, on a common neighborhood of zero for tests supported in each fixed $L$,

Use one common finite inventory on $U$. Assume:

1. the $H_{\lambda ,j}[\Phi ]$ are linear, with local finite-order bounds in the joint-test and translation-test arguments, and form the displayed homogeneous chains;

2. $W_{\Phi }$ is a continuous function near zero, linear in $\Phi$, and for some $a_L>0$,

   $$|W_{\Phi }(0)|+\underset{0<|h|<r_L}{\sup }\frac{|W_{\Phi }(h)-W_{\Phi }(0)|}{|h{|}^{a_L}}\le C_L\Vert \Phi {\Vert }_{m_L,L}.$$

Then Assumption ass:expansion follows with

and

where the degree-zero term is omitted when absent. The family remainder is $O_L({\delta }^{a_L})$.

A separately calculated logarithmic scale derivative of the subtracted family has the same power bound.

Proof. For small scales the profile samples only the specified germ. Each chain contributes its exact finite power–logarithm polynomial. The continuous remainder contributes

The Hölder estimate bounds the latter integral.

For the derivative statement use (eq:scaleprofile), not differentiation of a remainder inequality. The chain terms differentiate their exact polynomials, while the remaining term is

It has the same bound since $\int \theta =0$.