Normalized dyadic chart

Fix a dyadic band with scale $Q>0$ . For exponents $b_i$ , normalized coordinates are $y_i=x_i/Q^{b_i}$ ; for a time distance $\tau$ , one may also set $s=\tau/Q$ . The number $Q$ is constant within the chart.

Derivatives

For $F(x)=A\widetilde F(y)$ , where $A$ is constant on this chart,

\partial_x^I F=AQ^{-\sum_i b_i I_i}\partial_y^I\widetilde F.

Even if a variable scale $q(x)$ is comparable to $Q$ on the band, differentiating a normalized chart holds $Q$ fixed. Replacing $Q$ by $q(x)$ defines a different coordinate change and introduces chain-rule terms. Auxiliary parameters are held fixed unless their derivatives are explicitly included.